MATH Student Book. 5th Grade Unit 7

Transcription

1 MATH Student Book th Grade Unit

2 Unit FRACTION OPERATIONS MATH 0 FRACTION OPERATIONS Introduction. Like Denominators... Adding and Subtracting Fractions Adding and Subtracting Mixed Numbers 0 Estimating Sums and Differences Self Test : Like Denominators. Unlike Denominators... Adding Fractions Subtracting Fractions Adding Mixed Numbers Subtracting Mixed Numbers Self Test : Unlike Denominators LIFEPAC Test Pull-out. Multiplying and Dividing Fractions... Multiplying Fractions by a Whole Number Multiplying Fractions Multiplying Mixed Numbers 0 Dividing Fractions Self Test : Multiplying and Dividing Fractions. Review... Glossary

3 FRACTION OPERATIONS Unit Author: Glynlyon Staff Editor: Alan Christopherson, M.S. Media Credits: Page : Ingram Publishing, Thinkstock; : Maksymowicz, istock, Thinkstock; : Stephen Blair, istock, Thinkstock; : Naddiya, istock, Thinkstock; : Aliaksei_, istock, Thinkstock; : tashechka, istock, Thinkstock; : DAJ, Thinkstock; : ARTQU, istock, Thinkstock; : cenkeratila, istock, Thinkstock; elenabs, istock, Thinkstock; : Sylverarts, istock, Thinkstock; : Andres Rodriguez, Hemera, Thinkstock. 0 N. nd Ave. E. Rock Rapids, IA - MMXV by Alpha Omega Publications, a division of Glynlyon, Inc. All rights reserved. LIFEPAC is a registered trademark of Alpha Omega Publications, a division of Glynlyon, Inc. All trademarks and/or service marks referenced in this material are the property of their respective owners. Alpha Omega Publications, a division of Glynlyon, Inc. makes no claim of ownership to any trademarks and/or service marks other than their own and their affiliates, and makes no claim of affiliation to any companies whose trademarks may be listed in this material, other than their own.

4 Unit FRACTION OPERATIONS FRACTION OPERATIONS In this unit, you will begin computing with fractions. You will use models or a pencil and paper in order to solve real-life problems. To solve problems, you will learn how to add and subtract proper fractions and mixed numbers with both like and unlike denominators. You will also learn how to estimate a sum or difference of two fractions or mixed numbers. In addition, you will explore how to multiply with whole numbers, proper fractions, and mixed numbers. You will complete the unit by dividing with unit fractions and whole numbers. Objectives Read these objectives. The objectives tell you what you will be able to do when you have successfully completed this LIFEPAC. When you have finished this LIFEPAC, you should be able to: Add and subtract fractions and mixed numbers with like denominators. Estimate, add, and subtract fractions and mixed numbers with unlike denominators. Multiply with fractions and mixed numbers. Divide with unit fractions and whole numbers.

5 FRACTION OPERATIONS Unit. LIKE DENOMINATORS Do you remember how to represent a fraction on the number line? Proper fractions come between 0 and on the number line. To plot a proper fraction, evenly divide the length between 0 and into the number of pieces named by the denominator (bottom number). Then, put a point on the tick mark that is named by the numerator (top number). In this lesson, we ll use the number line and fraction models to help us learn how to add and subtract fractions. We ll also solve fraction word problems using addition and subtraction. Objectives Read these objectives. When you have completed this section, you should be able to: Add fractions that have like denominators. Subtract fractions that have like denominators. Add mixed numbers with like denominators. Subtract mixed numbers with like denominators. Estimate sums of fractions and mixed numbers. Estimate differences of fractions and mixed numbers. Vocabulary Study these new words. Learning the meanings of these words is a good study habit and will improve your understanding of this LIFEPAC. estimate. An approximate value that is close to the actual value. like denominators. Denominators that are the same number. Note: All vocabulary words in this LIFEPAC appear in boldface print the first time they are used. If you are unsure of the meaning when you are reading, study the definitions given. Section

6 Unit FRACTION OPERATIONS Adding and Subtracting Fractions Remember, to plot a proper fraction, evenly divide the length between 0 and into the number of pieces named by the denominator, and put a point on the tick mark that is named by the numerator. For example, to represent the fraction on the number line, first divide the length between 0 and into four equal pieces: 0 Vocabulary Then, draw a point on the first tick mark. 0 Adding and Subtracting Fractions That Have Like Denominators Let s use our number line again to help us add fractions that have like (or the same) denominators. We already plotted on the number line. Let s add to it. Since the number line is already divided into fourths, adding is the same as moving two tick marks to the right. 0 Remember that in a proper fraction, the numerator is less than the denominator. Now, our point is on the third tick mark. So + is equal to. Let s try another one. This time, we ll use a model to help us find the sum of and. Remember that to model a fraction, divide a whole amount evenly into the number of parts named by the denominator. Then, shade in the number of parts named by the numerator. So, to model the fraction, divide a rectangle into six equal parts. Then, shade two of them. Now, let s add to it. Because the model is already divided into sixths, adding is the same as shading in three more pieces. So, + is equal to. Section

7 FRACTION OPERATIONS Unit What did you notice in both problems about adding fractions? The numerators were added together, and the denominator stayed the same. Here s another example: Example: Kari ate ate of the pizza, Sam ate of the pizza, and Kristi of the pizza. How much did they eat altogether? Solution: To find the total amount that they ate, add the fractions together. The fractions have like denominators, so add the numerators and keep the denominator the same. + + = Notice that is not written in simplest form. and have a common factor of. = Kari, Sam, and Kristi ate of the pizza altogether. S-T-R-E-T-C-H... What fraction of the pizza is left over? Sometimes, the sum of two or more fractions is greater than. For example, let s add and, using a model. + = We added the numerators to get, and kept the denominators the same. Notice that the sum ( ) is an improper fraction. Another way to write an improper fraction is as a mixed number. As you can see from the model, is the same as. Remember, to add fractions with like denominators, add the numerators together and keep the denominator the same. Then, write the sum as a fraction or mixed number in simplest form. This might help... Remember that to convert an improper fraction to a mixed number, divide the numerator by the denominator. The quotient is the whole number part. The remainder is the numerator. And, the denominator stays the same. In this case, = R. Section

8 Unit FRACTION OPERATIONS Subtracting Fractions with Like Denominators Now, let s try subtraction with fractions. Subtracting fractions that have like denominators is similar to adding fractions that have like denominators. Subtract the numerators and keep the denominator the same. Then, represent the difference in simplest form. Like addition, subtraction can be represented on the number line or with a model. On the number line, move left to show subtraction. On a model, remove shading to show subtraction. Key point... Always write the final sum in simplest form by dividing the numerator and denominator of the sum by their GCF. Example: Find the difference. Solution: The fractions have like denominators. So, subtract the numerators, and keep the denominator the same. = Keep in mind... Any time the numerator of a fraction Now, write the difference in simplest form. is exactly half of the denominator, the and have a common factor of. fraction is equivalent to =. To represent this subtraction problem on the number line, divide the space between 0 and into eight equal parts. Then, start at and move three tick marks to the left. 0 To subtract fractions with like denominators, subtract the numerators and keep the denominator the same. Then, write the difference as a fraction in simplest form. Section

9 FRACTION OPERATIONS Unit Here s one more example: Example: Kari ate of the pizza, Sam ate of the pizza, and Kristi ate of the pizza. How much more did Sam eat than Kristi? Solution: Subtract the amount Kristi ate from the amount Sam ate. = Write the difference in simplest form. and have a common factor of. = Sam ate more of the pizza than Kristi. Let s Review! Before going on to the practice problems, make sure you understand the main points of this lesson. To add or subtract fractions with like denominators, add or subtract the numerators and keep the denominator the same. Write sums and differences in simplest form. Addition or subtraction with fractions can be represented on the number line or with a model. Circle the correct letter and answer.. A fraction where the numerator is the denominator is called a proper fraction. a. larger than b. equal to c. smaller than. Add. Write your answer in simplest form. + a. b. c. d. Section

10 Unit FRACTION OPERATIONS. Add. Write your answer in simplest form. a. b. 0. Add. Write your answer in simplest form. a. 0. Add. Write your answer in simplest form. a. b. b c c. + c. d. 0 d. d.. Mrs. Thomas bought yard of one fabric and yard of a different fabric. How many total yards of fabric did she buy? a. yard b. yard c. yards d. yards. Subtract. Write your answer in simplest form. a. b. c. 0 d. 0. Subtract. Write your answer in simplest form. a.. There was now? a. 0 b. c. 0 of a pie left over from dessert. Then, b. Complete these activities. c. d. of it got eaten. How much is left.0 Find the sum. Write your answer in simplest form. +. Find the difference. Write your answer in simplest form Match each addition or subtraction problem with its correct answer in simplest form. a. + b. + c. + d. 0 0 e. f d. Section

11 FRACTION OPERATIONS Unit Adding and Subtracting Mixed Numbers Adding and subtracting fractions with like denominators is a simple process. Add or subtract the numerators and keep the denominator the same. Then, represent the sum or difference as a fraction or mixed number in simplest form. For example, + =. Then, reduces to. In this lesson, we ll continue to add and subtract fractions with like denominators as we add and subtract mixed numbers. We ll use models to represent the addition and subtraction problems, as well as solve word problems. Adding Mixed Numbers with Like Denominators Brianna is making a double batch of muffins today. The recipe says she ll need cups of flour for each batch. How many total cups of flour will Brianna need? Let s use a model to help us solve this problem. A double batch means that Brianna will be making two batches. And, each batch calls for cups, so draw two models of. Notice that there are two whole rectangles shaded in and six-fourths shaded in, which is. The fourths can be combined to completely fill in the third rectangle. This leaves two-fourths shaded in the last rectangle. So, + is the same as, or. This might help... The sum is equal to because reduces to. Always write the fraction in simplest form. 0 Section

12 Unit FRACTION OPERATIONS We can also use a pencil and paper to add two mixed numbers together. To add mixed numbers, add the whole numbers together. Then, add the fractions together. Finally, write the mixed number in simplest form. + This is the original problem. + = Add the whole numbers together. + = Add the fractions together. = Rewrite the improper fraction as a mixed number. + = Add the whole numbers and the fractions together. = Write the fraction in simplest form. So, Brianna will need cups of flour to make her muffins. Example: This might help... Add. Remember that any fraction in which the + numerator and denominator are the same number has a value of. So, is equal to. Solution: Use a model to represent the addition problem. Next, combine the models. + So, the sum is. The sum can also be found using a pencil and paper. + This is the original problem. + = Add the whole numbers together. + = Add the fractions together. = Rewrite the improper fraction as a whole number. + = Add the whole numbers together. Section

13 FRACTION OPERATIONS Unit Subtracting Mixed Numbers with Like Denominators Now, let s try subtracting mixed numbers. To model a subtraction problem, use rectangles to represent the number being subtracted from. Then, remove the amount of shading indicated by the mixed number being subtracted. For example, let s model the subtraction problem. To model this subtraction problem, we begin by representing using rectangles. To subtract, we ll remove shading from of the rectangles. Notice though, that there is only to remove shading from. So we ll have to first represent one of the whole rectangles as a third. We can see that is actually the same as. Now remove shading from one of the whole rectangles, and two of the thirds. The remaining shaded amount represents the difference. So, =. = + + = = This is the original problem. Rename the mixed number being subtracted from. Subtract the whole numbers from each other and the fractions from each other. This might help... can be renamed as by taking from, representing it as thirds (remember, = ), and adding it to the fraction. Watch: = + + = + + = Section

14 Unit FRACTION OPERATIONS Example: Subtract. Solution: Let s subtract using the pencil and paper method. Notice that in this problem we do not have to rename the mixed number being subtracted from. This is the original problem. Subtract the whole numbers from each other and the fractions from each other. Write the fraction in simplest form. Let s try one more example. In this subtraction problem, we ll find the difference between a whole number and a mixed number. We ll use a model to help us understand what to do. Example: Subtract. Solution: Represent the number being subtracted from () using a model. Remember that to show subtraction, we unshade the amount that is being subtracted ( ). In order to subtract two fifths, let s represent the last rectangle in fifths. So, is the same as. Now, unshade. So, the difference is. Section

15 FRACTION OPERATIONS Unit Let s Review! Before going on to the practice problems, make sure you understand the main points of this lesson. To add or subtract mixed numbers, use a model or a pencil and paper. Add or subtract the whole number parts and the fraction parts separately. You may need to rename the mixed number being subtracted from in order to subtract the fraction part. Always write the sum or difference as a mixed number or fraction in simplest form. Circle the correct letter and answer.. Add. Write your answer in simplest form. a. b.. Add. Write your answer in simplest form. a. b.. Subtract. Write your answer in simplest form. a.. Subtract. Write your answer in simplest form. a. b. b c. d. + c. d. c. d. c.. When adding or subtracting mixed numbers with like denominators, the numerators. a. stay the same b. change. When adding or subtracting mixed numbers with like denominators, the denominators. a. stay the same b. change d. Section

16 Unit FRACTION OPERATIONS. Add. Write your answer in simplest form. + a. b. c. 0.0 Add. Write your answer in simplest form. + a. b. c.. Add. Write your answer in simplest form. + c. a. b.. Which of the following is another way to represent? a. b.. Sarah and Nathan each picked a bucket of strawberries. Sarah picked pounds, and Nathan picked pounds. How many pounds did they pick altogether? c. d. 0 d. d. d. a. pounds b. pounds c. pounds d. pounds. Subtract. Write your answer in simplest form. 0 0 a. 0 b. 0 c. 0. Subtract. Write your answer in simplest form. a. b. 0 c.. Subtract. Write your answer in simplest form. a. b. c.. Subtract. Write your answer in simplest form. a. b. c. d. 0 d. d. d. Section

17 FRACTION OPERATIONS Unit. Sarah and Nathan each picked a bucket of strawberries. Sarah picked pounds, and Nathan picked pounds. How much more did Sarah pick than Nathan? a. pounds b. pound c. pounds d. pound. Sarah and Nathan each picked a bucket of strawberries. Sarah picked pounds, and Nathan picked pounds. Which of the following is another way to represent pounds? a. pounds b. pounds c. pounds d. pounds Complete these activities..0 Match each addition or subtraction problem with its correct answer in simplest form. a. + b. c. + d..... Section

18 Unit FRACTION OPERATIONS Estimating Sums and Differences Kyle and Gina walk to school each morning. About how far does Kyle walk if he goes to Gina s house first? Notice that the question we were asked began with the word about. That means that we can estimate to find the answer. Remember that an estimate is an amount that is close to the actual value. Estimating is helpful when you need a quick answer, but not necessarily an exact answer. In this lesson, we ll practice estimating sums and differences of fractions and mixed numbers. miles Gina s house Kyle s house miles School miles School Estimating Sums of Fractions and Mixed Numbers From the map, we can see that Kyle walks miles to get to Gina s house and another miles to get to school. To estimate the total distance that Kyle walks, we ll use rounding and then add to find the total. We ll round the mixed numbers to the nearest whole or half. To round each mixed number, look at the fraction part of the mixed number and compare the numerator to the denominator. If the numerator is very small compared to the denominator, then the mixed number rounds down to the previous whole number. is an example of a mixed number that would round down to the previous whole number because is very little compared to. We can see this on the number line. Think about it... About how much further does Kyle walk than Gina each morning? If the numerator is close to half of the denominator, then the mixed number rounds to halfway between the whole numbers. is an example of a mixed number that would round to the nearest half because is about half of. Take a look: Section

19 FRACTION OPERATIONS Unit So, rounds to, and rounds to. Now that we ve rounded each mixed number to the nearest whole or half, we can estimate the sum. Sum: + Estimate: + = Kyle walks about miles in the morning, if he stops at Gina s house first. Let s try another one: Example: Estimate the sum. + 0 Solution: We ll round each mixed number to the nearest whole or half. Then, we ll add the rounded numbers together to find the estimate. In the fraction, the numerator is close to the amount of the denominator. When the numerator and denominator are almost the same number, the mixed number rounds up to the next whole number. So, rounds up to. In the fraction, the numerator is a little less than half of the denominator. Remember 0 that when the numerator is about half of the denominator, the mixed number rounds to the nearest half. So, rounds to 0. Now, estimate. 0 Estimate: + = Let s review what we ve learned about rounding mixed numbers. When the numerator of the fraction is very little compared to the denominator, round the mixed number down to the previous whole number. When the numerator of the fraction is about half of the denominator, round the mixed number to the nearest half. When the numerator of the fraction is close to the amount of the denominator, round the mixed number up to the next whole number. Section

20 Unit FRACTION OPERATIONS Estimating Differences of Fractions and Mixed Numbers Now, let s try some subtraction problems: Example: Samson is constructing a fence that will be feet long. He has already completed 0 feet of the fence. About how much does he have left to construct? Solution: To find the amount Samson has left to construct, we would subtract 0 from. To estimate, we ll round each mixed number to the nearest half or whole. Then, we ll estimate the difference using the rounded numbers. In the fraction, the numerator is close to the amount of the denominator. So, the mixed number rounds up to. In the fraction, the numerator is very little compared to the denominator. So, the mixed number 0 rounds down to 0. Estimate: 0 = Samson has about feet of fencing left to construct. Example: Estimate the difference. 0 Solution: Round each number to the nearest whole or half. Then, estimate using the rounded numbers. In the fraction 0, the numerator is about half of the denominator. So, the mixed number 0 rounds to. In the fraction, the numerator is very little compared to the denominator. So, the fraction rounds down to 0. Estimate: 0 = Section

21 FRACTION OPERATIONS Unit Let s Review! Before going on to the practice problems, make sure you understand the main points of this lesson. An estimate is an approximate value that is close to the actual value. Estimates are helpful for when you need a quick answer that is not exact. To estimate the sum or difference of two fractions or mixed numbers, round each fraction or mixed number to the nearest whole or half. Then, estimate with the rounded numbers. Fill in the blank.. The word in a word problem probably indicates you may estimate an answer. Circle the correct letter and answer.. Round this mixed number to the nearest whole or half. a. b. c. d.. Round this mixed number to the nearest whole or half. a. b. c. d.. Round this fraction to the nearest whole or half. a. 0 b. c. d.. Estimate the sum. 0 + a. b.. Estimate the sum. + a. b. c. d. c. d. 0 Section

22 Unit FRACTION OPERATIONS. This table shows the annual rainfall in four U.S. cities. What is the amount of rainfall that Miami receives, rounded to the nearest half or whole? a. inches b. inches c. inches d. inches. Use the table from Exercise.. About how much rainfall do San Francisco and Seattle get together? a. inches b. inches c. inches d. inches. Use the table from Exercise.. About how much more rainfall does New York receive each year than Seattle? a. inches b. inches c..0 Estimate the difference. 0 a. b.. Estimate the difference. 0 a. b. inches d. inches c. d. c. d.. Round this mixed number to the nearest whole or half. 0 a. b. c. d.. Round this mixed number to the nearest whole or half. a. b.. Estimate the sum. 0 + a. b. c.. Estimate the difference. 0 a. b. CITY c. d. d. c. d. ANNUAL RAINFALL San Francisco, CA 0 0 inches Seattle, WA inches Miami, FL 0 inches New York, NY inches Review the material in this section to prepare for the Self Test. The Self Test will check your understanding of this section. Any items you miss on this test will show you what areas you will need to restudy in order to prepare for the unit test. Section

23 FRACTION OPERATIONS Unit SELF TEST : LIKE DENOMINATORS Each numbered question = points Answer true or false..0 Angie and Kim are sharing a large sub sandwich. Angie ate of the sandwich, and Kim ate another of the sandwich. So, together they ate of the sandwich..0 Angie and Kim are sharing a large sub sandwich. Angie ate sandwich, and Kim ate another sandwich is left over. of the sandwich. So, of the of the Circle the correct letter and answer..0 The sum of and is. a. less than b. greater than c. equal to.0 Use a model or paper and pencil to add. Write your answer in simplest form. + a. b. c. d..0 Mason has to mow lawns today. So far, he has mowed of them. How many does he have left to do? a. b. c. d..0 Use a model or paper and pencil to subtract. Write your answer in simplest form. a. b. c. d. 0.0 Estimate the sum by rounding each mixed number to the nearest half or whole number. + a. b. c. d..0 Estimate the difference by rounding each mixed number to the nearest half or whole number. 0 a. b. c. d. Section

24 Unit FRACTION OPERATIONS Madison made the following table to record the height of each person in her family. Use the table to answer Questions.0 through.. NAME Dad Mom Madison Jade Ben HEIGHT (in feet).0 How much taller is her dad than her mom? a. foot b. feet c. foot d. feet.00 If Madison and Jade lay end to end, how far will they reach? a. feet b. feet c. 0 feet d. 0 feet.0 Round her mom s height to the nearest half or whole. a. feet b. feet c. feet.0 Round Jade s height to the nearest half or whole. a. feet b. feet c. feet.0 About how much taller is her mom than Jade? a. 0 feet b. foot c. foot d. feet Complete these activities..0 Find the difference. Write your answer in simplest form..0 Find the sum. Write your answer in simplest form Teacher check: Initials Score Date 0 Section

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