Investigate Multiply Using the Distributive Property

Transcription

1 LESSON 2. Investigate Multiply Using the Distributive Property FOCUS COHERENCE RIGOR LESSON AT A GLANCE F C R Focus: Common Core State Standards.NBT.B. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models. MATHEMATICAL PRACTICES MP1 Make sense of problems and persevere in solving them. MP7 Look for and make use of structure. F C R Coherence: Standards Across the Grades Before Grade After.OA.B..NBT.B..NBT.B. F C R Rigor: Level 1: Understand Concepts...Share and Show ( Checked Items) Level 2: Procedural Skills and Fluency...On Your Own Level : Applications...Think Smarter and Go Deeper Learning Objective Use the Distributive Property to multiply a 2-digit number by a 1-digit number. Language Objective Student pairs discuss how to use the Distributive Property to multiply a two-digit number by a one-digit number. Materials MathBoard, grid paper (see eteacher Resources), color pencils, base-ten blocks F C R For more about how GO Math! fosters Coherence within the Content Standards and Mathematical Progressions for this chapter, see page 1J. About the Math Professional Development Using a Rectangular or Area Model The Distributive Property is the underlying concept for multiplication of greater numbers. The Distributive Property is often illustrated by an algebraic equation. A model helps us visualize how the property works. Students learn to draw a model for multiplying a 2-digit number by a 1-digit number. They label the model with the partial products. They can see how multiplication is distributed over addition. In an upcoming lesson, students connect the model to the algorithm as they record the partial products and add them to find the product. Professional Development Videos 87A Chapter 2 a (b + c) = (a b) + (a c) a b c a b a c = ( ) + ( ) = 0 + = 9

2 Daily Routines Common Core Problem of the Day 2. Patrick has a stamp collection. He has stamps from 9 different countries. He has 17 stamps for each different country. Write an expression that would give an estimate of how many stamps Patrick has Vocabulary Distributive Property, partial product Interactive Student Edition Multimedia Glossary e Fluency Builder Write the following problems on the board. Have students practice basic division facts by writing quotients as fast as possible. 18 = 2 = 20 = = 9 9 = 2 = 8 8 = = 8 = = 1 = 0 = 1 ENGAGE with the Interactive Student Edition Essential Question How can you use the Distributive Property to multiply a 2-digit number by a 1-digit number? Making Connections Invite students to tell you what they know about parking meters. What is a parking meter? a machine that takes money so that you can park in a given spot How do parking meters work? You put coins in and the meter is set to allow you to park in the space legally for a certain amount of time. Learning Activity Connect the story to the problem. What are you trying to find? the number of minutes Shell s neighbor is parked at the diner in total over nights How long does Shell s neighbor park at the diner each night? 19 minutes How many nights per week does Shell s neighbor go to the diner? What expression will help you find the answer? 19 Literacy and Mathematics Choose one or more of the following activities. Ask students if they have ever heard the expression feed the meter. Have a volunteer who knows what it means share with the class; if no one knows, explain that it means to put coins or money into the parking meter so that it doesn t expire. Have students explain to a partner how they might solve the problem using addition. = = = 2 = How can you use the Distributive Property to multiply a 2-digit number by a 1-digit number? 87B

3 LESSON 2..NBT.B. Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, DO NOT EDIT--Changes made File info using strategies based onmust placebevalue andthrough the properties of operations. Illustrate and explain the calculation by using equations, CorrectionKey=B rectangular arrays, and/or area models. 2 EXPLORE Investigate Hands On Multiply Using the Distributive Property Materials color pencils, grid paper Use visuals to help students make a connection between the collocations they encounter and the concepts they represent. Collocations are expressions with more than one word that function as a single meaning. Group students into teams of two or three. Say, Some words work with other words to create a new meaning. Write on the board: break apart, hand in, put away, talk over. Have students draw pictures of these four collocations to demonstrate understanding. 87 Chapter 2 A. Outline a rectangle on the grid to model 1. B. Think of 1 as + 8. Break apart the model to show ( + 8). Label and shade the smaller rectangles. Use two different colors. Answers will vary. Possible answer is shown. Use the Distributive Property. Find the product each smaller rectangle represents. Then find the sum of the products. Record your answers. Answers will vary. Possible answer is shown. 0 = 8 = 8 8 = C. Model 1 again. Think of 1 as a different sum. Break apart the model to show ( + ). Find the product each smaller rectangle represents. Then find the sum of the products. Record your answers. Answers will vary. Possible answer is shown. 9 = = = 78 Chapter 2 87 Meeting Individual Needs MNLESE219C02L0.indd 87 Identify Relationships 8 The Distributive Property states that multiplying a sum by a number is the same as multiplying each addend by the number and then adding the products. 7. Differentiated Instruction You can use the Distributive Property to break apart numbers to make them easier to multiply. C G ive students an opportunity to share the different ways they broke apart 1. Describe the number of rows and columns in each of your rectangles. I used and ELL Strategy: Hands On Investigate Investigate break apart 1 into lesser numbers and use basic facts to multiply. Share with students that they have just used the Distributive Property. Read the definition of the Distributive Property again with students. MP Attend to precision. Have students restate the Distributive Property in their own words and explain how using the rectangle model demonstrates using the property. Students should recognize that the vertical line they drew broke the larger rectangle into two smaller rectangles, each representing a product. The sum of the products of both rectangles is the product of 1. Number and Operations in Base Ten.NBT.B. MATHEMATICAL PRACTICES MP, MP, MP Essential Question How can you use the Distributive Property to multiply a 2-digit number by a 1-digit number? MP Model with mathematics. Students will use grid paper and draw a rectangular model to represent 1. When the squares of an array are pushed together, it is sometimes referred to as an area model. A A rectangle can be drawn that is units down and 1 units across. B Students should draw a vertical line on the model to separate rows of 1 into rows of and rows of 8. How does breaking apart the model help you multiply 1? Possible answer: I can Reteach 2. 2 DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A 1 Reteach Enrich Multiply Using the Distributive Property Shading the Grids Use the Distributive Property. Shade and label each grid. You can use rectangular models to multiply 2-digit numbers by 1-digit numbers. Find 9 1. Differentiated Instruction Enrich 2. DO NOT EDIT--Changes must be made through "File info" CorrectionKey=A /7/1 7:21 PM Models will vary. Possible models are given Show 28 in two different ways Step 1 Draw a 9 by 1 rectangle on grid paper. Step 2 Use the Distributive Property and products you know to break apart the model into two smaller rectangles. Think: 1 1. Step Find the product each smaller rectangle represents Show 2 in two different ways Step Find the sum of the products Possible models are shown MNLEAN078C02R0.indd Stretch Your Thinking Find the partial products for one of your grids in Exercise 1. Then use the Distributive Property to find the product 28. Chapter Resources 1 So, Model the product on the grid. Record the product Possible answer: 20 0; 8 2; using the Distributive Property, 28 ( 20) 1 ( 8); so, , or 8. Reteach Chapter Resources 2/12/1 2:22 PM MNLEAN078C02E0.indd Enrich 2/12/1 2:21 PM

4 Draw Conclusions 1. Explain how you found the total number of squares in each model in Steps B and C. 2. Compare the sums of the products in Steps B and C with those of your classmates. What can you conclude?. To find 7 2, is it easier to break apart the factor, 2, as 20 + or 1 + 8? Explain. Another way to model the problem is to use base-ten blocks to show tens and ones. STEP 1 Possible answer: I divided the first model into a rectangle and a 8 rectangle, and added: = 78. Answers will vary for Step C. Possible answer: They are the same. This shows that you will get the same answer no matter how you break apart the rectangle. Possible explanation: 20 +, because I can use mental math to multiply by multiples of. Make Connections Use base-ten blocks to model 1. STEP 2 Hands On Break the model into tens and ones. Math Talk: Possible answer: I can break apart the model into tens and ones, and then use mental math to find the product. STEP Add the tens and the ones to find the product. Draw Conclusions Exercise 1 Help students connect the models they drew to the symbolic form of the Distributive Property. For the first model: 1 = ( + 8) = ( ) + ( 8) = = 78 Use Exercise to help students recognize that breaking apart a factor so that they can use mental math to multiply is more efficient. Make Connections Students are connecting the area model they draw on grid paper to an array model using base-ten blocks. A benefit of using base-ten blocks is that the greater factor will always be broken into and another addend. How does Step show the Distributive Property? Possible answer: it shows that multiplication was distributed over addition. Why do you think the word partial is used in the phrase partial products? Possible answer: the product of each part is only part of the whole product. rows of 1 ten ones So, 1 = 78. ( 1 ten) ( ones) ( ) ( ) In Step 2, the model is broken into two parts. Each part shows a partial product. The partial products are 0 and 18. See above. 88 Advanced Learners Math Talk ( ) + ( ) MATHEMATICAL PRACTICES Model Mathematics Why is this a good model for the problem? Visual Individual Materials 1-Centimeter Grid Paper (see eteacher Resources) What is the equation shown by the orange section of the model? = 0 What is the equation shown by the green section of the model? 7 = 28 Use the Distributive Property to write the equation represented by the whole model. ( ) + ( 7) = ( + 7) = 17 = 8 DEEPER MP Model with mathematics. To extend students thinking, discuss why it is helpful to use parentheses in Steps 2 and. Give students the opportunity to explain how the problem is clarified using parentheses. Math Talk Use MathTalk to focus on students understanding of breaking apart a factor to make finding the product easier. Is multiplying by easier than multiplying by a number from 2 to 9? Explain. Yes; Possible explanation: you don t need to recall a basic fact when you multiply by. You just write a zero after. COMMON ERRORS Error Students may add a number rather than the products. Example 1 = ( ) + rather than ( ) + ( ) Springboard to Learning Explain that they are breaking the original multiplication problem into two products. They must have both products to add for the total. 88

5 DO NOT EDIT--Changes must be made through File info CorrectionKey=A EXPLAIN Share and Show Share Share and and Show Show BOARD The first problem connects to the learning model. Have students use the MathBoard to explain their thinking. Use the checked exercises for Quick Check. Students should show their answers for the Quick Check on the MathBoard. Quick Check Check Quick a student misses the checked exercises Then Differentiate Instruction with Reteach 2. Personal Math Trainer.NBT.B. RtI Tier 1 Activity (online) Find the product = Rt I If If BOARD Model the product on the grid. Record the product. Models will vary. Possible models are given = 2. 1 = Hands On MATH MATH =. 1 = Use grid paper or base-ten blocks to model the product. Then record the product. Check students models = = = Problem Problem Solving Solving Applications Applications 9. MATHEMATICAL Explain how modeling partial products can be used PRACTICE to find the products of greater numbers. Possible explanation: using base-ten blocks, you can skip count by tens or multiply by multiples of to find the number of tens and multiply basic facts to find the number of MP Construct viable arguments and critique the reasoning of others. To extend students thinking, ask students to describe the similarities and differences between using grid paper and using base-ten blocks. They may wish to share which method they prefer and why. ELABORATE Problem Solving Thinking Applications MATHEMATICAL PRACTICES MP Attend to precision. Exercise 9 provides an opportunity for students to explain what they have been demonstrating with models. Exercise assesses students ability to multiply and model the Distributive Property. Check that student diagrams correctly reflect the Distributive Property. If diagrams are correct and students still have the wrong answer, check for regrouping errors. 89 Chapter 2 ones. Then, add the number of tens and ones to find the product. DEEPER. Use the Distributive Property to model the product on the grid. Record the product. 1 = Chapter 2 Lesson MNLESE219C02L0.indd /12/1 1:07 AM

6 MATHEMATICAL PRACTICES 11. COMMUNICA& t 1&E7&& t CONSTRUCT ARGUMENTS Kyle went to a fruit market. The market sells a wide variety of fruits and vegetables. The picture at the right shows a display of oranges. After students read the situation, discuss the information presented in the diagram. Students should identify that each of the rows has 12 oranges. They should also recognize that 2 rows are empty. Give students an opportunity to share their problems and solutions with the class. Write a problem that can be solved using the picture. Possible answers shown. Pose a problem. Solve your problem. One display can hold 8 rows of = 9 oranges. If 2 rows in the display are 2 12 = 2 empty, how many oranges are shown in 9 2 = 72 the display? or Math on the Spot Video Tutor Use this video to help students model and solve this type of Think Smarter problem. 8 rows 2 rows = rows Math on the Spot videos are in the Interactive Student Edition and at rows of 12 oranges = 72 oranges DEEPER Students should find that changing the information given in a problem will change the result. DEEPER Describe how you could change the problem by changing the number of rows of oranges and the number of empty spaces in the picture. Then solve the problem. Possible answer: one display can hold 9 rows of 12 oranges. If rows in the display are empty, how many oranges are shown in the display? 9 rows rows = rows; rows 12 oranges = 8 oranges. 90 DIFFERENTIATED INSTRUCTION D INDEPENDENT ACTIVITIES Differentiated Centers Kit Activities What s My Fact? Literature Tickle My Memory Students complete purple Activity Card by guessing a factor pair when given a product. Students read the book and learn poems and jokes to remember multiplication and division facts. EVALUATE Formative Assessment Essential Question Using the Language Objective Reflect Have student pairs discuss how to use the Distributive Property to answer the Essential Question. How can you use the Distributive Property to multiply a 2-digit number by a 1-digit number? Possible answer: the Distributive Property lets me break apart the greater number as a sum of two lesser numbers. Then I can multiply each of the two lesser numbers by the 1-digit factor, using basic facts I know. Then I add these partial products to find the total product. I can draw a rectangle to help decide how to break apart the 2-digit number. Math Journal WRITE Math Explain how you can use a model to find

7 Practice and Homework Use the Practice and Homework pages to provide students with more practice of the concepts and skills presented in this lesson. Students master their understanding as they complete practice items and then challenge their critical thinking skills with Problem Solving. Use the Write Math section to determine student s understanding of content for this lesson. Encourage students to use their Math Journals to record their answers. Multiply Using the Distributive Property Model the product on the grid. Record the product = 7 = 0 and 9 = 0 + = 7 Find the product = 9. 1 =. 17 = 1. 1 = 90 Practice and Homework COMMON CORE STANDARD.NBT.B. Use place value understanding and properties of operations to perform multi-digit arithmetic. Problem Solving. Michael arranged his pennies in the following display. 7. WRITE Math Explain how you can use a model to find 17. How many pennies does Michael have in all? 91 pennies Check students work. Chapter Chapter 2

8 Lesson Check (.NBT.B.) 1. The model shows how Maya planted flowers in her garden. How many flowers did Maya plant? 2. The model below represents the expression 18. How many tens will there be in the final product? Continue concepts and skills practice with Lesson Check. Use Spiral Review to engage students in previously taught concepts and to promote content retention. Common Core standards are correlated to each section. flowers 9 Spiral Review (.OA.A.2,.NBT.A.2,.NBT.B.,.NBT.B.). Center City has a population of twenty-one thousand, seventy people. Write the population in standard form.. Central School collected 12,1 pounds of newspaper to recycle. Eastland School collected 12,1 pounds of newspapers. How many more pounds of newspaper did Eastland School collect than Central School? 21, pounds. Allison has times as many baseball cards as football cards. In all, she has 120 baseball and football cards. How many baseball cards does Allison have? 92 0 baseball cards. A ruby-throated hummingbird beats its wings about times each second. About how many times does a ruby-throated hummingbird beat its wings in seconds? 20 times FOR MORE PRACTICE GO TO THE Personal Math Trainer 92