Mini-Lectures by Section

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1 Mini-Lectures by Section

3 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture Learn the definition of factor.. Write fractions in lowest terms.. Multiply and divide fractions.. Add and subtract fractions.. Solve applied problems that involve fractions. 6. Interpret data in a circle graph. Fractions 1. Identify each number as prime or composite. If the number is composite, write it in factored form. a) 1 b) 6 c) 97 d) 16. Simplify each fraction. a) b) c) 77. Change each improper fraction to a mixed number or a whole number, or change each mixed number to an improper fraction. 8 a) b) c) d) 6 9. Find each product and write in lowest terms. 9 a) b) c) d) Find each quotient and write in lowest terms. 9 a) b) c) d) 6 d) 6. Find each sum or difference and write in lowest terms. 1 1 a) + b) c) + d) e) f) g) h) Rachel is jogging for exercise. This week, she ran 1 miles on Monday, miles on Tuesday, and on 1 Thursday she ran times the distance that she ran on Monday. How many miles did she run this week? Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

$The circle graph below shows the approximate fractions of their responses. a) Which breed of dog was most popular? What was the fractional value for the most popular breed of dog?$ $Encourage students to simplify fractions by dividing numerator and denominator by the same number and by factoring into primes and dividing out common factors.$

4 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture In a recent survey 1000 people were asked about their favorite breed of dog. The circle graph below shows the approximate fractions of their responses. a) Which breed of dog was most popular? What was the fractional value for the most popular breed of dog? b) How many people chose the Yorkshire Terrier as their favorite breed of dog? c) How many people did not choose the Pug as their favorite breed of dog? Encourage students to simplify fractions by dividing numerator and denominator by the same number and by factoring into primes and dividing out common factors. Some students try to multiply/divide whole number parts together, and then multiply/divide fractional parts together when working with mixed numbers. Some students add/subtract the denominators when adding/subtracting fractions. Answers: 1a) prime, b) composite, 1, c) prime, d) composite, ; a) 1, b) 1, c) 6 11, d) ; 1 a) 1, b) 8, c) 11, d) 6 ; a) 1, b) 1 or 9 9, c) 8, d) or 8 8 ; a) or 7 9 9, b), c), d) ; 6a) 1, b) 1, c) 1 8, d) or 17 1, e) 11 60, f), g) , h) 8 ; 7) 7 miles ; 8a) Labrador Retriever, 91/00, b) 00, c) 9 Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley

5 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. Exponents, Order of Operations, and Inequality 1. Use exponents.. Use the rules for order of operations.. Use more than one grouping symbol.. Know the meaning of, <, >,, and.. Translate word statements to symbols. 6. Write statements that change the direction of inequality symbols. 1. Find the value of each exponential expression. a) 6 b) c). Find the value of each expression. a) 8 b) c) 10 + d) e) 6+ + f) g) ( ) ( ). Using inequality symbols: a) Determine whether the statement is true or false: 16 b) Write the statement in symbols: Seven is not equal to. c) Write the statement with the inequality symbol reversed: 1 > 8 d) Write the statement in words: > 1 Some students do not know how to say, or, or, etc., in words and need to see the words written. Students will often compute as. Illustrate the difference between the expressions with examples. Students should be reminded to work from inside out when evaluating expressions with grouping symbols. Students should be reminded that implies that either the < part or the = part can be true to satisfy the inequality (similarly for ). One way to remember the meaning of < and > is that the symbol always points to the lesser number. Refer students to the order of operations chart in the text. Answers: 1a) 6, b) 81, c) greater than one ; a) 0, b) 9, c), d) 1, e) 6, f) 1, g) 18; a) false, b) 7, c) 8< 1, d) Five is Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley

6 6 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. Variables, Expressions, and Equations 1. Evaluate algebraic expressions, given values for the variables.. Translate word phrases to algebraic expressions.. Identify solutions of equations.. Identify solutions of equations from a set of numbers.. Distinguish between expressions and equations. 1. Evaluate the expressions when a =, b =, and c = 6. ( a+ b) 1 ab a) a+ b b) c) + cb d) c 6 a + b 1 7 c. Write each word phrase as an algebraic expression, using x as the variable. a) The quotient of 6 less than a number and b) Fifty more than a number c) The product of 8 and the total of a number and. Decide whether the given number is a solution of the equation. a) Is 0 a solution of 6 x = 6 x? b) Is a solution of x + = 10? c) Is 9 a solution of 1 = + x?. Write each word statement as an equation. Use x as the variable. Find all solutions from the set { 1,,, 7, 9 }. a) A number plus equals 7. b) Two less than three times a number is thirteen. c) Four times a number added to three is equal to six more than the number. d) A number divided by fourteen is one-half.. Write each word sentence as an equation. Use x as the variable. a) The difference between a number and 1 is. b) The quotient of a number and is. c) Twice the difference between a number and 16 is times the number. d) Eight less than times a number is more than times the number. When a numerical coefficient is 1, the 1 is usually not written (e.g., 1x is usually written as x). Students often ignore a term when they do not see a coefficient. Remind students that an exponent refers only to the variable or number just before it (e.g., x means x, not x x ). When translating the difference between a and b, the order is kept the same: a b. Remind students that an expression evaluates to a number, while an equation forms a sentence (with an = symbol). Answers: 1a) 1, b), c) 9, d) ; a) x 6, b) x + 0, c) 8( x+ ) ; a) yes, b) no, c) yes; a), b), c) 1, d) 7; a) x 1 =, b) x x 16 = x, d) x 8 = x + =, c) ( ) Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

7 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. 7 Real Numbers and the Number Line 1. Classify numbers and graph them on number lines.. Tell which of two real numbers is less than the other.. Find the additive inverse of a real number.. Find the absolute value of a real number.. Interpret the meanings of real numbers from a table of data. 1. List all numbers from the set 10,,, 0,, 10 that are: a) natural numbers b) whole numbers c) integers d) rational numbers e) irrational numbers f) real numbers. Graph each of the numbers on a number line: 1 1,,, 1.. Select the lesser number in each pair. 1 a), 8 b), c) 10, 1. Find the additive inverse of each number. a) 6 b) c) 6. Simplify. a) b) 6 c) Decide whether the statement is true or false. a) 10 > b) c) 7 > 1 7. The table shows the percent change in real median income for selected age groups from 006 to 007 and 007 to 008. Use the table to answer the questions below. Age Groups Percent Change from 006 to 007 Percent Change from 007 to years years years years 1.9. Source: U.S. Census Bureau a) Which age group in which year represents the greatest percent decrease? b) Which age group in which year represents the greatest percent increase? c) Which age group in which year represents the smallest percent decrease? Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

8 8 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. Some students find it helpful to see each type of number in example 1 on a number line. Remind students that integers are rational numbers; any integer can be written as the ratio of itself and 1. Decimal numbers that terminate or repeat in a fixed block are rational numbers ask students to give examples of both. The decimal form of an irrational number neither terminates nor repeats. The number line is a good way to illustrate that opposite numbers are equidistant from 0 but on opposite sides of 0. Some students have never seen absolute value before and will need examples. Answers: 1a) 10, b) 0, 10, c) 10,, 0, 10, d) 10,,, 0, 10, e), f) 10,,, 0,, 10; ) ; a), b), c) 1; a) 6, b), c) ; a), b) 6, c) ; 6a) true, 6 b) true, c) false; 7a) -6 years from , b) - years from , c) - years from Copyright 01 Pearson Education,. Inc. Publishing as Addison-Wesley

BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. 9 Adding and Subtracting Real Numbers 1. Add two numbers with the same sign.. Add two numbers with different signs.

9 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. 9 Adding and Subtracting Real Numbers 1. Add two numbers with the same sign.. Add two numbers with different signs.. Use the definition of subtraction.. Use the rules for order of operations with real numbers.. Translate words and phrases involving addition and subtraction. 6. Use signed numbers to interpret data. 1. Find each sum or difference. a) 9 1 d) b) 16 + ( 10) c) 1 + ( 6) + e) 1 + ( 6) + 17 f) h) g) + + ( ). Write a numerical expression for each phrase, and simplify the expression. a) The sum of 6 and and 1 b) The sum of 10 and 1, increased by 1 c) 0.9 more than the sum of.6 and.1 Solve each problem. i) A scuba diver is at a depth of 16 feet below the surface. He descends another 8 feet. What is his new depth?. On January 1, in New Market, Indiana, the temperature rose 17º F in three hours. If the starting temperature was º F, what was the temperature three hours later?. The bar graph gives the median U.S. household income for the years 00 to 008. Source: U.S. Census Bureau a) Use a signed number to represent the change in household income from 00 to 006. b) Use a signed number to represent the change in household income from 007 to 008. Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

10 10 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1. Some students need to see addition problems done on a number line first. Caution students about the difference between the subtraction key and the change-of-sign key on a calculator. Refer students to the summary box for adding and subtracting signed numbers. Emphasize the uses of the symbol with students. Answers: 1a) 1, b) 6, c) 0, d) 1.1, e) 1, f), g) 10, h) 10, i) ; a) 6+ + ( 1);, 1 b) 10 + ( 1) + 1; 1, c) 6. + ( 1. ) ; 1. ; ) feet; ) 1ºF; a) $187, b) $70 Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

11 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture Multiplying and Dividing Real Numbers 1. Find the product of a positive number and a negative number.. Find the product of two negative numbers.. Identify factors of integers.. Use the reciprocal of a number to apply the definition of division.. Use the rules for order of operations when multiplying and dividing signed numbers. 6. Evaluate expressions involving variables. 7. Translate words and phrases involving multiplication and division. 8. Translate simple sentences into equations. 1. Find each product. Be sure to write your answer in simplest form. a) 0 ( ) b) ( 1) c) ( 0)( ) d) (.)(.) e) 8 9. Find each quotient. a) 16 8 b) 9 c).6 0 d) 1 e) 0 0. Find all of the integer factors of each number. a) 1 b) c) 9 d) 11. Perform each indicated operation. ( 7) a) 8 b) ( ) + ( ) c) 6 ( ) 1. Evaluate each expression if x =, y =, a = 1, and b =. a) ( y+ b) x b) y x b ya 6. Write each sentence as an expression. a) The product of and the difference between and 9. b) The quotient of 18 and the sum of 7 and. 7. Write each sentence with symbols, using x to represent the number. a) Seven times a number is. b) 1 less than a number is 6. Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

12 1 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1.6 Refer students to the rules for multiplying and dividing signed numbers. Give examples to show why division by zero is undefined but zero can be divided by any number except zero. x, x y, y and y x where y 0, all indicate division. Answers: 1a) 0, b) 60, c) 10, d) 7.6, e) ; a), b), c) undefined, d), e) 0; a) 1 and 1, and 6, and, 1 and 1, and 6, and, b) 1 and, and 16, and 8, 1 and, and 16, and 8, c) 1 and 9, 1 and 9, and ;d) 1 and 11, 1 and 11, a) 1, b) 7, c) ; a) 7, b) 8; 6a) ( ) 9 ; 6, b) 18 1 ;; 7a) 7x =, b) x = 7+ ( ) 6 Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

13 1. Use the commutative properties.. Use the associative properties.. Use the identity properties.. Use the inverse properties.. Use the distributive property. BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture Properties of Real Numbers 1. Name the property (commutative, associative, identity property, inverse property, or distributive property) illustrated by each statement. a) + ( 7) = 7+ b) c) + 6+ ( ) = ( + 6) + ( ) d) = = 0 e) ( c+ d) = c d f) ( + 7) + 10= 10+ ( + 7) g) 7 = h) 6 1 = 6. Use the distributive property to rewrite each expression. Simplify if possible. a) ( k + 6) b) ( h ) e) 7 + ( ) c) y z + d) ( a b c) Remind students that the commutative property deals with the order of addition (or multiplication), whereas the associative property deals with grouping. Students often confuse the additive identity (zero) and the multiplicative identity (one). The generalized distributive property covers multiplication over addition and/or subtraction. Have students provide examples to show whether or not the commutative / associative properties hold for subtraction and division. Answers: 1a) commutative, b) inverse, c) associative, d) inverse, e) distributive, f) commutative, g) commutative, h) identity; a) k + 18, b) h 10 y+ z, d) 10a + b + c, e) 7 ( + ); 0 +, c) ( ) Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

14 1 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture 1.8 Simplifying Expressions 1. Simplify expressions.. Identify terms and numerical coefficients.. Identify like terms.. Combine like terms.. Simplify expressions from word phrases. 1. Simplify each expression. a) 8x x + b) ( x 7) + x c) ( m + ). Give the numerical coefficient of each term. a) m b) 16x c) pq d) r. Simplify each expression. a) ( m+ 6) + m b) b+ 6b c) b b 7c c 8 e e e 1 d) ( ) + e) ( ) ( ). Translate each phrase into a mathematical expression. Use x as the variable. a) A number increased by the difference between and the number b) Six plus the product of more than a number and c) A number plus 7 added to the difference between 6 and twice the number Students have difficulty distinguishing between terms and factors terms are separated by a + or sign; factors are multiplied. Like terms not only have the same variables but also the same exponents. Remind students that expressions can be simplified, whereas equations are solved. Answers: 1a) 1x, b) 7x 1, c) m 9 ; a), b) 16, c) 1, d) ; a) m 1, b) 9b, c) 1 b, d) c 8, e) 7e 1e+ ; a) x+ ( x);, b) 6+ ( x+ ); x+ 1, c) ( 6 x) + ( x+ 7); x+ 1 Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley.

15 BEGINNING AND INTERMEDIATE ALGEBRA, Mini-Lecture.1 1 The Addition Property of Equality 1. Identify linear equations.. Use the addition property of equality.. Simplify, and then use the addition property of equality. 1. Decide whether each is an expression or an equation. If it is an expression, simplify it. If it is an equation, solve it. a) x+ 10 x b) 7y+ 1+ 8y = 6. Which pairs of equations are equivalent equations? a) x + = 8 and x = b) 1 = x 10 and x = c) x 1 = and x = 6 d) x + = 0 and x = 9. Which of the following are linear equations in one variable? a) x 7= x+ b) x = x+ c) x 8 = 0 d) x = 8x. Solve each equation, and check your solution. a) x = 16 b) 1 = x 1 c) x + 1 = 18 d) 19 = x + 16 e) 1 16 x x 6 = 18 = + f) ( ) g) 7= x h) x = 1 i) 6 + x = Solve each equation, and check your solution. 1 1 a) + x = b) + x = c) x = d) 9 1 x = + e). + x = 16 x f) 6x = x Encourage students to write all of the addition property steps and to avoid using shortcuts until they have mastered these types of equations. Encourage students to write the steps for solving the equations in a neat and organized manner. This habit will help immensely when the equations become more complex. Encourage students to simplify both sides of the equation before using the addition principle. Refer students to the addition property of equality definition in the text. Mention that letters besides x can be used for variables in equations. Answers: 1a) expression; x 7 +, b) equation; { 19} ; ) choices a and b; ) choices a and c; a) { } b) { 6 }, c) { }, d) { }, e) { 7 }, f) { 1 }, g) { 0 }, h) { 0 }, i) { 6} e) {. }, f) {. } Copyright 01 Pearson Education, Inc. Publishing as Addison-Wesley. 0, 1 ; a), b) 1, c) 16 1, d) 10,

(-,+) (+,+) Plotting Points

(-,+) (+,+) Plotting Points Algebra Basics +y (-,+) (+,+) -x +x (-,-) (+,-) Plotting Points -y Commutative Property of Addition/Multiplication * You can commute or move the terms * This only applies to addition and multiplication