Math 7 Notes Unit Three: Applying Rational Numbers

Math 7 Notes Unit Three: Applying Rational Numbers Strategy note to teachers: Typically students need more practice doing computations with fractions. You may want to consider teaching the sections on fractions first, and then introduce decimals as special fractions (rational numbers equivalent to fractions). This would allow you to reinforce the rules for fractions while teaching the sections on decimals. For instance, you could define a decimal as a special fraction and link all the rules introduced in fractions to decimals. That is, when you line up decimal points and fill in zeros when adding or subtracting decimals, that s the same as finding a common denominator and making equivalent fractions when adding or subtracting fractions, etc. Estimating with Decimals Syllabus Objective: (.8) The student will estimate using a variety of methods. Syllabus Objective: (.9) The student will round to the appropriate significant digit. An estimation strategy for adding and subtracting decimals is to round each number to the nearest integer and then perform the operation. Examples: 5.8 6 5. 5..7 would be our estimate would be our estimate 8.98 9 9. 9 8-8 would be our estimate An estimation strategy for multiplying and dividing decimals: When multiplying, round numbers to the nearest non-zero number or to numbers that are easy to multiply. When dividing, round to numbers that divide evenly (compatible numbers), leaving no remainders. Remember, the goal of estimating is to create a problem that can easily be done mentally. 8. 0.6. 8 Examples: 6.7 7 80 80 would be our estimate 8 would be our estimate Estimate by rounding to the indicated place value. Examples: 6.78 0.86; tenths 9.5 5.09; hundredths 6.78 6. 9.5 9. 0.86 0.9 5.09 5.0 5.5.08 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

Estimate a range for a sum..56 + 7.5 +.= Step. Use front-end estimation..58 7.5 7. + This is an underestimate. Step : Adjust the decimal part of the numbers. Round the decimals to 0.5 or..58.0.5.5..5.0 + = 6 This is an overestimate. The estimated range for the sum is between and 6. (Remember front-end estimation is always an underestimate since the whole number values of the decimals are less than the actual numbers.) Estimating can be used as a test-taking strategy. Use estimating to calculate your answer first, so you can eliminate any obviously wrong answers. A shopper buys items weighing. ounces, 7.89 ounces and.5 ounces. What is the total weight? A. 0.0095 B..995 C..0 D. 5.5 Round the individual values:. 7.89 8.5 5 The only two reasonable answers are C) and D). Eliminating unreasonable answers can help students to avoid making careless errors. NEW CCSS 7.NS.a - Describe situations in which opposite quantities combine to make 0. Marsha hiked Angel Falls Trail that gains 7 miles of 88 elevation up the mountain. She then hikes back to where she started from. Which best describes the total gain in elevation for the entire hike? A 7 88 B 0 C 7 88 D 7 Solution: B Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

Which is a solution of x 0 A B 0.5 C 0.5 D Solution: C NEW CCSS 7.EE.- Assess the reasonableness of answers to rational number problems using mental computation and estimation strategies. This should be an ongoing, everyday objective. Students should automatically estimate their answers to questions so they know they are in the right ballpark. For this to happen though, teachers must emphasize this skill over and over until students are able to see the ease and the value of this skill, and perform it with automaticity. Adding and Subtracting Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Remember, since decimals are special fractions, the rules for computing with fractions also work for decimals. Note: Choose numbers carefully that are not reducible as fractions, so as to eliminate that distraction. To add.7 and.7, follow the algorithm for adding fractions: Algorithm for Addition of Fractions. Find the common denominator.. Make equivalent fractions.. Add the numerators.. Bring down the denominator. 5. Simplify. 7 70 00,000 7 7,000,000 87,000.7.7. 70.7.87. The denominator for.7 is 00. The denominator for.7 is,000. The least common denominator is,000.. Add a zero after.7 so it will have a common denominator of,000. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

. Add the numerators by adding 70 and 7 = 87.. Bring down the common denominator of,000. This means there must be three digits to the right of the decimal point. Placing a decimal point gives.87 as the answer. After doing numerous decimal addition problems we can see a pattern which leads to a simpler algorithm that can be used to add/subtract decimals. Algorithm for Addition / Subtraction of Decimals. Rewrite the problems vertically, lining up the decimal points.. Fill in spaces with zeros.. Add or subtract the numbers.. Bring the decimal point straight down. ***When comparing the algorithm for adding/subtracting fractions and the algorithm for adding/subtracting decimals, we note that:. Lining up the decimal point is the same as finding the common denominator in fractions.. Filling in spaces with zeros is the same as making equivalent fractions.. Adding or subtracting the digits is the same as adding or subtracting the numerators.. Bringing the decimal point straight down is the same as bringing down the denominator....75 Rewrite vertically, lining up the decimal points....75. Fill in with zero to find the common denominator and make equivalent fractions.. Add and bring the decimal straight down..0.00.75.005.0.00.75.005 5.5 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

5. 5. 0 5. 0.5.5.5. 5 Multiplying Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Just as the algorithm for adding and subtracting decimals is related to the addition and subtraction of fractions, the algorithm for the multiplication of decimals also comes directly from the multiplication algorithm for fractions. Before we look at the fraction algorithm, let s first look at the algorithm for decimal multiplication. Algorithm for Multiplication of Decimals. Rewrite the numbers vertically.. Multiply normally, ignoring the decimal point.. Count the total number of digits to the right of the decimal points in the factors.. Count and place the decimal point that same number of places from right to left in the product (answer)...6.6. 6 65 686. Rewrite the problem vertically.. Multiply normally, ignoring the decimal point.. Count the number of digits to the right of the decimal points. There are two places to the right in the multiplicand (number on the top) and one place to the right in the multiplier (number on the bottom). 6. 8 6. Count and place the decimal point the same number of places () from right to left in the answer. Before going on, let s think of how this is related to the algorithm for multiplying fractions. The algorithm for multiplying fractions is to multiply the numerators, multiply the denominators, and then simplify. Multiplying the numbers while ignoring the decimal points is the same as multiplying the numerators. Counting the number of decimal places is the same as multiplying the denominators. In this problem, the denominators are 00 and 0. 00 0 000 which shows three zeros, which means decimal places. The point is the algorithm for multiplying Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 5 of Revised 0-CCSS

decimals comes from the algorithm from multiplication of fractions. That should almost be expected since decimals are special fractions. Remember to point out to students that if a number has no decimal point, the decimal point is understood to go after the number (behind the one s place). 5 5. So, when multiplying this number, it has NO decimal places after the decimal point. Multiplying by Positive Powers of 0 Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Examining the patterns discovered when multiplying by positive powers of 0 (0, 00,,000 ) we can do many multiplication problems mentally. What pattern do you see from these problems? 0 (.) =. 00 (567.) = 567.,000 (.56) = 560 0 (0.056) = 0.56 00 (5.7) = 570,000 (0.56) = 5.6 We can generalize a rule from the observed pattern: When multiplying by positive powers of 0, move the decimal point to the right, the same number of places as there are zeros. 0 (.75) 0 (.75) = 7.5 00 (5.7) 00 (5.7) = 5.7 One zero in 0, so move the decimal point one place to the right. Two zeros in 00, move the decimal point two places to the right. 000 (6.) Three zeros in,000; move the decimal point three places to the right. 000 (6.) = 6,00 Notice in this problem we had to fill in a couple of placeholders to move it three places to the right. Dividing Decimals by Integers Syllabus Objective: (.)The student will solve problems using operations on positive and negative numbers, including rationals. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 6 of Revised 0-CCSS

Algorithm for Dividing Decimals by Integers. Bring the decimal point straight up into the quotient.. Divide in the normal way and determine the sign. 7.9.56 7.9 5 Or use short division:.5 7.9.5 6 7.9 ) Place decimal point in quotient. ) 7 divides into 9, 5 times with left. ) Write remainder of, before. ) 7 divides into, 6 times evenly. Remember, when using short division, writing the remainder(s) in the dividend is showing work. Dividing Decimals and Integers by Decimals Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Algorithm for Dividing Decimals. In the divisor, move the decimal point as far to the right as possible.. In the dividend, move the decimal point the same number places to the right.. Bring the decimal point straight up into the quotient.. Divide in the normal way...56.56 5.6.8 5.6 6 96 96 Move the decimal point places to the right in the divisor. Move the decimal point places to the right in the dividend. Bring up the decimal point and divide normally. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 7 of Revised 0-CCSS

By moving the decimal the same number of places to the right in the divisor and the dividend, we are essentially multiplying our original expression by one. We are making equivalent fractions by multiplying the numerator and denominator by the same number. If we move the decimal point one place, we are multiplying the numerator and denominator by 0. By moving it two places, we are multiplying the numerator and denominator by 00, etc..56 00 5.6. 00 Dividing by Powers of 0 Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Examining the patterns discovered when dividing by powers of 0 (0, 00,,000 ) we can do many division problems mentally. What pattern do you see from these problems? 67.89 0 = 6.789 98.6 00 =.986 68 000 =.68 65 0 = 65. 78 00 =.78 8 000 =.008 We can generalize a rule from the above observed pattern: when dividing by positive powers of 0, move the decimal point to the left, the same number of places as there are zeros. 5.6 0 Since there is one zero in 0, move the decimal point one place to the left. 5.6 0 =.56 98 00 Since there are two zeros in 00, move the decimal point two places to the left. 98 00 = 98. 9 000 There are three zeros in,000, so move the decimal point three places to the left. 9. 000 =.009 Notice in this problem we had to fill in a couple of placeholders to move it three places to the left. Computing with Decimals and Signed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. The rules for adding, subtracting, multiplying and dividing decimals with signed numbers are the same as before, the only difference is you integrate the rules for integers. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 8 of Revised 0-CCSS

Compute: 6.7 (.) 6.70 Line up decimal points, fill in zeros, and add. +. Add negative numbers, the sum is negative. 7.9 Compute: 9.76 ( 7.) Since subtract means add the opposite, we add 7. 9.76 Line up decimal points, fill in zeros, and add. 7.00 Adding numbers with different signs, we subtract.76 the absolute values and take the sign of the greater absolute value so our sum is negative. Compute: ( 5.6)( 8) 5.6 Multiply, count one decimal place in problem and pl 8 Multiply negatives, the product is positive..8 Compute: 7.5.5 50..5 7.5 5 750. 75 00 00 0 Move decimal point two places to the right in the divisor to make it a whole number. Move decimal point two places to the right in the dividend. Bring decimal point straight up into the quotient. Divide as usual Determine the sign. Solving Equations Containing Decimals Syllabus Objective: (.5) The student will solve equations and inequalities in one variable with integer solutions. Syllabus Objective: (.) The student will solve one-step equations using mental math. Strategy for Solving Equations: To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. x +.5 =.5.5.5 Undo adding.5 by subtracting.5 from both sides. x = 5 Simplify and determine the sign. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 9 of Revised 0-CCSS

x +.5 = 7..5.50 Undo adding.5 by subtracting.5 from both sides. x = 8.8 Simplify and determine the sign. x.5 = 6. +.5 +.5 Undo subtracting.5 by adding.5to both sides. x =.8 Simplify and determine the sign..8 8n.8 8n 8 8 0.06 n y 0.7 5 y ( 5) ( 5)( 0.7) 5 y.5 Undo multiplying by 8 by dividing both sides by 8. Simplify and determine the sign. Undo dividing by 5, by multiplying both sides by 5. Simplify and determine the sign. Estimation with Fractions Syllabus Objective: (.8) The student will estimate using a variety of methods. Round to 0 if the numerator is much smaller than the denominator. 7 Examples:,, 0 8 00 Benchmarks for Rounding Fractions Round to ½ if the numerator is about half the denominator. Examples: 5 7,, 0 9 00 Round to if the numerator is nearly equal to the denominator. Examples: 7 7 87,, 0 8 00 Examples: Round to 0, ½ or. 7 7.. 9 50 0 5 Estimating Sums and Differences Round each fraction or mixed number to the nearest half, and then simplify using the rules for signed numbers. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 0 of Revised 0-CCSS

Estimate 7. 8 9 7 8 0 9 is our estimate Estimate 7. 8 5 7 8 5 is our estimate Estimate 5 8 7 5 6 8 8 7 is our estimate Estimating Products and Quotients Round each mixed number to the nearest integer, and then simplify. 5 7 8 6 5 88 is our estimate 5 7 ( ) 8 ( ) 6 5 6 is our estimate Adding and Subtracting Fractions Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

Common Denominators Let s say we have two cakes, one chocolate and the other vanilla. The chocolate cake was cut into fourths, the vanilla cake into thirds as shown below. You take one piece of each, as shown. Since you had pieces of cake, can you say you had of a cake? 7 Remember our definition of a fraction: the numerator indicates the number of equal size pieces you have while the denominator indicates how many equal sized pieces make one whole cake Since your pieces are not equally sized, we can t say you had of a cake. 7 And clearly 7 pieces does not make one whole cake. We can conclude:. 7 The key is to cut the cakes into equally sized pieces. We ll cut the first cake (already in fourths) the same way the second cake was cut. And we ll cut the second cake (already in thirds) the same way the first cake was cut. So each cake ends up being cut into twelve equally sized pieces. Cutting the cake into EQUAL size pieces illustrates the idea of common denominator. Let s look at several different methods of finding a common denominator. A common denominator is a denominator that all other denominators will divide into evenly. Methods of Finding a Common Denominator. Multiply the denominators. List multiples of each denominator, use a common multiple.. Find the prime factorization of the denominators, and find the Least Common Multiple. Use the Simplifying/Reducing Method, especially for larger denominators. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

In our cake illustration, the common denominator is the number of pieces that the cakes can be cut so that everyone has the same size piece. Method : To find a common denominator of and, multiply the denominators,. This technique is very useful when the denominators are prime or relatively prime (do not share any common factors other than ). Method : To find a common denominator of 5 6 and, list the multiples of each denominator. Multiples of 6: 6,, 8, Mulitples of :, 8,, Since is on each list of multiples, it is a common denominator. This technique is very useful when the greater denominator is a multiple of the smaller one. Such as, find the common denominator for and 5. 6 is a common 6 denominator since it is a multiple of. Method : To find a common denominator of 8 and 7, find the prime factorization of each denominator. Factoring, 8 Multiply the prime factors, using overlapping factors only once, we get. This technique can be tedious and may be the least desirable method; although it is good practice working with prime factors. Method : To find the common denominator of 8 and 5 using the Simplifying Method, create a fraction using the two denominators: 8, and then simplify: 8 6. 6 For 8, cross multiply: 8 7 or 7. The common denominator is 7. Note: It does not matter if you use 8 or. 8 This is an especially good way of finding common denominators for fractions that have large denominators or fractions whose denominators are not that familiar to you (large composite numbers). Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

Adding and Subtracting Fractions with Like Denominators To add or subtract fractions, you must have equal sized pieces. If a cake is cut into 8 equal pieces and you eat three pieces today, and then eat four pieces tomorrow, you would have eaten a total of 7 pieces of cake, or 7 8 of the cake. 7 + 8 8 8 The numerators are added to indicate the number of equal sized pieces that were eaten. The denominators are NOT added, because the denominator indicates the total number of equal sized pieces in the cake. If we added them, we would get 6, but there are only 8 pieces of cake. Examples: + 8 8 8 6 0 0 0 Adding and Subtracting Fractions with Unlike Denominators Let s add to. Do we get 7? What does the picture tell us? + Note that when creating your visuals, be sure to cut one figure vertically and the other horizontally. The pieces are NOT EQUAL sizes, so we cannot add them. We need to divide each shape into equally sized pieces. Do this by using the horizontal cuts from the second visual on the first. Then use the vertical cuts from the first visual on the second. (Both will now be divided into twelfths.) Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

+ 7 From the picture we can see that is the same as and has the same value as. Adding the numerators, a total of 7 equally sized pieces are shaded and pieces make one unit. If you did a number of these problems, you would be able to find a way of adding and subtracting fractions without drawing the picture. Algorithm for Adding / Subtracting Fractions. Find a common denominator.. Make equivalent fractions.. Add/subtract the numerators.. Bring down the denominator. 5. Simplify. 5. To find the common denominator, multiply the denominators, since they are relatively prime (have no common factors greater than ). (5 5). Make equivalent fractions. (shown). Add the numerators. ( + 0 = ). Bring down the denominator. (5) 5. Simplify. (not necessary in this problem) 5 5 0 5 5 Examples: 9 6 5 5 6 9 6 To find the least common denominator for this example, list the multiples or use the simplifying/reducing method. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 5 of Revised 0-CCSS

7 7 5 0 6 7 7 To find the least common denominator for this example, use the simplifying/reducing method. since 7, 6 the LCD is 7. Adding and Subtracting Mixed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Algorithm for Adding/Subtracting Mixed Numbers. Find a common denominator.. Make equivalent fractions.. Add/subtract the numerators.. Bring down the denominator. 5. Add/subtract the whole numbers. 6. Simplify the fraction answer part, combining it with the whole number part. 5 6 5 5 9 6 7 6 7 8 9 6 6 To find the common denominator, multiply the denominators, since they are relatively prime (have no common factors greater than ). 5 = 0 5 0 5 0 7 7 5 6 0 0 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 6 of Revised 0-CCSS

Borrowing From a Whole Number Subtracting fractions and borrowing is as easy as getting change for your money. You have 8 one dollar bills and you have to give your friend $.5. How much money would you have left? Since you don t have any coins, you would have to change one of the dollars into quarters. Why not ten dimes? Because you have to give your friend a quarter, so you get the change in terms of what you are working with, in this case quarters. 8 dollars 7 dollars quarters dollars quarter dollars quarter Subtract to get: dollars quarters Redoing this problem using fractions: 8 9 5 7 5 5 9 5 5 In the last problem, we borrowed quarters because we were working with quarters. Now we will borrow ths for the same reason. We change to 5 because we are 5 working with fifths. Borrowing from Mixed Numbers For this problem you have 6 dollars and quarter in your pocket, and you have to give your brother $.75. 6 dollars quarter 5 dollars 5 quarters dollars quarters dollars quarters Subtract to get: dollars quarters Since we don t have enough quarters, we change dollar for quarters, adding that to the quarter we already had, gives you 5 quarters. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 7 of Revised 0-CCSS

6 5 5 5 = We change 6 to when we simplify. 5 and add the to make 5 5. Algorithm for Borrowing with Mixed Numbers. Find a common denominator.. Make equivalent fractions.. Borrow, if necessary.. Subtract the whole numbers.. Subtract the numerators. 5. Bring down the denominator. 6. Simplify if needed. This example has unlike denominators. 6 6 8 5 5 5 8 8 = 7 Multiplying Fractions and Mixed Numbers NEW CCSS 7.NS. Apply and extend previous understandings of multiplication and division and of fractions to multiply and divide rational numbers. Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Before we learn how to multiply fractions, let s revisit the concept of multiplication using whole numbers. When I have an example like we can model that in several ways. \ Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 8 of Revised 0-CCSS

Since it can be read three groups of we can use We can show a rectangular array Show groups of the repeated addition model. 6 Each representation shows a total of 6. Mathematically, we say 6. Multiplication is defined as repeated addition. That won t change because we are using a different number set. In other words, to multiply fractions, I could also do repeated addition. 6 6 A visual representation of multiplication of fractions would look like the following. In this example, we want half of one third. So the visual begins with the one third we have. Since we want only half of it, we cut the visual in half and shade. Now we see part double shaded and 6 total parts or 6. In this example, we want to take one third of. So the visual begins with the one fourth. We want one third of it. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 9 of Revised 0-CCSS

Now we visually see part doubly shaded, and total pieces. So. Algorithm for Multiplying Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Cancel, if possible.. Multiply numerators.. Multiply denominators. 5. Simplify. 5 7 Since 5 5 it to 7. is not a fraction, we convert 7 can be written as 7 5 5 Using the commutative property, we can rewrite this as 7. 5 Using the associative property, we can rewrite this as 7. Simplify. 5 7 Then multiply and simplify, as a mixed number. 5 5 5 Rather than going through all those steps, we could take a shortcut and cancel. 5 9 8 0 Make sure you have proper or improper fractions. 5 9 8 0 Cancel 8 and 9 by common factor of 9. 5 9 Cancel 0 and 5 by common factor of 5. 8 8 Multiply numerators, multiply denominators, simplify. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 0 of Revised 0-CCSS

Dividing Fractions and Mixed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Before we learn how to divide fractions, let s revisit the concept of division using whole numbers. When I ask, how many s are there in 8, I can write that mathematically three ways. 8 8 8 To find out how many s there are in 8, we will use the subtraction model: 8 Now, how many times did we subtract? Count them: there are subtractions. So there are twos in eight. 6 Mathematically, we say 8 =. Division is defined as repeated subtraction. That won t change because we are using a different number set. In other words, to divide fractions, I could also do repeated subtraction. 0 Another way to look at this problem is using your experiences with money. How many quarters are there in $.50? Using repeated subtraction we have: 0 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

How many times did we subtract? Six. But this took a lot of time and space. A visual representation of division of fractions would look like the following. 8 We have. Representing that would be Since the question we need to answer is how many 8 s are there in, we need to cut this entire diagram into eighths. Then count each of the shaded one-eighths. As you can see there are four. So. 8 5 6 We have 5. Representing that would be 6 Since the question we need to answer is how many ' s are there in 5, we need to use the cuts 6 for thirds only. Then count each of the one thirds. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

As you can see there are. So 5. 6 Be careful to choose division examples that are easy to represent in visual form. Well, because some enjoy playing with numbers, they found a quick way of dividing fractions. They did this by looking at fractions that were to be divided and they noticed a pattern. And here is what they noticed. Algorithm for Dividing Fractions and Mixed Numbers. Make sure you have proper or improper fractions.. Invert the divisor ( nd number).. Cancel, if possible.. Multiply numerators. 5. Multiply denominators. 6. Determine your sign and simplify. The very simple reason we tip the divisor upside-down (use the reciprocal), then multiply (for division of fractions), is because it works. And it works faster than if we did repeated subtractions, not to mention it takes less time and less space. 5 5 (Invert the divisor.) 5 7 8 8 Multiply numerators and denominators, and simplify. Let s look at this same problem in the format of complex fractions. In this format we need to get the denominator to be equal to. To do that we multiply the top and bottom by 5, remember this is another name for whole divisor and multiply. 5 5. This is a much better explanation of why we invert the = 5 5 5 5 8 5 7 or 5 8 8 5 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

9 0 Make sure you have proper or improper fractions. 9 0 9 Invert the divisor. 5 0 9 Cancel 0 and by, and cancel 9 and by. 5 5 Multiply numerators and denominators. 5 7 Simplify. 7.EE.- Convert between forms of rational numbers as appropriate. Karen worked hours last week and earned $00. What was her hourly rate of pay? 00 00.5 or $8 per hour A radio station released 00 balloons at an outdoor celebration. Of these, were orange. How many were orange balloons? 00 or.75 00 5 orange balloons A town has raised of the $,000 it needs to furnish its new library. 8 How much more is it hoping to raise? 5 5 so,000 or.65(,000) $7,500 8 8 8 7.NS.c Apply properties of operations as strategies to multiply and divide rational numbers. 7.EE.- Apply properties of operations to calculate with rational numbers in any form. 5 5 5 56 Using the Distributive Property 5 5 5 5 Using the Commutative Property 5 5 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

5 5 5 5 5 5 5 0 8 0 0 8 9 0 0 Using the Commutative Property Using the Associative Property NEW CCSS 7.NS.b- Understand that integers can be divided, provided that the divisior is not zero, and every quotient of integers (with non-zero divisor) is a rational number. Be sure to include examples with both decimal and fractional numbers that include division by zero and into zero in numerous formats. Students need to be reminded once again that division by 0 is undefined under these subsets of the real numbers too. 0 undefined 0 0 0 6 0 0 75 undefined 0.67 0. 0 undefined 67 0 undefined 7 0 undefined 0 undefined 0 786 0 For this objective students also need to understand that the quotient when dividing integers is a rational number(with non-zero divisors). Use a variety of division problems like the following 8 6 6 0.5 or or.8 8 5 5 Teachers must overtly point out that each of these quotients,, 0.5 or, and or.8 5 are rational. Challenge students to find any division of integers problem involving non-zero divisors that do not have a rational number quotient. They will find there aren t any. Computing with Fractions and Signed Numbers Syllabus Objective: (.) The student will solve problems using operations on positive and negative numbers, including rationals. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 5 of Revised 0-CCSS

The rules for adding, subtracting, multiplying and dividing fractions with signed numbers are the same as before, the only difference is you integrate the rules for integers. Using a number line: - - 0 Procedurally: 5 or Using a number line: Procedurally: Procedurally: Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 6 of Revised 0-CCSS

7 Invert divisor 7 5 8 8 Multiply numerators and denominators, determine sign and simplify. 5 6 5 5 5 or 6 6 9 5 0 5 9 5 0 5 5 7 8 8 NEW CCSS 7.NS. Solve real-world and mathematical problems involving the four operations with rational numbers. (Computations with rational numbers extend the rules for manipulating fractions to complex fractions.) Mary is baking cookies for the holidays. The recipe calls for Invert then multiply numerators, multiply denominators, determine sign and simplify. cups of flour, cup of sugar and pound of butter. If she wants to cut the recipe in half, how much of each ingredient will she need? Solution: 7 8 cup flour, cup sugar and pound butter Cancel if possible, then multiply numerators, multiply denominators, determine sign and simplify. Metro Taxi Cab Company charges its customers as follows: $.00 for the st 9 mile $0.0 for each additional 9 mile How much would he pay for a mile taxi ride? Solution: $8.0 Mike paid $5.85 for 5 gallons of gasoline. What was the price per gallon? Solution: $5.85 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 7 of Revised 0-CCSS

Solving Equations Containing Fractions Syllabus Objective: (.5) The student will solve equations and inequalities in one variable with integer solutions. Strategy for Solving Equations: You solve equations containing fractions and signed numbers the same as you do with whole numbers; the strategy does not change. To solve linear equations, put the variable terms on one side of the equal sign, and put the constant (number) terms on the other side. To do this, use opposite (inverse) operations. Solve: x. 5 6 x 5 5 5 5 x 5 Undo adding one-third by subtracting onethird from both sides of the equation; make equivalent fractions with a common denominator of 5. Solve: x 5 x 5 5 x 5 5 0 x or Undo dividing by 5 by multiplying both sides by 5. Cancel. Multiply numerators and denominators, determine sign and simplify. x..6.. x. x. x. ( x.5) 9 ( x.5) 9 x.5.00.5.5 x.55 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 8 of Revised 0-CCSS

NEW CCSS 7.EE.- Solve multi-step real-life and mathematical problems posed with positive and negative rational numbers in any form (whole numbers, fractions and decimals), using tools strategically. A book has pages that are 7 inches wide and 9 inches high. The printed area measures 5 inches by 7 inches. The left margin is 5 in. and the 8 6 top margin is 9 in. How wide are the margins at the right and the bottom of each 6 page? Solution: Right: 5 in.; bottom: 6 6 in. A gasoline tank with a capacity of 5 gallons is full. How many gallons will it take to fill the tank? Solution: gallons Kevin s regular rate of pay is $8.00 per hour. When he worked overtime, he earns times as much per hour. How much will Kevin earn for 5 hours of overtime work? Solution: $6.00 An advertising sign is to have six lines of printing. The letters are to be inches high with inch between the lines. a. How much vertical space is required for the printing? Solution: 0 in. b. If there is a top margin of inches and a lower margin of inches, what will be the total height of the sign? Solution: 6 in. Mary s cell phone bill is automatically deducting $5.95 from her bank account every month. How much will the deductions total for the year? Solution: $659.0 Mr. McMillan plans to interview 8 applicants for a job. If he spends hours interviewing each day and spends hour interviewing each applicant, how many days will she need for the interviews? Solution: 5 days Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 9 of Revised 0-CCSS

NEW CCSS 7.EE. Use variables to represent quantities in a real-world or mathematical problem, and construct simple equations and inequalities to solve problems by reasoning about the quantities. NEW CCSS 7.EE.a- Solve word problems leading to equations of the form px+q=r and p(x+q)=r, where p, q, and r are specific rational numbers. Solve equations of these forms fluently. The youth group is going on a trip to the state fair. The trip costs $5. Included in that price is $ for a concert ticket and the cost of passes, one for the rides and one for the game booths. Each of the passes cost the same price. Write an equation representing the cost of the trip and determine the price of one pass. Solution: Let p = the cost of one pass p 5 p p p 0.5 Each pass costs $0.50 Check: (0.50) 5 5 5 5 Amy had $5 to spend on school supplies. After buying notebooks, she had $5. left. How much did each notebook cost? Write an equation representing the money Amy had to spend and determine the price of one notebook. Solution: Let n = the cost of one notebook n 5. 5.00 5. 5. n 9.77 n 9.77 n 6.59 Each notebook costs $6.59 Check: (6.59) 6. 5.00 9.77 6. 5.00 5.00 5.00 The perimeter of a rectangle is 5 cm. Its length is 6 cm. What is its width? Write an equation representing the perimeter and determine the width. Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page 0 of Revised 0-CCSS

Solution: Let w = the width of the rectangle Remember the formula for perimeter: lw P or (l + w) = P (6) w 5 (6 w) 5 w 5 (6 w) 5 w 6 w 7 w 6 6 w w The width is cm Checks: (6) () 5 or (6 ) 5 5 (7) 5 5 5 5 5 Solution: The sum of three consecutive numbers is 6. What are the three numbers? Write an equation and solve it to determine the three numbers. Let x = the st consecutive number x ( x) ( x) 6 x + = the nd consecutive number x 6 Check: x + = the rd consecutive number 0++=6 x 60 6=6 x 60 x 0 The numbers are 0, and x 0 x 0 The sum of three consecutive even numbers is 7. What are the three numbers? Write an equation and solve it to determine the three numbers. Solution: Let x = the st consecutive even number x + = the nd consecutive even number x + = the rd consecutive even number x ( x) ( x) 7 x 6 7 6 6 Check: x 66 x 66 ++6=7 7=7 x x x 6 The numbers are, and 6 Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS

NEW CCSS 7.EE.a- Compare an algebraic solution to an arithmetic solution, identifying the sequence of the operations used in each approach. In each of the examples above you notice an algebraic solution. The check steps represent an arithmetic solution. This standard asks for a comparison of the sequence of operations used in each approach. Notice that the arithmetic solution (or check) employs the Order of Operations and the algebraic solutions use the Order of Operations in reverse. In addition to the samples given Smarter Balanced Assessment Consortium (SBAC) has released some sample items. Items that fit this unit are included below. SBAC Identify the number(s) that makes each statement true. You may select more than one number for each statement.. = a positive number 5..9 = a negative number 9 7 9 = zero 9 9.5 = a negative number.75. Solutions:.9, SBAC 7, 9,.75 and. David wants to buy pineapples and some bananas. The price of pineapple is $.99 The price of bananas is $0.67 per pound. David wants to spend $0.00. Write an equation that represents the number of pounds of bananas, b, David can buy. Solution: b 6 On the number line below, draw a graph that represents the number of pounds of bananas David can buy. - - 0 5 6 7 8 9 0 x Math 7, Unit 0: Applying Rational Numbers Holt: Chapter Page of Revised 0-CCSS