UNIT 5 OPERATIONS WITH FRACTIONS

Transcription

1 Unit Media Lesson UNIT OPERATIONS WITH FRACTIONS INTRODUCTION In this Unit, we will use the multiple meanings and representations of fractions that we studied in the previous unit to develop understanding and processes for performing operations with fractions. You are likely familiar with algorithms for computing fraction operations. Please accept the challenge of really thinking through the meaning of operations and the different contexts of fractions so you understand why we perform operations as we do. If you accept this challenge, this unit will help you see the meaning of these processes not just the steps. The table below shows the learning objectives that are the achievement goal for this unit. Read through them carefully now to gain initial exposure to the terms and concept names for the lesson. Refer back to the list at the end of the lesson to see if you can perform each objective. Learning Objective Media Examples You Try Add fractions with like denominators 4 Add fractions with unlike denominators 4 Add improper fractions and mixed numbers 4 Subtract fractions with like denominators 8 Subtract fractions with unlike denominators 6 8 Subtract improper fractions and mixed numbers 7 8 Multiply a unit fraction times a whole number 9 Multiply a composite fraction times a whole number 0 Multiply a whole number times a fraction Multiply two fractions Multiply mixed numbers 4 Divide a whole number by a fraction 6 9 Divide fraction with common denominators 7 9 Divide fraction with uncommon denominators 8 9 Divide mixed numbers 0 Perform +,,, on signed fraction 4 Use the order of operations with signed fractions 4

2 Unit Media Lesson UNIT MEDIA LESSON SECTION.: ADDING FRACTIONS In this section, we will learn to visualize the addition of fractions using an area model and number lines. Recall that the operation of addition is combining two amounts or adding one amount on to another. We will achieve this with fractions by ensuring we have a common unit fraction (common denominator) for the numbers we are combining. We can then add the number of copies of each fraction while retaining the common denominator. Problem MEDIA EXAMPLE Adding Fractions with Like Denominators Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the operation and represent the sum using the symbolic representation of the algorithm.. Jon had have? of a pepperoni pizza and of a mushroom pizza. How much of one whole pizza did Jon is copies of is copies of Combined, we have a total of copies of or the fraction.. Christianne sprinted 8 of a mile and then jogged another 7 8 of a mile. How far did she run in total? 8 is copies of 7 8 is copies of Combined, we have a total of copies of or the fraction. Simplified result as an improper fraction: Simplified result as a mixed number:

3 Problem MEDIA EXAMPLE Adding Fractions with Unlike Denominators Unit Media Lesson Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the operation and represent the sum using the symbolic representation of the algorithm. a) Step : Rewrite the given fractions with a common denominator. is equivalent to is equivalent to Step : Rewrite the equivalent fractions with the common denominator using copies of language using a common unit fraction. : is copies of : is copies of Step : Combine (add). Combined, we have a total of copies of or the fraction. Step 4: Simplify if necessary. b) 4 8

4 c) Unit Media Lesson Problem MEDIA EXAMPLE Adding Improper Fractions and Mixed Numbers Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the operation and represent the sum using the symbolic representation of the algorithm. a)

5 b) 4 Unit Media Lesson Step : Rewrite the fractional parts of the mixed numbers with a common denominator Step : Rewrite the equivalent fractions with the common denominator using copies of language using a common unit fraction. : is copies of : is copies of Step : Combine the fractional parts and the whole number parts. Fractional Parts: Combined, we have a total of copies of or the fraction. Whole Numbers plus fractional parts: Combined we have a total of Step 4: Simplify if necessary. c) 9

6 Problem 4 YOU TRY - Adding Numbers with Fractions Unit Media Lesson Use the diagrams given to represent the values in the addition problem and find the sum. Then perform the operation and represent the sum using the symbolic representation of the algorithm. a) 4 Step : Rewrite the given fractions with a common denominator. is equivalent to 4 is equivalent to Step : Rewrite the equivalent fractions with the common denominator using copies of language using a common unit fraction. : is copies of 4 : is copies of Step : Combine (add). Combined, we have a total of copies of or the fraction. Step 4: Simplify if necessary. b) 4 8 6

7 Unit Media Lesson SECTION.: SUBTRACTING FRACTIONS In this section, we will learn to visualize subtracting fractions using an area model and number lines. We will investigate this idea using our two models of subtraction; subtraction as taking away a part of a whole, and subtraction as comparing two quantities. Problem MEDIA EXAMPLE Subtracting Fractions with Like Denominators Use the diagrams given to represent the values in the subtraction problem and find the difference. Then perform the operation and represent the difference using the symbolic representation of the algorithm. a) The day after Thanksgiving, there was breakfast. How much pie is leftover now? of a pumpkin pie remaining. Lara ate of a pie for b) Billy ran 8 of a mile. Roberta ran 8 of a mile. How much further did Roberta run? 7

8 Problem 6 MEDIA EXAMPLE Subtracting Fractions with Unlike Denominators Unit Media Lesson Use the diagrams given to represent the values in the subtraction problem and find the difference. Then perform the operation and represent the difference using the symbolic representation of the algorithm. a) b) 6 c) Perform the operation and represent the difference using the symbolic representation of the algorithm

9 Unit Media Lesson Problem 7 MEDIA EXAMPLE Subtracting Improper Fractions and Mixed Numbers Use the diagrams given to represent the values in the subtraction problem and find the difference. Then perform the operation and represent the difference using the symbolic representation of the algorithm. a) 8 6 b) 4 c) 0 9

10 Unit Media Lesson Problem 8 YOU TRY - Subtracting Numbers with Fractions Use the diagrams given to represent the values in the subtraction problem and find the difference. Then perform the operation and represent the difference using the symbolic representation of the algorithm. a) Michael lives 8 school does Rachel live? of a mile from school. Rachel lives 7 8 of a mile from school. How much closer to b) c) Barney had 7 8 of his weekly salary left after paying his bills. He then spent a weekend trip. What fraction of his weekly salary remains? 4 of his weekly salary on 0

11 Unit Media Lesson SECTION.: MULTIPLYING FRACTIONS In this section, we will examine multiplying fractions using the idea that a b or a b is equivalent to a copies of b. Problem 9 MEDIA EXAMPLE Multiplying a Unit Fraction and a Whole Number Use the diagrams given to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Ebony ate of a pizza. The pizza had slices. How many slices did she eat? b) Kimber was 0 miles from home. She travelled How many miles did she travel while listening to her favorite song? of this distance while listening to her favorite song. c) Logan has a gallon bucket. He fills it 4 of the way to the top. How much water is in the bucket?

12 Problem 0 Unit Media Lesson MEDIA EXAMPLE Multiplying a Composite Fraction and a Whole Number Use the diagrams given to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Ebony ate of a pizza. The pizza had slices. How many slices did she eat? b) Kimber was 0 miles from home. She travelled How many miles did she travel while listening to her favorite song? of this distance while listening to her favorite song. c) Logan has a gallon bucket. He fills it 4 of the way to the top. How much water is in the bucket?

13 Problem YOU TRY Multiplying Unit or Composite Fractions with Whole Numbers Unit Media Lesson Use the diagrams given to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Carese was running a 0 kilometer race. She ran did she run in minutes? of the race in minutes. How many kilometers b) Casey s gas tank holds gallons. The gas gage says it is tank? full. How many gallons of gas are in her

14 Problem MEDIA EXAMPLE Multiplying Whole Numbers and Fractions Unit Media Lesson Use the diagrams given to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Ashley has packs of cupcakes. There are 4 cupcakes per pack. How many cupcakes does Ashley have? b) Anderson jogged around a lake times this week. The distance around the lake is did Anderson jog in total over the week? of a mile. How far c) Knia is having a dinner party with a total of 7 people. She bought of a pound of cold cuts per person. How many pounds of cold cuts did she buy in total? 4

15 Problem MEDIA EXAMPLE Multiplying Two Fractions Unit Media Lesson Use the diagrams given to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Yesterday, Cameron walked had walked of a mile to school. Today, Cameron s friend picked him up after he of the way to school. How far to Cameron walk today? b) Jassey bought a rectangular piece of land to grow vegetables. The land is mile wide. How many square miles of land did Jassey buy? of a mile long and 4 of a c) Ray made a tray of brownies. He ate of the tray after they had cooled. The next day, he ate 4 of what was left over in the tray. How much of the whole tray did Ray eat the next day?

16 Unit Media Lesson Problem 4 MEDIA EXAMPLE Multiplying Mixed Numbers Create a diagram to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) According to the Bureau of Labor Statistics, the buying power of the dollar is 4 times larger in 06 when compared to 99. Determine the comparable buying power in 06 of $0 in 99. b) J Von bought a rectangular piece of land in Alaska. The land was How many square miles of land did he buy? 4 miles wide and 8 miles long. 6

17 Unit Media Lesson Problem YOU TRY Multiplying Mixed Numbers, Fractions, and Whole Numbers Create a diagram to represent the multiplication problem and find the product. Then perform the operation and represent the product using the symbolic representation of the algorithm. a) Rick walked around a track 6 times this month. The distance around the track is did Rick walk in total over the month? 4 of a mile. How far b) Kate has a rectangular piece a land to raise horses. The land is wide. How many square miles of land does Kate own? of a mile long and 8 of a mile c) Javier makes times as much an hour as when he first started his job. If he made $9 an hour when he first started, how much does Javier make now? 7

18 Unit Media Lesson SECTION.4: DIVIDING FRACTIONS In this section, we will learn to visualize dividing fractions using an area model and number lines. We will investigate this idea using two models of division; dividing as partitioning the dividend into a known number of copies, and dividing as determining how many copies of a given size are in the dividend. In general, we do not usually find a common denominator to divide fractions. However, we will begin with examples by finding common denominators to illustrate a general process for dividing fractions. Problem 6 MEDIA EXAMPLE Dividing Whole Numbers by Fractions Use the diagrams given to represent the division problem and find the quotient. a) Tia made cakes for her guests. If each guest receives of a cake, how many guests can Tia serve? Symbolic Representation: b) Elaine ran 6 miles this month. If she ran of a mile every day she ran, how many days did Elaine run? Symbolic Representation: 8

19 Unit Media Lesson Problem 7 MEDIA EXAMPLE Dividing Fractions Using Common Denominators Use the diagrams given to represent the division problem and find the quotient. a) 9 Symbolic Representation: b) 8 Symbolic Representation: c) 8 4 Symbolic Representation: d) What patterns do you observe in problems a through c? 9

20 Unit Media Lesson Results: Based on the previous examples, we can divide two fractions by dividing their numerators (left to right) and their denominators (left to right) just like we multiply the numerators and denominators of fractions to multiply them. The last two problems illustrate this fact below or 4 4 Problem: The previous two problems worked out nicely because dividing the numerators and denominators resulted in an integer in both the resulting numerator and denominator. Consider this problem that doesn t work out as nicely Using the division method actually made the problem worse! We want the result as a fraction of one integer over another integer, not a fraction with fractions as numerators and denominators. Solution: We will find a common denominator for the two fractions we are dividing and perform the division like we did for the first problems. Then we will check for patterns to simplify the process. 7. Original Problem. Rewrite the two fractions with a common denominator of.. Divide the numerators and the denominators Rewrite the division in the numerator as a fraction. 0 0 Pattern: Let s look at the original problem and the final answer. Final Solution: 7 Comparable Method: Rule: To divide two fractions, multiply the first fraction by the reciprocal of the second fraction. (Note: The reciprocal of a fraction a b is b a. In this example, the reciprocal of is. 0

21 Unit Media Lesson Problem 8 MEDIA EXAMPLE Dividing Fractions with Unlike Denominators Divide the fractions by multiplying the first fraction by the reciprocal of the second fraction. Simplify your result if necessary. Write any answers greater than as both an improper fraction and a mixed number. a) b) 8 0 c) Problem 9 YOU TRY Dividing Fractions a) A snail crawled meters this week. If he crawled did he crawl? of a meter every day he crawled, how many days Symbolic Representation: b) Divide the fractions by dividing the numerators and the denominators. Simplify your result if necessary. i. 8 9 ii. iii c) Divide the fractions by multiplying the first fraction by the reciprocal of the second fraction. Simplify your result if necessary. i. 9 ii. 6 4 iii

22 Problem 0 MEDIA EXAMPLE Dividing Mixed Numbers Unit Media Lesson Rewrite any mixed numbers as improper fractions. Then perform the division by multiplying the first fraction by the reciprocal of the second fraction. Simplify your result if necessary and rewrite your result as a mixed number when possible. a) b) 4 8 c) Problem YOU TRY Dividing Mixed Numbers Rewrite any mixed numbers as improper fractions. Then perform the division by multiplying the first fraction by the reciprocal of the second fraction. Simplify your result if necessary and rewrite your result as a mixed number when possible. a) b) 6 c) 6 4

23 Unit Media Lesson SECTION.: SIGNED FRACTIONS AND THE ORDER OF OPERATIONS In this section, we will use our knowledge of operations on integers and the order of operations to perform operations with signed fractions and the order of operations with fractions. Problem MEDIA EXAMPLE Operations on Signed Fractions Perform the indicated operations on the fractions and/or mixed numbers using your knowledge of signed numbers. a) 7 b) 7 c) 4 4 d) e) f) Problem MEDIA EXAMPLE The Order of Operations and Signed Fractions Perform the indicated operations on the fractions and/or mixed numbers using your knowledge of signed numbers and the order of operations. a) b) 9 7 c) 4

24 Problem 4 YOU TRY The Order of Operations and Signed Fractions Unit Media Lesson Perform the indicated operations on the fractions and/or mixed numbers using your knowledge of signed numbers and the order of operations. a) 4 4 b) 9 4 c) 4 4