# Lesson 1: Interpreting Division of a Whole Number by a Fraction Visual Models

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1 Lesson 1 Lesson 1: Interpreting Division of a Whole Number by a Fraction Visual Models Classwork Opening Exercise Draw a model of the fraction. Describe what the fraction means. Example 1 Maria has 3 lb. of trail mix. She needs to share it equally among 6 friends. How much will each friend be given? What is 4 this question asking us to do? How can this question be modeled? Lesson 1: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/8/13 S.1

2 Lesson 1 Example 2 Imagine that you have 2 of a cup of frosting that you need to share equally between three desserts. How would we 5 write this as a division question? We can start by drawing a model of two-fifths. How can we show that we are dividing two-fifths into three equal parts? What does this part represent? Exercises 1 5 For each question below, rewrite the problem as a multiplication question. Then model the answer = Lesson 1: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/8/13 S.2

3 Lesson = = = = Lesson 1: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/8/13 S.3

4 Lesson 1 Problem Set Rewrite each problem as a multiplication question. Model the answer Lesson 1: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/8/13 S.4

5 Lesson 2 Lesson 2: Interpreting Division of a Whole Number by a Fraction Visual Models Classwork Example 1 Question # Write it as a division question. Write it as a multiplication question. Make a rough draft of a model to represent the question: Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.5

6 Lesson 2 As you travel to each model, be sure to answer the following questions: Original Questions Write the division question that was answered in each model. What multiplication question could the model also answer? Write the question given to each group as a multiplication question. 1. How many 1 2 miles are in 12 miles? 2. How many quarter hours are in 5 hours? 3. How many 1 3 cups are in 9 cups? 4. How many 1 8 pizzas are in 4 pizzas? 5. How many one-fifths are in 7 wholes? Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.6

7 Lesson 2 Example 2 Molly used 9 cups of flour to bake bread. If this was 3 of the total amount of flour she started with, what was the 4 original amount of flour? a. Create a model to represent what the question is asking for. b. Explain how you would determine the answer using the model. Exercises A construction company is setting up signs on 4 miles of the road. If the company places a sign every 1 of a mile, 8 how many signs will it need? Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.7

8 Lesson 2 2. George bought 12 pizzas for a birthday party. If each person will eat 3 of a pizza, how many people can George feed 8 with 12 pizzas? 3. The Lopez family has adopted 6 miles of trail on the Erie Canal. If each family member cleans up 3 of a mile, how 4 many family members will be needed to clean the adopted section? Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.8

9 Lesson 2 4. Margo is freezing 8 cups of strawberries. If this is 2 of the total strawberries that were picked, how many cups of 3 strawberries did Margo pick? 5. Regina is chopping up wood. She has chopped 10 logs so far. If the 10 logs represent 5 8 be chopped, how many logs need to be chopped in all? of all the logs that need to Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.9

10 Lesson 2 Problem Set Rewrite each problem as a multiplication question. Model the answer. 1. Nicole has used 6 feet of ribbon. This represents 3 of the total amount of ribbon she started with. How much 8 ribbon did Nicole have at the start? 2. How many quarter hours are in 5 hours? Lesson 2: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting Division of a Whole Number by a Fraction Visual Models 9/16/13 S.10

11 Lesson 3 Lesson 3: Interpreting and Computing Division of a Fraction by a Fraction More Models Classwork Opening Exercise Draw a model to represent How could we reword this question? Example Draw a model to show the division problem. Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.11

12 Lesson 3 Example Be sure to draw a model to support your answer. Example Be sure to create a model to support your answer. Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.12

13 Lesson 3 Exercises 1 6 For the following exercises, rewrite the division problem. Then be sure to draw a model to support your answer. 1. How many fourths are in three fourths? Draw a model to support your answer. How are Example 2 and Exercise 1 similar? How are the divisors and dividends related? What conclusions can you draw from these observations? Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.13

14 Lesson Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.14

15 Lesson Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.15

16 Lesson 3 Problem Set For the following exercises, rewrite the division problem. Then be sure to draw a model to support your answer Lesson 3: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/9/13 S.16

17 Lesson 4 Lesson 4: Interpreting and Computing Division of a Fraction by a Fraction More Models Classwork Opening Exercise Write at least three equivalent fractions for each fraction below. Be sure to show how the two fractions are related. a. 2 3 b Example 1 Molly purchased 11 8 cups of strawberries. She eats 2 in a serving. How many servings did Molly purchase? 8 Use a model to prove your answer. Lesson 4: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/8/13 S.17

18 Lesson 4 Example 2 Now imagine that Molly s friend Xavier purchased 11 8 cups of strawberries and that he eats 3 cup servings. How many 4 servings has he purchased? Use a model to prove your answer. Example 3 Find the quotient: 3 2. Use a model to show your answer. 4 3 Lesson 4: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/8/13 S.18

19 Lesson 4 Exercises 1 5 A model should be included in your solution Lesson 4: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/8/13 S.19

20 Lesson Lesson 4: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/8/13 S.20

21 Lesson 4 Problem Set Lesson 4: Date: 2013 Common Core, Inc. All rights reserv ed. commoncore.org Interpreting and Computing Division of a Fraction by a Fraction More Models 9/8/13 S.21

22 Lesson 5 Lesson 5: Creating Division Stories Classwork Opening Exercises Fraction bar: Number Line: Molly s friend Xavier purchased 11 8 cups of strawberries, and he eats 3 cup servings. How many servings has he 4 purchased? Area Model: Lesson 5: Creating Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

23 Lesson 5 Example Step 1: Decide on an interpretation. Step 2: Draw a model. Step 3: Find the answer. Step 4: Choose a unit. Step 5: Set up a situation. Lesson 5: Creating Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

24 Lesson 5 Exercise 1 Using the same dividend and divisor, work with a partner to create your own story problem. You may use the same unit, but your situation must be unique. You could try another unit such as ounces, yards, or miles if you prefer. Example Step 1: Decide on an interpretation. Step 2: Draw a diagram. Lesson 5: Creating Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

25 Lesson 5 Step 3: Find the answer. Step 4: Choose a unit. Step 5: Set up a situation. Exercise 2 Using the same dividend and divisor, work with a partner to create your own story problem. You may use the same unit, but your situation must be unique. You could try another unit such as ounces, yards, or miles if you prefer. Lesson 5: Creating Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

26 Lesson 5 Lesson Summary The method of creating division stories has five steps, to be followed in order: Step 1: Decide on an interpretation (measurement or partitive). Today we used only measurement division. Step 2: Draw a model. Step 3: Find the answer. Step 4: Choose a unit. Step 5: Set up a situation. This means writing a story problem that is interesting, realistic, short, and clear and that has all the information necessary to solve it. It may take you several attempts before you find a story that works well with the given dividend and divisor. Problem Set Please use each of the five steps of the process you learned. Label each step. 1. Write a measurement division story problem for Write a measurement division story problem for Lesson 5: Creating Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

27 Lesson 6 Lesson 6: More Division Stories Classwork Example 1 Divide Step 1: Decide on an interpretation. Step 2: Draw a model. Step 3: Find the answer. Step 4: Choose a unit. Step 5: Set up a situation. Lesson 6: More Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

28 Lesson 6 Exercise 1 Using the same dividend and divisor, work with a partner to create your own story problem. You may use the same unit, dollars, but your situation must be unique. You could try another unit, such as miles, if you prefer. Example 2 Divide Step 1: Decide on an interpretation. Step 2: Draw a model. Lesson 6: More Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

29 Lesson 6 Step 3: Find the answer. Step 4: Choose a unit. Step 5: Set up a situation. Exercise 2 Using the same dividend and divisor, work with a partner to create your own story problem. Try a different unit. Remember spending money gives a before and after word problem. If you use dollars, you are looking for a situation where 3 of some greater dollar amount is \$45. 8 Lesson 6: More Division Stories Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

30 Lesson 6 Problem Set 1. Write a partitive division story problem for Write a partitive division story problem for Lesson 6: Date: More Division Stories 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

31 Lesson 7 Lesson 7: The Relationship Between Visual Fraction Models and Equations Classwork Example Shade 2 of the 5 sections 2 5. Label the part that is known 3 4. Make notes below on the math sentences needed to solve the problem. Lesson 7: The Relationship Between Visual Fraction Models and Equations Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.

32 Lesson 7 Exercise Show the number sentences below. Exercise Show the number sentences below. Lesson 7: The Relationship Between Visual Fraction Models and Equations Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

33 Lesson 7 Lesson Summary Connecting models of fraction division to multiplication through the use of reciprocals helps in understanding the invert and multiply rule. Problem Set 1. Draw a model that shows Find the answer as well Draw a model that shows Find the answer as well. 2 Lesson 7: The Relationship Between Visual Fraction Models and Equations Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

34 Lesson 8 Lesson 8: Dividing Fractions and Mixed Numbers Classwork Example 1: Introduction to Calculating the Quotient of a Mixed Number and a Fraction a. Carli has 4 1 walls left to paint in order for all the bedrooms in her house to have the same color paint. However, 2 she has used almost all of her paint and only has 5 of a gallon left. How much paint can she use on each wall in 6 order to have enough to paint the remaining walls? b. Calculate the quotient Lesson 8: Dividing Fractions and Mixed Numbers Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

35 Lesson 8 Exercise 1 Show your work for the memory game in the boxes provided below. A. B. C. D. E. F. G. H. I. J. K. L. Lesson 8: Date: Dividing Fractions and Mixed Numbers 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

36 Lesson 8 Problem Set Calculate each quotient Lesson 8: Dividing Fractions and Mixed Numbers Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

37 Lesson 9 Lesson 9: Sums and Differences of Decimals Classwork Example Example Exercises 1 5 Calculate each sum or difference. 1. Samantha and her friends are going on a road trip that is How much further do they have to drive? 50 miles long. They have already driven Lesson 9: Sums and Differences of Decimals Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

38 Lesson 9 2. Ben needs to replace two sides of his fence. One side is meters long, and the other is meters long. 10 How much fence does Ben need to buy? 3. Mike wants to paint his new office with two different colors. If he needs gallons of red paint and 3 1 gallons of 10 brown paint, how much paint does he need in total? 4. After Arianna completed some work, she figured she still had pictures to paint. If she completed another pictures, how many pictures does Arianna still have to paint? 25 Use a calculator to convert the fractions into decimals before calculating the sum or difference. 5. Rahzel wants to determine how much gasoline he and his wife use in a month. He calculated that he used gallons of gas last month. Rahzel s wife used 41 3 gallons of gas last month. How much total gas did Rahzel 8 and his wife use last month? Round your answer to the nearest hundredth. Lesson 9: Sums and Differences of Decimals Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

39 Lesson 9 Problem Set 1. Find each sum or difference. a b c d e Use a calculator to find each sum or difference. Round your answer to the nearest hundredth. a b Lesson 9: Sums and Differences of Decimals Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

40 Lesson 10 Lesson 10: The Distributive Property and the Product of Decimals Classwork Opening Exercise Calculate the product Example 1: Introduction to Partial Products Use partial products and the distributive property to calculate the product Example 2: Introduction to Partial Products Use partial products and the distributive property to calculate the area of the rectangular patio shown below ft 500 ft Lesson 10: Date: The Distributive Property and the Product of Decimals 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

41 Lesson 10 Exercise Use the boxes below to show your work for each station. Make sure you are putting the solution for each station in the correct box. Station One: Station Two: Station Three: Station Four: Station Five: Lesson 10: The Distributive Property and the Product of Decimals Date: 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

42 Lesson 10 Problem Set Calculate the product using partial products Lesson 10: Date: The Distributive Property and the Product of Decimals 9/9/13 S Common Core, Inc. All rights reserve d. commoncore.org

43 Lesson 11 Lesson 11: Fraction Multiplication and the Product of Decimals Classwork Exploratory Challenge You will not only solve each problem, but your groups will also need to prove to the class that the decimal in the product is located in the correct place. As a group, you will be expected to present your informal proof to the class. 1. Calculate the product Xavier earns \$11.50 per hour working at the nearby grocery store. Last week, Xavier worked for 13.5 hours. How much money did Xavier earn last week? Remember to round to the nearest penny. Discussion Record notes from the discussion in the box below. Lesson 11: Fraction Multiplication and the Product of Decimals Date: 9/16/13 S Common Core, Inc. All rights reserve d. commoncore.org

44 Lesson 11 Exercises Calculate the product Kevin spends \$11.25 on lunch every week during the school year. If there are 35.5 weeks during the school year, how much does Kevin spend on lunch over the entire school year? Remember to round to the nearest penny. 3. Gunnar s car gets 22.4 miles per gallon, and his gas tank can hold gallons of gas. How many miles can Gunnar travel if he uses all of the gas in the gas tank? 4. The principal of East High School wants to buy a new cover for the sand pit used in the long jump competition. He measured the sand pit and found that the length is 29.2 feet and the width is 9.8 feet. What will the area of the new cover be? Lesson 11: Fraction Multiplication and the Product of Decimals Date: 9/16/13 S Common Core, Inc. All rights reserve d. commoncore.org

45 Lesson 11 Problem Set Solve each problem. Remember to round to the nearest penny when necessary. 1. Calculate the product: Deprina buys a large cup of coffee for \$4.70 on her way to work every day. If there are 24 works days in the month, how much does Deprina spend on coffee throughout the entire month? 3. Krego earns \$2, every month. He also earns an extra \$4.75 every time he sells a new gym membership. Last month, Krego sold 32 new gym memberships. How much money did Krego earn last month? 4. Kendra just bought a new house and needs to buy new sod for her backyard. If the dimensions of her yard are 24.6 feet by 14.8 feet, what is the area of her yard? Lesson 11: Date: Fraction Multiplication and the Product of Decimals 9/16/13 S Common Core, Inc. All rights reserve d. commoncore.org

46 Lesson 12 Lesson 12: Estimating Digits in a Quotient Classwork Opening Exercise Show an example of how you would solve 5, You can use any method or model to show your work. Just be sure that you can explain how you got to your solution. Example 1 We can also use estimates before we divide to help us solve division problems. In this lesson we will be using estimation to help us divide two numbers using the division algorithm. Estimate the quotient of 8, Then divide. Create a model to show the division of 8,085 and 33. Lesson 12: Estimating Digits in a Quotient Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

47 Lesson 12 Example 2 Use estimation and the standard algorithm to divide. 1, Exercises 1. 1, a. Estimate the quotient. b. Use the algorithm to divide. Draw a model to show how the steps relate to the steps used in the algorithm. c. Check your work. Lesson 12: Estimating Digits in a Quotient Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

48 Lesson , a. Estimate the quotient. b. Use the algorithm to divide. Draw a model to show how the steps relate to the steps used in the algorithm. c. Check your work. 3. 2, a. Estimate the quotient. b. Use the algorithm to divide. c. Check your work. Lesson 12: Estimating Digits in a Quotient Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

49 Lesson , a. Estimate the quotient. b. Use the algorithm to divide. c. Check your work. Lesson 12: Estimating Digits in a Quotient Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

50 Lesson 12 Problem Set Complete the following steps for each problem: a. Estimate the quotient. b. Use the division algorithm to solve. c. Show a model that supports your work with the division algorithm. d. Be sure to check your work. 1. 3, , , Lesson 12: Date: Estimating Digits in a Quotient 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

51 Lesson 13 Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm Classwork Example 1 a. Create a model to divide: 1, b. Use the division algorithm to show 1, c. Check your work Example 2 Find the quotient of 205, Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

52 Lesson 13 Example 3 Find the quotient of 17,216, Exercises For each question you need to do the following: a. Solve the question. Next to each line explain your work using place value. b. Evaluate the reasonableness of your answer , , Lesson 13: Date: Dividing Multi-Digit Numbers Using the Algorithm 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

53 Lesson , ,295, , ,823, Lesson 13: Dividing Multi-Digit Numbers Using the Algorithm Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

54 Lesson 13 Problem Set , , ,183, Lesson 13: Date: Dividing Multi-Digit Numbers Using the Algorithm 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

55 Lesson 14 Lesson 14: The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions Classwork Example 1 Divide 31, Example 2 Divide Lesson 14: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions 9/17/13 S.55

56 Lesson 14 Example 3 A plane travels 3, miles in 6.9 hours. What is the plane s unit rate? Exercises Estimate the quotient first. Use the estimate to justify the reasonableness of your answer. 1. Daryl spent \$4.68 on each pound of trail mix. He spent a total of \$ How many pounds of trail mix did he purchase? 2. Kareem purchased several packs of gum to place in gift baskets for \$1.26 each. He spent a total of \$8.82. How many packs of gum did he buy? Lesson 14: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions 9/17/13 S.56

57 Lesson Jerod is making candles from beeswax. He has ounces of beeswax. If each candle uses 8.4 ounces of beeswax, how many candles can he make? Will there be any wax left over? 4. There are 20.5 cups of batter in the bowl. If each cupcake uses 0.4 cups of batter, how many cupcakes can be made? 5. In Exercises 3 and 4, how were the remainders, or extra parts, interpreted? Lesson 14: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions 9/17/13 S.57

58 Lesson Lesson 14: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions 9/17/13 S.58

59 Lesson 14 Problem Set 1. Asian purchased 3.5 lbs. of his favorite mixture of dried fruits to use in a trail mix. The total cost was \$ How much did the fruit cost per pound? 2. Divide: Lesson 14: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Division Algorithm Converting Decimal Division into Whole Number Division Using Fractions 9/17/13 S.59

60 Lesson 15 Lesson 15: The Division Algorithm Converting Decimal Division to Whole Number Division Using Mental Math Classwork Opening Exercise Start by finding the quotient of 1,728 and 32. What would happen if we multiplied the divisor by 10? What would happen if we multiplied the dividend by 10? Lesson 15: The Division Algorithm Converting Decimal Division to Whole Number Division Using Mental Math Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

61 Lesson 15 What would happen if we multiplied both the divisor and dividend by 10? What would happen if we multiplied both the divisor and dividend by 100? What would happen if we multiplied both the divisor and the dividend by 1,000, 10,000 or 100,000? What do you predict will happen? How can we use this to help us divide when there are decimals in the divisor? How can we use this to help us divide and 3.2? Lesson 15: The Division Algorithm Converting Decimal Division to Whole Number Division Using Mental Math Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

62 Lesson 15 Example 1 Using our discoveries from the discussion, let s divide by 8.2. How can we rewrite this problem using what we learned in Lesson 14? How could we use the short cut from our discussion to change the original numbers to 5,271 and 82? Example 2 Now, let s divide by How can we rewrite this division problem so that the divisor is a whole number and the quotient remains the same? Exercises Students will participate in a game called pass the paper. Students will work in groups of no more than four. There will be a different paper for each player. When the game starts, each student solves the first problem on his paper then passes the paper clockwise to the second person who uses multiplication to check the work that was done by the previous student. Then the paper is passed clockwise again to the third person who solves the second problem. The paper is then passed to the fourth person who checks the second problem. This process continues until all of the questions on every paper are complete or time runs out. Lesson 15: The Division Algorithm Converting Decimal Division to Whole Number Division Using Mental Math Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

63 Lesson 15 Problem Set , Lesson 15: Date: The Division Algorithm Converting Decimal Division to Whole Number Division Using Mental Math 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

64 Lesson 16 Lesson 16: Even and Odd Numbers Classwork Opening Exercise What is an even number? List some examples of even numbers. What is an odd number? List some examples of odd numbers. What happens when we add two even numbers? Will we always get an even number? Lesson 16: Even and Odd Numbers Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

65 Lesson 16 Exercises 1. Why is the sum of two even numbers even? a. Think of the problem Draw dots to represent each number. b. Circle pairs of dots to determine if any of the dots are left over. c. Will this be true every time two even numbers are added together? Why or why not? 2. Why is the sum of two odd numbers even? a. Think of the problem Draw dots to represent each number. b. Circle pairs of dots to determine if any of the dots are left over. c. Will this be true every time two odd numbers are added together? Why or why not? Lesson 16: Even and Odd Numbers Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

66 Lesson Why is the sum of an even number and an odd number odd? a. Think of the problem Draw dots to represent each number. b. Circle pairs of dots to determine if any of the dots are left over. c. Will this be true every tie an even number and an odd number are added together? Why or why not? d. What if the first addend was odd and the second was even. Would the sum still be odd? Why or why not? For example, if we had , would the sum be odd? Let s sum it up: Lesson 16: Even and Odd Numbers Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

67 Lesson 16 Exploratory Challenge Each group will determine on its poster paper whether one of the following products is odd or even: The product of two even numbers, the product of two odd numbers, or the product of an even number and an odd number. Use previous knowledge about even and odd numbers, the connection between addition and multiplication, and visual methods (e.g., dots) in your proofs. 1. The product of two even numbers is even. 2. The product of two odd numbers is odd. 3. The product of an even number and an odd number is even. Lesson 16: Even and Odd Numbers Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

68 Lesson 16 Lesson Summary Adding: The sum of two even numbers is even. The sum of two odd numbers is odd. The sum of an even number and an odd number is odd. Multiplying: The product of two even numbers is even. The product of two odd numbers is odd. The product of an even number and an odd number is even. Problem Set Tell whether each sum or product is even or odd. Explain your reasoning , ,462, , , ,481, Lesson 16: Date: Even and Odd Numbers 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

69 Lesson 17 Lesson 17: Divisibility Tests for 3 and 9 Classwork Opening Exercise Below is a list of 10 numbers. Place each number in the circle(s) that is a factor of the number. You will place some numbers more than once. For example if 32 were on the list, you would place it in the circles with 2, 4, and 8 because they are all factors of ; 36; 80; 115; 214; 360; 975; 4,678; 29,785; 414, Lesson 17: Divisibility Tests for 3 and 9 Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

70 Lesson 17 Discussion Divisibility rule for 2: Divisibility rule for 4: Divisibility rules for 5: Divisibility rule for 8: Divisibility rule for 10: Example 1 This example will show you how to apply the two new divisibility rules we just discussed. Is 378 divisible by 3 or 9? Why or why not? a. What are the three digits in the number 378? b. What is the sum of the three digits? c. Is 18 divisible by 9? d. Is the entire number 378 divisible by 9? Why or why not? e. Is the number 378 divisible by 3? Why or why not? Lesson 17: Divisibility Tests for 3 and 9 Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

71 Lesson 17 Example 2 Is 3,822 divisible by 3 or 9? Why or why not? Exercises Circle ALL the numbers that are factors of the given number. Complete any necessary work in the space provided. 1. Is 2,838 divisible by Explain your reasoning for your choices. 2. Is 34,515 divisible by Explain your reasoning for your choices. Lesson 17: Divisibility Tests for 3 and 9 Date: 9/16/13 S Common Core, Inc. Some rights reserved. commoncore.org

72 Lesson Is 10,534,341 divisible by Explain your reasoning for your choices. 4. Is 4,320 divisible by Explain your reasoning for your choices. 5. Is 6,240 divisible by Explain your reasoning for your choices. Lesson 17: Divisibility Tests for 3 and 9 Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

73 Lesson 17 Lesson Summary To determine if a number is divisible by 3 or 9: Calculate the sum of the digits. If the sum of the digits is divisible by 3, the entire number is divisible by 3. If the sum of the digits is divisible by 9, the entire number is divisible by 9. Note: If a number is divisible by 9, the number is also divisible by 3. Problem Set 1. Is 32,643 divisible by both 3 and 9? Why or why not? 2. Circle all the factors of 424,380 from the list below Circle all the factors of 322,875 from the list below Write a 3 digit number that is divisible by both 3 and 4. Explain how you know this number is divisible by 3 and Write a 4 digit number that is divisible by both 5 and 9. Explain how you know this number is divisible by 5 and 9. Lesson 17: Divisibility Tests for 3 and 9 Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

74 Lesson 18 Lesson 18: Least Common Multiple and Greatest Common Factor Classwork The Greatest Common Factor of two whole numbers a and b, written GCF (a, b), is the greatest whole number, which is a factor of both a and b. The Least Common Multiple of two nonzero numbers a and b, written LCM (a, b), is the least whole number (larger than zero), which is a multiple of both a and b. Example 1 Greatest Common Factor: Find the greatest common factor of 12 and 18. GCF 12, 18 Listing these factors as pairs can help you not miss any. Start with one times the number. Circle all factors that appear on both lists. Place a triangle around the greatest of these common factors Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

75 Lesson 18 Example 2 (5 minutes) Least Common Multiple: Find the least common multiple of 12 and 18. LCM 12, 18 Write the first 10 multiples of 12. Write the first 10 multiples of 18. Circle the multiples that appear on both lists. Put a triangle around the least of these common multiples. Exploratory Challenge 1: Factors and GCF Choose one of these problems that have not yet been solved. Solve it together on your student page. Then use your marker to copy your work neatly on the chart paper. Use your marker to cross out your choice so the next group solves a different problem. GCF 30, 50 GCF 30, 45 GCF 45, 60 GCF 42, 70 GCF 96, 144 Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

76 Lesson 18 Next, choose one of these problems that have not yet been solved: a. There are 18 girls and 24 boys who want to participate in a Trivia Challenge. If each team must have the same number of girls and boys, what is the greatest number of teams that can enter? How many boys and girls will be on each team? b. The Ski Club members are preparing identical welcome kits for the new skiers. They have 60 hand warmer packets and 48 foot warmer packets. What is the greatest number of kits they can prepare using all of the hand warmer and foot warmer packets? c. There are 435 representatives and 100 senators serving in the United States Congress. How many identical groups with the same numbers of representative and senators could be formed from all of Congress, if we want the largest groups possible?? d. Is the GCF of a pair of numbers ever equal to one of the numbers? Explain with an example. e. Is the GCF of a pair of numbers ever greater than both numbers? Explain with an example. Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

78 Lesson 18 d. Is the LCM of a pair of numbers ever equal to one of the numbers? Explain with an example. e. Is the LCM of a pair of numbers ever less than both numbers? Explain with an example. Exploratory Challenge 3: Using Prime Factors to Determine GCF Choose one of these problems that have not yet been solved. Solve it together on your student page. Then use your marker to copy your work neatly on the chart paper. Use your marker to cross out your choice so the next group solves a different problem. GCF 30, 50 GCF 30, 45 GCF 45, 60 GCF 42, 70 GCF 96, 144 Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

79 Lesson 18 Next, choose one of these problems that has not yet been solved: a. Would you rather find all the factors of a number or find all the prime factors of a number? Why? b. Find the GCF of your original pair of numbers. c. Is the product of your LCM and GCF less than, greater than, or equal to the product of your numbers? d. Glenn s favorite number is very special because it reminds him of the day his daughter, Sarah, was born. The factors of this number do not repeat and are all the prime numbers less than 12. What is Glenn s number? When was Sarah born? Exploratory Challenge 4: Applying Factors to the Distributive Property Choose one of these problems that have not yet been solved. Solve it together on your student page. Then use your marker to copy your work neatly on the chart paper. Use your marker to cross out your choice so the next group solves a different problem. Find the GCF from the two numbers and rewrite the sum using the Distributive Property = = = = = Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. Some rights reserved. commoncore.org

80 Lesson 18 Next, add another new example to one of these two statements applying factors to the Distributive Property. Choose any numbers for n, a, and b. n(a) + n(b) = n(a + b) n(a) n(b) = n(a b) Problem Set Complete the remaining exploratory challenge problems from class. Lesson 18: Least Common Multiple and Greatest Common Factor Date: 9/16/13 S Common Core, Inc. All rights reserved. commoncore.org

81 Lesson 19 Lesson 19: The Euclidean Algorithm as an Application of the Long Division Algorithm Classwork Opening Exercise Euclid s Algorithm is used to find the greatest common factor (GCF) of two whole numbers. 1. Divide the larger of the two numbers by the smaller one. 2. If there is a remainder, divide it into the divisor. 3. Continue dividing the last divisor by the last remainder until the remainder is zero. 4. The final divisor is the GCF of the original pair of numbers = = = Example 1: Euclid s Algorithm Conceptualized 100 units 20 units 60 units 20 units Lesson 19: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Euclidean Algorithm as an Application of the Long Division Algorithm 9/17/13 S.81

82 Lesson 19 Example 2: Lesson 18 Classwork Revisited a. Let s apply Euclid s Algorithm to some of the problems from our last lesson. i. What is the GCF of 30 and 50? ii. Using Euclid s Algorithm, we follow the steps that are listed in the opening exercise. b. What is the GCF of 30 and 45? Example 3: Larger Numbers GCF (96, 144) GCF (660, 840) Lesson 19: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Euclidean Algorithm as an Application of the Long Division Algorithm 9/17/13 S.82

83 Lesson 19 Example 4: Area Problems The greatest common factor has many uses. Among them, the GCF lets us find out the maximum size of squares that will cover a rectangle. Whenever we solve problems like this, we cannot have any gaps or any overlapping squares. Of course, the maximum size squares will be the minimum number of squares needed. A rectangular computer table measures 30 inches by 50 inches. We need to cover it with square tiles. What is the side length of the largest square tile we can use to completely cover the table, so that there is no overlap or gaps? c. If we use squares that are 10 by 10, how many will we need? d. If this were a giant chunk of cheese in a factory, would it change the thinking or the calculations we just did? e. How many 10 inch 10 inch squares of cheese could be cut from the giant 30 inch 50 inch slab? Lesson 19: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Euclidean Algorithm as an Application of the Long Division Algorithm 9/17/13 S.83

84 Lesson 19 Problem Set 1. Use Euclid s Algorithm to find the greatest common factor of the following pairs of numbers: a. GCF of 12 and 78 b. GCF of 18 and Juanita and Samuel are planning a pizza party. They order a rectangular sheet pizza which measures 21 inches by 36 inches. They tell the pizza maker not to cut it because they want to cut it themselves. a. All pieces of pizza must be square with none left over. What is the length of the side of the largest square pieces into which Juanita and Samuel can cut the pizza? b. How many pieces of this size will there be? 3. Shelly and Mickelle are making a quilt. They have a piece of fabric that measures 48 inches by 168 inches. a. All pieces of fabric must be square with none left over. What is the length of the side of the largest square pieces into which Shelly and Mickelle can cut the fabric? b. How many pieces of this size will there be? Lesson 19: Date: 2013 Common Core, Inc. All rights reserved. commoncore.org The Euclidean Algorithm as an Application of the Long Division Algorithm 9/17/13 S.84

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