Division of Fractions Study Guide - Grade 5 PDF

STUDY GUIDE GRADE 5 | UNIT 7 Division of Fractions Table of Contents Introduction....................................................................

STUDY GUIDE GRADE 5 | UNIT 7 Division of Fractions Table of Contents Introduction..........................................................................................................................2 Test Your Prerequisite Skills..............................................................................................3 Objectives............................................................................................................................4 Lesson 1: Visualizing Division of Fractions - Warm Up!..................................................................................................................5 - Learn about It!..........................................................................................................6 - Let’s Practice!............................................................................................................8 - Check Your Understanding!................................................................................. 12 Lesson 2: Dividing a Fraction by Another Fraction - Warm Up!............................................................................................................... 13 - Learn about It!....................................................................................................... 14 - Let’s Practice!......................................................................................................... 15 - Check Your Understanding!................................................................................. 19 Lesson 3: Dividing Whole Numbers by a Fraction and Vice Versa - Warm Up!............................................................................................................... 20 - Learn about It!....................................................................................................... 21 - Let’s Practice!......................................................................................................... 22 - Check Your Understanding!................................................................................. 26 Challenge Yourself!.......................................................................................................... 27 Performance Task............................................................................................................ 28 Wrap-up............................................................................................................................ 31 Key to Let’s Practice!......................................................................................................... 32 References........................................................................................................................ 32 1  STUDY GUIDE Click Home icon to go back to Table of Contents GRADE 5|MATHEMATICS UNIT 7 Division of Fractions We have learned how to multiply fractions in the previous unit. In this unit, we will be dealing with the division of fractions and its application in our daily lives. We have the value of sharing what we have with others. For instance, you have half of a cake left, and you still want to divide this equally among your four friends. What fractional part of the cake will each of them eat? The skill of dividing fractions is essential to answer this. Another scenario in which you can utilize your skill in dividing fractions is when you cook. Suppose a dish 1 5 requires cup of melted butter, but you only have 2 6 cup left. How many servings can you make out of the available portion of the ingredient? Division of fractions can be applied in our day-to-day activities. It is only but essential to learn and master this skill. In this unit, you will learn how to divide fractions by another fraction, by a whole number, or vice versa. 2 STUDY GUIDE Test Your Prerequisite Skills Multiplying and dividing whole numbers Finding the factors of a whole number Finding the reciprocal of a fraction Multiplying fractions Before you get started, answer the following items on a separate sheet of paper. This will help you assess your prior knowledge and practice some skills that you will need in studying the lessons in this unit. Show your complete solution. 1. Perform the indicated operations. a. 15 × 5 b. 26 × 2 c. 72 ÷ 9 d. 42 ÷ 6 2. List all the factors of the given whole number. a. 12 b. 28 c. 18 3. Find the reciprocal of the following numbers. 5 a. 6 3 STUDY GUIDE 11 b. 12 c. 7 d. 10 4. Multiply the following fractions. 1 1 a. ×4 2 3 5 b. ×6 4 1 3 c. ×7 9 Objectives At the end of this unit, you should be able to show that multiplying a fraction by its reciprocal is equal to 1; visualize division of fractions; divide simple fractions and whole numbers by a fraction and vice versa; and solve routine or nonroutine problems involving division without or with any of the other operations of fractions and whole numbers using appropriate problem solving strategies and tools. 4 STUDY GUIDE Lesson 1: Visualizing Division of Fractions Warm Up! Divide and Count! Materials Needed: sheets of paper, pen Instructions: 1. Form groups with five members each. 2. Each member will be provided with a sheet of paper with equal dimensions. Divide each paper into eight equal parts. 3. Since a sheet of paper is divided into eight equal parts, one small 1 box is of the paper. 8 4. Count the total number of small boxes that you have in your group. Number of small boxes: ________ 5. Write a mathematical expression that represents the number of sheets provided per group, the fractional part of one small box with the whole sheet of paper, and the total number of small boxes you have in your group. Mathematical expression: ________________________ 6. The first group to answer correctly will be given a reward. 5 STUDY GUIDE Learn about It! The activity in Warm Up! shows how to divide a whole number by a fraction by folding the papers into a number of equal parts. Recall that division is splitting a quantity into equal parts or groups. We use the symbol ÷ to denote division. Moreover, the line between the numerator and the denominator represents the same thing. How do we represent division of fractions? Let us take the following example: 1 Jenny bought of a kilogram of baking powder. If 2 1 she puts of a kilogram into each cup, how many 8 cups can she fill? 1 Solution: To determine the number of cups that Jenny can fill, we divide 2 1 by. Let us illustrate the given problem. 8 6 STUDY GUIDE We can represent 1 kilogram of baking powder using the illustration below. Divide the illustration by 2 to represent the amount of baking powder that 1 Jenny bought, that is, of a kilogram. 2 1 Using the same figure, divide the whole unit by 8 since she has to put of a 8 kilogram of baking powder into each cup. 7 STUDY GUIDE 1 There are 4 one-eighths in. Thus, Jenny can fill four cups of baking powder. 2 Let’s Practice! 3 1 Example 1: Divide by. 4 8 Solution: 3 Step 1: Illustrate. 4 8 STUDY GUIDE 1 Step 2: Represent by dividing the same figure into 8 equal parts. 8 3 3 1 There are 6 one-eighths in. Thus, ÷ = 6. 4 4 8 Try It Yourself! 5 1 What is ÷ 12? 6 1 1 Example 2: What is divided by ? 3 3 1 1 1 Solution: Illustrate and observe how many you find in. 3 3 3 1 1 1 1 There is one in. Thus, ÷ = 1. 3 3 3 3 9 STUDY GUIDE Try It Yourself! 4 4 What is divided by ? 5 5 5 1 Example 3: Divide by. 8 8 Solution: 5 Step 1: Illustrate. 8 1 Step 2: Represent by dividing the same figure into 8 equal parts. 8 𝟏 𝟏 𝟏 𝟖 𝟖 𝟖 𝟏 𝟏 𝟖 𝟖 5 5 1 There are 5 one-eights in. Thus, ÷ = 5. 8 8 8 10 STUDY GUIDE Try It Yourself! 1 1 Divide by. 2 4 Real-World Problems Example 4: A vacant space in a bookshelf is 6 cm 1 long. Each book on the shelf is cm thick. 2 How many books will fit on the shelf? 1 Solution: Divide 6 by. 2 Step 1: Illustrate the dividend. 1 Step 2: Represent in each of the given figures. 2 Since there are 12 one-halves in 6, twelve books will fit on the shelf. 11 STUDY GUIDE Try It Yourself! Joy brought 4 pizzas to her grandmother’s house. Each pizza is divided into 12 equal parts. How many slices of pizza are there in all? Check Your Understanding! 1. Divide the following fractions: 1 1 a. ÷ 16 2 7 1 b. ÷8 8 11 11 c. ÷ 12 12 1 d. 25 ÷ 5 3 e. 16 ÷ 8 2. Chris will make a slogan in a cartolina. He bought 6 cartolinas which he will each be dividing into three. How many pieces of cartolina does he have in all? 1 3. Joseph has of a pizza. He wants to share it equally with his 4 friends. 2 How much of the pizza does each person get? 12 STUDY GUIDE Lesson 2: Dividing a Fraction by Another Fraction Warm Up! Flip and Multiply! Materials Needed: paper, pen Instructions: 1. This activity may be done individually or in pairs. 2. Use what you have learned in the previous lesson to find the quotient of the following fractions: 𝟏 𝟏 ÷ = _____ 𝟑 𝟑 𝟓 𝟓 ÷ = _____ 𝟔 𝟔 𝟕 𝟕 ÷ = _____ 𝟑 𝟑 3. Flip the divisor in such a way that the numerator becomes the denominator and vice versa. Then, multiply the dividend and the new divisor. 13 STUDY GUIDE 𝟏 𝟏 𝟏 ÷ = _____ × = _____ 𝟑 𝟑 𝟑 𝟓 𝟓 𝟓 ÷ = _____ × = _____ 𝟔 𝟔 𝟔 𝟕 𝟕 𝟕 ÷ = _____ × = _____ 𝟑 𝟑 𝟑 4. What have you observed about the answers in the first column and the second column? Learn about It! The activity in Warm Up! shows another way of dividing fractions. Recall that in dividing a number by itself, the result will always be equal to 1. You were then asked to flip the divisor, then multiply it to the dividend. In doing so, you obtain the same answer as in the first column. When flipping a fraction, or when the numerator becomes the denominator and vice versa, the new fraction is called the reciprocal of the original one. Note that multiplying a number by its reciprocal will result in 1. Definition 2.1: The reciprocal of a number is the number obtained by dividing one by the fraction; a fraction in which the numerator becomes the denominator and vice versa. 14 STUDY GUIDE Dividing a Fraction by Another Fraction Aside from using figures, there is another way to divide fractions. We use the concept of reciprocals to divide fractions. 2 1 Example: Divide by. 5 10 Step 1: Write the reciprocal of the divisor. 1 10 The reciprocal of is or 10. 10 1 Step 2: Multiply the dividend and the reciprocal of the divisor. 2 10 20 × = 5 1 5 Step 3: Simplify the quotient. 20 =4 5 2 1 Thus, divided by is 4. 5 10 Let’s Practice! Example 1: What number will you multiply by 8 to get 1? Solution: Multiplying a number by its reciprocal will result in 1. The 1 reciprocal of 8 is. Thus, the number we can multiply by 8 to 8 1 get 1 is. 8 15 STUDY GUIDE Try It Yourself! 15 What number will you multiply to to get 1? 23 6 6 Example 2: Divide by. 7 10 Solution: Step 1: Write the reciprocal of the divisor. 6 10 The reciprocal of is. 10 6 Step 2: Multiply the dividend and the reciprocal of the divisor. 6 10 60 × = 7 6 42 Step 3: Simplify the quotient. 60 10 3 = or 1 42 7 7 6 10 3 Thus, ÷ =1. 7 6 7 16 STUDY GUIDE Try It Yourself! 15 5 Divide: ÷6 23 1 9 4 Example 3: Simplify: ×( ÷ ). 9 10 5 Solution: Using the GEMDAS rule, first perform the operation inside the 1 parentheses. Then, multiply the answer to. 9 Step 1: Write the reciprocal of the divisor. 4 5 The reciprocal of is. 5 4 Step 2: Multiply the dividend and the reciprocal of the divisor. 9 5 45 × = 10 4 40 Step 3: Simplify the quotient. 45 9 1 = or 18 40 8 1 1 Step 4: Multiply and 1. 9 8 1 1 1 9 1 ×1 = × = 9 8 9 8 8 17 STUDY GUIDE Try It Yourself! 1 7 1 Simplify: × (8 ÷ 4) 2 Real-World Problems 8 1 Example 4: Ellen has gallon of milk left. She needs 9 9 gallon of milk per bottle. How many bottles of milk can she use? 8 1 Solution: To determine how many bottles of milk she used, divide by. 9 9 Step 1: Write the reciprocal of the divisor. 1 9 The reciprocal of is or 9. 9 1 Step 2: Multiply the dividend and the reciprocal of the divisor. 8 9 × =8 9 1 Therefore, Ellen can use 8 bottles of milk. 18 STUDY GUIDE Try It Yourself! 7 1 Roxanne bought kg of flour. She put kg in each 9 9 container. How many containers did she fill? Check Your Understanding! 1. Divide the following fractions: 13 3 5 1 8 7 a. ÷ b. ÷ c. ÷ 15 5 8 4 9 14 2. Determine the number such that when it is multiplied to the given fraction, the result will be 1. 7 18 16 a. b. c. 14 19 21 19 3. Enna bought kg of candies. She plans to pack the candies such that 8 5 each bag contains kg of candies. How many bags does she need? 8 7 1 4. A race covers of a mile. A team needs runners who will each run of a 8 8 mile. How many runners are needed for a team? 19 STUDY GUIDE Lesson 3: Dividing Whole Numbers by a Fraction and Vice Versa Warm Up! Divide It More! Materials Needed: paper, pen Instructions: 1. This activity may be done individually or in pairs. 3 2. Illustrate in the figure that follows: 4 3 3. Divide the portion representing into 6 equal parts. 4 4. How will you represent the fractional part of one small rectangle in terms of the whole? 3 5. What mathematical sentence can you make from , 6, and the 4 fractional part of one small rectangle? 20 STUDY GUIDE Learn about It! How do we divide fractions by a whole number and vice versa? 3 The activity in Warm Up! introduced you on how to divide a fraction, that is , 4 by a whole number 6 using a visual representation. Upon doing so, you were 3 1 able to find out that when is divided by 6, the result is. 4 8 Just like in dividing fractions, you can also use the concept of reciprocals in dividing fractions by a whole number and vice versa. Recall that in expressing a whole number into a fraction, it is written with a denominator of 1. Let us take the following example: 7 Armando ran of a mile around a tracking field. He made 5 10 laps in all. How many miles are there in each lap? 7 Solution: Divide by 5 to determine the number of miles Armando ran 10 in each lap. Step 1: Write the reciprocal of the divisor. 1 The reciprocal of 5 is. 5 21 STUDY GUIDE Step 2: Multiply the dividend and the reciprocal of the divisor. 7 1 7 × = 10 5 50 Step 3: Simplify the quotient. The quotient is in simplified form already. 7 Thus, the number of miles Armando ran in each lap is. 50 Let’s Practice! 3 Example 1: Divide by 6. 5 Solution: Step 1: Write the reciprocal of the divisor. 1 The reciprocal of 6 is. 6 Step 2: Multiply the dividend and the reciprocal of the divisor. 3 1 3 × = 5 6 30 22 STUDY GUIDE Step 3: Simplify the quotient. 3 1 = 30 10 Try It Yourself! 7 Divide by 16. 8 5 Example 2: How many are there in 5? 7 5 Solution: Divide 5 by. 7 Step 1: Write the reciprocal of the divisor. 5 7 The reciprocal of is. 7 5 Step 2: Multiply the dividend and the reciprocal of the divisor. 5 7 × =7 1 5 5 Thus, there are seven in 5. 7 23 STUDY GUIDE Try It Yourself! How many one-eighths are there in 24? 1 Example 3: Simplify: ( ÷ 4) ÷ 10 2 Solution: Using the GEMDAS rule, first perform the operation inside the parentheses. Then, divide the answer by 10. 1 Step 1: Divide by 4. Write the reciprocal of the divisor. 2 1 The reciprocal of 4 is. 4 Step 2: Multiply the dividend and the reciprocal of the divisor. 1 1 1 × = 2 4 8 1 Step 3: Divide by 10. 8 1 1 1 1 ÷ 10 = × = 8 8 10 80 1 1 Thus, ( ÷ 4) ÷ 10 =. 2 80 24 STUDY GUIDE Try It Yourself! 1 Simplify: (6 ÷ ) ÷ 12 3 Real-World Problems Example 4: Three-fifths of a class are girls. If there were 24 girls, how many children are there in the class? Solution: To determine the number of children in the class, find out what 3 number should be multiplied by to obtain 24. To do so, 5 3 simply divide 24 by. 5 Step 1: Write the reciprocal of the divisor. 3 5 The reciprocal of is. 5 3 Step 2: Multiply the dividend and the reciprocal of the divisor. 24 5 × = 40 1 3 Therefore, there are 40 students in a class. 25 STUDY GUIDE Try It Yourself! Joy got her allowance from her mother. If she saves one-third of her allowance, which is ₱83.00, to buy a discounted CD of her favorite music icon, how much is her allowance? Check Your Understanding! 1. Divide the following: 3 2 3 a. 15 ÷ c. 18 ÷ e. 12 ÷ 8 9 4 16 7 b. ÷4 d. ÷ 49 19 10 2. Simplify. 1 2 4 4 a. ( ÷ 2) ÷ 10 b. 15 ÷ ( ÷ ) c. 26 × ( ÷ 8) 6 3 5 9 3 3. Enna collected 1 728 seashells. She wants to share of these to her 8 friends. How many of her friends will receive? 3 4. Peter has a 10-foot string. He wants to cut it into of a foot each. Then, 4 each piece will be cut further into 5 pieces. How long will the shortest piece of a string be? 26 STUDY GUIDE Challenge Yourself! 1 1. Peter received a discount of ₱500.00. This is of the price of what he 10 bought. How much is the original price of the item? 2. For what values of divisors will you get quotients that are less than the dividend? How about quotients that are greater than the dividend? 7 1 3. Ms. Ann asked her class to divide by. The solutions of two of her 12 4 students, Danica and Francine, are shown below. Find out who got the answer correctly and what mistake the other one did. Danica’s Solution Francine’s Solution 7 1 ÷ 7 1 12 4 7 4 7 ÷ 12 4 × = 12 1 3 12 1 3 7 4 1 × = 7 4 7 × =2 12 1 3 27 STUDY GUIDE Performance Task You are a baker in a coffee shop. You wanted to try a new recipe called Choco Peanut Butter Cookies, and these are the ingredients needed: 1 kg unsalted butter 10 1 kg light brown sugar 10 1 egg, beaten 1 kg caster sugar 10 3 kg self-raising flour 20 2 tbsp. cocoa powder 1 teaspoon salt 4 3 kg of milk chocolate chopped into chunks 20 1 kg of milk chocolate melted for drizzling 20 3 kg peanut butter 40 The ingredients as mentioned above make a serving of 12. You would only want 6 of your team to taste these cookies before serving it to the public. Thus, each of the ingredients should be divided by two. Your solutions should show two ways of dividing each ingredient by 2. One is using a pictorial model (visually), and the other one is through the use of reciprocals. 28 STUDY GUIDE Complete the table as follows: Serves 12 Serves 6 1 kg unsalted butter 10 1 kg light brown sugar 10 1 egg, beaten 1 kg caster sugar 10 3 kg self-raising flour 20 2 tbsp cocoa powder 1 teaspoon salt 4 3 kg of milk chocolate 20 chopped into chunks 1 kg of milk chocolate 20 melted for drizzling 3 kg peanut butter 40 29  STUDY GUIDE Performance Task Rubric Below Needs Successful Exemplary Criteria Expectation Improvement Performance Performance (0–49%) (50–74%) (75–99%) (99+%) There are more There are 6-10 There are 1-5 100% of the Accuracy 10 errors in the errors in the errors in the computations computations. computations. computations. are correct. Solutions are Solutions are Solutions are organized organized Organization Solutions are organized properly. All properly but of the not organized properly. Only a information some necessary Solutions properly. few information needed in the parts are is missing. analysis is missing. present. Report is Report is Report is Report is Timeliness of submitted submitted 3-4 submitted 1-2 submitted on submission more than 4 days late. days late. time. days late. 30 STUDY GUIDE Wrap-up DIVISION OF FRACTIONS Find the RECIPROCAL of the divisor. Multiply the DIVIDEND and the RECIPROCAL. SIMPLIFY your answer. Key Terms/ Formulas Key Term Description The reciprocal of a number is the number obtained by dividing one by the fraction; a Reciprocal fraction in which the numerator becomes the denominator and vice versa. 31 STUDY GUIDE Key to Let’s Practice! Lesson 1 Lesson 2 Lesson 3 1. 10 1. 23 1. 7 15 128 2. 1 2. 18 2. 192 3. 2 23 1 4. 48 slices of 3 3. 1 3. 1 2 4 4. ₱249.00 pizza 4. 7 containers References BBC Good Food. “How to Bake Choco Peanut Butter Cookies.” Accessed March 14, 2018. https://www.bbcgoodfood.com/recipes/double-choc- peanut-butter-cookies Davison, David M., et al. Pre-Algebra. Philippines: Pearson Education, Inc., 2005. Helping with Math. “How to Divide Fractions.” Accessed March 6, 2018. https://www.helpingwithmath.com/by_subject/fractions/fra_dividing.ht m Math is Fun. ”Dividing Fractions.” Accessed March 6, 2018. https://www.mathsisfun.com/fractions_division.html 32