For more information, see the Math Notes box in Lesson of the Core Connections, Course 1 text.

Transcription

1 Number TYPES OF NUMBERS When two or more integers are multiplied together, each number is a factor of the product. Nonnegative integers that have eactly two factors, namely, one and itself, are called prime numbers. Ecept for one and zero, the other non-prime numbers are composite. One has only one factor, so it is neither prime nor composite. It is also the Multiplicative Identity since one multiplied by any number does not change the value. Written in symbols,! n = n. Zero is the Additive Identity since adding zero to any number does not change the value. Written in symbols, 0 + n = n. For more information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Eample Identify each integer as prime, composite, or neither. 6 6 has factors of, 6,, and so 6 is composite. 7 7 has factors of and 7 so 7 is prime. has only as a factor so is neither. Eample For each composite number, factor it into primes and write the number as a product of primes using eponents as possible. 4 4 = 4! 6 =!!! =! = 9! 5 =!! 5 =! 5 Problems Identify each integer as prime, composite, or neither. For each composite number, factor it into primes and write the number as a product of primes, using eponents as possible Answers.!! 5.! ! 5 5. prime 6.!9 7. prime 8.! 5 9. prime 0. neither.!. 5! Parent Guide with Etra Practice

2 PROPERTIES OF ADDITION AND MULTIPLICATION In addition and multiplication, the order of the numbers can be reversed: + 5 = 5 + and! 5 = 5!. This is called the Commutative Property. In symbols: The Commutative Property of Addition states: a + b = b + a and The Commutative Property of Multiplication states: a!b = b! a. When adding three numbers or multiplying three numbers, the grouping can be changed: ( + ) + 5 = + ( + 5) and (! )! 5 =!(! 5). This is the Associative Property. In symbols: The Associative Property of Addition states: (a + b) + c = a + (b + c) and The Associative Property of Multiplication states: (a! b)! c = a!(b! c). The Distributive Property distributes one operation over another. So far in these courses, students have seen multiplication distributed over addition. In symbols: For all numbers a,!b,!and c,!!a(b + c) = a!b + a! c. For eample, ( + 5) =! +! 5. Properties of numbers are also discussed in the Math Notes boes in Lessons..4,.., and 7.. of the Core Connections, Course tet. For additional information see the Math Notes boes in Lesson 4.. of the Core Connections, Course tet or Lessons..5 and..7 of the Core Connections, Course tet. The properties of multiplication and addition allow calculations to be rearranged. Doing this is helpful when doing calculations mentally. Name the property or reason that justifies each step. Eample Calculate mentally: 4! (7! 5) Step = 4!(5!7) Commutative Property of Multiplication Step = (4! 5)!7 Associative Property of Multiplication Step = (00)!7 mental math Step 4 = 700 mental math Eample Calculate mentally: 8(56) Step = 8(50 + 6) by renaming 56 as Step = 8(50) + 8(6) Distributive Property Step = mental math Step 4 = 448 mental math Core Connections, Courses

3 Number Problems Listed below are possible steps used to mentally calculate a problem. Give the missing reasons that justify the steps.. 5(9) = 5(0 + ( )) renamed 9 as 0 + ( ) 5(0 ) = 5(0) + 5( ) a 50 + ( 5) = 50 + ( 0 + 5) renamed 5 as 0 + ( 5) 50 + ( 0) + ( 5) = (50 + ( 0)) + ( 5) b 40 + ( 5) = 5 mental math = a = ( ) b = ( ) + 77 c = 777 mental math. 49() = (49) a (49) = (50 ) renamed 49 as 50 (50 ) = (50) () b (50) = (6 )(50) renamed as 6! (6 )(50) = 6( 50) c 6( 50) = 6(00) mental math 600 = 588 mental math Answers. a. Distributive b. Associative. a. Commutative b. Associative c. Associative. a. Commutative b. Distributive c. Associative Parent Guide with Etra Practice

4 MULTIPLICATION WITH GENERIC RECTANGLES If a large rectangle is cut into a number of smaller rectangles, then the area of the large rectangle is equal to the sum of the areas of all the smaller rectangles. The idea of breaking a product up into parts is the basis for multiplication using generic rectangles. We say generic because the dimensions are not to scale. We use it to help us visualize multiplication and as a way to multiply numbers by using a diagram. Multiplication using rectangle models reinforces the multiplication algorithm and will continue to be used and etended through Algebra and Algebra. For additional information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet and Lesson.. of the Core Connections, Course tet. Eample 0 Multiply! 5 using a generic rectangle. Since we are multiplying a two-digit number by a two-digit number we need a generic rectangle as shown at right. The numbers to be multiplied are separated (decomposed) based on place value. In this case, for eample, has two tens (0) and ones. The area of the product (the large rectangle) is equal to the sum of the areas of each of the smaller rectangles. The area of each of the smaller rectangles is found by multiplying its dimensions. Find the area of each of the smaller rectangles and then sum them together.! 5 = (0 + )(0 + 5) = = Eample Multiply 4! 5 using a generic rectangle Since we are multiplying a three-digit number by a two-digit number we need si sections in our rectangle. Fill in the areas and add them together to get: 4! 5 = = Core Connections, Courses

5 Number Problems Use a generic rectangle to find each product.. 47! 5. 8! 84. 6! ! ! ! 85 What multiplication problem does each generic rectangle represent and what is the product? Answers , ! 5 = ! 4 = ! 8 = 0,46 Parent Guide with Etra Practice 5

6 EQUIVALENT FRACTIONS Fractions that name the same value are called equivalent fractions, such as = 6 9. One method for finding equivalent fractions is to use the Multiplicative Identity (Identity Property of Multiplication), that is, multiplying the given fraction by a form of the number such as, 5, etc. In this course we call these fractions a Giant One. Multiplying by does 5 not change the value of a number. For additional information, see the Math Notes boes in Lesson.. of the Core Connections, Course tet or Lesson..8 of the Core Connections, Course tet. Eample Find three equivalent fractions for.! = 4! = 6! 4 4 = 4 8 Eample Use the Giant One to find an equivalent fraction to 7 using 96ths: 7!!!!!!=? 96 Which Giant One do you use? Since 96 = 8, the Giant One is 8 8 : 7! 8 8 = Problems Use the Giant One to find the specified equivalent fraction. Your answer should include the Giant One you use and the equivalent numerator Answers. 5 5, , , , , 0 6., 8 6 Core Connections, Courses

7 Number SIMPLIFYING FRACTIONS The Giant One is also useful when simplifying or reducing fractions to lowest terms. Use the greatest common factor of the numerator and denominator for the Giant One. Divide the numerator and denominator by the greatest common factor and write the resulting fraction as a product with the Giant One. What remains is the simplified version of the fraction. Eample Eample Simplify 0 4. The greatest common factor of 0 and 4 is 4. Simplify The greatest common factor of 45 and 60 is 5. Problems Simplify each fraction to lowest terms Answers Parent Guide with Etra Practice 7

8 FRACTION-DECIMAL-PERCENT EQUIVALENTS Fractions, decimals, and percents are different ways to represent the same portion or number. fraction words or pictures decimal percent Representations of a Portion For additional information, see the Math Notes boes in Lessons..4 and..5 of the Core Connections, Course tet, Lesson.. of the Core Connections, Course tet, or Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 5 materials or the Core Connections, Course Checkpoint materials. Eamples Decimal to percent: Multiply the decimal by 00. (0.8)(00) = 8% Fraction to percent: Write a proportion to find an equivalent fraction using 00 as the denominator. The numerator is the percent. 4 5 = 00 so 4 5 = = 80% Decimal to fraction: Use the digits in the decimal as the numerator. Use the decimal place value name as the denominator. Simplify as needed. a. 0. = 0 = 5 b. 0.7 = 7 00 Percent to decimal: Divide the percent by 00. 4% 00 = 0.4 Percent to fraction: Use 00 as the denominator. Use the percent as the numerator. Simplify as needed. % = 00 = 50 56% = = 4 5 Fraction to decimal: Divide the numerator by the denominator. 8 = 8 = = 5 8 = 0.65 = = 0.77 = 0.7 To see the process for converting repeating decimals to fractions, see problem - in the Core Connections, Course tet, problem 9-05 in the Core Connections, Course tet, or the Math Notes boes referenced above. 8 Core Connections, Courses

9 Number Problems Convert the fraction, decimal, or percent as indicated.. Change 4 to a decimal.. Change 50% into a fraction in lowest terms.. Change 0.75 to a fraction in lowest terms. 4. Change 75% to a decimal. 5. Change 0.8 to a percent. 6. Change 5 7. Change 0. to a fraction. 8. Change 8 to a percent. to a decimal. 9. Change to a decimal. 0. Change 0.08 to a percent.. Change 87% to a decimal.. Change 5 to a percent.. Change 0.4 to a fraction in lowest terms. 4. Change 65% to a fraction in lowest terms. 5. Change 9 to a decimal. 6. Change 5% to a fraction in lowest terms. 7. Change 8 5 to a decimal. 8. Change.5 to a percent. 9. Change to a decimal. 6 Change the decimal to a percent. 0. Change 7 to a decimal.. Change 4% to a fraction. Change the fraction to a decimal.. Change 7 to a decimal. 8 Change the decimal to a percent.. Change 0.75 to a percent. Change the percent to a fraction. 4. Change 0. to a fraction 5. Change 0.75 to a fraction Parent Guide with Etra Practice 9

10 Answers % 6. 0% % % or % ; 6.5% ; %; ; 87.5% = Core Connections, Courses

11 Number OPERATIONS WITH FRACTIONS ADDITION AND SUBTRACTION OF FRACTIONS Before fractions can be added or subtracted, the fractions must have the same denominator, that is, a common denominator. We will present two methods for adding or subtracting fractions. AREA MODEL METHOD Step : Copy the problem.!+! Step : Draw and divide equal-sized rectangles for each fraction. One rectangle is cut vertically into an equal number of pieces based on the first denominator (bottom). The other is cut horizontally, using the second denominator. The number of shaded pieces in each rectangle is based on the numerator (top). Label each rectangle, with the fraction it represents. + + Step : Step 4: Step 5: Superimpose the lines from each rectangle onto the other rectangle, as if one rectangle is placed on top of the other one. Rename the fractions as siths, because the new rectangles are divided into si equal parts. Change the numerators to match the number of siths in each figure. Draw an empty rectangle with siths, then combine all siths by shading the same number of siths in the new rectangle as the total that were shaded in both rectangles from the previous step Parent Guide with Etra Practice

12 Eample!+! 5 can be modeled as: so Thus,!+! 5!=! 9 0. Eample!+! 4 5 would be: = 0 Problems Use the area model method to add the following fractions.. 4!+! 5.!+! 7.!+! 4 Answers = 5 Core Connections, Courses

13 Number IDENTITY PROPERTY OF MULTIPLICATION (GIANT ONE) METHOD The Giant One, known in mathematics as the Identity Property of Multiplication or Multiplicative Identity, uses a fraction with the same numerator and denominator (, for eample) to write an equivalent fraction that helps to create common denominators. For additional information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint materials. Eample Add!+! 4 using the Giant One. Step : Multiply both and by Giant s 4 to get a common denominator.! ! = 4 + Step : Add the numerators of both fractions to get the answer. 4!+!!=! 7 Problems Find each sum or difference. Use the method of your choice..!+! 5. 6!+!. 8!+! !+! !+! !+! !! 8. 4!! !! 0. 4!+!. 5 6!+!. 7 8!+! !! 4. 4!! 5. 5!+! !! 4 7.!! !+! !! !! Parent Guide with Etra Practice

14 Answers =. =. 9 8 = ! = ! 8 7.! ! 5 0.! To summarize addition and subtraction of fractions:. Rename each fraction with equivalents that have a common denominator.. Add or subtract only the numerators, keeping the common denominator.. Simplify if possible. 4 Core Connections, Courses

15 Number ADDITION AND SUBTRACTION OF MIXED NUMBERS To subtract mied numbers, change the mied numbers into fractions greater than one, find a common denominator, then subtract. Other strategies are also possible. For additional information, see the Math Notes bo in Lesson 4.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 4 materials. Eample Find the difference: 5!!!!!!!!!!!! 5!=!6 5!!=!!!! 48 5!!!!!!!!!"!=!! 5! 5!=!" 5 5 5!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!=! 5 =! 8 5 Problems Find each difference..!!! 4. 4!!! !!! !!! 5. 7!!! !!! Answers. 5!!! 4 7!"!0 4!!! 4 7 "! 4.!!! 6!"! 6 6!!! 6!"! 6!or. 7 6!!! 4!"!4!!! 9!"! !!! 5. 7!!! 5!"!!!! 5!"!6!or !"! 8 5 8!!! 8!"!9 4!!! 55 5!!! 64 4!"! 6 5!or 5!"! 65 4!or 7 4 Parent Guide with Etra Practice 5

16 To add mied numbers, it is possible to change the mied numbers into fractions greater than one, find a common denominator, then add. Many times it is more efficient to add the whole numbers, add the fractions after getting a common denominator, and then simplify the answer. For additional information, see the Math Notes bo in Lesson 4.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 4 materials. Eample Find the sum: 8 4!+!4 5!!!!!!!!!!8 4 = 8 + 4!!!! 5!!!=!!! !!!!!!!!+4 5!=!4 + 5!!! 4 4!!!= !!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!!! 0 = 0 Problems Find each sum.. 5 4!+! 6. 5!+! !+!5 4. 5!+! !+! !+! Answers = = = = Core Connections, Courses

17 Number MULTIPLICATION OF FRACTIONS MULTIPLYING FRACTIONS WITH AN AREA MODEL Multiplication of fractions is reviewed using a rectangular area model. Lines that divide the rectangle to represent one fraction are drawn vertically, and the correct number of parts are shaded. Then lines that divide the rectangle to represent the second fraction are drawn horizontally and part of the shaded region is darkened to represent the product of the two fractions. Eample! 5 8 (that is,!of! 5 8 ) Step : Draw a generic rectangle and divide it into 8 pieces vertically. Lightly shade 5 of those pieces. Label it 5 8. Step : Use a horizontal line and divide the generic rectangle in half. Darkly shade!of! 5 and label it. 8 Step : Write a number sentence.! 5 8 = 5 6 The rule for multiplying fractions derived from the models above is to multiply the numerators, then multiply the denominators. Simplify the product when possible. For additional information, see the Math Notes boes in Lesson 5..4 of the Core Connections, Course tet or Lesson..5 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 7A materials. Parent Guide with Etra Practice 7

18 Eample a.!!!"!!!!! 7!!!7!!"!! 4 b. 4! 6!!"!!!!!6!!"!! 8 7 4!!!7 8!!"!! 9 4 Problems Draw an area model for each of the following multiplication problems and write the answer..! 6. 4! 5.! 5 9 Use the rule for multiplying fractions to find the answer for the following problems. Simplify when possible. 4.! 5 5.! ! ! 8.! ! ! 4. 5!. 7!. 8! ! ! ! ! ! ! ! 5 4 Answers = = = 5. 0 = = = = = = = = Core Connections, Courses

19 Number MULTIPLYING MIXED NUMBERS There are two ways to multiply mied numbers. One is with generic rectangles. For additional information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Eample Find the product:!!!. Step : Draw the generic rectangle. Label the top plus. Label the side plus. + + Step : Write the area of each smaller rectangle in each of the four parts of the drawing. + Find the total area:!+!!+!!+! 4!=! 4 = + = = = 4 Step : Write a number sentence:! = 4 Eample Find the product:! 4. 6!+! 4!+!!+!! 6!+! 9!+! 8!+!!! 6 8! = 6 4 = 4 4 = = 4 Problems Use a generic rectangle to find each product.. 4!!!. 6!!!. 4!!! 4.!!! 6 5.!!! Parent Guide with Etra Practice 9

20 Answers You can also multiply mied numbers by changing them to fractions greater than, then multiplying the numerators and multiplying the denominators. Simplify if possible. For more information, see the Math Notes bo in Lesson 5.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 7A materials. Eample!!! 4! 5!!! 5 4! 5!!!5!!!4! 5 8! 8 Problems Find each product, using the method of your choice. Simplify when possible.. 4!!! 8. 5!!! !!! !!! !!! !!! !!! !!! !!! !!! 7 0 Answers Core Connections, Courses

21 Number DIVISION BY FRACTIONS Division by fractions introduces three methods to help students understand how dividing by fractions works. In general, think of division for a problem like 8 as, In 8, how many groups of are there? Similarly,!! 4 means, In, how many fourths are there? For more information, see the Math Notes boes in Lessons 7.. and 7..4 of the Core Connections, Course tet or Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 8B materials. The first two eamples show how to divide fractions using a diagram. Eample Use the rectangular model to divide:!! 4. Step : Step : Using the rectangle, we first divide it into equal pieces. Each piece represents. Shade of it. Then divide the original rectangle into four equal pieces. Each section represents 4. In the shaded section,, there are fourths. 4 Step : Write the equation.!! 4 = Eample In 4, how many s are there? In 4 there is one full That is, 4 =? shaded and half of another one (that is half of one-half). Start with So: 4 = (one and one-half halves) Parent Guide with Etra Practice

22 Problems Use the rectangular model to divide..!! 6.!! 4.!! !! 5.!! 9 Answers. 8 rds 6ths. halves quarters. 4 one fourths 8 siths three fourths 4 fourths 4. quarters halves 5. 4 rds 9ths halves 4 ninths The net two eamples use common denominators to divide by a fraction. Epress both fractions with a common denominator, then divide the first numerator by the second. Eample Eample 4 4 5!=>! 5 0 5!=>! 0!=>! 6 5 or 5 6!=>! 4 6 =>! 8 6 6!=>! 8!or 8 Core Connections, Courses

23 Number One more way to divide fractions is to use the Giant One from previous work with fractions to create a Super Giant One. To use a Super Giant One, write the division problem in fraction form, with a fraction in both the numerator and the denominator. Use the reciprocal of the denominator for the numerator and the denominator in the Super Giant One, multiply the fractions as usual, and simplify the resulting fraction when possible. Eample 5 Eample 6! = 4 = 4 = 4! = 8 4 = 9 = 4 Eample 7 Eample 8 = 4! = 8 9 = 8 9!! 5!!!0 5!!!! 9 5!!!!0 9 Compared to:! = 0 9 = 0 9 = 9 Problems Complete the division problems below. Use any method Answers Parent Guide with Etra Practice

24 OPERATIONS WITH DECIMALS ARITHMETIC OPERATIONS WITH DECIMALS ADDING AND SUBTRACTING DECIMALS: Write the problem in column form with the decimal points in a vertical column. Write in zeros so that all decimal parts of the number have the same number of digits. Add or subtract as with whole numbers. Place the decimal point in the answer aligned with those above. MULTIPLYING DECIMALS: Multiply as with whole numbers. In the product, the number of decimal places is equal to the total number of decimal places in the factors (numbers you multiplied). Sometimes zeros need to be added to place the decimal point. DIVIDING DECIMALS: When dividing a decimal by a whole number, place the decimal point in the answer space directly above the decimal point in the number being divided. Divide as with whole numbers. Sometimes it is necessary to add zeros to the number being divided to complete the division. When dividing decimals or whole numbers by a decimal, the divisor must be multiplied by a power of ten to make it a whole number. The dividend must be multiplied by the same power of ten. Then divide following the same rules for division by a whole number. For additional information, see the Math Notes boes in Lesson 5.. of the Core Connections, Course tet, or Lessons.. and.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint, Checkpoint 7A, and Checkpoint 8B materials. Eample Add 47.7, 8.9, 4.56, and Eample 4 Multiply 0.7 by ( decimal places ) (4 decimal places) (6 decimal places) Eample Subtract from ! Eample 5 Divide.4 by ) Eample Multiply 7. by ( decimal places ) 4. 5 ( decimal places ) (4 decimal places) Eample 6 Divide 7.4 by.. First multiply each number by 0 or ! 74.! Core Connections, Courses

25 Number Problems Parent Guide with Etra Practice 5

26 Divide. Round answers to the hundredth, if necessary Answers or or or , , , Core Connections, Courses

27 Number MULTIPLYING DECIMALS AND PERCENTS Understanding how many places to move the decimal point in a decimal multiplication problem is connected to the multiplication of fractions and place value. Computations involving calculating a percent of a number are simplified by changing the percent to a decimal. Eample Eample Multiply (0.)! (0.). In fractions this means 0! 0 " Knowing that the answer must be in the hundredths place tells you how many places to move the decimal point (to the left) without using the fractions. Multiply (.7)! (0.0). In fractions this means 7 0! 00 " Knowing that the answer must be in the thousandths place tells you how many places to move the decimal point (to the left) without using the fractions. (tenths)(tenths) = hundredths Therefore move two places (tenths)(hundredths) = thousandths Therefore move three places Eample Calculate 7% of.5 without using a calculator. Since 7% = 7 00 = 0.7, 7% of.5 (0.7)!(.5) Parent Guide with Etra Practice 7

28 Problems Identify the number of places to the left to move the decimal point in the product. Do not compute the product.. (0.)! (0.5). (.5)! (0.). (.)!(.6) 4. (0.6)!(.4) 5. 7! (.06) 6. (4.)!(.46) Compute without using a calculator. 7. (0.8)!(0.0) 8. (.)! (0.) 9. (.75)! (0.09) 0. (4.5)!(.). (.8)! (0.0). (7.89)!(6.). 8% of % of % of % of % of % of 4 Answers Core Connections, Courses

29 Number OPERATIONS WITH INTEGERS ADDITION OF INTEGERS Students review addition of integers using two concrete models: movement along a number line and positive and negative integer tiles. To add two integers using a number line, start at the first number and then move the appropriate number of spaces to the right or left depending on whether the second number is positive or negative, respectively. Your final location is the sum of the two integers. To add two integers using integer tiles, a positive number is represented by the appropriate number of (+) tiles and a negative number is represented by the appropriate number of ( ) tiles. To add two integers start with a tile representation of the first integer in a diagram and then place into the diagram a tile representative of the second integer. Any equal number of (+) tiles and ( ) tiles makes zero and can be removed from the diagram. The tiles that remain represent the sum. For additional information, see the Math Notes boes in Lesson.. of the Core Connections, Course tet, or Lesson..4 of the Core Connections, Course tet. Eample!4 + 6 Eample! + (!4)!6!5!4!!! 0 4 5!6!5!4!!! 0 4 5!4 + 6 = Eample 5 + (!6) Eample 4! + (!4) =!6 + 7 Start with tiles representing the first number Add to the diagram tiles representing the second number ! + 7 = Circle the zero pairs. is the answer. +! +! +! +! +! 5 + (!6) =! Parent Guide with Etra Practice 9

30 ADDITION OF INTEGERS IN GENERAL When you add integers using the tile model, zero pairs are only formed if the two numbers have different signs. After you circle the zero pairs, you count the uncircled tiles to find the sum. If the signs are the same, no zero pairs are formed, and you find the sum of the tiles. Integers can be added without building models by using the rules below. If the signs are the same, add the numbers and keep the same sign. If the signs are different, ignore the signs (that is, use the absolute value of each number.) Subtract the number closest to zero from the number farthest from zero. The sign of the answer is the same as the number that is farthest from zero, that is, the number with the greater absolute value. Eample For 4 +, 4 is farther from zero on the number line than, so subtract: 4 =. The answer is, since the 4, that is, the number farthest from zero, is negative in the original problem. Problems Use either model or the rules above to find these sums (!). 6 + (!). 7 + (!7) 4.! !8 + 6.! !5 + (!8) 8.!0 + (!) 9.!+ (!6) 0.!8 + 0.!7 + 5.!6 +.! (!0) + (!) 6.! (!6) ! (!65) !6 + + (!4) (!) + (!) + (!8).!6 + (!) + (!) + 9.!6 + (!) (!70) 5. + (!7) + (!8) (!) 6.!6 + (!) 7.!6 + (!8) (!) (!6) (!70) (!) + (!5) Core Connections, Courses

31 Number Answers SUBTRACTION OF INTEGERS Subtraction of integers may also be represented using the concrete models of number lines and (+) and ( ) tiles. Subtraction is the opposite of addition so it makes sense to do the opposite actions of addition. When using the number line, adding a positive integer moves to the right so subtracting a positive integer moves to the left. Adding a negative integer move to the left so subtracting a negative integer moves to the right. When using the tiles, addition means to place additional tile pieces into the picture and look for zeros to simplify. Subtraction means to remove tile pieces from the picture. Sometimes you will need to place zero pairs in the picture before you have a sufficient number of the desired pieces to remove. For additional information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Eample 4! 6 6 Eample!! (!4) ( 4)!6!5!4!!! 0 4 5!6!5!4!!! ! 6 =!!! (!4) = Parent Guide with Etra Practice

32 Eample!6! (!) Build the first integer. Remove three negatives. Three negatives are left.!6! (!) =! Eample 4!! (!) Build the first integer It is not possible to remove three negatives so add some zeros. Now remove three negatives and circle any zeros. One positive remains !! (!) = Problems Find each difference. Use one of the models for at least the first five differences..!6! (!).! (!). 6! (!) 4.! ! (!) 6. 7! 7. 5! () 8.!! (!0) 9.!! 0 0.! (!0).!6! (!)! 5. 6! (!)! 5. 8! (!8) 4.!9! 9 5.!9! 9! (!9) Answers (and possible models). 4. 5!6!5!4!!! !6!5!4!!! Core Connections, Courses

33 Number CONNECTING ADDITION AND SUBTRACTION In the net si eamples, compare (a) to (b), (c) to (d), and (e) to (f). Notice that eamples (a), (c), and (e) are subtraction problems and eamples (b), (d), and (f) are addition problems. The answers to each pair of eamples are the same. Also notice that the second integers in the pairs are opposites (that is, they are the same distance from zero on opposite sides of the number line) while the first integers in each pair are the same a.! (!6)! (!6) = 8 b = 8 c.!! (!4)!! (!4) = + + d.! ! + 4 = e.!4! (!)!4! (!) =! + f.! !4 + =! You can conclude that subtracting an integer is the same as adding its opposite. This fact is summarized below. SUBTRACTION OF INTEGERS IN GENERAL To find the difference of two integers, change the subtraction sign to an addition sign. Net change the sign of the integer you are subtracting, and then apply the rules for addition of integers. For more information on the rules for subtracting integers, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Parent Guide with Etra Practice

34 Eamples Use the rule for subtracting integers stated above to find each difference (that is, subtract). a. 9! (!) becomes 9 + (+) = b.!9! (!) becomes!9 + (+) = c.!9! becomes!9 + (!) =! d. 9! becomes 9 + (!) =! Problems Use the rule stated above to find each difference.. 9! (!). 9!.!9! 4.!9! (!) 5.!4! 5 6.!6! (!5) 7.!40! 6 8.!40! (!6) 9. 40! ! (!6).!5! (!)! 5! 6.!5!! (!5)! (!6). 5!! (!5)! ! (!4)! 6! (!7) 5.!5! (!5)! ! (!6) 7.! ! 5 9.!! 0.!7!.!0! 5.!0! 7.!! (!) 4.!! (!4) 5. 0! (!) 6. 5! (!9) 7. 7! (!7) 8. 5! 9.!58! 7 0.!79! (!).!6! 8.!06! 4. 47! (!55) 4. 57! 49 5.!00! (!00) Answers Core Connections, Courses

35 Number MULTIPLICATION AND DIVISION OF INTEGERS Multiply and divide integers two at a time. If the signs are the same, their product will be positive. If the signs are different, their product will be negative. Follow the same rules for fractions and decimals. Remember to apply the correct order of operations when you are working with more than one operation. For additional information, see the Math Notes bo in Lesson..4 of the Core Connections, Course tet. Eamples a.! = 6 or! = 6 b.! " (!) = 6 or (+)! (+) = 6 c. = or = d. (!) (!) = or (!) (!) = e. (!) " =!6 or! (") = "6 f. (!) =! or (!) =! g. 9!("7) = "6 or!7 " 9 =!6 h.!6 9 =!7 or 9 (!6) =! 7 Problems Use the rules above to find each product or quotient.. ( 4)(). ( )(4). ( )(5) 4. ( )(8) 5. (4)( 9) 6. ()( 8) 7. (45)( ) 8. (05)( 7) 9. ( 7)( 6) 0. ( 7)( 9). ( )( 8). ( 7)( 4). ( 8)( 4)() 4. ( )( )( ) 5. ( 5)( )(8)(4) 6. ( 5)( 4)( 6)( ) 7. ( )( 5)(4)(8) 8. ( )( 5)( 4)( 8) 9. ( )( 5)(4)( 8) 0. ( 5)(4)( 8). 0 ( 5). 8 ( ). 96 ( ) 4. 8 ( 6) ( 4) ( 5). 08 ( ). 6. ( ) ( 6) ( 4) ( 7) ( 5) ( 4) Parent Guide with Etra Practice 5

36 Answers Core Connections, Courses

37 Number DIAMOND PROBLEMS In every Diamond Problem, the product of the two side numbers (left and right) is the top number and their sum is the bottom number. Diamond Problems are an ecellent way of practicing addition, subtraction, multiplication, and division of positive and negative integers, decimals and fractions. They have the added benefit of preparing students for factoring binomials in algebra. product ab a b a + b sum Eample 0 0 The top number is the product of 0 and 0, or 00. The bottom number is the sum of 0 and 0, or = Eample 8 The product of the right number and is 8. Thus, if you divide 8 by you get 4, the right number. The sum of and 4 is 6, the bottom number Eample 6 4 To get the left number, subtract 4 from 6, 6 4 =. The product of and 4 is 8, the top number Eample 4 8 The easiest way to find the side numbers in a situation like this one is to look at all the pairs of factors of 8. They are: and 8, and 4, 4 and, and 8 and. Only one of these pairs has a sum of : and 4. Thus, the side numbers are and Parent Guide with Etra Practice 7

38 Problems Complete each of the following Diamond Problems y a 8b b a 7a Answers. and 4. 4 and 6. 6 and and and and and and and! and 50. and 7 5. and. y and + y 4. a and a 5. 6b and 48b 6. 4a and a 8 Core Connections, Courses

39 Number ABSOLUTE VALUE The absolute value of a number is the distance of the number from zero. Since the absolute value represents a distance, without regard to direction, absolute value is always non-negative The symbol for absolute value is. On the number line above, both 5 and 5 are 5 units from zero. This distance is displayed as!5 = 5 and is read, The absolute value of negative five equals five. Similarly, 5 = 5 means, The absolute value of five is five. = 5 means that could be either 5 or 5 because both of those points are five units from zero. The problem =!5 has no solution because the absolute value of a number has to be positive. Do not confuse this fact when a negative sign appears outside the absolute value symbol. For additional information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Eamples a.!6 = 6 b. 7 = 7 c. = 9! = 9 or 9 d. =! no solution e.! =!! = or f.! 8 =!5 = 5 Part (d) has no solution, since any absolute value is positive. In part (e), the problem asks for the opposite of, which is negative. Problems Determine the absolute value or the values of..!.. = 4 4. = 6 5. = 4 6. = 7.!9 8. =! 9.! =! 0.! 7. = 7.!7. 5! 8 4.!6! 5.!6 + Make a table using -values from 4 to 4 to draw a graph of each equation. 6. y = 7. y =! 8. y = + 9. y = + 0. y =! Parent Guide with Etra Practice 9

40 Answers... 4, , , 4 6., no solution 9., , y y y y y ORDER OF OPERATIONS When students are first given epressions like + 4, some students think the answer is 4 and some think the answer is. This is why mathematicians decided on a method to simplify an epression that uses more than one operation so that everyone can agree on the answer. There is a set of rules to follow that provides a consistent way for everyone to evaluate epressions. These rules, called the Order of Operations, must be followed in order to arrive at a correct answer. As indicated by the name, these rules state the order in which the mathematical operations are to be completed. For additional information, see the Math Notes boes in Lesson 6.. of the Core Connections, Course tet, Lesson.. of the Core Connections, Course tet, or Lesson..4 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 5 materials or the Core Connections, Course Checkpoint materials. The first step is to organize the numerical epression into parts called terms, which are single numbers or products of numbers. A numerical epression is made up of a sum or difference of terms. Eamples of numerical terms are: 4, (6), 6(9 4),, (5 + ), and 6!4 6. For the problem above, + 4, the terms are circled at right. + 4 Each term is simplified separately, giving + 8. Then the terms are added: + 8 =. Thus, + 4 =. 40 Core Connections, Courses

41 Number Eample To evaluate an epression:! + (6 " ) + 0 Circle each term in the epression. Simplify each term until it is one number by: Simplifying the epressions within the parentheses. Evaluating each eponential part (e.g., ). Multiplying and dividing from left to right. Finally, combine terms by adding or subtracting from left to right. + (6 ) () () Eample 5! ( )! 5 a. Circle the terms. b. Simplify inside the parentheses. c. Simplify the eponents. d. Multiply and divide from left to right. Finally, add and subtract from left to right. a. 5!!!8!+!6 ( )!!!5 b. 5!!!8!+!6 ( 9)!!!5 c. 5!!!8 4!+!6 ( 9)!!!5 d Eample a. Circle the terms. b. Multiply and divide left to right, including eponents. Add or subtract from left to right. a. 0!+! 5+7!!!4!+!! 4 b Parent Guide with Etra Practice 4

42 Problems Circle the terms, then simplify each epression.. 5! (9 4) (7 + ) (8 + ) ! " (4 " 5) ! ! (7 7) (5 ) + (9 + ) () 6 + (6 ) ! 5 5. (7 ) (9 ) ( + 4) (6 + 4) + (5 ) ( 5 ) +! ( 5 ") 8 Answers Core Connections, Courses

43 Number CHOOSING A SCALE The ais (or aes) of a graph must be marked with equal-sized spaces called intervals. Marking the uniform intervals on the aes is called scaling the aes. The difference between consecutive markings tells the size (scale) of each interval. Note that each ais of a two-dimensional graph may use a different scale. Sometimes the ais or set of aes is not provided. A student must count the number of usable spaces on the graph paper. How many spaces are usable depends in part on how large the graph will be and how much space will be needed for labeling beside each ais. Follow these steps to scale each ais of a graph.. Find the difference between the smallest and largest numbers (the range) you need to put on an ais.. Count the number of intervals (spaces) you have on your ais.. Divide the range by the number of intervals to find the interval size. 4. Label the marks on the ais using the interval size. Sometimes dividing the range by the number of intervals produces an interval size that makes it difficult to interpret the location of points on the graph. The student may then eercise judgment and round the interval size up (always up, if rounded at all) to a number that is convenient to use. Interval sizes like,, 5, 0, 0, 5, 50, 00, etc., work well. For more information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. Eample. The difference between 0 and 60 is 60.. The number line is divided into 5 equal intervals.. 60 divided by 5 is. 4. The marks are labeled with multiples of the interval size Eample. The difference between 00 and 0 is 00.. There are 4 intervals = The ais is labeled with multiples of Parent Guide with Etra Practice 4

44 Eample. The difference on the vertical ais is = 750. (The origin is (0, 0).) On the horizontal ais the range is 6 0 = 6.. There are 5 spaces vertically and spaces horizontally.. The vertical interval size is = 50. The horizontal interval is 6 =. 4. The aes are labeled appropriately Eample 4 Sometimes the aes etend in the negative direction.. The range is 0 ( 5) = 5.. There are 7 intervals along the line = 5 4. Label the aes with multiples of five Problems Scale each ais: Core Connections, Courses

45 Number 7. y 8. y Use fractions. y y Answers., 4, 6, 8, 0,. 9, 6,, 0,, 6. 86, 0, 8, 4 4., 8, 8, 8, 8 5.,, 0, 9, , 6, 4,, 0, 8 7. :, 4, 6, 8, y: 4, 8,, 6, 4 9. : 60, 0, 80, 40, 60 y: 40, 80, 0, 60, :, 6, 9, 5, 8 y: 4, 8,, 0, 4 0. :!! 4,, 4, 4, y :!!,,,, Parent Guide with Etra Practice 45

46 FOUR-QUADRANT GRAPHING The graphing that was started in earlier grades is now etended to include negative values, and students will graph algebraic equations with two variables. For addition information, see problem - in the Core Connections, Course tet and problem -40 in the Core Connections, Course tet. GRAPHING POINTS Points on a coordinate graph are identified by two numbers in an ordered pair written as (, y). The first number is the coordinate of the point and the second number is the y coordinate. Taken together, the two coordinates name eactly one point on the graph. The eamples below show how to place a point on an y coordinate graph. Eample Graph point A(, ). Go right units from the origin (0, 0), then go down units. Mark the point. y Eample Plot the point C( 4, 0) on a coordinate grid. Go to the left from the origin 4 units, but do not go up or down. Mark the point. y C A(, ) 46 Core Connections, Courses

47 Number Problems. Name the coordinate pair for each point shown on the grid below.. Use the ordered pair to locate each point on a coordinate grid. Place a dot at each point and label it with its letter name. Z y U S K(0, 4) L( 5, 0) M(, ) N(, ) y W O(, ) P( 4, 6) V Q(4, 5) R( 5, 4) T T(, 6) Answers. S(, ) T(, 6) U(0, 6) V(, 4) W( 6, 0) Z( 5, ). L N y M R P T K O Q Parent Guide with Etra Practice 47

48 DESCRIBING AND EXTENDING PATTERNS Students are asked to use their observations and pattern recognition skills to etend patterns and predict the number of dots that will be in a figure that is too large to draw. Later, variables will be used to describe the patterns. Eample Eamine the dot pattern at right. Assuming the pattern continues: a. Draw Figure 4. b. How many dots will be in Figure 0? Figure Figure Figure Solution: The horizontal dots are one more than the figure number and the vertical dots are even numbers (or, twice the figure number). Figure 4 Figure has dots, Figure has 6 dots, and Figure has 9 dots. The number of dots is the figure number multiplied by three. Figure 0 has 0 dots. Problems For each dot pattern, draw the net figure and determine the number of dots in Figure 0... Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure Figure 48 Core Connections, Courses

49 Epressions, Equations, and Functions Answers. 50 dots. dots. 0 dots Figure 4 Figure 5 Figure 4 4. dots dots dots Figure 5 Figure 4 Figure 4 DISTRIBUTIVE PROPERTY The Distributive Property shows how to epress sums and products in two ways: a(b + c) = ab + ac. This can also be written (b + c)a = ab + ac. To simplify: Factored form Distributed form Simplified form a(b + c) a(b) + a(c) ab + ac Multiply each term on the inside of the parentheses by the term on the outside. Combine terms if possible. For additional information, see the Math Notes boes in Lessons..4 and 7.. of the Core Connections, Course tet, Lesson 4.. of the Core Connections, Course tet, or Lesson..5 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 8A materials. Eample Eample Eample (47) = (40 + 7) = (! 40) + (! 7) = = 94 ( + 4) = (! ) + (! 4) = + 4( + y + ) = (4! ) + (4! y) + 4() = 4 + y + 4 Parent Guide with Etra Practice 49

50 Problems Simplify each epression below by applying the Distributive Property.. 6(9 + 4). 4(9 + 8). 7(8 + 6) 4. 5(7 + 4) 5. (7) = (0 + 7) 6. 6(46) = 6(40 + 6) 7. 8(4) 8. 6(78) 9. ( + 6) 0. 5( + 7). 8(! 4). 6(! 0). (8 + )4 4. ( + )5 5.!7( + ) 6.!4(y + ) 7.!(y! 5) 8.!5(b! 4) 9.!( + 6) 0.!( + 7).!(! 4).!(!! ). ( + ) 4. 4( + ) 5.!(5! 7) 6.!(! 6) Answers. (6 9) + (6 4) = = 78. (4 9) + (4 8) = 6 + = = = = = = = ! !7! 7 6.!4y! 7.!y !5b !! 6 0.!! 7.! ! ! Core Connections, Courses

51 Epressions, Equations, and Functions When the Distributive Property is used to reverse, it is called factoring. Factoring changes a sum of terms (no parentheses) to a product (with parentheses.) ab + ac = a(b + c) To factor: Write the common factor of all the terms outside of the parentheses. Place the remaining factors of each of the original terms inside of the parentheses. For additional eamples and practice, see the Core Connections, Course Checkpoint 8A materials. Eample = 4! + 4! = 4( + ) Eample 5 6! 9 = "! " = (! ) Eample 6 6 +y + =! +! 4y +! = ( + 4y +) Problems Factor each epression below by using the Distributive Property in reverse y! z 4. + y 5. 8m y m! 8. 5y!0 9.!0 0.! 6.! 6. 5y y + 4z y ! ! + y Answers. 6( + ). 5(y! ). 4( + 5z) 4. ( + y) 5. 8(m + ) 6. 8(y + 5) 7. 4(m!) 8. 5(5y! ) 9. (! 5) 0. (! ). (! ). 5(y + 7). 4( + y + z) 4. 6( + y +) 5. 7(! 7 + 4) 6. (!+ y) Parent Guide with Etra Practice 5

52 VARIABLE EXPRESSIONS A variable is a symbol used to represent one or more numbers. It is common to use letters of the alphabet for variables. The value of a variable used several times in one epression must be the same. For additional practice and more eamples, see the Core Connections, Course Checkpoint 6 materials. Eample If the unknown distance of Cecil s hop is represented by the variable h, write an epression for: a. Three equal hops!!!h + h + h or h b. Five equal hops!!!h + h + h + h + h or 5h c. Two equal hops and walking feet!!!h + h + or h + Eample If the unknown cost of a banana is b, and the unknown cost of an apple is a, write an epression for the cost of: a. Three bananas and two apples!!!b + b + b + a + a or b + a b. One banana and apples!!!b + a + a + a or b + a c. One banana, one apple, a $ item, and a $ item!!!b + a + + or b + a Core Connections, Courses

53 Epressions, Equations, and Functions Problems If the unknown distance of Cecil s jump is represented by J, write an epression for:. Three jumps. Si jumps. Four jumps and walking 5 feet 4. Walking feet, two jumps, walking feet If the unknown distance of Cecil s jump is represented by J, and the unknown distance of Cecil s hop is represented by H, write an epression for: 5. Two jumps and two hops 6. One jump, three hops, and two jumps 7. One jump, three hops and walking 7 feet 8. Walking 6 feet, three hops, and two jumps If the unknown cost of a taco is T, and the unknown cost of a carton of milk is M, write an epression for the cost of: 9. Three tacos and two milks 0. One taco and four milks. One taco, one milk and two tacos, one milk. Two tacos, one milk, and a $ item Answers. J + J + J = J. J + J + J + J + J + J = 6J. J + J + J + J + 5 = 4J J + J + = J J + J + H + H = J + H 6. J + H + H + H + J + J = J + H 7. J + H + H + H + 7 = J + H H + H + H + J + J = H + J T + M 0. T + 4M. T + M + T + M = T + M. T + M + Parent Guide with Etra Practice 5

54 USING VARIABLES TO GENERALIZE Previously, students etended patterns and predicted subsequent figures. Now students use their knowledge of variables to generalize the pattern they observe. For additional information, see the Math Notes bo in Lesson 4.. of the Core Connections, Course tet. Eample Eamine the dot pattern below. Draw the net figure, state the number of dots in Figure 5, and give a variable epression for the number of dots in figure n. Figure Figure Figure The net figure is: Figure 4 Each figure is a square with a side length that is the same as the figure number, so Figure 5 has 5!5 = 5 dots and figure n would have n! n = n dots. Eample Eamine the dot pattern below. Draw the net figure, state the number of dots in Figure 5, and give a variable epression for the number of dots in figure n. Figure Figure Figure The net figure is: Figure 4 Each figure is an L shape with the number of dots one more than twice the figure number of dots so figure 5 has = dots. Figure n would have n + dots. Another way to see the pattern is as the figure number plus the figure number plus. This pattern then gives = for Figure 5 and n + (n + ) for Figure n. Of course, n + (n + ) = n Core Connections, Courses

55 Epressions, Equations, and Functions Problems Eamine each dot pattern below. Draw the net figure, tell the number of dots in figure 5, and give a variable epression for the number of dots in figure n... Figure Figure Figure. 4. Figure Figure Figure Figure Figure Figure Figure Figure Figure 5. Figure Figure Figure Answers Note: In each answer, n represents the figure number.. 60 dots. 45 dots. Figure 4 4! n = 4n dots 9 dots 4. Figure 4! n = n dots 40 dots Figure 4 n + 4 dots The base of each figure is 4 dots, plus the number of dots that matches the figure number. Figure 4 n! (n + ) dots 5. 0 dots Figure 4 n!(n+) dots The figures in this pattern are half of the figures in problem 4. Parent Guide with Etra Practice 55

56 SUBSTITUTION AND EVALUATION OF EXPRESSIONS Substitution is replacing one symbol with an equivalent symbol (a number, a variable, or an epression). One application of the substitution property is replacing a variable name with a number in an epression or equation. A variable is a letter used to represent one or more numbers (or other algebraic epression). The numbers are the values of the variable. A variable epression has numbers and variables with arithmetic operations performed on it. In general, if a = b, then a may replace b and b may replace a. After numerical substitutions have been made, following the order of operations and doing the arithmetic will correctly evaluate the epression. For additional information, see the Math Notes bo in Lesson 4.. of the Core Connections, Course tet, Lesson.. of the Core Connections, Course tet, or Lesson..4 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 8A materials or the Core Connections, Course Checkpoint 6 materials. Eamples Evaluate each variable epression for =. a. 5! 5()! 5 b. + 0! () + 0! c. 8! 8! 6 d.!! e.! 5 " ()! 5 " 9! 5 " 4 f. 5 +! 5() + ()! 5 + 9! 4 Problems Evaluate each of the variable epressions below for = 4 and y =. Be sure to follow the Order of Operations as you simplify each epression !. + y + 4. y! + 5.! 5 6.! ! 8.! y 0. y +!. + y +. + y! Evaluate the epressions below using the values of the variables in each problem. These problems ask you to evaluate each epression twice, once with each of the values..! + 5 for =! and = 4 4.! + 6 for =! and = 5 5.! + 7 for =! and = 6.! + 5 for =!4 and = 5 56 Core Connections, Courses

57 Epressions, Equations, and Functions Evaluate the variable epressions for = 4 and y =. ( ) 7. ( + ) 8. ( + ) 9. ( + y) + 4 y+ ( ( )). y +! y 0. y + +7 ( ). + y ( ) ( + 4y) Answers ; ; ; ; ALGEBRA TILES AND PERIMETER Algebraic epressions can be represented by the perimeters of algebra tiles (rectangles and squares) and combinations of algebra tiles. The dimensions of each tile are shown along its sides and the tile is named by its area as shown on the tile itself in the figures at right. When using the tiles, perimeter is the distance around the eterior of a figure. For additional information, see the Math Notes bo in Lesson 6..4 of the Core Connections, Course tet. Eample Eample P = units P = units Problems Parent Guide with Etra Practice 57

58 Determine the perimeter of each figure Answers un un un un un un un un. 58 Core Connections, Courses

59 Epressions, Equations, and Functions COMBINING LIKE TERMS Algebraic epressions can also be simplified by combining (adding or subtracting) terms that have the same variable(s) raised to the same powers, into one term. The skill of combining like terms is necessary for solving equations. For additional information, see the Math Notes bo in Lesson 6..4 of the Core Connections, Course tet, Lesson 4.. of the Core Connections, Course tet, or Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 7A materials. Eample Combine like terms to simplify the epression All these terms have as the variable, so they are combined into one term, 5. Eample Simplify the epression The terms with can be combined. The terms without variables (the constants) can also be combined Note that in the simplified form the term with the variable is listed before the constant term. Eample Simplify the epression Note that terms with the same variable but with different eponents are not combined and are listed in order of decreasing power of the variable, in simplified form, with the constant term last. Parent Guide with Etra Practice 59

60 Eample 4 The algebra tiles, as shown in the Perimeter Using Algebra Tiles section, are used to model how to combine like terms. The large square represents, the rectangle represents, and the small square represents one. We can only combine tiles that are alike: large squares with large squares, rectangles with rectangles, and small squares with small squares. If we want to combine: and , visualize the tiles to help combine the like terms: ( large squares) + ( rectangles) + 4 (4 small squares) + ( large squares) + 5 (5 rectangles) + 7 (7 small squares) The combination of the two sets of tiles, written algebraically, is: Eample 5 Sometimes it is helpful to take an epression that is written horizontally, circle the terms with their signs, and rewrite like terms in vertical columns before you combine them: Problems Combine the following sets of terms. ( 5 + 6) + ( + 4 9) ! ! 9 5!! This procedure may make it easier to identify the terms as well as the sign of each term.. ( ) + (4 + + ). ( + + 4) + ( ). (8 + ) + ( ) 4. ( ) ( + + 4) 5. (4 7 + ) + ( 5) 6. ( 7) ( + 9) 7. (5 + 6) + ( ) c + 4c ( 4c ) a + a 4a + 6a + 4a + Answers c + 4c + 0. a a + a Core Connections, Courses

61 Epressions, Equations, and Functions GRAPHING AND SOLVING INEQUALITIES GRAPHING INEQUALITIES The solutions to an equation can be represented as a point (or points) on the number line. If the epression comparison mat has a range of solutions, the solution is epressed as an inequality represented by a ray or segment with solid or open endpoints. Solid endpoints indicate that the endpoint is included in the solution ( or ), while the open dot indicates that it is not part of the solution (< or >). For additional information, see the Math Notes bo in Lesson 7..4 of the Core Connections, Course tet. Eample > 6 Eample y < Eample 0 Eample 4 y Problems Graph each inequality on a number line.. m <.. y < < 6. < 7. m > Answers Parent Guide with Etra Practice 6

62 SOLVING INEQUALITIES To solve an inequality, eamine both of the epressions on an epression comparison mat. Use the result as a dividing point on the number line. Then test a value from each side of the dividing point on the number line in the inequality. If the test number is true, then that part of the number line is part of the solution. In addition, if the inequality is or, then the dividing point is part of the solution and is indicated by a solid dot. If the inequality is > or <, then the dividing point is not part of the solution, indicated by an open dot. For additional information, see the Math Notes bo in Lesson 6..4 of the Core Connections, Course tet. Eample 9 m + Solve the equation: 9 = m + 7 = m Draw a number line. Put a solid dot at 7. Test a number on each side of 7 in the original inequality. We use 0 and 0. TRUE 7 FALSE Eample < + 6 Solve the equation: = + 6 = + 9 = 9 = Draw a number line. Put an open dot at. Test 0 and 4 in the original inequality. FALSE TRUE m = 0 7 m = 0 = 4 = 0 9 > > 0 + ( 4) < (0) < > 9 > 8 < < 6 TRUE FALSE 5 < TRUE The solution is m 7. FALSE The solution is >. 7 6 Core Connections, Courses

63 Epressions, Equations, and Functions Problems Solve each inequality.. + >. y m < y m + 7. < (m + ) m 9. m + m + 7 Answers. > 4. y m 4 5. y < 5 6. m 7. < 8. m 5 9. m SIMPLIFYING EXPRESSIONS (ON AN EXPRESSION MAT) Single Region Epression Mats (Core Connections, Course ) Algebra tiles and Epression Mats are concrete organizational tools used to represent algebraic epressions. Pairs of Epression Mats can be modified to make Epression Comparison Mats (see net section) and Equation Mats. Positive tiles are shaded and negative tiles are blank. A matching pair of tiles with one tile shaded and the other one blank represents zero (0). Eample Eample Represent! +. Represent (! ). = + = = + = Note that (! ) =! 6. Parent Guide with Etra Practice 6

64 Eample Eample 4 This epression makes zero. Simplify (!) + (!). = + = = + = After removing zeros,! remains. Problems Simplify each epression. = + = ! ! ! + +! 0. + (!) + 5!.! + + (!)! 4. ( + )!.! +! + 4. (! ) +! 5. ( + ) +! 64 Core Connections, Courses

65 Epressions, Equations, and Functions Answers..!.! 4.! +! 4 5.! ! 7.! 8.! ! +.!! !! ! Two Region Epression Mats (Core Connections, Course ) An Epression Mat is an organizational tool that is used to represent algebraic epressions. Pairs of Epression Mats can be modified to make an Equation Mat. The upper half of an Epression Mat is the positive region and the lower half is the negative region. Positive algebra tiles are shaded and negative tiles are blank. A matching pair of tiles with one tile shaded and the other one blank represents zero (0). Tiles may be removed from or moved on an epression mat in one of three ways: () removing the same number of opposite tiles in the same region; () flipping a tile from one region to another. Such moves create opposites of the original tile, so a shaded tile becomes un-shaded and an un-shaded tile becomes shaded; and () removing an equal number of identical tiles from both the + and - regions. See the Math Notes bo in Lesson..6 of the Core Connections, Course tet. Eamples can be represented various ways = + = _ The Epression Mats at right all represent zero. + _ + _ + _ = + = Parent Guide with Etra Practice 65

66 Eample +! (! ) Epressions can be simplified by moving tiles to the top (change the sign) and looking for zeros. + _ + _ + _ = + = Eample! (y! ) + y! + y + y + _ y y _ y y _ y = + =! (y! ) + y!! y + + y!!y + Problems Simplify each epression _ y y y y y y _ y 66 Core Connections, Courses

67 Epressions, Equations, and Functions ! 4! 7 8.!! 4! 7 9.!(! + ) 0. 4! ( + ). 5! (! + ).! 5! (! ).! y! y 4.! ! 5.! (y + 5) 6.!( + y) y 7.! 7! (! 7) 8.!( + y + )! + y Answers y + 4.! y! 6.!y !! 8.!5! 7 9.! 0.!. 8!.! 7.!4y !y! 6. + y !4! y! COMPARING QUANTITIES (ON AN EXPRESSION MAT) Combining two Epression Mats into an Epression Comparison Mat creates a concrete model for simplifying (and later solving) inequalities and equations. Tiles may be removed or moved on the mat in the following ways: () Removing the same number of opposite tiles (zeros) on the same side; () Removing an equal number of identical tiles (balanced set) from both the left and right sides; () Adding the same number of opposite tiles (zeros) on the same side; and (4) Adding an equal number of identical tiles (balanced set) to both the left and right sides. These strategies are called legal moves. After moving and simplifying the Epression Comparison Mat, students are asked to tell which side is greater. Sometimes it is only possible to tell which side is greater if you know possible values of the variable. Parent Guide with Etra Practice 67

68 Eample Determine which side is greater by using legal moves to simplify. Step Remove balanced set Mat A Mat B Step Remove zeros Mat A Mat B Step Remove balanced set Mat A Mat B = + =??? The left side is greater because after Step : 4 > 0. Also, after Step : 6 >. Note that this eample shows only one of several possible strategies. Eample Use legal moves so that all the -variables are on one side and all the unit tiles are on the other. Step Add balanced set Step Add balanced set Step Remove zeros Mat A Mat B Mat A Mat B Mat A Mat B??? What remains is on Mat A and 4 on Mat B. There are other possible arrangements. Whatever the arrangement, it is not possible to tell which side is greater because we do not know the value of. Students are epected to record the results algebraically as directed by the teacher. One possible recording is shown at right. Mat A!! + +! + Mat B! +! + +! Core Connections, Courses

69 Epressions, Equations, and Functions Problems For each of the problems below, use the strategies of removing zeros or simplifying by removing balanced sets to determine which side is greater, if possible. Record your steps. = + =. Mat A Mat B.. Mat A Mat B?? Mat A? Mat B 4. Mat A: 5 + (!8) Mat B:! Mat A: ( + )! Mat B: 4!! Mat A: 4 + (!) + 4 Mat B: + +! For each of the problems below, use the strategies of removing zeros or adding/removing balanced sets so that all the -variables are on one side and the unit tiles are on the other. Record your steps. 7. Mat A Mat B 8. Mat A Mat B 9.?? Mat A? Mat B 0. Mat A:! Mat B: +. Mat A: (!5) Mat B: + + (!8). Mat A: + Mat B:!! Answers (Answers to problems 7 through may vary.). A is greater. B is greater. not possible to tell 4. B is greater 5. not possible to tell 6. A is greater 7. A: ; B: 8. A: ; B: 9. A: ; B: 0. A: ; B:. A: ; B:. A: ; B: 6 Parent Guide with Etra Practice 69

70 SOLVING WORD PROBLEMS (THE 5-D PROCESS) The 5-D process is one method that students can use to solve various types of problems, especially word problems. The D s stand for Describe, Define, Do, Decide, and Declare. When students use the 5-D process, it provides a record of the student s thinking. The patterns in the table lead directly to writing algebraic equations for the word problems. Writing equations is one of the most important algebra skills students learn. Using the 5-D process helps to make this skill accessible to all students. In order to help students see the relationships in a word problem, we require them to include at least four entries (rows) in their tables. The repetition of the operations is needed to see how the columns are related. After students have had practice using the 5-D process to solve problems, we begin generalizing from the patterns in the table to write an equation that represents the relationships in the problem. We also believe that writing the answer in a sentence after the table is complete is important because many students forget what the question actually was. The sentence helps the student see the big picture and brings closure to the problem. See the Math Notes bo in Lesson 5.. of the Core Connections, Course tet Lesson.. of the Core Connections, Course tet for a detailed, step-by-step demonstration of a word problem similar to the one below that is solved using the 5-D process. Eample A bo of fruit has three times as many nectarines as grapefruit. Together there are 6 pieces of fruit. How many pieces of each type of fruit are there? Step : Describe: Number of nectarines is three times the number of grapefruit. Number of nectarines plus number of grapefruit equals 6. Step : Define: Set up a table with columns. The first column should be the item you know the least about. Choose any easy amount for that column. Define # of Grapefruit Trial : What else do we need to know? The number of nectarines, which is three times the number of grapefruit. Define # of Grapefruit # of Nectarines Trial : () = Eample continues on net page 70 Core Connections, Courses

71 Epressions, Equations, and Functions Eample continued from previous page. Step : Do: What is the total number of fruit? Define Do # of Grapefruit # of Nectarines Total Pieces of Fruit Trial : 44 Step 4: Decide: We need to check the total pieces of fruit based on trial # of grapefruit and compare it to the total given in the problem. Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? Trial : 44 too high Start another trial. Our total was 44; the total needed is 6, so our trial started too high and our net trial should start lower. Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? Trial : 44 too high Trial : too high Start another trial. Our total was 40; the total needed is 6, so our trial started too high and our net trial should start still lower. Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? Trial : 44 too high Trial : too high Trial : 8 4 too low Start another trial. Our total was ; the total needed is 6, so our trial started too low and our net one should be higher than 8 but lower than 0. Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? Trial : 44 too high Trial : too high Trial : 8 4 too low Trial 4: correct Step 5: Declare: The answer was found. Answer the question in a sentence. There are 9 grapefruit and 7 nectarines in the bo. Parent Guide with Etra Practice 7

72 Eample The perimeter of a rectangle is 0 feet. If the length of the rectangle is ten feet more than the width, what are the dimensions (length and width) of the rectangle? Describe/Draw: width + 0 width Start with the width because, of the two required answers, it is the one we know the least about. The length is 0 feet more than the width, so add 0 to the first trial. Define Do Decide Width Length Perimeter 0? Trial : 0 0 (0 + 0) = 60 too low Since the trial of 0 resulted in an answer that is too low, we should increase the number in the net trial. Pay close attention to the result of each trial. Each result helps determine the net trial as you narrow down the possible trials to reach the answer. Note: As students get more eperience with using the 5-D process, they learn to make better-educated trials from one step to the net to solve problems quickly or to establish the pattern they need to write an equation. Define Do Decide Width Length Perimeter 0? Trial : 0 0 (0 + 0) = 60 too low Trial : too low Trial : too high Trial 4: correct Declare: The dimensions are 5 and 5 feet. 7 Core Connections, Courses

73 Epressions, Equations, and Functions Eample Jorge has some dimes and quarters. He has 0 more dimes than quarters and the collection of coins is worth $.40. How many dimes and quarters does Jorge have? Note: This type of problem is more difficult than others because the number of things asked for is different than their value. Separate columns for each part of the problem must be added to the table as shown below. Students often neglect to write the third and fourth columns. Describe: The number of quarters plus 0 equals the number of dimes. The total value of the coins is $.40. Define Do Decide # # Value of Value of Quarters Dimes Quarters Dimes Total Value $.40? Trial : too high Trial : too high Trial : too high Trial 4: correct Declare: Jorge has four quarters and 4 dimes. HELPFUL QUESTIONS TO ASK YOUR STUDENT If your student is having difficulty with a 5-D problem, it may be because he/she does not understand the problem, not because he/she does not understand the 5-D process. Here are some helpful questions to ask when your child does not understand the problem. (These are useful in non-word problem situations, too.). What are you being asked to find?. What information have you been given?. Is there any unneeded information? If so, what is it? 4. Is there any necessary information that is missing? If so, what information do you need? TIPS ABOUT COLUMN TITLES. You may select any number for the first trial. Ten or the student s age are adequate numbers for the first trial. The result will help you to determine the number to use for the second trial.. Continue establishing columns by asking, What else do we need to know to determine whether the number we used for our trial is correct or too low or too high?. Put the answer to one calculation in each column. Students sometimes try to put the answer to several mental calculations in one column. (See the note in Eample.) Parent Guide with Etra Practice 7

74 Problems Solve these problems using the 5-D process. Write each answer in a sentence.. A wood board 00 centimeters long is cut into two pieces. One piece is 6 centimeters longer than the other. What are the lengths of the two pieces?. Thu is five years older than her brother Tuan. The sum of their ages is 5. What are their ages?. Tomas is thinking of a number. If he triples his number and subtracts, the result is 05. What is the number that Tomas is thinking about? 4. Two consecutive numbers have a sum of. What are the two numbers? 5. Two consecutive even numbers have a sum of 46. What are the numbers? 6. Joe s age is three times Aaron s age and Aaron is si years older than Christina. If the sum of their ages is 49, what is Christina s age? Joe s age? Aaron s age? 7. Farmer Fran has 8 barnyard animals, consisting of only chickens and goats. If these animals have 6 legs, how many of each type of animal are there? 8. A wood board 56 centimeters long is cut into three parts. The two longer parts are the same length and are 5 centimeters longer than the shortest part. How long are the three parts? 9. Juan has 5 coins, all nickels and dimes. This collection of coins is worth 90. How many nickels and dimes are there? (Hint: Create separate column titles for, Number of Nickels, Value of Nickels, Number of Dimes, and Value of Dimes. ) 0. Tickets to the school play are $ 5.00 for adults and $.50 for students. If the total value of all the tickets sold was $57.50 and 00 more students bought tickets than adults, how many adults and students bought tickets?. A wood board 50 centimeters long is cut into five pieces: three short ones of equal length and two that are both 5 centimeters longer than the shorter ones. What are the lengths of the boards?. Conrad has a collection of three types of coins: nickels, dimes, and quarters. There is an equal amount of nickels and quarters but three times as many dimes. If the entire collection is worth $ 9.60, how many nickels, dimes, and quarters are there? 74 Core Connections, Courses

75 Epressions, Equations, and Functions Answers. The lengths of the boards are 7 cm and 6 cm.. Thu is 8 years old and her brother is years old.. Tomas is thinking of the number The two consecutive numbers are 6 and The two consecutive numbers are 4 and Farmer Fran has 0 goats and 8 chickens. 6. Christine is 5, Aaron is, and Joe is 9 years old. 8. The lengths of the boards are 4, 57, and 57 cm. 9. Juan has nickels and dimes. 0. There were 55 adult and 55 student tickets purchased for the play.. The lengths of the boards are 44 and 59 cm.. Conrad has 6 nickels and quarters and 48 dimes. WRITING EQUATIONS FOR WORD PROBLEMS (THE 5-D PROCESS) At first students used the 5-D Process to solve problems. However, solving complicated problems with the 5-D Process can be time consuming and it may be difficult to find the correct solution if it is not an integer. The patterns developed in the 5-D Process can be generalized by using a variable to write an equation. Once you have an equation for the problem, it is often more efficient to solve the equation than to continue to use the 5-D Process. Most of the problems here will not be comple so that you can practice writing equations using the 5-D Process. The same eample problems previously used are used here ecept they are now etended to writing and solving equations. Eample A bo of fruit has three times as many nectarines as grapefruit. Together there are 6 pieces of fruit. How many pieces of each type of fruit are there? Describe: Number of nectarines is three times the number of grapefruit. Number of nectarines plus number of grapefruit equals 6. Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? Trial : 44 too high Trial : too high Eample continues on net page Parent Guide with Etra Practice 75

76 Eample continued from previous page. After several trials to establish a pattern in the problem, you can generalize it using a variable. Since we could try any number of grapefruit, use to represent it. The pattern for the number of oranges is three times the number of grapefruit, or. The total pieces of fruit is the sum of column one and column two, so our table becomes: Define Do Decide # of Grapefruit # of Nectarines Total Pieces of Fruit 6? + = 6 Since we want the total to agree with the check, our equation is + = 6. Simplifying this yields 4 = 6, so = 9 (grapefruit) and then = 7 (nectarines). Declare: There are 9 grapefruit and 7 nectarines. Eample The perimeter of a rectangle is 0 feet. If the length of the rectangle is 0 feet more than the width, what are the dimensions (length and width) of the rectangle? Describe/Draw: width width + 0 Define Do Decide Width Length Perimeter 0? Trial : 0 5 (0 + 5) = 70 too low Trial : too low Again, since we could guess any width, we labeled this column. The pattern for the second column is that it is 0 more than the first: + 0. The perimeter is found by multiplying the sum of the width and length by. Our table now becomes: Define Do Decide Width Length Perimeter 0? + 0 ( + + 0) = 0 Solving the equation: ( + + 0) = = = 0 4 = 00 So = 5 (width) and + 0 = 5 (length). Declare: The width is 5 feet and the length is 5 feet. 76 Core Connections, Courses

77 Epressions, Equations, and Functions Eample Jorge has some dimes and quarters. He has 0 more dimes than quarters and the collection of coins is worth $.40. How many dimes and quarters does Jorge have? Describe: The number of quarters plus 0 equals the number of dimes. The total value of the coins is $.40. Define Do Decide Quarters Dimes Value of Value of Quarters Dimes Total Value $.40? Trial : too high Trial : too high ( + 0) ( + 0) Since you need to know both the number of coins and their value, the equation is more complicated. The number of quarters becomes, but then in the table the Value of Quarters column is 0.5. Thus the number of dimes is + 0, but the value of dimes is 0.0( + 0). Finally, to find the numbers, the equation becomes ( + 0) =.40. Solving the equation: = = =.40 = 4.00 Declare: There are 4 quarters worth $.00 and 4 dimes worth $.40 for a total value of $.40. Problems Start the problems using the 5-D Process. Then write an equation. Solve the equation.. A wood board 00 centimeters long is cut into two pieces. One piece is 6 centimeters longer than the other. What are the lengths of the two pieces?. Thu is five years older than her brother Tuan. The sum of their ages is 5. What are their ages?. Tomás is thinking of a number. If he triples his number and subtracts, the result is 05. Of what number is Tomás thinking? 4. Two consecutive numbers have a sum of. What are the two numbers? 5. Two consecutive even numbers have a sum of 46. What are the numbers? 6. Joe s age is three times Aaron s age and Aaron is si years older than Christina. If the sum of their ages is 49, what is Christina s age? Joe s age? Aaron s age? Parent Guide with Etra Practice 77

78 7. Farmer Fran has 8 barnyard animals, consisting of only chickens and goats. If these animals have 6 legs, how many of each type of animal are there? 8. A wood board 56 centimeters long is cut into three parts. The two longer parts are the same length and are 5 centimeters longer than the shortest part. How long are the three parts? 9. Juan has 5 coins, all nickels and dimes. This collection of coins is worth 90. How many nickels and dimes are there? (Hint: Create separate column titles for, Number of Nickels, Value of Nickels, Number of Dimes, and Value of Dimes. ) 0. Tickets to the school play are $ 5.00 for adults and $.50 for students. If the total value of all the tickets sold was $57.50 and 00 more students bought tickets than adults, how many adults and students bought tickets?. A wood board 50 centimeters long is cut into five pieces: three short ones of equal length and two that are both 5 centimeters longer than the shorter ones. What are the lengths of the boards?. Conrad has a collection of three types of coins: nickels, dimes, and quarters. There is an equal amount of nickels and quarters but three times as many dimes. If the entire collection is worth $ 9.60, how many nickels, dimes, and quarters are there? Answers (Equations may vary.). + ( + 6) = 00 The lengths of the boards are 7 cm and 6 cm.. = 05 Tomás is thinking of the number ( + ) = 46 The two consecutive numbers are 4 and ( + 5) = 5 Thu is 8 years old and her brother is years old ( + ) = The two consecutive numbers are 6 and ( + 6) + ( + 6) = 49 Christine is 5, Aaron is, and Joe is 9 years old (8 ) = 6 Farmer Fran has 0 goats and 8 chickens (5 ) = 0.90 Juan has nickels and dimes.. + ( + 5) = 50 The lengths of the boards are 44 and 59 cm ( + 5) + ( + 5) = 56 The lengths of the boards are 4, 57, and 57 cm. 0. $5 + $.50( + 00) = There were 55 adult and 55 student tickets purchased for the play () = 9.60 Conrad has 6 quarters, 6 nickels, and 48 dimes. Core Connections, Courses

79 Epressions, Equations, and Functions SOLVING EQUATIONS Using a Two Region Equation Mat (Core Connections, Course ) Students combined two Epression Mats to figure out what value(s) of the variable make(s) one epression greater than the other. Now two Epression Mats are combined into an Equation Mat as a concrete model for solving equations. Practicing solving equations using this model will help students transition to solving equations abstractly with better accuracy and understanding. In general, and as shown in the eample below, start by simplifying the Epression Mat. Net, isolate the variables on one side of the Equation Mat and the non-variables (unit tiles) on the other by adding/removing balanced sets and zeros. Then determine the value of the variable. Students are epected to be able to record and eplain their steps. For additional information, see the Math Notes bo in Lesson 6.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 8 materials. Procedure and Eample Solve: + (!4) + =! + = + = First build the equation on the Equation Mat. Second, simplify by removing zeros ( and + on the right side of the mat). Third, remove a balanced set () from both sides. 4! 4 = + Isolate the variable by adding a balanced set (+4) to both sides and remove the zeros on the left side.! 4 = Eample continues on net page Parent Guide with Etra Practice 79

80 Eample continued from previous page. Finally, since both sides of the equation are equal, determine the value of by dividing. = 6 = Once the students understand how to solve equations using an Equation Mat, they may use the visual eperience of moving the tiles to solve equations with variables and numbers. The procedures for moving variables and numbers in the solving process follow the same rules. Note: When the process of solving an equation ends with different numbers on each side of the equal sign (for eample = 4), there is no solution to the problem. When the result is the same epression or number on each side of the equation (for eample, + = + ) it means that all numbers are solutions. For more information about these special cases, see the Math Note bo in Lesson 6..6 of the Core Connections, Course tet. Eample Solve +! = Solution +! = ! = = 0 = 5 Eample Solve! + + (! ) =!4 +!! Solution! + + (! ) =!4 +!!! + +! =!! 6! =!! 6 =!4 =! 80 Core Connections, Courses

81 Epressions, Equations, and Functions Problems Solve each equation = !! =! 5.! =! ! =! ! =! ! + = !! =! 6 8.!4 +! = ! + = ! + = + +.! 7 =!!.! + =!! 4.! + 7 =! 4. +! 4 =! + 5. ( + ) = + 6. (! ) + = = ! + =! 5! 9. (4 + ) = !! = 4(! ) Answers Parent Guide with Etra Practice 8

82 Using a Four Region Equation Mat (Core Connections, Course ) Combining two Epression Mats into an Equation Mat creates a concrete model for solving equations. Practice solving equations using the model will help students transition to solving equations abstractly with better accuracy and understanding. In general, and as shown in the first eample below, the negative in front of the parenthesis causes everything inside to flip from the top to the bottom or the bottom to the top of an Epression Mat, that is, all terms in the epression change signs. After simplifying the parentheses, simplify each Epression Mat. Net, isolate the variables on one side of the Equation Mat and the non-variables on the other side by removing matching tiles from both sides. Then determine the value of the variable. Students should be able to eplain their steps. See the Math Notes boes in Lessons..9 and.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 5 materials. Procedure and Eample + + Solve +! (!) = + 5! (! ). First build the equation on the Equation Mat. Second, simplify each side using legal moves on each Epression Mat, that is, on each side of the Equation Mat. + _ + _ + + Isolate -terms on one side and non--terms on the other by removing matching tiles from both sides of the equation mat. Finally, since both sides of the equation are equal, determine the value of Core Connections, Courses

83 Epressions, Equations, and Functions Once students understand how to solve equations using an Equation Mat, they may use the visual eperience of moving tiles to solve equations with variables and numbers. The procedures for moving variables and numbers in the solving process follow the same rules. Note: When the process of solving an equation ends with different numbers on each side of the equal sign (for eample, = 4), there is no solution to the problem. When the result is the same epression or number on each side of the equation (for eample, + = + ) it means that all numbers are solutions. See the Math Notes bo in Lesson..4 of the Core Connections, Course tet. Eample Solve +! = Solution +! = ! = = 0 = 5 problem simplify add, subtract 4 on each side divide Eample Solve! +! (! + ) =!4 + (!! ) Solution Problems! +! (! + ) =!4 + (!! )! + +! =!4!!! =!! 6 =!4 =! problem remove parenthesis (flip) simplify add, add to each side divide Solve each equation..! =! +. +! =! ! =! ! (! ) = !( + ) =! 6 6.!4 +! = ! + = ! + = y! 8! y = ! (! y) = 4 + y! (! y).! 7 =!!.!! =!! 4.! + 7 =! 4. +! 4 =!! (!) 5.!! =!! (!5 + ) 6.!4! =!! = !(! ) =! 5! 9. 6!! = 4! ! (! + ) =! 5 Parent Guide with Etra Practice 8

84 Answers. =. = 5. = 4. = 5. = 6. = 6 7. = 7 8. no solution 9. = 6 0. = 7. =. all numbers. = 4. = 0 5. = 6. no solution 7. = 8. = 7 9. = 5 0. = 4 SOLVING EQUATIONS IN CONTEXT Initially, equations are solved either by applying math facts (for eample, 4 =, since 4! =,! = ) or by matching equal quantities, simplifying the equation, and using math facts as shown in the eamples below. Equations are often written in the contet of a geometric situation. Write an equation that represents each situation and find the value of the variable. Eample 0 Eample 8 +0 = = = 44 + = 6 = 6 = Eample Eample 4 y y y 5 y 40 y = 5 + y = 80 y = 5 + = 40 y =.5 5 = 40 = 8 84 Core Connections, Courses

85 Epressions, Equations, and Functions Problems Write an equation that represents each situation and then find the value of the variable n n n n 5 n Solve each equation = 9 8. y = 9. y = 4 0. m = 6. + = = 5. m + m + 7 = m = 0 5. y = k + = = 5 8. m + 7 = m + 9. (y + ) = 0. (c + ) + c + = 57 Parent Guide with Etra Practice 85

86 Answers. + = 5;! =. + 4 = + 6;! =. + 7 = 5;! = n + = n + 8;!n = = 80;! = 58! = 80;! = 70! 7. = 6 8. y = 9. y = 8 0. m =. =. = 4. m = 4. = 7 5. y = 6 6. k = 7. = 9 8. m = 9. y = 9 0. c =.5 86 Core Connections, Courses

87 Epressions, Equations, and Functions TABLES, GRAPHS, AND RULES Three ways to write relationships for data are tables, words (descriptions), and rules. The pattern in tables between input () and output (y) values usually establishes the rule for a relationship. If you know the rule, it may be used to generate sets of input and output values. A description of a relationship may be translated into a table of values or a general rule (equation) that describes the relationship between the input values and output values. Each of these three forms of relationships may be used to create a graph to visually represent the relationship. For additional information, see the Math Notes boes in Lessons..,..4,..5, and.. of the Core Connections, Course tet. Eample Complete the table by determining the relationship between the input () values and output (y) values, write the rule for the relationship, then graph the data. input () output (y) Begin by eamining the four pairs of input values: 4 and 8, 5 and 0, 0 and 0, and 4. Determine what arithmetic operation(s) are applied to the input value of each pair to get the second value. The operation(s) applied to the first value must be the same in all four cases to produce each given output value. In this eample, the second value in each pair is twice the first value. Since the pattern works for all four points, make the conjecture that the rule is y(output) = (input). This makes the missing values and 6, and 4, and, and 6. The rule is y =. Finally, graph each pair of data on an y-coordinate system, as shown at right. 4 y Parent Guide with Etra Practice 87

88 Eample Complete the table by determining the relationship between the input () and output (y) values, then write the rule for the relationship. input () 4 0 output (y) Use the same approach as Eample. In this table, the relationship is more complicated than simply multiplying the input value or adding (or subtracting) a number. Use a Guess and Check approach to try different patterns. For eample, the first pair of values could be found by the rule +, that is, + =. However, that rule fails when you check it for - and :! + "!. From this guess you know that the rule must be some combination of multiplying the input value and then adding or subtracting to that product. The net guess could be to double. Try it for the first two or three input values and see how close each result is to the known output values: for and, () = 4 ; for and, (!) =! ; and for 4 and 7, (4) = 8. Notice that each result is one more than the actual output value. If you subtract from each product, the result is the epected output value. Make the conjecture that the rule is y(output) = (input)! and test it for the other input values: for and 7, (!)! =!7 ; for 0 and, (0)! =!; for and 5, (!)! =!5 ; and for and, ()! =. So the rule is y =!. Eample Complete the table below for y =! +, then graph each of the points in the table. input () output (y) Replace with each input value, multiply by, then add. The results are ordered pairs: ( 4, 9), (, 7), (, 5), (, ), (0, ), (, ), (, ), (, 5), and (4, 7). Plot these points on the graph (see Chapter if you need help with the fundamentals of graphing). y Core Connections, Courses

89 Epressions, Equations, and Functions Eample 4 Complete the table below for y =! +, then graph the pairs of points and connect them with a smooth curve. input () 0 4 output (y)! + Replace in the equation with each input value. Square the value, multiply the value by, then add both of these results and to get the output (y) value for each input () value. The results are ordered pairs: (, 9), (, 4), (0, ), (, 0), (, ), (, 4), and (4, 9). y Eample 5 y Make an y table for the graph at right, then write a rule for the table. input () output (y) 5 5!! 4 Working left to right on the graph, read the coordinates of each point and record them in the table. 4 6 Guess and check by multiplying the input value, then adding or subtracting numbers to get the output value. For eample you could start by multiplying the input value by : (!4) =!8, (!4) =!, (!) =!6, etc. The results are not close to the correct output value. The product is also the opposite sign (+ -) of what you want. Your net choice could be to multiply by :!(!4) = 8,!(!) = 6,!(!) = 4. Each result is three more than the epected output value, so make the conjecture that the rule is y =!!. Test it for the remaining points:!(!)! =!,!(0)! =!, and!()! =!5. The rule is y =!!. Parent Guide with Etra Practice 89

90 Problems Complete each table. Then write a rule relating and y... Input() Input() 5 9 Output(y) Output(y) Input() Input() Output(y) Output(y) Input() Input() 4 6 Output(y) Output(y) Input() Input() 0 4 Output(y) 9 Output(y) Input() Input() 0 Output(y) 4 Output(y) Input() Input() 0 Output(y) Output(y) Complete a table for each rule, then graph and connect the points. For each rule, start with a table like the one below. Input () 0 Output (y). y =! 4. y = + 5. y =! + 6. y =! 6 90 Core Connections, Courses

91 Epressions, Equations, and Functions Answers. 4, 8, 7; y = , 5, 9; y =!. 40,, 6; y =! 5 4., 5, 45; y =! 5. 8,, 8; y =! + 6., 5, ; y = , 7, 5; y =! 8., 7, 6; y =! + 9., 0, ; y = 0. y = +. y = + 4. y = +. y 8 6 input () 0 output (y) y input () 0 output (y) input () 0 output (y) y input () 0 output (y) y Parent Guide with Etra Practice 9

92 MULTIPLE REPRESENTATIONS The first part of Chapter 4 of Core Connections, Course ties together several ways to represent the same relationship. The basis for any relationship is a consistent pattern that connects input and output values. This course uses tile patterns to help visualize algebraic relationships. (Note: In this course we consider tile patterns to be continuous relationships and graph them with a continuous line or curve.) These relationships may also be displayed on a graph, in a table, or as an equation. In each situation, all four representations show the same relationship. Students learn how to use each format to display relationships as well as how to Table switch from one representation to another. We use the diagram at right to show the connections between the Graph various ways to display a relationship and call it the representations web. See the Math Notes bo in Lesson 4..7 of the Core Connections, Course tet. Pattern For additional eamples and practice see the Core Connections, Course Checkpoint 6 materials. Rule Eample At this point in the course we use the notion of growth to help understand linear relationships. For eample, a simple tile pattern may start with two tiles and grow by three tiles in each successive figure as shown below. 9 Fig. 0 Fig. Fig. Fig. Fig. 4 The picture of the tile figures may also be described by an equation in y = m + b form, where and y are variables and m represents the growth rate and b represents the starting value of the pattern. In this eample, y = +, where represents the number of tiles in the original figure (usually called Figure 0 ) and is the growth factor that describes the rate at which each successive figure adds tiles to the previous figure. This relationship may also be displayed in a table, called an y table, as shown below. The rule is written in the last column of the table. Figure number () 0 4 Number of tiles (y) Finally, the relationship may be displayed on an y-coordinate graph by plotting the points in the table as shown at right. The highlighted points on the graph represent the tile pattern. The line represents all of the points described by the equation y = +. 4 Core Connections, Courses y

93 Epressions, Equations, and Functions Eample Draw Figures 0, 4, and 5 for the tile pattern below. Use the pattern to predict the number of tiles in Figure 00, describe the figure, write a rule that will give the number of tiles in any figure, record the data for the first si tiles (Figures 0 through 5) in a table, and graph the data. Fig. 0 Fig. Fig. Fig. Fig. 4 Fig. 5 Each figure adds four tiles: two tiles to the top row and two tiles to the lower portion of the figure. Figure 0 has two tiles, so the rule is y = 4 + and Figure 00 has 4(00) + = 40 tiles. There are 0 tiles in the top row and 00 tiles in the lower portion of figure 00. The table is: Figure number () Number of tiles (y) The graph is shown at right y Eample 4 Use the table below to determine the rule in y = m + b form that describes the pattern. input () output (y) The constant difference between the output values is the growth rate, that is, the value of m. The output value paired with the input value = 0 is the starting value, that is, the value of b. So this table can be described by the rule: y =!. Note: If there is not a constant difference between the output values for consecutive integer input values, then the rule for the pattern is not in the form y = m + b. Eample 4 y 4 Use the graph at right to create an y table, then write a rule for the pattern it represents. First transfer the coordinates of the points into an y table. input () 0 output (y) Using the method described in Eample, that is, noting that the growth rate between the output values is 4 and the value of y at = 0 is 5, the rule is: y =! Parent Guide with Etra Practice 9

94 Problems. Based on the tile pattern below, draw Figures 0, 4, and 5. Then find a rule that will give the number of tiles in any figure and use it to find the number of tiles in Figure 00. Finally, display the data for the first si figures (numbers 0-5) in a table and on a graph. Fig. 0 Fig. Fig. Fig. Fig. 4 Fig. 5. Based on the tile pattern below, draw Figures 0, 4, and 5. Then find a rule that will give the number of tiles in any figure and use it to find the number of tiles in Figure 00. Finally, display the data for the first si figures (numbers 0-5) in a table and on a graph. Fig. 0 Fig. Fig. Fig. Fig. 4 Fig. 5 Use the patterns in the tables and graphs to write rules for each relationship.. input () output (y) input () output (y) y 6. 4 y Core Connections, Courses

95 Epressions, Equations, and Functions Answers. y 5 Fig. 0 Fig. 4 Fig. 5 0 The rule is y = + 5. Figure 00 will have 05 tiles. It will have a base of three tiles, with 0 tiles etending up from the right tile in the base and 00 tiles etending to the right of the top tile in the vertical etension above the base. 5 4 Figure number () Number of tiles (y) y 0 Fig. 0 Fig. 4 Fig The rule is y = 4 +. Figure 00 will have 40 tiles in the shape of an X with 00 tiles on each branch of the X, all connected to a single square in the middle. Figure number () Number of tiles (y) y =! 4. y =! y =! 6. y =! Parent Guide with Etra Practice 95

96 EQUATIONS WITH FRACTIONAL COEFFICIENTS Students used scale factors (multipliers) to enlarge and reduce figures as well as increase and decrease quantities. All of the original quantities or lengths were multiplied by the scale factor to get the new quantities or lengths. To reverse this process and scale from the new situation back to the original, we divide by the scale factor. Division by a scale factor is the same as multiplying by a reciprocal. This same concept is useful in solving one-step equations with fractional coefficients. To remove a fractional coefficient you may divide each term in the equation by the coefficient or multiply each term by the reciprocal of the coefficient. To remove fractions in more complicated equations students use Fraction Busters. Multiplying all of the terms of an equation by the common denominator will remove all of the fractions from the equation. Then the equation can be solved in the usual way. For additional information, see the Math Notes boes in Lesson 7..6 of the Core Connections, Course tet or Lesson 5.. of the Core Connections, Course tet. For additional eamples and practice see the Core Connections, Course Checkpoint 7 materials. Eample of a One-Step Equation Solve: = Method : Use division and common denominators = = = = = 6 = 6 = 8 Method : Use reciprocals = ( ) = = 8 ( ) Eample of Fraction Busters Solve: + 5 = 6 Multiplying by 0 (the common denominator) will eliminate the fractions. 0( + 5 ) = 0(6) 0( ) +0( 5 ) = 0(6) 5 + = 60 7 = 60! = 60 7 " Core Connections, Courses

97 Epressions, Equations, and Functions Problems Solve each equation.. 4 = = 4. 5 y = 40 4.! 8 m = = 5 6.! 5 = 7. y+7 = y 5 8. m! m 5 = 5 9.! 5 = 0. +! 5 =. + 4 = ! = 4 Answers. = 80. = 05. y = m =! y = 6. =.5 7. y =!7 8. m = 9. =! = 6 7. = = 65 Parent Guide with Etra Practice 97

98 EQUATIONS WITH MULTIPLE VARIABLES Solving equations with more than one variable uses the same process as solving an equation with one variable. The only difference is that instead of the answer always being a number, it may be an epression that includes numbers and variables. The usual steps may include: removing parentheses, simplifying by combining like terms, removing the same thing from both sides of the equation, moving the desired variables to one side of the equation and the rest of the variables to the other side, and possibly division or multiplication. Eample Eample Solve for y Subtract Divide by! Simplify! y = 6!y =! + 6 y =!+6! y =! Solve for y Subtract 7 Distribute the Subtract Divide by Simplify 7 + ( + y) = ( + y) = 4 + y = 4 y =! + 4 y =!+4 y =! + Eample Eample 4 Solve for y =! 4 Add 4 y + 4 = Divide by y + 4 = Solve for t Divide by pr I = prt I pr = t Problems Solve each equation for the specified variable.. y in 5 + y = 5. in 5 + y = 5. w in l + w = P 4. m in 4n = m! 5. a in a + b = c 6. a in b! a = c 7. p in 6! (q! p) = 4 p 8. in y = r in 4(r! s) = r! 5s 98 Core Connections, Courses

99 Epressions, Equations, and Functions Answers (Other equivalent forms are possible.). y =! =! 5 y +. w =!l + P 4. m = 4n+ 5. a = c!b 6. a = c!b!or! b!c! 7. p = q! 8. = 4y! 4 9. r = 7s LINEAR GRAPHS USING y = m + b Slope (rate of change) is a number that indicates the steepness (or flatness) of a line, that is, its rate of change, as well as its direction (up or down) left to right. Slope (rate of change) is determined by the ratio: vertical change horizontal change = change in y change in between any two points on a line. Some books and teachers refer to this ratio as the rise (y) over the run (). For lines that go up (from left to right), the sign of the slope is positive. For lines that go down (left to right), the sign of the slope is negative. Any linear equation written as y = m + b, where m and b are any real numbers, is in slope-intercept form. m is the slope of the line. b is the y-intercept, that is, the point (0, b) where the line intersects (crosses) the y-ais. Eample Write the slope of the line containing the points (, ) and (4, ). y First graph the two points and draw the line through them. Look for and draw a slope triangle using the two given points. Write the ratio triangle: 5. vertical change in y horizontal change in using the legs of the right (, ) 5 (4, ) Assign a positive or negative value to the slope depending on whether the line goes up (+) or down ( ) from left to right. The slope is! 5. Parent Guide with Etra Practice 99

100 Eample Write the slope of the line containing the points ( 9, 5) and (5, ). Since the points are inconvenient to graph, use a generic slope triangle, visualizing where the points lie with respect to each other and the aes. Make a sketch of the points. ( 9, 5) y (5, ) 8 Draw a slope triangle and determine the length of each leg. Write the ratio of y to : 8 54 =. The slope is. 54 Eample Given a table, determine the rate of change (slope) and the equation of the line. y rate of change = y - intercept = (0, 4) The equation of the line is y = + 4. Eample 4 y Graph the linear equation y =!. Using y = m + b, the slope in y =! is and the y-intercept is the point (0, ). To graph, begin at the vertical!change y-intercept (0, ). Remember that slope is, so go horizontal!change up units (since is positive) from (0, ) and then move right units. This gives a second point on the graph. To create the graph, draw a straight line through the two points. 00 Core Connections, Courses

101 Epressions, Equations, and Functions Problems Determine the slope of each line using the highlighted points. y... y y Find the slope of the line containing each pair of points. Sketch a slope triangle to visualize the vertical and horizontal change. 4. (, ) and (5, 7) 5. (, 5) and (9, 4) 6. (, ) and (7, 4) 7. (, ) and (, ) 8. (, 5) and (4, 5) 9. (5, 8) and (, 5) Use a slope triangle to find the slope of the line containing each pair of points: 0. (50, 40) and (0, 75). (0, 9) and (44, 80). (5, ) and ( 5, 0) Identify the slope and y-intercept in each equation.. y =! 4. y =! y = 4 6. y =! + 7. y =! 7 8. y = 5 Draw a graph to find the equation of the line with: 9. slope = and passing through (, ). 0. slope = and passing through (, ).. slope =! and passing through (, ).. slope = 4 and passing through (, 8). Parent Guide with Etra Practice 0

102 For each table, determine the rate of change and the equation. Be sure to record whether the rate of change is positive or negative for both and y.. y y y Using the slope and y-intercept, determine the equation of the line. y y y y Graph the following linear equations on graph paper. 0. y = +. y =! +. y =!4 5. y = y = 0 Core Connections, Courses

103 Epressions, Equations, and Functions Answers.! ! ! ! 5 0 =! ! 7. ; (0, ) 4. -; (0, 5) 5. 4; (0, 0) 6.! ; (0, ) 7. ; (0, 7) 8. 0; (0, 5) 9. y = + 0. y =! 4. y =!. y =!4! 4. ; y = + 4. ; y =! + 5. ; y = + 6. y =! 7. y =! + 8. y = + 9. y =! y = +. y =! + 5. y =!4 y y y. y =! + 4. y =! + 6 y y Parent Guide with Etra Practice 0

104 SYSTEMS OF LINEAR EQUATIONS Two lines on an y-coordinate grid are called a system of linear equations. They intersect at a point unless they are parallel or the equations are different forms of the same line. The point of intersection is the only pair of (, y) values that will make both equations true. One way to find the point of intersection is to graph the two lines. However, graphing is both time-consuming and, in many cases, not eact, because the result may only be a close approimation of the coordinates. When two linear equations are written equal to y (in general, in the form y = m + b ), we can take advantage of the fact that both y values are the same (equal) at the point of intersection. For eample, if two lines are described by the equations y =! + 5 and y =!, and we know that both y values are equal, then the other two sides of the equations must also be equal to each other. We say that both right sides of these equations have equal values at the point of intersection and write! + 5 =!, so that the result looks like the work we did with equation mats. We can solve this equation in the usual way and find that =. Now we know the -coordinate of the point of intersection. Since this value will be the same in both of the original equations at the point of intersection, we can substitute = in either equation to solve for y: y =!() + 5 so y = or y =! and y =. So the two lines in this eample intersect at (, ). For additional information, see the Math Notes boes in Lessons 5.., 5.., and 5..4 of the Core Connections, Course tet. Eample Find the point of intersection for y = 5 + and y =!! 5. Substitute the equal parts of the equations. 5 + =!!5 8 =!6 Solve for. =! Replace with! in either original equation and solve for y. y = 5(!) + y =!0 + or y =!(!)!5 y = 6!5 The two lines intersect at (, 9). y =!9 y =!9 04 Core Connections, Courses

105 Epressions, Equations, and Functions Eample The Mathematical Amusement Park is different from other amusement parks. Visitors encounter their first decision involving math when they pay their entrance fee. They have a choice between two plans. With Plan they pay $5 to enter the park and $ for each ride. With Plan they pay $ to enter the park and $ for each ride. For what number of rides will the plans cost the same amount? The first step in the solution is to write an equation that describes the total cost of each plan. In this eample, let equal the number of rides and y be the total cost. Then the equation to represent Plan for rides is y = 5 +. Similarly, the equation representing Plan for rides is y = +. We know that if the two plans cost the same, then the y value of y = 5 + and y = + must be the same. The net step is to write one equation using, then solve for. 5 + = = = 7 Use the value of to find y. y = 5 + (7) = 6 The solution is (7, 6). This means that if you go on 7 rides, both plans will have the same cost of $6. Parent Guide with Etra Practice 05

106 Problems Find the point of intersection (, y) for each system of linear equations.. y =! 6 y =! 4. y =! 5 y = +4. y =! 5 y = + 5. y = + 7 y = 4! 5. y = +6 y = y = 7! y =! 8 Write a system of linear equations for each problem and use them to find a solution. 7. Jacques will wash the windows of a house for $5.00 plus $.00 per window. Ray will wash them for $5.00 plus $.00 per window. Let be the number of windows and y be the total charge for washing them. Write an equation that represents how much each person charges to wash windows. Solve the system of equations and eplain what the solution means and when it would be most economical to use each window washer. 8. Elle has moved to Hawksbluff for one year and wants to join a health club. She has narrowed her choices to two places: Thigh Hopes and ABSolutely fabulus. Thigh Hopes charges a fee of $95 to join and an additional $5 per month. ABSolutely fabulus charges a fee of $5 to join and a monthly fee of $. Write two equations that represent each club's charges. What do your variables represent? Solve the system of equations and tell when the costs will be the same. Elle will only live there for one year, so which club will be less epensive? 9. Misha and Nora want to buy season passes for a ski lift but neither of them has the $5 needed to purchase a pass. Nora decides to get a job that pays $6.5 per hour. She has nothing saved right now but she can work four hours each week. Misha already has $80 and plans to save $5 of her weekly allowance. Who will be able to purchase a pass first? 0. Ginny is raising pumpkins to enter a contest to see who can grow the heaviest pumpkin. Her best pumpkin weighs pounds and is growing at the rate of.5 pounds per week. Martha planted her pumpkins late. Her best pumpkin weighs 0 pounds but she epects it to grow 4 pounds per week. Assuming that their pumpkins grow at these rates, in how many weeks will their pumpkins weigh the same? How much will they weigh? If the contest ends in seven weeks, who will have the heavier pumpkin at that time?. Larry and his sister, Betty, are saving money to buy their own laptop computers. Larry has $5 and can save $5 each week. Betty has $80 and can save $0 each week. When will Larry and Betty have the same amount of money? 06 Core Connections, Courses

107 Epressions, Equations, and Functions Answers. (9, ). (4, 7). (4, 4) 4. (9, 5) 5. (4, ) 6. (, ) 7. Let = number of windows, y = cost. Jacques: y = 5 + ; Roy: y = 5 +. The solution is (0, 5), which means that the cost to wash 0 windows is $5. For fewer than 0 windows use Roy; for more than 0 windows, use Jacques. 8. Let = weeks, y = total charges. Thigh Hopes: y = ; ABSolutely fabulus: y = 5 +. The solution is (0, 45). At 0 months the cost at either club is $45. For months use ABSolutely fabulus. 9. Let = weeks, y = total savings. Misha: y = ; Nora: y = 5. The solution is (8, 00). Both of them will have $00 in 8 weeks, so Nora will have $5 in 9 weeks and be able to purchase the lift pass first. An alternative solution is to write both equations, then substitute 5 for y in each equation and solve for. In this case, Nora can buy a ticket in 9 weeks, Misha in 9.67 weeks. 0. Let = weeks and y = weight of the pumpkin. Ginny: y =.5 + ; Martha: y = The solution is (8, 4), so their pumpkins will weigh 4 pounds in 8 weeks. Ginny would win (9.5 pounds to 8 pounds for Martha).. Let = weeks, y = total money saved. Larry: y = ; Betty: y = The solution is (, 600). They will both have $600 in weeks. Parent Guide with Etra Practice 07

108 EXPONENTS AND SCIENTIFIC NOTATION EXPONENTS In the epression 5, 5 is the base and is the eponent. For a, is the base and a is the eponent. 5 means 5 5 and 5 means 5 5 5, so you can write 55 5 (which means 55 5 ) or you can write it like this: 5! 5! 5! 5! 5. 5! 5 You can use the Giant One to find the numbers in common. There are two Giant Ones, namely, 5 5! 5! 5! 5! 5 5 twice, so = 5 5! 5 or 5. Writing 5 is usually sufficient. When there is a variable, it is treated the same way. 7 means!!!!!!.!! The Giant One here is!(three of them). The answers is means (5 5)(5 5 5), which is 5 5. (5 ) means (5 )(5 )(5 ) or (5 5)(5 5)(5 5), which is 5 6. When the problems have variables such as 4 5, you only need to add the eponents. The answer is 9. If the problem is ( 4 ) 5 ( 4 to the fifth power) it means The answer is 0. You multiply eponents in this case. If the problem is 0, you subtract the bottom eponent from the top eponent (0 4). 4 The answer is 6. You can also have problems like 0. You still subtract, 0 ( 4) is 4,!4 and the answer is 4. You need to be sure the bases are the same to use these laws. 5 y 6 cannot be further simplified. In general the laws of eponents are: a b = (a + b) ( a ) b = ab 0 =!n = n These rules hold if 0 and y 0. a = (a b) b ( a y b ) c = ac y bc For additional information, see Math Notes bo in Lesson 8..4 of the Core Connections, Course tet. 08 Core Connections, Courses

109 Epressions, Equations, and Functions Eamples a. 8! 7 = 5 b. 9 = 6 c. (z 8 ) = z 4 d. ( y ) 4 = 8 y e. 4! = 7 f. ( y ) = 4 4 y 6 g. ( y! ) = 7 6 y!6 or i.! = = y 6 h. 8 y 5 z y 6 z! = 5 z 4 y or 5 y! z 4 j. 5! 5 "4 = 5 " = 5 = 5 Problems Simplify each epression (5 ) 6. ( 4 ) 7. (4 y ) (y ). (4a b ). 5 y 4 z 4 y z. 6 y z! y z! ! 6.! 7. 6! 6 " 8. (! ) Answers y y 6 or. 64a 6 b 6 or 64a6 b 6. y z 4 y or 8 y z y 6 Parent Guide with Etra Practice 09

110 SCIENTIFIC NOTATION Scientific notation is a way of writing very large and very small numbers compactly. A number is said to be in scientific notation when it is written as the product of two factors as described below. The first factor is less than 0 and greater than or equal to. The second factor has a base of 0 and an integer eponent (power of 0). The factors are separated by a multiplication sign. A positive eponent indicates a number whose absolute value is greater than one. A negative eponent indicates a number whose absolute value is less than one. Scientific Notation Standard Form ,000,000, It is important to note that the eponent does not necessarily mean to use that number of zeros. The number 5. 0 means 5. 00,000,000,000. Thus, two of the places in the standard form of the number are the and the in 5.. Standard form in this case is 5,000,000,000. In this eample you are moving the decimal point to the right places to find standard form. The number means You are moving the decimal point to the left 5 places to find standard form. Here the standard form is For additional information, see the Math Notes bo in Lesson 8.. of the Core Connections, Course tet. Eample Write each number in standard form. 7.84!0 8! 784,000,000 and.7!0 "! When taking a number in standard form and writing it in scientific notation, remember there is only one digit before the decimal point, that is, the number must be between and 9, inclusive. 0 Core Connections, Courses

111 Epressions, Equations, and Functions Eample 5,050,000! 5.05!0 7 and !.7!0 "4 The eponent denotes the number of places you move the decimal point in the standard form. In the first eample above, the decimal point is at the end of the number and it was moved 7 places. In the second eample above, the eponent is negative because the original number is very small, that is, less than one. Problems Write each number in standard form !0..5! ! !0 " !0 "4 Write each number in scientific notation. 6. 9,000,000, ,600, ,700,000,000,000 Note: On your scientific calculator, displays like 4.57 and.65 are numbers epressed in scientific notation. The first number means 4.57!0 and the second means.65!0 ". The calculator does this because there is not enough room on its display window to show the entire number. Answers. 785,000,000,000.,5,000, ! !0 " ! !0 " !0..5!0 ". 5.6! ! !0 " !0 4 Parent Guide with Etra Practice

112 RATIOS A ratio is a comparison of two quantities by division. It can be written in several ways: 65 miles hour, 65 miles: hour, or 65 miles to hour For additional information see the Math Notes boes in Lesson 4..4 of the Core Connections, Course tet or Lesson 5.. of the Core Connections, Course tet. Eample A bag contains the following marbles: 7 clear, 8 red and 5 blue. The following ratios may be stated: a. Ratio of blue to total number of marbles 5 0 = 4. b. Ratio of red to clear 8 7. c. Ratio of red to blue 8 5. d. Ratio of blue to red 5 8. Problems. Molly s favorite juice drink is made by miing cups of apple juice, 5 cups of cranberry juice, and cups of ginger ale. State the following ratios: a. Ratio of cranberry juice to apple juice. b. Ratio of ginger ale to apple juice. c. Ratio of ginger ale to finished juice drink (the miture).. A 40-passenger bus is carrying 0 girls, 6 boys, and teachers on a field trip to the state capital. State the following ratios: a. Ratio of girls to boys. b. Ratio of boys to girls. c. Ratio of teachers to students. d. Ratio of teachers to passengers.. It is important for Molly (from problem one) to keep the ratios the same when she mies larger or smaller amounts of the drink. Otherwise, the drink does not taste right. If she needs a total of 0 cups of juice drink, how many cups of each liquid should be used? 4. If Molly (from problem one) needs 5 cups of juice drink, how many cups of each liquid should be used? Remember that the ratios must stay the same. Core Connections, Courses

113 Ratios and Proportional Relationships Answers. a. 5 b. c. 0 = 5. 9 c. apple, 5 c. cranberry, 6 c. ginger ale. a. 0 6 = 5 4 b. 6 0 = 4 5 c. 6 d c. apple, c. cranberry, 5 c. ginger ale SCALING FIGURES AND SCALE FACTOR Geometric figures can be reduced or enlarged. When this change happens, every length of the figure is reduced or enlarged equally (proportionally), and the measures of the corresponding angles stay the same. The ratio of any two corresponding sides of the original and new figure is called a scale factor. The scale factor may be written as a percent or a fraction. It is common to write new figure measurements over their original figure measurements in a scale ratio, that is, NEW ORIGINAL. For additional information, see the Math Notes boes in Lesson 4.. of the Core Connections, Course tet or Lesson 6..6 of the Core Connections, Course tet. Eample using a 00% enlargement F C 6 mm mm 5 mm 0 mm B mm A E 4 mm original triangle new triangle D Side length ratios: DE AB = 4 = FD CA = 6 = FE CB = 0 5 = The scale factor for length is to. Eample Figures A and B at right are similar. Assuming that Figure A is the original figure, find the scale factor and find the lengths of the missing sides of Figure B. The scale factor is = 4. The lengths of the missing sides of Figure B are: 4 (0) =.5, 4 (8) = 4.5, and 4 (0) = 5. 0 A 0 8 B Parent Guide with Etra Practice

114 Problems Determine the scale factor for each pair of similar figures in problems through 4.. Original New. Original New D A 8 6 C B H E 4 G F Original New 4. Original New A triangle has sides 5,, and. The triangle was enlarged by a scale factor of 00%. a. What are the lengths of the sides of the new triangle? b. What is the ratio of the perimeter of the new triangle to the perimeter of the original triangle? 6. A rectangle has a length of 60 cm and a width of 40 cm. The rectangle was reduced by a scale factor of 5%. a. What are the dimensions of the new rectangle? b. What is the ratio of the perimeter of the new rectangle to the perimeter of the original rectangle? Answers. 4 8 =. 8 = a. 5, 6, 9 b. 6. a. 5 cm and 0 cm b. 4 4 Core Connections, Courses

115 Ratios and Proportional Relationships PROPORTIONAL RELATIONSHIPS A proportion is an equation stating the two ratios (fractions) are equal. Two values are in a proportional relationship if a proportion may be set up to relate the values. For more information, see the Math Notes boes in Lessons 4.., 4..4, and 7.. of the Core Connections, Course tet or Lessons.. and 7..5 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 9 materials or the Core Connections, Course Checkpoint materials. Eample The average cost of a pair of designer jeans has increased $5 in 4 years. What is the unit growth rate (dollars per year)? Solution: The growth rate is 5 dollars! 4!years = dollars!!year 5 dollars! 4!years. Using a Giant One: 5!dollars! 4!years. To create a unit rate we need a denominator of one. = 4 4!!dollars!".75 dollars.!year year Eample Ryan s famous chili recipe uses tablespoons of chili powder for 5 servings. How many tablespoons are needed for the family reunion needing 40 servings? Solution: The rate is!tablespoons 5!servings so the problem may be written as a proportion: 5 = t 40. One method of solving the proportion is to use the Giant One: Another method is to use cross multiplication: Finally, since the unit rate is 5 tablespoon per serving, the equation t = 5 s represents the general proportional situation and one could substitute the number of servings needed into the equation: t =! 40 = 4. Using any method the answer is 4 tablespoons. 5 Parent Guide with Etra Practice 5

116 Eample Based on the table at right, what is the unit growth rate (meters per year)? Solution: + height (m) 5 7 years Problems For problems through 0 find the unit rate. For problems through 5, solve each problem.. Typing 7 words in 7 minutes (words per minute). Reading 58 pages in 86 minutes (pages per minute). Buying 5 boes of cereal for $4.5 (cost per bo) 4. Scoring 98 points in a 40 minute game (points per minute) 5. Buying 4 pounds of bananas cost $.89 (cost per pound) 6. Buying pounds of peanuts for $.5 (cost per pound) 7. Mowing acres of lawn in 4 of a hour (acres per hour) 8. Paying $.89 for.7 pounds of chicken (cost per pound) 9. weight (g) length (cm) What is the weight per cm? 0. For the graph at right, what is the rate in miles per hour?. If a bo of 00 pencils costs $4.75, what should you epect to pay for 5 pencils?. When Amber does her math homework, she finishes 0 problems every 7 minutes. How long will it take for her to complete 5 problems? Distance (miles) Time (hours). Ben and his friends are having a TV marathon, and after 4 hours they have watched 5 episodes of the show. About how long will it take to complete the season, which has 4 episodes? 4. The ta on a $600 vase is $54. What should be the ta on a $700 vase? 6 Core Connections, Courses

117 Ratios and Proportional Relationships 5. Use the table at right to determine how long it will take the Spirit club to wa 60 cars. cars waed While baking, Evan discovered a recipe that required cups of walnuts for every cups of 4 flour. How many cups of walnuts will he need for 4 cups of flour? 7. Based on the graph, what would the cost to refill 50 bottles? 8. Sam grew 4 inches in 4 months. How much should he grow in one year? 9. On his afternoon jog, Chris took 4 minutes to run miles. How many miles can he run in 4 60 minutes? 0. If Caitlin needs cans of paint for each room in her house, how many cans of paint will she need to paint the 7-room house?. Stephen receives 0 minutes of video game time every 45 minutes of dog walking he does. If he wants 90 minutes of game time, how many hours will he need to work?. Sarah s grape vine grew 5 inches in 6 weeks, write an equation to represent its growth after t weeks.. On average Ma makes 45 out of 60 shots with the basketball, write an equation to represent the average number of shots made out of attempts. 4. Write an equation to represent the situation in problem 4 above. 5. Write an equation to represent the situation in problem 7 above. $ hours bottles refilled Answers. 4 words minute $ ! pound 6..8! $ 9. 5! grams centimeter.! pages minute..89! $ bo pound 0.! 7! miles hour 4..45! points minute 7.! acre hour 8..9! $ pound. $ min.. 9. hours 4. $ hours cup 7. $ inches 9.! 5.6 miles 0. 9 cans. 8 hours. g = 5 t. s = 4 5. C =.5b 4. t = 0.09c Parent Guide with Etra Practice 7

118 RATES AND UNIT RATES Rate of change is a ratio that describes how one quantity is changing with respect to another. Unit rate is a rate that compares the change in one quantity to a one-unit change in another quantity. Some eamples of rates are miles per hour and price per pound. If 6 ounces of flour cost $0.80 then the unit cost, that is the cost per one once, is $ = $0.05. For additional information see the Math Notes boes in Lesson 7.. of the Core Connections, Course tet, Lesson 4..4 of the Core Connections, Course tet, or Lesson 7..5 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 9 materials or the Core Connections, Course Checkpoint materials. Eample A rice recipe uses 6 cups of rice for 5 people. At the same rate, how much rice is needed for 40 people? The rate is: 6 cups 5 people so we need to solve 6 5 = 40. The multiplier needed for the Giant One is 40 5!or. Using that multiplier yields 6 Note that the equation 6 5 = 40 5! = 6 40 so 6 cups of rice is needed. can also be solved using proportions. Eample Arrange these rates from least to greatest: 0 miles in 5 minutes 60 miles in one hour 70 miles in hr Changing each rate to a common denominator of 60 minutes yields: 0 mi 5 min = => !.4.4 = 7 60 mi min 60 mi hr = 60 mi 70 mi 60 min = 70 mi hr 00 min = 60 => 70 00! 0.6 = 4 mi min So the order from least to greatest is: 70 miles in hr < 60 miles in one hour < 0 miles in 5 minutes. Note that by using 60 minutes (one hour) for the common unit to compare speeds, we can epress each rate as a unit rate: 4 mph, 60 mph, and 7 mph. 8 Core Connections, Courses

119 Ratios and Proportional Relationships Eample A train in France traveled 9 miles in 5 hours. What is the unit rate in miles per hour? 9 mi = 5 hr hr Unit rate means the denominator needs to be hour so: One of 0. or simple division yields = 86.4 miles per hour. 0.. Solving by using a Giant Problems Solve each rate problem below. Eplain your method.. Balvina knows that 6 cups of rice will make enough Spanish rice to feed 5 people. She needs to know how many cups of rice are needed to feed 5 people.. Elaine can plant 6 flowers in 5 minutes. How long will it take her to plant 0 flowers at the same rate?. A plane travels 400 miles in 8 hours. How far would it travel in 6 hours at this rate? 4. Shane rode his bike for hours and traveled miles. At this rate, how long would it take him to travel miles? 5. Selina s car used 5.6 gallons of gas to go 4 miles. At this rate, how many gallons would it take her to go 480 miles? 6. Arrange these readers from fastest to slowest: Abel read 50 pages in 45 minutes, Brian read 90 pages in 75 minutes, and Charlie read 75 pages in hours. 7. Arrange these lunch buyers from greatest to least assuming they buy lunch 5 days per week: Alice spends $ per day, Betty spends $5 every two weeks, and Cindy spends $75 per month. 8. A train in Japan can travel 8.5 miles in 5 hours. Find the unit rate in miles per hour. 9. An ice skater covered 500 meters in 06 seconds. Find his unit rate in meters per second. 0. A cellular company offers a price of $9.95 for 00 minutes. Find the unit rate in cost per minute.. A car traveled 00 miles on 8 gallons of gas. Find the unit rate of miles per gallon and the unit rate of gallons per mile.. Lee s paper clip chain is feet long. He is going to add paper clips continually for the net eight hours. At the end of eight hours the chain is 80 feet long. Find the unit rate of growth in feet per hour. Parent Guide with Etra Practice 9

120 Answers. 54 cups. 75 min. 550 miles 4. hr 5. gallons 6. C, B, A 7. C, A, B mi/hr 9.! 4.5 m/s 0.! $0.0 /min. 5 m/g; g/m. 6 ft/hr 5 DISTANCE, RATE, AND TIME Distance (d) equals the product of the rate of speed (r) and the time (t). This relationship is shown below in three forms: d = r!t!!!!!!!!!r = d t!!!!!!!!!t = d r It is important that the units of measure are consistent. For additional information see the Math Notes bo in Lesson 8.. of the Core Connections, Course tet. Eample Find the rate of speed of a passenger car if the distance traveled is 57 miles and the time elapsed is hours. 57 miles = r! hours! = r! 5 miles/hour = rate 57 miles hours Eample Find the distance traveled by a train at 5 miles per hour for 40 minutes. 40 The units of time are not the same so we need to change 40 minutes into hours. 60 = hour. d = (5 miles/hour)( hour)! d = 90 miles 0 Core Connections, Courses

121 Ratios and Proportional Relationships Eample The Central Middle School hamster race is fast approaching. Fred said that his hamster traveled 60 feet in 90 seconds and Wilma said she timed for one minute and her hamster traveled yards. Which hamster has the fastest rate? rate = distance time but all the measurements need to be in the same units. In this eample, we use feet and minutes. Fred s hamster: rate = 60 feet Wilma s hamster: Fred s hamster is faster. rate =.5 minutes 6 feet minute! rate = 40 feet/minute! rate = 6 feet/minute Problems Solve the following problems.. Find the time if the distance is 57.5 miles and the speed is 6 mph.. Find the distance if the speed is 67 mph and the time is.5 hours.. Find the rate if the distance is 47 miles and the time is.8 hours. 4. Find the distance if the speed is 60 mph and the time is hour and 45 minutes. 5. Find the rate in mph if the distance is.5 miles and the time is 0 minutes. 6. Find the time in minutes if the distance is miles and the rate is 0 mph. 7. Which rate is faster? A: 60 feet in 90 seconds or B: 60 inches in 5 seconds 8. Which distance is longer? A: 4 feet/second for a minute or B: inches/min for an hour 9. Which time is shorter? A: 4 miles at 60 mph or B: 6 miles at 80 mph Answers..5 hr. 4.5 mi. 65 mph miles mph 6. 4 min 7. B 8. A 9. A Parent Guide with Etra Practice

122 CALCULATING AND USING PERCENTS Students also calculate percentages by composition and decomposition, that is, breaking numbers into parts, and then adding or subtracting the results. This method is particularly useful for doing mental calculations. A percent ruler is also used for problems when you need to find the percent or the whole. For additional information, see the Math Notes bo in Lesson 9..4 of the Core Connections, Course tet. Knowing quick methods to calculate 0% of a number and % of a number will help you to calculate other percents by composition. Use the fact that 0% = 0 and % = 00. Eample To calculate % of 40, you can think of (0% of 40) + (% of 40). 0% of 40 0 of 40 = 4 and % of % of 40 (4) + (0.4) =.8 of 40 = 0.4 so Eample To calculate 9% of 750, you can think of 0% of 750 % of % of of 750 = 75 and % of % of = 67.5 of 750 = 7.5 so Other common percents such as 50% =, 5% = 4, 75% = 4, 0% = 5 may also be used. Students also use a percent ruler to find missing parts in percent problems. Eample Jana saved $7.50 of the original price of a sweater when it was on sale for 0% off. What was the original price of the sweater? $0 $7.50? 0% 0% 0% 0% 00% If every 0% is $7.50, the other four 0% parts ( 4! ) find that 00% is $7.50. Core Connections, Courses

123 Ratios and Proportional Relationships Eample 4 To calculate 7% of.4 convert the percent to a decimal and then use direct computation by hand or with a calculator. 7% of.4! 7 (.4)! 0.7(.4) = Problems Solve each problem without a calculator. Show your work or reasoning.. What is % of 60?. What is 9% of 500?. What is 4% of 8? 4. What is 80% of 65? is 0% of what number? 6. $.50 is 5% of what amount? is what % of 80? 8. What is 5% of? 9. What percent is 6 out of 0? 0. $0 is what percent of $5?. 0% of what number is 7. 5% of what number is 4? Compute by any method. Round to the nearest cent as appropriate.. 7% of % of % of % of % of % of $ % of $ % of $00..65% of $0,500. 0% of % of % of $ % of 6..5% of % of 450 Answers $ % % 0. 40% $ $ $48. $ $ Parent Guide with Etra Practice

124 PERCENT PROBLEMS USING DIAGRAMS A variety of percent problems described in words involve the relationship between the percent, the part and the whole. When this is represented using a number line, solutions may be found using logical reasoning or equivalent fractions (proportions). These linear models might look like the diagram at right. part of the whole part of 00% whole 00% For additional information, see the Math Notes bo in Lesson 5.. of the Core Connections, Course tet. Eample Sam s Discount Tires advertises a tire that originally cost $50 on sale for $5. What is the percent discount? A possible diagram for this situation is shown at right: part of the whole part of 00% $5 off $50 tire? % 00% In this situation it is easy to reason that since the percent number total (00%) is twice the cost number total ($50), the percent number saved is twice the cost number saved and is therefore a 0% discount. The problem could also be solved using a proportion 5 50 =? 00. whole Eample Martin received 808 votes for mayor of Smallville. If this was % of the total votes cast, how many people voted for mayor of Smallville? A possible diagram for this situation is shown at right: In this case it is better to write a pair of equivalent fractions as a proportion: 808 = If using the Giant One, the multiplier is 00 = so!.5.5 = A total of 55 people voted for mayor of Smallville. Note that the proportion in this problem could also be solved using cross-multiplication. 4 part of the whole part of 00% 808 votes? total votes % 00% 00. whole Core Connections, Courses

125 Ratios and Proportional Relationships Problems Use a diagram to solve each of the problems below.. Sarah s English test had 90 questions and she got 8 questions wrong. What percent of the questions did she get correct?. Cargo pants that regularly sell for $6 are now on sale for 0% off. How much is the discount?. The bill for a stay in a hotel was $88 including $5 ta. What percent of the bill was the ta? 4. Alicia got 60 questions correct on her science test. If she received a score of 75%, how many questions were on the test? 5. Basketball shoes are on sale for % off. What is the regular price if the sale price is $4? 6. Sergio got 80% on his math test. If he answered 4 questions correctly, how many questions were on the test? 7. A $65 coat is now on sale for $5. What percent discount is given? 8. Ellen bought soccer shorts on sale for $6 off the regular price of $40. What percent did she save? 9. According to school rules, Carol has to convince 60% of her classmates to vote for her in order to be elected class president. There are students in her class. How many students must she convince? 0. A sweater that regularly sold for $5 is now on sale at 0% off. What is the sale price?. Jody found an $88 pair of sandals marked 0% off. What is the dollar value of the discount?. Ly scored 90% on a test. If he answered 5 questions correctly, how many questions were on the test?. By the end of wrestling season, Mighty Ma had lost seven pounds and now weighs 8 pounds. What was the percent decrease from his starting weight? 4. George has 45 cards in his baseball card collection. Of these, 85 of the cards are pitchers. What percent of the cards are pitchers? 5. Julio bought soccer shoes at a 5% off sale and saved $4. What was the regular price of the shoes? Answers. 80%. $0.80. about 8% questions 5. $ questions 7. 0% 8. 5% 9. 0 students 0. $6.40. $ questions. about 5% 4. about 5% 5. $0 Parent Guide with Etra Practice 5

126 SCALING TO SOLVE PERCENT AND OTHER PROBLEMS Students used scale factors (multipliers) to enlarge and reduce figures as well as increase and decrease quantities. All of the original quantities or lengths were multiplied by the scale factor to get the new quantities or lengths. To reverse this process and scale from the new situation back to the original, we divide by the scale factor. Division by a scale factor is the same as multiplying by a reciprocal. This same concept is useful in solving equations with fractional coefficients. To remove a fractional coefficient you may divide each term in the equation by the coefficient or multiply each term by the reciprocal of the coefficient. Recall that a reciprocal is the multiplicative inverse of a number, that is, the product of the two numbers is. For eample, the reciprocal of is, is, and 5 is 5. Scaling may also be used with percentage problems where a quantity is increased or decreased by a certain percent. Scaling by a factor of does not change the quantity. Increasing by a certain percent may be found by multiplying by ( + the percent) and decreasing by a certain percent may be found by multiplying by ( the percent). For additional information, see the Math Notes boes in Lesson 7..4 of the Core Connections, Course tet. Eample The large triangle at right was reduced by a scale factor of to create a similar triangle. If the side labeled now 5 has a length of 80' in the new figure, what was the original length? To undo the reduction, multiply 80' by the reciprocal of 5, namely 5, or divide 80' by 5.!!!! 5 80' 80 ' 5!is the same as 80 '! 5, so = 00'. Eample Solve: = Method : Use division and a Giant One = = = = = 6 = 6 = 8 Method : Use reciprocals = ( ) = = 8 ( ) 6 Core Connections, Courses

127 Ratios and Proportional Relationships Eample Samantha wants to leave a 5% tip on her lunch bill of $.50. What scale factor should be used and how much money should she leave? Since tipping increases the total, the scale factor is ( + 5%) =.5. She should leave (.5)(.50) = $4.8 or about $4.50. Eample 4 Carlos sees that all DVDs are on sales at 40% off. If the regular price of a DVD is $4.95, what is the scale factor and how much is the sale price? If items are reduced 40%, the scale factor is ( 40%) = The sale price is (0.60)(4.95) = $4.97. Problems. A rectangle was enlarged by a scale factor of 5 original width? and the new width is 40 cm. What was the. A side of a triangle was reduced by a scale factor of. If the new side is now 8 inches, what was the original side?. The scale factor used to create the design for a backyard is inches for every 75 feet ( 75 ). If on the design, the fire pit is 6 inches away from the house, how far from the house, in feet, should the fire pit be dug? 4. After a very successful year, Cheap-Rentals raised salaries by a scale factor of. If Luan 0 now makes $4.0 per hour, what did she earn before? 5. Solve: 4 = Solve: 5 = 4 7. Solve: 5 y = Solve:! 8 m = 6 9. What is the total cost of a $9.50 family dinner after you add a 0% tip? 0. If the current cost to attend Magicland Park is now $9.50 per person, what will be the cost after a 8% increase?. Winter coats are on clearance at 60% off. If the regular price is $79, what is the sale price?. The company president has offered to reduce his salary 0% to cut epenses. If she now earns $75,000, what will be her new salary? Parent Guide with Etra Practice 7

128 Answers. 6 cm. 7 inches. 8 4 feet 4. $ ! 4 9. $ $.86. $.60. $57,500 PERCENT INCREASE OR DECREASE A percent increase is the amount that a quantity has increased based on a percent of the original amount. A percent decrease is the amount that a quantity has decreased based on a percent of the original amount. An equation that represents either situation is: amount of increase or decrease = (% change)(original amount) For additional information see the Math Notes bo in Lesson 7.. of the Core Connections, Course tet. Eample A town s population grew from 879 to 746 over five years. What was the percent increase in the population? Subtract to find the change: = 5547 Put the known numbers in the equation: 5547 = ()(879) The scale factor becomes, the unknown: = Divide: = !.95 Change to percent: = 95.% The population increased by 95.%. Eample A sumo wrestler retired from sumo wrestling and went on a diet. When he retired he weighed 85 pounds. After two years he weighed 8 pounds. What was the percent decrease in his weight? Subtract to find the change: 85 8 = 47 Put the known numbers in the equation: 47 = ()(85) The scale factor becomes, the unknown: = Divide: = 47 85! 0.8 Change to percent:! 8.% His weight decreased by about 8. %. Core Connections, Courses

129 Ratios and Proportional Relationships Problems Solve the following problems.. Forty years ago gasoline cost $0.0 per gallon on average. Ten years ago gasoline averaged about $.50 per gallon. What is the percent increase in the cost of gasoline?. When Spencer was 5, he was 8 inches tall. Today he is 5 feet inches tall. What is the percent increase in Spencer s height?. The cars of the early 900s cost $500. Today a new car costs an average of $7,000. What is the percent increase of the cost of an automobile? 4. The population of the U.S. at the first census in 790 was,99 people. By 000 the population had increased to 84,000,000! What is the percent increase in the population? 5. In 000 the rate for a first class U.S. postage stamp increased to $0.4. This represents a $0. increase since 97. What is the percent increase in cost since 97? 6. In 906 Americans consumed an average of 6.85 gallons of whole milk per year. By 998 the average consumption was 8. gallons. What is the percent decrease in consumption of whole milk? 7. In 984 there were 5 students for each computer in U.S. public schools. By 998 there were 6. students for each computer. What is the percent decrease in the ratio of students to computers? 8. Sara bought a dress on sale for $0. She saved 45%. What was the original cost? 9. Pat was shopping and found a jacket with the original price of $0 on sale for $9.99. What was the percent decrease in the cost? 0. The price of a pair of pants decreased from $49.99 to $9.95. What was the percent decrease in the price? Answers. 400%. 5%. 500% 4. 7,8,0.4% 5. 9.% % % 8. $ % % Parent Guide with Etra Practice 9

130 SIMPLE AND COMPOUND INTEREST In Course students are introduced to simple interest, the interest is paid only on the original amount invested. The formula for simple interest is: I = Prt and the total amount including interest would be: A = P + I. In Course, students are introduced to compound interest using the formula: A = P( + r) n. Compound interest is paid on both the original amount invested and the interest previously earned. Note that in these formulas, P = principal (amount invested), r = rate of interest, t and n both represent the number of time periods for which the total amount A, is calculated and I = interest earned. For additional information, see the Math Notes boes in Lesson 7..8 of the Core Connections, Course tet or Lesson 8.. of the Core Connections, Course tet. Eample Wayne earns 5.% simple interest for 5 years on $000. How much interest does he earn and what is the total amount in the account? Put the numbers in the formula I = Prt. Change the percent to a decimal. Multiply. Add principal and interest. I = 000(5.%)5 = 000(0.05)5 = 795 Wayne would earn $795 interest. $000 + $795 = $795 in the account Eample Use the numbers in Eample to find how much money Wayne would have if he earned 5.% interest compounded annually. Put the numbers in the formula A = P( + r) n. Change the percent to a decimal. Multiply. Wayne would have $ A = 000( + 5.%) 5 = 000( ) 5!or 000(.05) 5 = Students are asked to compare the difference in earnings when an amount is earning simple or compound interest. In these eamples, Wayne would have $88.86 more with compound interest than he would have with simple interest: $88.86 $795 = $ Core Connections, Courses

131 Ratios and Proportional Relationships Problems Solve the following problems.. Tong loaned Jody $50 for a month. He charged 5% simple interest for the month. How much did Jody have to pay Tong?. Jessica s grandparents gave her $000 for college to put in a savings account until she starts college in four years. Her grandparents agreed to pay her an additional 7.5% simple interest on the $000 for every year. How much etra money will her grandparents give her at the end of four years?. David read an ad offering 8 % simple interest on accounts over $500 left for a minimum 4 of 5 years. He has $500 and thinks this sounds like a great deal. How much money will he earn in the 5 years? 4. Javier s parents set an amount of money aside when he was born. They earned 4.5% simple interest on that money each year. When Javier was 5, the account had a total of $0.50 interest paid on it. How much did Javier s parents set aside when he was born? 5. Kristina received $5 for her birthday. Her parents offered to pay her.5% simple interest per year if she would save it for at least one year. How much interest could Kristina earn? 6. Kristina decided she would do better if she put her money in the bank, which paid.8% interest compounded annually. Was she right? 7. Suppose Jessica (from problem ) had put her $000 in the bank at.5% interest compounded annually. How much money would she have earned there at the end of 4 years? 8. Mai put $450 in the bank at 4.4% interest compounded annually. How much was in her account after 7 years? 9. What is the difference in the amount of money in the bank after five years if $500 is invested at.% interest compounded annually or at.9% interest compounded annually? 0. Ronna was listening to her parents talking about what a good deal compounded interest was for a retirement account. She wondered how much money she would have if she invested $000 at age 0 at.8% interest compounded quarterly (four times each year) and left it until she reached age 65. Determine what the value of the $000 would become. Parent Guide with Etra Practice

132 Answers. I = 50(0.05) = $.50; Jody paid back $ I = 000(0.075)4 = $600. I = $500(0.0875)5 = $ $0.50 = (0.045)5; = $ I = 5(0.05) = $ A = 5( ) = $8.50; No, for one year she needs to take the higher interest rate if the compounding is done annually. Only after one year will compounding earn more than simple interest. 7. A = 000( ) 4 = $ A = 450( ) 7 = $ A = 500( + 0.0) 5 500( ) 5 = $96.4 $884.4 = $ A = 000( ) 80 (because 45 4 = 80 quarters) = $88,64.5 Core Connections, Courses

133 Ratios and Proportional Relationships SLOPE The slope of a line is the ratio of the change in y to the change in between any two points on a line. Slope indicates the steepness (or flatness) of a line, as well as its direction (up or down) left to right. vertical change Slope is determined by the ratio horizontal change between any two points on a line. For lines that go up (from left to right), the sign of the slope is positive (the change in y is positive). For lines that go down (left to right), the sign of the slope is negative (the change is y is negative). A horizontal line has zero slope while the slope of a vertical line is undefined. For additional information see the Math Notes bo in Lesson 7..4 of the Core Connections, Course tet. Eample Write the slope of the line containing the points (, ) and (4, 5). First graph the two points and draw the line through them. Look for and draw a slope triangle using the two given points. (, ) y 5 ( 4, 5) Write the ratio triangle: 5. vertical change in y horizontal change in using the legs of the right Assign a positive or negative value to the slope (this one is positive) depending on whether the line goes up (+) or down ( ) from left to right. Eample y If the points are inconvenient to graph, use a generic slope triangle, visualizing where the points lie with respect to each other. For eample, to find the slope of the line that contains the points (, ) and (7, 4), sketch the graph at right to approimate the position of the two points, draw a slope triangle, find the length of the leg of each triangle, and write the ratio y = 6, then simplify. The slope is! since the change in y is negative (decreasing). 6 (, ) 8 (7, 4) Parent Guide with Etra Practice

134 Problems Write the slope of the line containing each pair of points.. (, 4) and (5, 7). (5, ) and (9, 4). (, ) and ( 4, 7) 4. (, ) and (, ) 5. (, ) and (4, ) 6. (, ) and (, 0) Determine the slope of each line using the highlighted points. 7. y 8. y 9. y Answers...! 4.! ! 5 7.! Core Connections, Courses

135 Geometry AREA OF POLYGONS AND COMPLEX FIGURES Area is the number of non-overlapping square units needed to cover the interior region of a twodimensional figure or the surface area of a three-dimensional figure. For eample, area is the region that is covered by floor tile (two-dimensional) or paint on a bo or a ball (threedimensional). For additional information about specific shapes, see the boes below. For additional general information, see the Math Notes bo in Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint materials or the Core Connections, Course Checkpoint 4 materials. AREA OF A RECTANGLE To find the area of a rectangle, follow the steps below.. Identify the base.. Identify the height.. Multiply the base times the height to find the area in square units: A = bh. A square is a rectangle in which the base and height are of equal length. Find the area of a square by multiplying the base times itself: A = b. Eample 4 square units 8 base = 8 units height = 4 units A = 8 4 = square units Parent Guide with Etra Practice 5

136 Problems Find the areas of the rectangles (figures -8) and squares (figures 9-) below mi 4 mi 5 cm 7 in. 8 m 6 cm in. m miles miles units 6.8 cm 8.7 units 7.5 miles.5 cm miles 8.6 feet 8 cm. cm.5 feet Answers. 8 sq. miles. 0 sq. cm. sq. in sq. m 5. sq. miles sq. feet 7..8 sq. cm sq. miles sq. cm sq. cm..5 sq. feet sq. feet 6 Core Connections, Courses

137 Geometry AREA OF A PARALLELOGRAM A parallelogram is easily changed to a rectangle by separating a triangle from one end of the parallelogram and moving it to the other end as shown in the three figures below. For additional information, see the Math Notes bo in Lesson 5.. of the Core Connections, Course tet. height base height base base base base height base parallelogram move triangle rectangle Step Step Step To find the area of a parallelogram, multiply the base times the height as you did with the rectangle: A = bh. Eample 6 cm 9 cm base = 9 cm height = 6 cm A = 9 6 = 54 square cm Problems Find the area of each parallelogram below feet 8 cm 8 feet 0 cm cm 7.5 in. cm in m m. ft 5 ft 9.8 cm 8.4 cm. cm 5.7 cm Parent Guide with Etra Practice 7

138 Answers. 48 sq. feet. 80 sq. cm. 44 sq. m 4. 9 sq. cm sq. in sq. ft sq. cm sq. cm AREA OF A TRIANGLE The area of a triangle is equal to one-half the area of a parallelogram. This fact can easily be shown by cutting a parallelogram in half along a diagonal (see below). For additional information, see Math Notes bo in Lesson 5..4 of the Core Connections, Course tet. height height base parallelogram Step base height base draw a diagonal Step height base match triangles by cutting apart or by folding Step As you match the triangles by either cutting the parallelogram apart or by folding along the diagonal, the result is two congruent (same size and shape) triangles. Thus, the area of a triangle has half the area of the parallelogram that can be created from two copies of the triangle. To find the area of a triangle, follow the steps below.. Identify the base.. Identify the height.. Multiply the base times the height. 4. Divide the product of the base times the height by : A = bh or bh. Eample Eample base = 6 cm 8 cm height = 8 cm 6 cm A = 6 8 = 8 = 64cm base = 7 cm height = 4 cm 4 cm A = 7 4 = 8 = 4cm 7 cm 8 Core Connections, Courses

139 Geometry Problems... 6 cm ft 8 cm 4 ft in. 5 ft 7 in. 7 ft m 6 cm.5 m cm 9 cm.5 ft cm 7 ft Answers. 4 sq. cm. 84 sq. ft. 9 sq. cm sq. in sq. ft sq. m sq. cm sq. ft AREA OF A TRAPEZOID A trapezoid is another shape that can be transformed into a parallelogram. Change a trapezoid into a parallelogram by following the three steps below. height top (t) base (b) height top (t) base (b) base (b) top (t) height height top (t) base (b) base (b) top (t) height Trapezoid duplicate the trapezoid and rotate put the two trapezoids together to form a parallelogram Step Step Step To find the area of a trapezoid, multiply the base of the large parallelogram in Step (base and top) times the height and then take half of the total area. Remember to add the lengths of the base and the top of the trapezoid before multiplying by the height. Note that some tets call the top length the upper base and the base the lower base. A = (b + t)h or A = b + t h For additional information, see the Math Notes bo in Lesson 6.. of the Core Connections, Course tet. Parent Guide with Etra Practice 9

140 Eample 8 in. in. 4 in. top = 8 in. base = in. height = 4 in. A = 8+ 4 = 0 4 = 0 4 = 40 in. Problems Find the areas of the trapezoids below in. cm 5 cm cm 5 in. 8 in cm 7 in. 8 cm 5 in. 8 m 5 cm 0 in cm feet 4 feet 5 feet m 8 m 8.4 cm 4 cm 0.5 cm cm 6.5 cm Answers. 4 sq. cm. 00 sq. in.. 4 sq. feet sq. cm sq. in sq. m 7. 5 sq. cm 8..5 sq. cm. 40 Core Connections, Courses

141 Geometry CALCULATING COMPLEX AREAS USING SUBPROBLEMS Students can use their knowledge of areas of polygons to find the areas of more complicated figures. The use of subproblems (that is, solving smaller problems in order to solve a larger problem) is one way to find the areas of complicated figures. Eample 9 " Find the area of the figure at right. 8 " 4 " " Method # Method # Method # 9 " 9 " 9 " 8 " A B 4 " 8 " A B 4 " 8 " 4 " " " " Subproblems: Subproblems: Subproblems:. Find the area of rectangle A: 8 9 = 7 square inches. Find the area of rectangle B: 4 ( 9) = 4 = 8 square inches. Add the area of rectangle A to the area of rectangle B: = 80 square inches. Find the area of rectangle A: 9 (8 4) = 9 4 = 6 square inches. Find the area of rectangle B: 4 = 44 square inches. Add the area of rectangle A to the area of rectangle B: = 80 square inches. Make a large rectangle by enclosing the upper right corner.. Find the area of the new, larger rectangle: 8 = 88 square inches. Find the area of the shaded rectangle: (8 4) ( 9) = 4 = 8 square inches 4. Subtract the shaded rectangle from the larger rectangle: 88 8 = 80 square inches Parent Guide with Etra Practice 4

142 Eample Find the area of the figure at right. 0 cm 6 cm 0 cm Subproblems: 8 cm 8 cm 6 cm 0 cm. Make a rectangle out of the figure by enclosing the top.. Find the area of the entire rectangle: 8 0 = 80 square cm. Find the area of the shaded triangle. Use the formula A = bh. b = 8 and h = 0 6 = 4, so A = (8 4) = = 6 square cm. 4. Subtract the area of the triangle from the area of the rectangle: 80 6 = 64 square cm. Problems Find the areas of the figures below... 7 m. 5" 0 ' 7 ' 6 ' 8 m m 9" 9" 0 ' 6 m 7" yds 8 m yds 5 m 0 m yds 8 m 5 m 0 yds 4 m 5 m ' 7 cm 5 cm ' cm 0 cm 7 cm cm 0 ' 0 ' 6 cm cm 4 cm 8 ' 4 ' 4 cm Core Connections, Courses

143 Geometry 0.. Find the area of the shaded region. m 6 m 8 m 9 " 8 m 7" 5 " Answers ". Find the area of the shaded region. ' 8 ' 4 ' 7 '. 58 sq. ft.. 5 sq. m.. 0 sq. in sq. yd sq. m sq. m sq. cm sq. ft. 9. sq. cm sq. m sq. in.. sq. ft. PRISMS VOLUME AND SURFACE AREA SURFACE AREA OF A PRISM The surface area of a prism is the sum of the areas of all of the faces, including the bases. Surface area is epressed in square units. For additional information, see the Math Notes boes in Lessons 9.. and 9.. of the Core Connections, Course tet and Lesson 9..4 of the Core Connections, Course tet. Eample Find the surface area of the triangular prism at right. Step : Area of the bases: (6 cm)(8 cm) = 48 cm Step : Area of the lateral faces Area of face : (6 cm)(7 cm) = 4 cm Area of face : (8 cm)(7 cm) = 56 cm Area of face : (0 cm)(7 cm) = 70 cm Step : Surface Area of Prism = sum of bases and lateral faces: SA = 48 cm + 4 cm + 56 cm + 70 cm = 6 cm 6 cm 0 cm 8 cm 7 cm Parent Guide with Etra Practice 4

144 Problems Find the surface area of each prism.... 5mm 5' ' 0 cm ' 9mm 8mm 5' 4 cm 4 cm The pentagon is equilateral. 0 cm 6. 0 cm cm 6 cm 6 cm 8 cm 5 ft 6 ft 0 cm 6 cm 6 cm 8 ft Answers. 4 mm. 9 cm. 0 ft 4. 9 cm ft cm 44 Core Connections, Courses

145 Geometry VOLUME OF A PRISM Volume is a three dimensional concept. It measures the amount of interior space of a threedimensional figure based on a cubic unit, that is, the number of by by cubes that will fit inside a figure. The volume of a prism is the area of either base (B) multiplied by the height (h) of the prism. V = (Area of base) (height) or V = Bh For additional information, see the Math Notes boes in Lesson 9.. of the Core Connections, Course tet and Lesson 9..4 of the Core Connections, Course tet. Eample Find the volume of the square prism below Eample Find the volume of the triangular prism below The base is a square with area (B) 8 8 = 64 units. Volume = B(h) = 64(5) = 0 units Eample Find the volume of the trapezoidal prism below The base is a right triangle with area (5)(7) = 7.5 units. Volume = B(h) = 7.5(9) = 57.5 units Eample 4 Find the height of the prism with a volume of.5 cm and base area of 5 cm. Volume = B(h).5 = 5(h) h =.5 5 h = 5. cm The base is a trapezoid with area (7 + 5) 8 = 88 units. Volume = B(h) = 88(0) = 880 units Parent Guide with Etra Practice 45

146 Problems Calculate the volume of each prism. The base of each figure is shaded.. Rectangular Prism. Right Triangular Prism. Rectangular Prism 5 in. 6 in. 4 ft 6 cm 8 cm ft ft 7 cm 8.5 in. 4. Right Triangular Prism 5. Trapezoidal Prism 6. Triangular Prism with 7. cm B = 5 cm 6' 4.5 cm 0' 6' 4 cm 8' 5 cm 7. Find the volume of a prism with base area cm and height.5 cm. 8. Find the height of a prism with base area cm and volume 76 cm. 9. Find the base area of a prism with volume 47.0 cm and height. cm. 8 4 cm Answers. ft. 68 cm. 40 in cm 5. 4 ft cm cm cm cm 46 Core Connections, Courses

147 Geometry NAMING QUADRILATERALS AND ANGLES A quadrilateral is any four-sided polygon. There are si special cases of quadrilaterals with which students should be familiar. Trapezoid A quadrilateral with at least one pair of parallel sides. Parallelogram A quadrilateral with both pairs of opposite sides parallel. Rectangle A quadrilateral with four right angles. Rhombus A quadrilateral with four sides of equal length. Square A quadrilateral with four right angles and four sides of equal length. Kite A quadrilateral with two distinct pairs of consecutive sides of equal length. Names of Basic Angles Acute angles are angles with measures between (but not including) 0º and 90º, right angles measure 90º, and obtuse angles measure between (but not including) 90ºand 80º. A straight angle measures 80º acute right obtuse straight Students use protractors to measure the size of angles. For more information see the Math Notes bo in Lesson 8.. of the Core Connections, Course tet. Parent Guide with Etra Practice 47

148 Eample For the figure at right, describe the quadrilateral using all terms that are appropriate. It is a rectangle since it has four right angles. It is a parallelogram since it has two pairs of parallel sides. It is also a trapezoid since it has at least one pair of parallel sides. Eample Describe the angle at right as right, obtuse or acute. Estimate the size of the angle and then use a protractor to measure the angle. It may be necessary to trace the angle and etend the sides. This angle opens narrower than a right angle so it is acute. Using a protractor shows that the angle measures 60º. Problems For each figure, describe the quadrilateral using all terms that are appropriate. Assume that sides that look parallel are parallel Core Connections, Courses

149 Geometry Describe the angles below as right, obtuse or acute. Estimate the size of the angle and then use a protractor to measure the angle Answers. parallelogram, trapezoid 4. rectangle, parallelogram, trapezoid. rhombus, parallelogram, trapezoid 5. square, rhombus, rectangle, parallelogram, trapezoid. trapezoid 6. not a quadrilateral 7. acute, 5 8. obtuse, 0 9. obtuse, 5 0. right, 90. acute, 80. acute, 8 Parent Guide with Etra Practice 49

150 ANGLE PAIR RELATIONSHIPS Properties of Angle Pairs Intersecting lines form four angles. The pairs of angles across from each other are called vertical angles. The measures of vertical angles are equal. w z y and y are vertical angles w and z are vertical angles If the sum of the measures of two angles is eactly 80º, then the angles are called supplementary angles. c and d are supplementary c d c = 0 d = 70 If the sum of the measures of two angles is eactly 90º, then the angles are called complementary angles. a b a = 0 b = 60 a and b are complementary Angles that share a verte and one side but have no common interior points (that is, do not overlap each other) are called adjacent angles. m n m and n are adjacent angles For additional information, see the Math Notes boes in Lesson 8.. of the Core Connections, Course tet and Lesson 9.. of the Core Connections, Course tet. 50 Core Connections, Courses

151 Geometry Eample Find the measure of the missing angles if m = 50º. Eample Classify each pair of angles below as vertical, supplementary, complementary, or adjacent m = m (vertical angles) m = 50º and (supplementary angles) m = 80º 50º = 0º m = m 4 (vertical angles) m 4 = 0º a. and are adjacent and supplementary b. and are complementary c. and 5 are adjacent d. and 4 are adjacent and supplementary e. and 4 are vertical Problems Find the measure of each angle labeled with a variable.... a 80º b 5º c d 75º e 4. f 40º 5. h 0º g i 0º 6. 75º n j p m l k 5º Answers. m a = 00º. m b = 55º. m c = 05º m d = 75º m e = 05º 4. m f = 50º 5. m g = 60º m h = 50º m i = 70º 6. m j = 75º m k = 65º m l = 40º m m = 40º m n = 05º m p = 05º Parent Guide with Etra Practice 5

152 PROPERTIES OF ANGLES, LINES, AND TRIANGLES Students learn the relationships created when two parallel lines are intersected by a transversal. They also study angle relationships in triangles. Parallel lines Triangles corresponding angles are equal: alternate interior angles are (eterior angle = sum remote interior angles) equal: Also shown in the above figures: vertical angles are equal: m = m linear pairs are supplementary: m + m 4 = 80 and m 6 + m 7 = 80 In addition, an isosceles triangle, ΔABC, has BA = BC and m A = m C. An equilateral triangle, ΔGFH, has GF = FH = HG and m G = m F = m H = 60. For more information, see the Math Notes boes in Lessons 9.., 9.., and 9..4 of the Core Connections, Course tet. A B C G F H Eample Solve for. Use the Eterior Angle Theorem: = = 08 = 08 6 = Eample Solve for. There are a number of relationships in this diagram. First, and the 7 angle are supplementary, so we know that m + 7 = 80 so m = 5. Using the same idea, m = 47. Net, m = 80, so m = 80. Because angle forms a vertical pair with the angle marked 7 +, 80 = 7 +, so = Core Connections, Courses

153 Geometry Eample Find the measure of the acute alternate interior angles. Parallel lines mean that alternate interior angles are equal, so = + 46 = 8 = 6. Use either algebraic angle measure: (6 ) + 46 = 58 for the measure of the acute angle Problems Use the geometric properties you have learned to solve for in each diagram and write the property you use in each case Parent Guide with Etra Practice 5

154 in. 8 in. 8 8 cm cm Answers Core Connections, Courses

155 Geometry CIRCLES CIRCUMFERENCE AND AREA CIRCUMFERENCE The radius of a circle is a line segment from its center to any point on the circle. The term is also used for the length of these segments. More than one radius are called radii. A chord of a circle is a line segment joining any two points on a circle. A diameter of a circle is a chord that goes through its center. The term is also used for the length of these chords. The length of a diameter is twice the length of a radius. radius The circumference of a circle is similar to the perimeter of a polygon. The circumference is the length of a circle. The circumference would tell you how much string it would take to go around a circle once. diameter chord Circumference is eplored by investigating the ratio of the circumference to the diameter of a circle. This ratio is a constant number, pi (π). Circumference is then found by multiplying π by the diameter. Students may use,.4, or the π button on their calculator, depending on the 7 teacher s or the book s directions. C = π r or C = πd For additional information, see the Math Notes boes in Lessons 8.. and 9.. of the Core Connections, Course tet or Lesson.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 4 materials. Eample Find the circumference of a circle with a diameter of 5 inches. d = 5 inches C = π d = π(5) or.4(5) = 5.7 inches Eample Find the circumference of a circle with a radius of 0 units. r = 0, so d = (0) = 0 C =.4(0) = 6.8 units Eample Find the diameter of a circle with a circumference of 6.8 inches. C = π d 6.8 = π d 6.8 =.4d d = = 5 inches Parent Guide with Etra Practice 55

156 Problems Find the circumference of each circle given the following radius or diameter lengths. Round your answer to the nearest hundredth.. d = in.. d =.4 cm. r =. ft 4. d = 5 m 5. r =.54 mi Find the circumference of each circle shown below. Round your answer to the nearest hundredth ' 0 cm Find the diameter of each circle given the circumference. Round your answer to the nearest tenth. 8. C = 48.6 yds 9. C = 5.6 ft 0. C = mm Answers in cm...9 ft m mi ft cm yds 9.. ft 0. 6 mm 56 Core Connections, Courses

157 Geometry AREA OF A CIRCLE In class, students have done eplorations with circles and circular objects to discover the relationship between circumference, diameter, and pi (π). To read more about the in-class eploration of area, see problems 9- through 9-6 (especially 9-6) in the Core Connections, Course tet. In order to find the area of a circle, students need to identify the radius of the circle. The radius is half the diameter. Net they will square the radius and multiply the result by π. Depending on the teacher s or book s preference, students may use for π when the radius or diameter is a 7 fraction,.4 for π as an approimation, or the π button on a calculator. When using the π button, most teachers will want students to round to the nearest tenth or hundredth. The formula for the area of a circle is: A = r π. Eample Find the area of a circle with r = 7 feet. A = ( 7) π = (7 7) (.4) = square feet Eample Find the area of a circle with d = 84 cm. r = 4 cm A = ( 4) π = (4 4) (.4) = square cm Problems Find the area of the circles with the following radius or diameter lengths. Use.4 for the value of π. Round to the nearest hundredth.. r = 6 cm. r =. in.. d = 6 ft 4. r = m 5. d = 4 5 cm 6. r = 5 in. 7. r =.6 cm 8. r = 4 9. d = 4.5 ft 0. r =.0 m Answers..04 cm..5 in ft 4. 4 m cm in cm or 5.90 in ft m Parent Guide with Etra Practice 57

158 RIGID TRANSFORMATIONS Studying transformations of geometric shapes builds a foundation for a key idea in geometry: congruence. In this introduction to transformations, the students eplore three rigid motions: translations, reflections, and rotations. A translation slides a figure horizontally, vertically or both. A reflection flips a figure across a fied line (for eample, the -ais). A rotation turns an object about a point (for eample, (0, 0)). This eploration is done with simple tools that can be found at home (tracing paper) as well as with computer software. Students change the position and/or orientation of a shape by applying one or more of these motions to the original figure to create its image in a new position without changing its size or shape. Transformations also lead directly to studying symmetry in shapes. These ideas will help with describing and classifying geometric shapes later in the course. For additional information, see the Math Notes bo in Lesson 6.. of the Core Connections, Course tet. Eample Decide which transformation was used on each pair of shapes below. Some may be a combination of transformations. a. b. c. d. e. f. 58 Core Connections, Courses

159 Geometry Identifying a single transformation is usually easy for students. In part (a), the parallelogram is reflected (flipped) across an invisible vertical line. (Imagine a mirror running vertically between the two figures. One figure would be the reflection of the other.) Reflecting a shape once changes its orientation, that is, how its parts sit on the flat surface. For eample, in part (a), the two sides of the figure at left slant upwards to the right, whereas in its reflection at right, they slant upwards to the left. Likewise, the angles in the figure at left switch positions in the figure at right. In part (b), the shape is translated (or slid) to the right and down. The orientation is the same. Part (c) shows a combination of transformations. First the triangle is reflected (flipped) across an invisible horizontal line. Then it is translated (slid) to the right. The pentagon in part (d) has been rotated (turned) clockwise to create the second figure. Imagine tracing the first figure on tracing paper, then holding the tracing paper with a pin at one point below the first pentagon, then turning the paper to the right (that is, clockwise) 90. The second pentagon would be the result. Some students might see this as a reflection across a diagonal line. The pentagon itself could be, but with the added dot, the entire shape cannot be a reflection. If it had been reflected, the dot would have to be on the corner below the one shown in the rotated figure. The triangles in part (e) are rotations of each other (90 clockwise again). Part (f) shows another combination. The triangle is rotated (the horizontal side becomes vertical) but also reflected since the longest side of the triangle points in the opposite direction from the first figure. Eample Translate (slide) ΔABC right si units and up three units. Give the coordinates of the new triangle. The original vertices are A( 5, ), B(, ), and C(0, 5). The new vertices are A' (, ), B ' (, 4), and C ' (6, ). Notice that the change to each original point (, y) can be represented by ( + 6, y + ). A B y A' C B' C' Eample Reflect (flip) ΔABC with coordinates A(5, ), B(, 4), and C(4, 6) across the y-ais to get ΔA' B'C '. The key is that the reflection is the same distance from the y-ais as the original figure. The new points are A '( 5, ), B'(, 4), and C '( 4, 6). Notice that in reflecting across the y-ais, the change to each original point (, y) can be represented by (, y). If you reflect ΔABC across the -ais to get ΔPQR, then the new points are P(5, ), Q(, 4), and R(4, 6). In this case, reflecting across the -ais, the change to each original point (, y) can be represented by (, y). A' C' y B' B Q C A P R Parent Guide with Etra Practice 59

160 Eample 4 Rotate (turn) ΔABC with coordinates A(, 0), B(6, 0), and C(, 4) 90 C' A' counterclockwise about the origin (0, 0) to get ΔA' B'C ' with coordinates B'' A (0, ), B'(0, 6), and C'( 4, ). Notice that for this 90 A'' A B counterclockwise rotation about the origin, the change to each original point (, y) can be represented by ( y, ). C'' Rotating another 90 (80 from the starting location) yields ΔA"B"C" with coordinates A"(, 0), B"( 6, 0), and C"(, 4). For this 80 counterclockwise rotation about the origin, the change to each original point (, y) can be represented by (, y). Similarly a 70 counterclockwise or 90 clockwise rotation about the origin takes each original point (, y) to the point (y, ). ' y B' C Problems For each pair of triangles, describe the transformation that moves triangle A to the location of triangle B... A A B B. 4. A B A B For the following problems, refer to the figures below: Figure A Figure B Figure C y y y C B C A B A B A C 60 Core Connections, Courses

161 Geometry State the new coordinates after each transformation. 5. Slide figure A left units and down units. 6. Slide figure B right units and down 5 units. 7. Slide figure C left unit and up units. 8. Flip figure A across the -ais. 9. Flip figure B across the -ais. 0. Flip figure C across the -ais.. Flip figure A across the y-ais.. Flip figure B across the y-ais.. Flip figure C across the y-ais. 4. Rotate figure A 90 counterclockwise about the origin. 5. Rotate figure B 90 counterclockwise about the origin. 6. Rotate figure C 90 counterclockwise about the origin. 7. Rotate figure A 80 counterclockwise about the origin. 8. Rotate figure C 80 counterclockwise about the origin. 9. Rotate figure B 70 counterclockwise about the origin. 0. Rotate figure C 90 clockwise about the origin. Answers ( 4 may vary; 5 0 given in the order A ', B', C'). translation. rotation and translation. reflection 4. rotation and translation 5. (, ) (, ) (, ) 6. (, ) (, ) (, 0) 7. ( 5, 4) (, 4) (, ) 8. (, 0) (, 4) (5, ) 9. ( 5, ) (, ) (0, 5) 0. ( 4, ) (4, ) (, ). (, 0) (, 4) ( 5, ). (5, ) (, ) (0, 5). (4, ) ( 4, ) (, ) 4. (0, ) ( 4, ) (, 5) 5. (, 5) ( 5, 0) (, ) 6. (, 4) (, 4) (, ) 7. (, 0) (, 4) ( 5, ) 8. (4, ) ( 4, ) (, ) 9. (, 5) (, ) (5, 0) 0. (, 4) (, 4) (, ) Parent Guide with Etra Practice 6

162 SIMILAR FIGURES Two figures that have the same shape but not necessarily the same size are similar. In similar figures the measures of the corresponding angles are equal and the ratios of the corresponding sides are proportional. This ratio is called the scale factor. For information about corresponding sides and angles of similar figures see Lesson 4.. in the Core Connections, Course tet or the Math Notes bo in Lesson 6.. of the Core Connections, Course tet. For information about scale factor and similarity, see the Math Notes boes in Lesson 4.. of the Core Connections, Course tet or Lesson 6..6 of the Core Connections, Course tet. Eample Determine if the figures are similar. If so, what is the scale factor? cm 9 cm cm 9 cm cm 7 cm 9 = = 7 9 = or The ratios of corresponding sides are equal so the figures are similar. The scale factor that compares the small figure to the large one is or to. The scale factor that compares the large figure to the small figure is or to. Eample Determine if the figures are similar. If so, state the scale factor. 8 ft 6 ft 9 ft 6 4 = 8 = 9 6 and all equal. 4 ft 6 ft 6 ft ft 8 6 = 4 so the shapes are not similar. 8 ft 6 Core Connections, Courses

163 Geometry Eample Determine the scale factor for the pair of similar figures. Use the scale factor to find the side length labeled with a variable. original 8 cm 5 cm new cm scale factor = 5 original 5 new 8 5 = ; = 4 5 = 4.8 cm Problems Determine if the figures are similar. If so, state the scale factor of the first to the second... Parallelograms Kites Determine the scale factor for each pair of similar figures. Use the scale factor to find the side labeled with the variable y y t a b z c 5 Parent Guide with Etra Practice 6

164 Answers. similar;. similar; 8 =.6. not similar ; = ; y = ; = 0 = 6, y = 6 = 5, t = 8, z = 5 = ; a = 6 5 =., b = 4 5 = 4.8, c = 6 PYTHAGOREAN THEOREM A right triangle is a triangle in which the two shorter sides form a right angle. The shorter sides are called legs. Opposite the right angle is the third and longest side called the hypotenuse. The Pythagorean Theorem states that for any right triangle, the sum of the squares of the lengths of the legs is equal to the square of the length of the hypotenuse. (leg ) + (leg ) = (hypotenuse) leg leg hypotenuse leg hypotenuse leg For additional information, see Math Notes bo in Lesson 9.. of the Core Connections, Course tet. Eample Use the Pythagorean Theorem to find. a. b = = 69 = = = = 00 = 6 = 6 64 Core Connections, Courses

165 Geometry Eample Not all problems will have eact answers. Use square root notation and your calculator. 4 m m = m = 00 m = 84 m = Eample A guy wire is needed to support a tower. The wire is attached to the ground five meters from the base of the tower. How long is the wire if the tower is 0 meters tall? First draw a diagram to model the problem, then write an equation using the Pythagorean Theorem and solve it. 0 5 = = = 5 = 5.8 cm Problems Write an equation and solve it to find the length of the unknown side. Round answers to the nearest hundredth a y c 6 8 b 0 0 Parent Guide with Etra Practice 65

166 Draw a diagram, write an equation, and solve it. Round answers to nearest hundredth. 7. Find the diagonal of a television screen 0 inches wide by 5 inches tall. 8. A 9-meter ladder is one meter from the base of a building. How high up the building will the ladder reach? 9. Sam drove eight miles south and then five miles west. How far is he from his starting point? 0. The length of the hypotenuse of a right triangle is si centimeters. If one leg is four centimeters, how long is the other leg?. Find the length of a path that runs diagonally across a 55-yard by 00-yard field.. How long an umbrella will fit in the bottom of a suitcase.5 feet by.5 feet? Answers in m mi cm. 4. yd..9 ft 66 Core Connections, Courses

167 Geometry CYLINDERS VOLUME AND SURFACE AREA VOLUME OF A CYLINDER The volume of a cylinder is the area of its base multiplied by its height: V = B h Since the base of a cylinder is a circle of area A = r π, we can write: V = r πh For additional information, see the Math Notes bo in Lesson 0.. of the Core Connections, Course tet. Eample ft Eample? 4 ft SODA cm Find the volume of the cylinder above. Use a calculator for the value of π. Volume = r πh = () π (4) = 6π =.0 ft The soda can above has a volume of 55 cm and a height of cm. What is its diameter? Use a calculator for the value of π. Volume = r πh 55 = r π () 55 π = r 9.4 = r radius =.07 diameter = (.07) = 6.4 cm Problems Find the volume of each cylinder.. r = 5 cm h = 0 cm. r = 7.5 in. h = 8. in.. diameter = 0 cm h = 5 cm 4. base area = 50 cm h = 4 cm 5. r = 7 cm h = 0 cm 6. d = 9 cm h = cm Parent Guide with Etra Practice 67

168 Find the missing part of each cylinder. 7. If the volume is 575 ft and the height is ft, find the diameter. 8. If the volume is 6,0.07 inches and the radius is 7. inches, find the height. 9. If the circumference is 6 cm and the height is 5 cm, find the volume. Answers cm. 4.9 in cm cm cm cm ft 8. 8 inches 9. 8, cm SURFACE AREA OF A CYLINDER The surface area of a cylinder is the sum of the two base areas and the lateral surface area. The formula for the surface area is: SA = r π + πdh or SA = r π + πrh where r = radius, d = diameter, and h = height of the cylinder. For additional information, see the Math Notes bo in Lesson 0.. of the Core Connections, Course tet. Eample Find the surface area of the cylinder at right. Use a calculator for the value of π. 8 cm 5 cm Step : Area of the two circular bases [(8 cm) π] = 8π cm 5 cm Step : Area of the lateral face 68 π(6)5 = 40π cm Step : Surface area of the cylinder 8π cm + 40π cm = 68π cm 56. cm 5 cm circumference of base = 6π cm lateral face rectangle Core Connections, Courses

169 Geometry Eample Eample 0 cm 5 ft 0 cm SA = r π + πrh = (5) π + π 5 0 = 50π + 00π = 50π 47.4 cm If the volume of the tank above is 500π ft, what is the surface area? V = π r h 500π = π r (5) 500π 5π = r 00 = r 0 = r SA = r π + πrh = 0 π + π(0)(5) = 00π + 00π = 00π ft Problems Find the surface area of each cylinder.. r = 6 cm, h = 0 cm. r =.5 in., h = 5 in.. d = 9 in., h = 8.5 in. 4. d = 5 cm, h =0 cm 5. base area = 5, height = 8 6. Volume = 000 cm, height = 5 cm Answers cm in in cm un cm Parent Guide with Etra Practice 69

170 PYRAMIDS AND CONES VOLUME The volume of a pyramid is one-third the volume of the prism with the same base and height and the volume of a cone is onethird the volume of the cylinder with the same base and height. The formula for the volume of the pyramid or cone with base B and height h is: V = Bh h base area (B) h For the cone, since the base is a circle the formula may also be written: V = r πh For additional information, see the Math Notes bo in Lesson 0..4 of the Core Connections, Course tet. Eample Find the volume of the cone below. Eample Find the volume of the pyramid below. Eample If the volume of a cone is cm and its radius is 9 cm, find its height. 0 ' Volume = r π h 7 5' 8' = (9) π h = π(8) h Volume = (7) π 0 490π = 5. units Problems Find the volume of each cone.. r = 4 cm h = 0 cm 4. d = 9 cm h = 0 cm 70 Base is a right triangle B = 5 8 = 0 Volume = ft. r =.5 in. h = 0.4 in. 5. r = 6 ft h = ft π = h 5 cm = h. d = in. h = 6 in. 6. r = 4 ft h = 6 ft Core Connections, Courses

171 Geometry Find the volume of each pyramid. 7. base is a square with side 8 cm h = cm 8. base is a right triangle with legs 4 ft and 6 ft h = 0 ft 9. base is a rectangle with width 6 in., length 8 in. h = 5 in. Find the missing part of each cone described below. 0. If V = 000 cm and r = 0 cm, find h.. If V = 000 cm and h = 5 cm, find r.. If the circumference of the base = 6 cm and h = 0 cm, find the volume. Answers cm in in cm ft ft cm 8. 4 ft in cm..8 cm. 4.4 cm Parent Guide with Etra Practice 7

172 SPHERES VOLUME For a sphere with radius r, the volume is found using: V = 4 π r. For more information, see the Math Notes bo in Lesson 0..5 of the Core Connections, Course tet. radius center Eample Find the volume of the sphere at right. V = 4 π r = 4 π = π ft eact answer or using π.4 (.4 ).49 ft approimate answer feet Eample A sphere has a volume of 97π. Find the radius. Use the formula for volume and solve the equation for the radius. V = 4 π r = 97π Substituting 4πr = 96π Multiply by to remove the fraction r = 96π 4π = 79 Divide by 4π to isolate r. r = 79 = 9 To undo cubing, take the cube root 7 Core Connections, Courses

173 Geometry Problems Use the given information to find the eact and approimate volume of the sphere.. radius = 0 cm. radius = 4 ft. diameter = 0 cm 4. diameter = miles 6. circumference of great circle = π 6. circumference of great circle = π Use the given information to answer each question related to spheres. 7. If the radius is 7 cm, find the volume. 8. If the diameter is 0 inches, find the volume. 9. If the volume of the sphere is 6π, find the radius. 0. If the volume of the sphere is 56π, find the radius. Answers. 4000π cm. 56π ft. 500π 5. cm 4. 9π 4. mi 5. 88π 904. un 6. 9π 4. un 7. 7π cm π 5.60 in. 9. r = units 0. r = 4 units Parent Guide with Etra Practice 7

174 SIMPLE PROBABILITY Outcome: Any possible or actual result of the action considered, such as rolling a 5 on a standard number cube or getting tails when flipping a coin. Event: A desired (or successful) outcome or group of outcomes from an eperiment, such as rolling an even number on a standard number cube. Sample space: All possible outcomes of a situation. For eample, the sample space for flipping a coin is heads and tails; rolling a standard number cube has si possible outcomes (,,, 4, 5, and 6). Probability: The likelihood that an event will occur. Probabilities may be written as fractions, decimals, or percents. An event that is guaranteed to happen has a probability of, or 00%. An event that has no chance of happening has a probability of 0, or 0%. Events that might happen have probabilities between 0 and or between 0% and 00%. In general, the more likely an event is to happen, the greater its probability. Eperimental probability: The probability based on data collected in eperiments. number of successful outcomes in the eperiment Eperimental probability = total number of outcomes in the eperiment Theoretical probability is a calculated probability based on the possible outcomes when they all have the same chance of occurring. Theoretical probability = number of successful outcomes (events) total number of possible outcomes In the contet of probability, successful usually means a desired or specified outcome (event), such as rolling a on a number cube (probability of 6 ). To calculate the probability of rolling a, first figure out how many possible outcomes there are. Since there are si faces on the number cube, the number of possible outcomes is 6. Of the si faces, only one of the faces has a on it. Thus, to find the probability of rolling a, you would write: P() = number of ways to roll number of possible outcomes =. or 0.6.or approimately 6.7% 6 Eample If you roll a fair, 6-sided number cube, what is P(), that is, the probability that you will roll a? Because the si sides are equally likely to come up, and there is only one, P() = Core Connections, Courses

175 Statistics and Probability Eample There are marbles in a bag: clear, 4 green, 5 yellow, and blue. If one marble is chosen randomly from the bag, what is the probability that it will be yellow? P(yellow) = 5 (yellow) (outcomes) = 5 Eample Joe flipped a coin 50 times. When he recorded his tosses, his result was 0 heads and 0 tails. Joe s activity provided data to calculate eperimental probability for flipping a coin. a. What is the theoretical probability of Joe flipping heads? The theoretical probability is 50% or, because there are only two possibilities (heads and tails), and each is equally likely to occur. b. What was the eperimental probability of flipping a coin and getting heads based on Joe s activity? The eperimental probability is 0 when he flipped the coin. 50, 5, or 60%. These are the results Joe actually got Eample 4 Decide whether these statements describe theoretical or eperimental probabilities. a. The chance of rolling a 6 on a fair die is 6. This statement is theoretical. b. I rolled the die times and 5 came up three times. This statement is eperimental. c. There are 5 marbles in a bag; 5 blue, 6 yellow, and 4 green. The probability of getting a blue marble is. This statement is theoretical. d. When Veronika pulled three marbles out of the bag she got yellow and blue, or yellow, blue. This statement is eperimental. Parent Guide with Etra Practice 75

176 Problems. There are 4 crayons in a bo: 5 black, white, 7 red, yellow, blue, and 4 green. What is the probability of randomly choosing a green? Did you respond with an eperimental or theoretical probability?. A spinner is divided into four equal sections numbered, 4, 6, and 8. What is the probability of spinning an 8?. A fair number cube marked,,, 4, 5, and 6 is rolled. Tyler tossed the cube 40 times, and noted that 6 times an even number showed. What is the eperimental probability that an even number will be rolled? What is the theoretical probability? 4. Sara is at a picnic and reaches into an ice chest, without looking, to grab a can of soda. If there are 4 cans of orange, cans of fruit punch, and 0 cans of cola, what is the probability that she takes a can of fruit punch? Did you respond with an eperimental probability or a theoretical one? 5. A baseball batting average is the probability a baseball player hits the ball when batting. If a baseball player has a batting average of 66, it means the player s probability of getting of getting a hit is Is a batting average an eperimental probability or theoretical? 6. In 0, 9 people died by being struck by lightning, and 4 people were injured. There were 0,000,000 people in the United States. What is the probability of being one of the people struck by lightning? 7. In a medical study, 07 people were given a new vitamin pill. If a participant got sick, they were removed from the study. Ten of the participants caught a common cold, came down with the flu, 8 got sick to their stomach, and 77 never got sick. What was the probability of getting sick if you participated in this study? Did you respond with an eperimental probability or a theoretical one? 8. Insurance companies use probabilities to determine the rate they will charge for an insurance policy. In a study of 00 people that had life insurance policies, an insurance company found that people were over 80 years old when they died, 8 people died when they were between 70 and 80 years old, 5 died between 60 and 70 years old, and 55 died when they were younger than 60 years old. In this study what was the probability of dying younger than 70 years old? Did you respond with an eperimental probability or a theoretical one? Answers. 6 ; theoretical ; 6 4. ;theoretical 5. eperimental ,000,000! ! 0.8 eperimental ! 5.7% eperimental 76 Core Connections, Courses

177 Statistics and Probability INDEPENDENT AND DEPENDENT EVENTS Two events are independent if the outcome of one event does not affect the outcome of the other event. For eample, if you draw a card from a standard deck of playing cards but replace it before you draw again, the outcomes of the two draws are independent. Two events are dependent if the outcome of one event affects the outcome of the other event. For eample, if you draw a card from a standard deck of playing cards and do not replace it for the net draw, the outcomes of the two draws are dependent. Eample Juan pulled a red card from the deck of regular playing cards. This probability is 6 5 or. He puts the card back into the deck. Will his chance of pulling a red card net time change? No, his chance of pulling a red card net time will not change, because he replaced the card. There are still 6 red cards out of 5. This is an eample of an independent event; his pulling out and replacing a red card does not affect any subsequent selections from the deck. Eample Brett has a bag of 0 multi-colored candies. 5 are red, 6 are blue, 5 are green, are yellow, and are brown. If he pulls out a yellow candy and eats it, does this change his probability of pulling any other candy from the bag? Yes, this changes the probability, because he now has only 9 candies in the bag and only yellow candy. Originally, his probability of yellow was 0 or ; it is now. Similarly, red 5 9 was 5 0 or 5 and now is 9, better than. This is an eample of a dependent event. Parent Guide with Etra Practice 77

178 Problems Decide whether these events are independent or dependent events.. Flipping a coin, and then flipping it again.. Taking a black 7 out of a deck of cards and not returning it, then taking out another card.. Taking a red licorice from a bag and eating it, then taking out another piece of licorice. Answers. independent. dependent. dependent 78 Core Connections, Courses

179 Statistics and Probability COMPOUND PROBABILITY AND COUNTING METHODS COMPOUND PROBABILITY Sometimes when you are finding a probability, you are interested in either of two outcomes taking place, but not both. For eample, you may be interested in drawing a king or a queen from a deck of cards. At other times, you might be interested in one event followed by another event. For eample, you might want to roll a one on a number cube and then roll a si. The probabilities of combinations of simple events are called compound events. To find the probability of either one event or another event that has nothing in common with the first, you can find the probability of each event separately and then add their probabilities. Using the eample above of drawing a king or a queen from a deck of cards: P(king) = 5 4 and P(queen) = 4 5 so P(king or queen) = = 5 8 = For two independent events, to find the probability of both one and the other event occurring, you can find the probability of each event separately and then multiply their probabilities. Using the eample of rolling a one followed by a si on a number cube: P() = 6 and P(6) = 6 so P( then 6) = 6! 6 = 6 Note that you would carry out the same computation if you wanted to know the probability of rolling a one on a green cube, and a si on a red cube, if you rolled both of them at the same time. Eample A spinner is divided into five equal sections numbered,,, 4, and 5. What is the probability of spinning either a or a 5? Step : Determine both probabilities: P() = 5 and P(5) = 5 Step : Since these are either-or compound events, add the fractions describing each probability: = 5 The probability of spinning a or a 5 is 5 : P( or 5) = 5 Parent Guide with Etra Practice 79

180 Eample If each of the regions in each spinner at right is the same size, what is the probability of spinning each spinner and getting a green t-shirt? white green red blue sweater t-shirt sweatshirt Step : Step : Determine both possibilities: P(green) = 4 and P(t-shirt) = Since you are interested in the compound event of both green and a t-shirt, multiply both probabilities: 4! = The probability of spinning a green t-shirt is : P(green t-shirt) = Problems Assume in each of the problems below that events are independent of each other.. One die, numbered,,, 4, 5, and 6, is rolled. What is the probability of rolling either a or a 6?. Mary is playing a game in which she rolls one die and spins a spinner. What is the probability she will get both the and black she needs to win the game? blue red. A spinner is divided into eight equal sections. The sections are numbered,,, 4, 5, 6, 7, and 8. What is the probability of spinning a,, or a 4? 4. Patty has a bo of colored pencils. There are blue, black, gray, red, green, orange, purple, and yellow in the bo. Patty closes her eyes and chooses one pencil. She is hoping to choose a green or a red. What is the probability she will get her wish? black 5. Use the spinners at right to tell Paul what his chances are of getting the silver truck he wants. scooter car blue truck black silver 6. On the way to school, the school bus must go through two traffic signals. The first light is green for 5 seconds out of each minute, and the second light is green for 5 seconds out of each minute. What is the probability that both lights will be green on the way to school? 80 Core Connections, Courses

181 Statistics and Probability 7. There are 50 students at South Lake Middle School. 5 enjoy swimming, 50 enjoy skateboarding, and 75 enjoy playing softball. What is the probability a student enjoys all three sports? 8. John has a bag of jellybeans. There are 00 beans in the bag. of the beans are cherry, 4 4 of the beans are orange, 4 of the beans are licorice, and of the beans are lemon. 4 What is the probability that John will chose one of his favorite flavors, orange, or cherry? 9. A nationwide survey showed that only 4% of children liked eating lima beans. What is the probability that any two children will both like lima beans? Answers. 6 or ! 5 60 " ! 50 50! = or 9. 6 Parent Guide with Etra Practice 8

182 COUNTING METHODS There are several different models you can use to determine all possible outcomes for compound events when both one event and the other occur: a systematic list, a probability table, and a probability tree. See the Math Notes bo in Lesson 5.5. of the Core Connections, Course tet for details on these three methods. Not only can you use a probability table to help list all the outcomes, but you can also use it to help you determine probabilities of independent compound events when both one event and the other occur. For eample, the following probability table (sometimes called an area model) helps determine the probabilities from Eample above: sweater sweatshirt t-shirt white red blue green Each bo in the rectangle represents the compound event of both a color and the type of clothing (sweater, sweatshirt, or t-shirt). The area of each bo represents the probability of getting each combination. For eample, the shaded region represents the probability of getting a green t-shirt: 4! =. Eample At a class picnic Will and Jeff were playing a game where they would shoot a free throw and then flip a coin. Each boy only makes one free throw out of three attempts. Use a probability table (area model) to find the probability that one of the boys makes a free throw, and then flips a head. What is the probability that they miss the free throw and then flip tails? Make Miss Miss H T By finding the area of the small rectangles, the probabilities are: P(make and heads) =! = 6, and P(miss and tails) =! = 6 8 Core Connections, Courses

183 Statistics and Probability Eample 4 Chris owns a coffee cart that he parks outside the downtown courthouse each morning. 65% of his customers are lawyers; the rest are jury members. 60% of Chris s sales include a muffin, 0% include cereal, and the rest are coffee only. What is the probability a lawyer purchases a muffin or cereal? The probabilities could be represented in an area model as follows: muffin 0.60 cereal 0.0 coffee only 0.0 lawyer 0.65 jury 0.5 Probabilities can then be calculated: The probability a lawyer purchases a muffin or cereal is = or 45.5%. lawyer 0.65 jury 0.5 muffin cereal coffee only Eample 5 The local ice cream store has choices of plain, sugar, or waffle cones. Their ice cream choices are vanilla, chocolate, bubble gum, or frozen strawberry yogurt. The following toppings are available for the ice cream cones: sprinkles, chocolate pieces, and chopped nuts. What are all the possible outcomes for a cone and one scoop of ice cream and a topping? How many outcomes are possible? Probability tables are useful only when there are two events. In this situation there are three events (cone, flavor, topping), so we will use a probability tree. There are four possible flavors, each with three possible cones. Then each of those outcomes can have three possible toppings. There are 6 outcomes for the compound event of choosing a flavor, cone, and topping. Note that the list of outcomes, and the total number of outcomes, does not change if we change the order of events. We could just as easily have chosen the cone first. Vanilla Chocolate Bubble Gum Frozen Yogurt plain sugar waffle plain sugar waffle plain sugar waffle plain sugar waffle sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts sprinkles chocolate pieces chopped nuts Parent Guide with Etra Practice 8

184 Problems Use probability tables or tree diagrams to solve these problems.. How many different combinations are possible when buying a new bike if the following options are available: mountain bike or road bike black, red, yellow, or blue paint speed, 5 speed, or 0 speed. A new truck is available with: standard or automatic transmission wheel or 4 wheel drive regular or king cab long or short bed How many combinations are possible?. A ta assessor categorizes 5% of the homes in how city as having a large backyard, 65% as having a small backyard, and 0% as having no backyard. 0% of the homes have a tile roof, the rest have some other kind of roof. What is the probability a home with a tile roof has a backyard? 4. There is space for only 96 students at University High School to enroll in a shop class: 5 students in woodworking, 5 students in metalworking, and the rest in print shop. Threefourths of the spaces are reserved for seniors, and one-fourth are for juniors. What is the probability that a student enrolled in shop class is a senior in print shop? What is the probability that a student enrolled in shop class is a junior in wood or metal shop? 5. Insurance companies use probabilities to determine the rate they will charge for an insurance policy. In a study of 000 people that had life insurance policies, an insurance company collected the following data of how old people were when they died, compared to how tall they were. In this study, what was the probability of being tall (over 6ft) and dying young under 50 years old? What was the probability of being tall and dying under 70 years old? What was the probability of being between 50 and 70 years old? <50 years old yrs old yrs old 70 80yrs old >80 years old over 6ft tall under 6ft tall Core Connections, Courses

185 Statistics and Probability Answers. There are 4 possible combinations as shown below. Mountain Road black red yellow blue black red yellow blue speed 5 speed 0speed speed 5 speed 0speed speed 5 speed 0speed speed 5 speed 0 speed speed 5 speed 0speed speed 5 speed 0 speed speed 5 speed 0speed speed 5 speed 0 speed. There are 6 possible combinations as shown below. Standard Automatic - wheel drive 4 - wheel drive - wheel drive 4 - wheel drive regular cab king cab regular cab king cab regular cab king cab regular cab king cab long bed short bed long bed short bed long bed short bed long bed short bed long bed short bed long bed short bed long bed short bed long bed short bed Parent Guide with Etra Practice 85

186 . The probability is = 0.05 or 0.5%. large yard 5% small yard 65% no yard 0% tile roof 0% other roof 70% 4. The probability of a senior in print shop is about 0.59%. The probability of a junior in wood or metal shop is seniors 4 juniors 4 woodworking metalworking print shop The probability of being tall (over 6ft) and dying young under 50 years old is = 0.0. The probability of being tall and dying under 70 years old is ! The probability of being between 50 and 70 years old is ! Core Connections, Courses

187 Statistics and Probability CIRCLE GRAPHS A circle graph (or pie chart) is a diagram that represents proportions of categorized data as parts of a circle. Each sector or wedge represents a percent or fraction of the circle. The fractions or percents must total, or 00%. Since there are 60 degrees in a circle, the size of each sector (in degrees) is found by multiplying the fraction or percent by 60 degrees. For additional information, see the Math Notes bo in Lesson 7.. of the Core Connections, Course tet. Eample Ms. Sallee s class of 0 students was surveyed about the number of hours of homework done each night and here are the results: less than hour students to hours 9 students to hours students to 4 hours 4 students more than 4 hours students The proper size for the sectors is found as follows: < : 0! 60! = 6! ; : 9 0! 60! = 08! : 0! 60! = 44! ; 4: 4 0! 60! = 40! > 4: 0! 60! = 0! The circle graph is shown at right. Eample - 4 hours more than 4 hours - hours less than hour - hours The 800 students at Central Middle School were surveyed to determine their favorite school lunch item. The results are shown below. hamburger chicken tacos other salad bar pizza Use the circle graph at left to answer each question. a. Which lunch item was most popular? b. Approimately how many students voted for the salad bar? c. Which two lunch items appear to have equal popularity? Answers: a. pizza is the largest sector; b.! 800 = 00 ; 4 c. hamburger and chicken tacos have the same size sectors Parent Guide with Etra Practice 87

188 Problems For problems through use the circle graph at right. The graph shows the results of the 00 votes for prom queen. Dominique. Who won the election?. Did the person who won the election get more than half of the votes? Camille Alicia. Approimately how many votes did Camille receive? Barbara 4. Of the milk consumed in the United States, 0% is whole, 50% is low fat, and 0% is skim. Draw a circle graph to show this data. 5. On an average weekday, Sam s time is spent as follows: sleep 8 hours, school 6 hours, entertainment hours, homework hours, meals hour, and job 4 hours. Draw a circle graph to show this data. 6. Records from a pizza parlor show the most popular one-item pizzas are: pepperoni 4%, sausage 5%, mushroom 0%, olive 9% and the rest were others. Draw a circle graph to show this data. 7. To pay for a 00 billion dollar state budget, the following monies were collected: income taes 90 billion dollars, sales taes 74 billion dollars, business taes 0 billion dollars, and the rest were from miscellaneous sources. Draw a circle graph to show this data. 8. Greece was the host country for the 004 Summer Olympics. The Greek medal count was 6 gold, 6 silver, and 4 bronze. Draw a circle graph to show this data. 88 Core Connections, Courses

189 Statistics and Probability Answers. Alicia. No Alicia s sector is less than half of a circle.. Approimately 40 votes job skim whole meals sleep homework low fat entertainment school other business taes misc. olive mushroom pepperoni income taes sausage sales taes 8. bronze gold silver Parent Guide with Etra Practice 89

190 MEASURES OF CENTRAL TENDENCY Measures of central tendency are numbers that locate or approimate the center of a set of data that is, a typical value that describes the set of data. Mean and median are the most common measures of central tendency. (Mode will not be covered in this course.) The mean is the arithmetic average of a data set. Add all the values in a set and divide this sum by the number of values in the set. The median is the middle number in a set of data arranged numerically. An outlier is a number that is much smaller or larger than most of the others in the data set. The range of a data set is the difference between the highest and lowest values of the data set. For additional information, see the Math Notes boes in Lesson 8.. of the Core Connections, Course tet or Lessons.. and..4 of the Core Connections, Course tet. The mean is calculated by finding the sum of the data set and dividing it by the number of elements in the set. Eample Find the mean of this set of data: 4,, 7, 44, 8, 4, 4, 4, 4, and = = 7.8 The mean of this set of data is 7.8. Eample Find the mean of this set of data: 9, 8, 80, 9, 78, 75, 95, and = = 8. The mean of this set of data is 8.. Problems Find the mean of each set of data.. 9, 8, 4, 0,, 6, and 4.. 5, 4, 5, 7,, and , 89, 79, 84, 95, 79, 78, 89, 76, 8, 76, 9, 89, 8, and. 4. 6, 04, 0,, 00, 07,, 8,, 0, 08, 09, 05, 0, and Core Connections, Courses

191 Statistics and Probability The median is the middle number in a set of data arranged in numerical order. If there is an even number of values, the median is the mean (average) of the two middle numbers. Eample Find the median of this set of data: 4,, 7, 44, 8, 4, 4, and 4. Arrange the data in order:, 4, 4, 4, 7, 8, 4, 4, 44. Find the middle value(s): 7 and 8. Since there are two middle values, find their mean: = 75,! 75 = 7.5. Therefore, the median of this data set is 7.5. Eample 4 Find the median of this set of data: 9, 8, 80, 9, 78, 75, 95, 77, and 77. Arrange the data in order: 75, 77, 77, 78, 80, 8, 9, 9, and 95. Find the middle value(s): 80. Therefore, the median of this data set is 80. Problems Find median of each set of data. 5. 9, 8, 4, 0,, 6, and , 4, 7, 5,, and , 89, 79, 84, 95, 79, 78, 89, 76, 8, 76, 9, 89, 8, and. 8. 6, 04, 0,, 00, 07,, 8,, 0, 08, 09, 05, 0, and 9. The range of a set of data is the difference between the highest value and the lowest value. Eample 5 Find the range of this set of data: 4, 09,, 96, 40, and 8. The highest value is 40. The lowest value is ! 96 = 44. The range of this set of data is 44. Eample 6 Find the range of this set of data: 7, 44, 6, 9, 78, 5, 57, 54, 6, 7, and 48. The highest value is 78. The lowest value is 7. 78! 7 = 5. The range of this set of data is 5. Parent Guide with Etra Practice 9

192 Problems Find the range of each set of data in problems 5 through 8. Outliers are numbers in a data set that are either much higher or much lower that the other numbers in the set. Eample 7 Find the outlier of this set of data: 88, 90 96, 9, 87,, 85, and 94. The outlier is. Eample 8 Find the outlier of this set of data: 67, 54, 49, 76, 64, 59, 60, 7,, 44, and 66. The outlier is. Problems Identify the outlier in each set of data , 77, 75, 68, 98, 70, 7, and ,, 7, 6, 0, 6, and , 645, 78, 455, 754, 790, 84, 64, 49, and , 65, 9, 5, 55, 4, 79, 85, 55, 7, 78, 8, 9, and 76. Answers median 0; range 8 6. median 8.5; range 9 7. median 8; range median 07; range Core Connections, Courses

193 Statistics and Probability GRAPHICAL REPRESENTATIONS OF DATA Students represent distributions of single-variable data numerical data using dot plots, stem-andleaf plots, bo plots, and histograms. They represent categorical one-variable data on bar graphs. Each representation communicates information in a slightly different way. STEM-AND-LEAF-PLOTS A stem-and-leaf plot is a way to display data that shows the individual values from a set of data and how the values are distributed. The stem part on the graph represents all of the digits ecept the last one. The leaf part of the graph represents the last digit of each number. Read more about stem-and-leaf plots, and how they compare to dot plots and histograms, in the Math Notes in Lesson.. and.. in the Core Connections, Course tet, and in the Math Notes in Lesson 7.. of the Core Connections, Course tet. Eample Make a stem-and-leaf plot of this set of data: 4,, 7, 44, 8, 9, 4, 4, 4, 4, 5, and Eample Make a stem-and-leaf plot of this set of data: 9, 8, 80, 9, 78, 75, 95, 77, and Problems Make a stem-and-leaf plot of each set of data.. 9, 8, 4, 0,, 6, 8, and 4.. 5, 4, 7, 5, 9,, 4, and , 89, 79, 84, 95, 79, 89, 67, 8, 76, 9, 89, 8, and. 4. 6, 04, 0,, 00, 07,, 8,, 0, 08, 09, 05, 0, and 9. Parent Guide with Etra Practice 9

194 HISTOGRAMS AND BAR GRAPHS Histograms and bar graphs are visual ways to represent data. Both consist of vertical bars (called bins) with heights that represent the number of data points (called the frequency) in each bin. Histograms are for displaying distributions of numerical data. In a histogram each bar represents the number of data elements within a certain range of values. All the bars touch each other. Values at the left side of a bin s range are included in that bin. Each range of values should have the same width. Bar graphs are for displaying categorical data. In a bar graph each bar represents the number of data elements in a certain category. All the bars are the same width and are separated from each other. For additional information and eamples, see the Math Notes boes in Lessons.. and.. of the Core Connections, Course tet, or Lesson 7.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 9A materials or Connections, Course Checkpoint 7B materials in the back of those tets. Eample The scores for a 5-point quiz are listed below arranged from least to greatest. 7, 7,,, 5, 6, 6, 6, 8, 9, 0, 0, 0,,,,, 4 Using intervals of five points, create a histogram for the class. Frequency See histogram at right. Scores on the right end of the interval are included in the net interval. The interval between 0 and 5 only includes the two scores of and. The interval between 5 and 0 only includes the si scores of 5, 6, 6, 6, 8, and 9. 0 Score Eample 4 Ms. Lim asked each of her students about their favorite kind of pet. Based on their responses, she drew the bar graph at right. Use the bar graph to answer each question. 94 a. What is the favorite pet? b. How many students chose a bird as their favorite pet? c. What was the least favorite pet? d. If every student voted once, how many students are in the class? Answers: a. dog b. 6 c. fish d. 8 Frequency 0 cat dog fish bird Favorite pet Core Connections, Courses

195 Statistics and Probability Problems 5. Mr. Diaz surveyed his employees about the time it takes them to get to work. The results are shown in the histogram at right. a. How many employees completed the survey? b. How many employees get to work in less than 0 minutes? c. How many employees get to work in less than 40 minutes? d. How many employees take 60 minutes to get to work? Frequency 0 Minutes to work 6. The two sith grade classes at Vista Middle School voted for their favorite dessert. The results are shown in the bar graph at right for the five favorite choices. a. What was the favorite dessert and how many students made that choice? Frequency b. How many students selected cake as their favorite dessert? c. How many students selected yogurt as their favorite? d. How many more students selected ice cream than pudding? 0 yogurt cake pudding fruit ice cream 7. Mr. Fernandez asked 0 people at work how many pets they owned. The results are shown at right. Make a histogram to display this data. Use intervals of one pet. 8. During the fist week of school Ms. Chan asked her students to name the county where they were born. There were so many different countries she grouped them by continent: North America: 4 students, South America: students, Europe: students, Asia: 0 students, Africa: student, Australia: 0 students. Make a bar graph to display this information. Favorite dessert 0 pets 5 people pet 8 people pets 0 people pets people 4 pets people 5 pets person 9 pets person 9. Three coins were tossed 0 times and the number of results that were heads each time is shown below:,,, 0,,,,,,,,,,, 0,,, 0,, Make a histogram to show the results. 0. The physical education teacher at West Middle School asked the class about their favorite winter activity. Here were the results: reading: 8 students, ice skating: 4 students, skiing: 6 student, snowboarding: students, computer activities: 4 students. Make a bar graph to show the results. Parent Guide with Etra Practice 95

196 BOX PLOTS Another way to display a distribution of one-variable numerical data is with a bo plot. A bo plot is the only display of data that clearly shows the median, quartiles, range, and outliers of a data set. For additional information, see the Math Notes boes in Lessons 8..4 and 8..5 of the Core Connections, Course tet, or Lessons 7.. and 7..5 of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 9A materials or the Core Connections, Course Checkpoint 7B materials in the back of the tet. Eample 5 Display this data in a bo plot: 5, 55, 55, 6, 65, 7, 76, 78, 79, 8, 8, 85, 9, and 9. Since this data is already in order from least to greatest, the range is 9 5 = 4. Thus you start with a number line with equal intervals from 50 to 00. The median of the set of data is 77. A vertical segment is drawn at this value above the number line. The median of the lower half of the data (the first quartile) is 6. A vertical segment is drawn at this value above the number line. The median of the upper half of the data (the third quartile) is 8. A vertical segment is drawn at this value above the number line. A bo is drawn between the first and third quartiles. Place a vertical segment at the minimum value (5) and at the maimum value (9). Use a line segment to connect the minimum to the bo and the maimum to the bo. Eample 6 Display this data in a bo plot: 6, 65, 9, 5,, 79, 85, 55, 7, 78, 8, 9, and 76. Place the data in order from least to greatest:, 5, 55, 6, 65, 7, 76, 78, 79, 8, 85, 9, 9. The range is 9 = 8. Thus you want a number line with equal intervals from 0 to 00. Find the median of the set of data: 76. Draw the line segment. Find the first quartile: = 7; 7 = Draw the line segment. Find the third quartile: = 68; 68 = 84. Draw the line segment. Draw the bo connecting the first and third quartiles. Place a line segment at the minimum value () and a line segment at the maimum value (9). Connect the minimum and maimum values to the bo Core Connections, Courses

197 Statistics and Probability Problems Create a stem-and-leaf plot and a bo plot for each set of data in problems 5 through , 47, 5, 85, 46,, 8, 80, and , 6, 56, 80, 7, 55, 54, and , 54, 5, 58, 6, 7, 7, 78, 7, 8, 8, 7, 6, 67, and , 5, 48, 9, 57, 87, 94, 68, 86, 7, 58, 74, 85, 9, 88, and Given a set of data: 65, 6, 69, 59, 67, 64, 5, 75, 64, 60, 7, 57, and 9. a. Make a stem-and-leaf plot of this data. b. Find the mean and median of this data. c. Find the range of this data. d. Make a bo plot for this data. 6. Given a set of data: 48, 4, 7, 9, 49, 46, 8, 8, 45, 45, 5, 46.5, 4, 46, 46.5, 4, 46.5, 48, 4.5, 9, and a. Make a stem-and-leaf plot of this data. b. Find the mean and median of this data. c. Find the range of the data. d. Make a bo plot for this data. Parent Guide with Etra Practice 97

198 Answers a. 4 b. 6 c. 4 d a. ice cream 0 b c. d. 5 Frequency 0 Number of pets Frequency Frequency Frequency 0 S. America N. America Europe Asia Africa Austrailia Continent of birth 0 Number of heads 0 read ice ski snowboard computer skate Favorite winter activity 98 Core Connections, Courses

199 Statistics and Probability Mean: 66.5 Median: 64 Range: Mean: Median: 4 Range: Parent Guide with Etra Practice 99

200 SCATTERPLOTS, ASSOCIATION, AND LINE OF BEST FIT Data that is collected by measuring or observing naturally varies. A scatterplot helps students decide is there is a relationship (an association) between two numerical variables. If there is a possible linear relationship, the trend can be shown graphically with a line of best fit on the scatterplot. In this course, students use a ruler to eyeball a line of best fit. The equation of the best-fit line can be determined from the slope and the y-intercept. An association is often described by its form, direction, strength, and outliers. See the Math Notes boes in Lessons 7.., 7.., and 7.. of the Core Connections, Course tet. For additional eamples and practice, see the Core Connections, Course Checkpoint 9 materials. Eample Sam collected data by measuring the pencils of her classmates. She recorded the length of the painted part of each pencil and its weight. Her data is shown on the graph at right. a. Describe the association between weight and length of the pencil. b. Create a line of best fit where y is the weight of the pencil in grams and is the length of the paint on the pencil in centimeters. c. Sam s teacher has a pencil with.5 cm of paint. Predict the weight of the teacher s pencil using the equation found in part (b). Weight (g) Length of Paint (cm) Answer: a. There is a strong positive linear association with one apparent outlier at.cm. b. The equation of the line of best fit is approimately: y = See graph at right. c. 4 (.5) +.5! 4.4 g. Weight (g) Length of Paint (cm) 00 Core Connections, Courses

201 Statistics and Probability Problems In problems through 4 describe (if they eist), the form, direction, strength, and outliers of the scatterplot... Age of Owner Number of Times Test Taken Number of Cars Owned Number of Test Items Correct. 4. Chapter 5 Test Score Distance From Light Bulb Height Brightness of Light Bulb 5. Dry ice (frozen carbon dioide) evaporates at room temperature. Giulia s father uses dry ice to keep the glasses in the restaurant cold. Since dry ice evaporates in the restaurant cooler, Giulia was curious how long a piece of dry ice would last. She collected the data shown in the table at right. Draw a scatterplot and a line of best fit. What is the approimate equation of the line of best fit? # of hours after noon Weight of dry ice (oz) Parent Guide with Etra Practice 0