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Session 1: The Meaning of Multiplication and Division Jennifer Suh July 23 – 25, 2015 Chicago Institute https://www.youtube.com/watch?v=aU4pyi.

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Presentation on theme: "Session 1: The Meaning of Multiplication and Division Jennifer Suh July 23 – 25, 2015 Chicago Institute https://www.youtube.com/watch?v=aU4pyi."— Presentation transcript:

1 Session 1: The Meaning of Multiplication and Division Jennifer Suh jsuh4@gmu.edu July 23 – 25, 2015 Chicago Institute https://www.youtube.com/watch?v=aU4pyi B-kq0

2 What comes in these sets? Introductions (Name tent) What is your name? Grade you teach? Chat with your neighbor- Where are you from? What do you do there? Finding sets Work in your teams to find at least one interesting things that come in these sets.

3 Dr. Jennifer Suh Jennifer M. Suh, PhD, jsuh4@gmu.edu Associate Professor, Mathematics Education George Mason University Interests: Developing students’ mathematics proficiency & teachers' mathematics knowledge through Lesson Study and representational fluency through mathematics tools and emerging technologies Highlight of my summer: Spending the summer with my boys (Two sons) mom who traveled here with me. Will head to the beach in 3 weeks

4 SETS by NUMBERS What comes in these sets? 2345678910 Like Scategory-add only new ideas to the category for points or to generate many examples

5 Revisiting…Sets by Numbers

6 Adapted from Fundamentals, Creative Publications, 4-5 How to play: 1.Players take turns rolling a number cube 2.After each roll, a player decides which column to place the digit. 3.That player then adds the value to his/her total. 4.The player who is closest to the target (in the last total) without going over the target wins.

7 Choose 1 to discuss with a partner How might you use this game in your mathematics classroom? How might you modify this game for your students? What would you look for while your students played the game?

8 Target 100 with tools

9 Target 1 TenthsHundredthsTotal 6.06 2.26 7.33 5.38 3.68 2.70 5.75 1.85

10 Other versions… Target 1 --> tenths, hundredths Target 10 --> tenths, ones Target 1,000 --> hundreds, tens Target 10,000 --> thousands, hundreds But can we still use tools?

11 Decimals on a Hundred Chart.01.02.03.04.05.06.07.08.09.10.11.12.13.14.15.16.17.18.19.20.21.22.23.24.25.26.27.28.29.30.31.32.33.34.35.36.37.38.39.40.41.42.43.44.45.46.47.48.49.50.51.52.53.54.55.56.57.58.59.60.61.62.63.64.65.66.67.68.69.70.71.72.73.74.75.76.77.78.79.80.81.82.83.84.85.86.87.88.89.90.91.92.93.94.95.96.97.98.991.00

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13 Putting Essential Understanding of Multiplication and Division into Practice The way in which you teach a foundational concept or skill has an impact on the way in which students will interact with and learn later related content. For example, the types of representations that you include in your introduction of multiplication and division are the ones that your students will use to evaluate other representations and ideas in later grades.

14 About the Institute… 1. Explore the meaning of multiplication/division 1. Examine problem solving situations of multiplication/division 2. Apply the properties of multiplication and division 3. Establish the concepts of multi-digit computation 4. Identify strategies for developing mental computation 4. Revisit approaches to basic facts

15 Principles to Action Take a look at the 8 practices. Consider which practice is easiest for you to implement in your classroom. Consider which practice is most challenging for you to implement. Think-Pair-Share We will circle back to these practices throughout the institute.

16 Our focus this session is on:

17

18 Our focus for this section:

19

20 Standards in this Section

21 Problem Solving Structures of Multiplication and Division

22 Write a multiplication or division word problem that has the solution 24 golf balls.

23 With a partner… Review the problem solving cards. Sort the cards by the type of problem they represent. Arrange them according to the grid on the next slide.

24 Multiplication and Division Structures Unknown ProductNumber of Groups Unknown (How many groups?) Size of Group Unknown (How many in each group?) Equal GroupsMark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Area/ArraysMark’s bookshelf has 3 shelves with 6 books on each shelf. How many books does Mark have? Mark has 18 books. They are on shelves with 6 books on each shelf. How many shelves are there? Mark has 18 books on 3 shelves. How many books are on each shelf? CompareIn June, Mark saved 5 times as much money as May. In May, he saved $7. How much money did he save in June? In June, Mark saved 5 times as much money as he did in May. If he saved $35.00 in June, how much did he save in May? In June, Mark saved $35.00. In May, he saved $7.00. How many times as much money did he save in June as May?

25 Multiplication and Division Structures Unknown ProductNumber of Groups Unknown (How many groups?) Size of Group Unknown (How many in each group?) Equal GroupsMark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Area/ArraysMark’s bookshelf has 3 shelves with 6 books on each shelf. How many books does Mark have? Mark has 18 books. They are on shelves with 6 books on each shelf. How many shelves are there? Mark has 18 books on 3 shelves. How many books are on each shelf? CompareIn June, Mark saved 5 times as much money as May. In May, he saved $7. How much money did he save in June? In June, Mark saved 5 times as much money as he did in May. If he saved $35.00 in June, how much did he save in May? In June, Mark saved $35.00. In May, he saved $7.00. How many times as much money did he save in June as May?

26 Multiplication and Division Structures Unknown ProductNumber of Groups Unknown (How many groups?) Size of Group Unknown (How many in each group?) Equal GroupsMark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Area/ArraysMark’s bookshelf has 3 shelves with 6 books on each shelf. How many books does Mark have? Mark has 18 books. They are on shelves with 6 books on each shelf. How many shelves are there? Mark has 18 books on 3 shelves. How many books are on each shelf? CompareIn June, Mark saved 5 times as much money as May. In May, he saved $7. How much money did he save in June? In June, Mark saved 5 times as much money as he did in May. If he saved $35.00 in June, how much did he save in May? In June, Mark saved $35.00. In May, he saved $7.00. How many times as much money did he save in June as May?

27 Multiplication and Division Structures Unknown ProductNumber of Groups Unknown (How many groups?) Size of Group Unknown (How many in each group?) Equal GroupsMark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Area/ArraysMark’s bookshelf has 3 shelves with 6 books on each shelf. How many books does Mark have? Mark has 18 books. They are on shelves with 6 books on each shelf. How many shelves are there? Mark has 18 books on 3 shelves. How many books are on each shelf? CompareIn June, Mark saved 5 times as much money as May. In May, he saved $7. How much money did he save in June? In June, Mark saved $35.00. In May, he saved $7.00. How many times as much money did he save in June as May? In June, Mark saved 5 times as much money as he did in May. If he saved $35.00 in June, how much did he save in May?

28 Multiplication and Division Structures Unknown ProductNumber of Groups Unknown (How many groups?) Size of Group Unknown (How many in each group?) Equal GroupsMark has 4 bags of apples. There are 6 apples in each bag. How many apples does Mark have altogether? Mark has 24 apples. He put them into bags containing 6 apples each. How many bags did Mark use? Mark has 24 apples. He wants to share them equally among his 4 friends. How many apples will each friend receive? Area/ArraysMark’s bookshelf has 3 shelves with 6 books on each shelf. How many books does Mark have? Mark has 18 books. They are on shelves with 6 books on each shelf. How many shelves are there? Mark has 18 books on 3 shelves. How many books are on each shelf? CompareIn June, Mark saved 5 times as much money as May. In May, he saved $7. How much money did he save in June? In June, Mark saved $35.00. In May, he saved $7.00. How many times as much money did he save in June as May? In June, Mark saved 5 times as much money as he did in May. If he saved $35.00 in June, how much did he save in May?

29 Let’s look back at the problems we wrote. Which problem solving structure does your problem represent?

30 What might our results tell us about problem solving structures in our classrooms?

31 Thinking vs Getting Answers

32 The clown gave my little brother 7 red balloons and some green balloons. Altogether my brother got 13 balloons. How many green balloons did he get? Using “KEY” words Clement, L. and Bernhard, J. (2005) “A Problem-Solving Alternative to Using Key Words” Mathematics Teaching in the Middle School, March 2005, 10-7 p. 360, NCTM

33 The clown gave my little brother 7 red balloons and some green balloons. Altogether my brother got 13 balloons. How many green balloons did he get? Using “KEY” words Elliott ran 6 times as far as Andrew. Elliott ran 4 miles. How far did Andrew run? Clement, L. and Bernhard, J. (2005) “A Problem-Solving Alternative to Using Key Words” Mathematics Teaching in the Middle School, March 2005, 10-7 p. 360, NCTM

34 The clown gave my little brother 7 red balloons and some green balloons. Altogether my brother got 13 balloons. How many green balloons did he get? Using “KEY” words Elliott ran 6 times as far as Andrew. Elliott ran 4 miles. How far did Andrew run? How many legs do 6 elephants have? Clement, L. and Bernhard, J. (2005) “A Problem-Solving Alternative to Using Key Words” Mathematics Teaching in the Middle School, March 2005, 10-7 p. 360, NCTM

35 The reason many of us have used a key word or a steps approach to teaching problem solving is that we have not had any alternative instructional strategies!

36 K-W-S for Problem Solving The store has 13 cans of tennis balls on the shelf. Each can has 3 balls in it. How many tennis balls does the store have?

37 KNOWWANTSOLVE What do I KNOW about the problem? What do I WANT to find out? How will I SOLVE the problem?

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39 More on this later… Understanding through Context, Connection, and Children’s Literature

40 The Meaning of Multiplication and Division

41 Think to yourself… Which representation of 4 x 6 do you think is “best”? Why? 6 + 6 + 6 + 6 AB C D E

42 Let’s start with multiplication What can multiplication look like?

43 What might it look like? Each chocolate chip cookie had 6 chocolate chips. The mouse ate 4 cookies. How many chocolate chips did he eat?

44 So Multiplication can describe equal groups. 6 x 4 would tell the total number of chocolate chips 9 x 4 would tell the total number in 9 groups of 4 penguins.

45 Tricycles There are some tricycles in the park. How many WHEELS might there be altogether? How many wheels will there not be? If we each write a number sentence to describe our choice, what will be the same about all of them? Why?

46 Did you notice? Were your numbers even or odd? Why? Could it [?] have been 20? Why or why not? What was common about all the numbers selected for [?] Why does that make sense?

47 Bikes and trikes 1. Yesterday, Skip counted all of the cycles in the shop. There were seven bikes and four trikes in the shop. How many wheels were there on these eleven cycles? Show your work. 2. Today Skip counted all of the wheels of all of the cycles in the shop and he found that there were 30 wheels in all. There were the same number of bikes as there were trikes. How many bikes were there? How many trikes were there? Show your work.

48 Bikes and trikes

49 Performance Task Alice counted 7 bike and tricycle riders and 19 cycle wheels going past her house. How many tricycles were there?

50 More bikes and trikes Sarah founds 32 seats and 72 wheels in her shop. How many bikes and trikes were at her store? There are similar to Farmer Ben’s Chicken and Pig problem. He has 78 feet and 27 heads. How many pigs and chicken are there?

51 What might it look like? Sammy ate 6 crayons during each of his first 4 classes. How many crayons did Sammy eat?

52 So multiplication… Can also describe repeated addition. For example, 6 x 4 would mean 4 + 4 + 4 + 4 + 4 + 4.

53 Number Line Jumps Each jump is a whole number amount. All jumps are equal length. What number could [?] to be? What number could [?] not be? 0 [?]

54 What Might It Look Like? 24 ants went off on their own. They were marching in rows and columns. How many ants were in a row? How many were in a column? How do you know?

55 Maybe

56 What might it look like? In class, the worms built rectangles with exactly 24 color tiles. What might the length and width of their rectangles have been?

57 Maybe

58 So.. Multiplication can be represented by areas of rectangles. For example, 6 x 4 describes the number of square units in a rectangle 6 units long by 4 units wide. If the tiles did not touch, the arrangement is called an array.

59 Modeling through Literature Connections If I can hop like a Frog (Comparison Models) Frogs are champion jumpers. A 3-inchfrog can hop 60 inches. That means the frog is jumping 20 times its body’s length. If you hopped like a frog, How far could you hop?

60 What might it look like? Jackson caught 6 meatballs during a storm. Lea caught 6 times more than Jackson. How many meatballs did Lea catch?

61 Stickers Ian has 4 times as many stickers as Rachel. Do you think Ian had 35 stickers? Why or why not? Suppose he had 36 stickers. What number sentence would you write to describe this?

62 Was this an array? Multiplication can also describe a comparison. For example, 24 = 6 x 4 since 24 is 4 times as much as 6.

63 Think to yourself… Which representation of 4 x 6 do you think is “best”? Why? 6 + 6 + 6 + 6 AB C D E

64 Essential Understanding Essential Understanding 1a In the multiplicative expression A × B, A can be defined as a scaling factor. Multiplication is a scalar process involving two quantities, with one quantity—the multiplier—serving as a scaling factor and specifying how the operation resizes, or rescales, the other quantity—the multiplicative unit. The rescaled result is the product of the multiplication.

65 Multiply or divide to solve word problems involving multiplicative comparison, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem, distinguishing multiplicative comparison from additive comparison. (4.OA.A2, p. 29)

66 Additive or Muliplicative?

67 Assessing Students Understanding

68

69 Journal: What do you think? Many teachers tell 3 rd grade students that multiplication is a shortcut for adding. Do you think that is an important reason for multiplying?

70 Understanding through Context, Connection, and Children’s Literature The Doorbell Rang- “Fair Share”- Partitive Division

71 Understanding through Context, Connection, and Children’s Literature Divide and Ride-Measurement/Quotitive Model with Sailor Overboard Game

72 And what about division? Write a division problem for the picture.

73 Write a division equation to represent this picture. Partitive Division: Sharing Equally or Dealing Out What is the size of each group? There are 4 bowls with fish. There are 24 fish in all. Each bowl has the same number of fish. How many fish are in each bowl?

74 Write a division equation to represent this picture. Measurement Division: Making Groups of an Equal Size How many groups? There are 24 fish to put in bowls. There will be 6 fish in each bowl. How many bowls are needed?

75 Write a division equation to represent this picture. Did your question ask for the size of each group? (partitive division) Did your question ask about the size of each group? (measurement division)

76 So 24 ÷ 4 can either mean There are 24 items placed in 4 equal groups and the question is how many are in each group. [This is often called partitive or sharing division.] OR There are 24 items grouped in groups of 4 and the question is how many groups. [This is often called quotative or measuring division.]

77 Modeling Division

78

79 Setting up chairs Suppose you asked this question: Jennifer set up chairs in the gym in equal rows. There were 60 chairs. How many rows and how many chairs in each row do you think she might have had? What would you hope to see? Hear?

80 Comparing times Carson practiced piano for 60 minutes. That is ___ times as much as Lisa practiced. [Choose one of these numbers to put in the blank: 2, 3, 4, 5, 6, or 10.] How much time did Lisa practice?

81 AND division is the opposite of multiplication. 30 ÷ 6 = 5 since 5 x 6 = 30. How about extend this idea of Inverse Operation to get more Algebraic and Introduce the notion of DOING and UnDOING- One of the important Algebraic Habits of Mind

82 Journal: A Challenge What do you think? Is it possible to draw a multiplication picture that is not also a division picture or not?

83 Cookies There are 24 cookies on a plate. Each child is only allowed to take 3 cookies. How many children can be served? Solve this problem four times- once using only +, once only -, once only x and once only ÷. Write a number sentence each time.

84 Using Bar Diagrams to Solve Multiplication/Division Problems

85 Using Bar Diagrams Equal Groups: Size of Groups Unknown Jackson has 4 folders. Each folder has 85 apps. How many apps are in his 4 folders? 85 ? ?

86 Equal Groups: Group Size Unknown Mrs. Smith had 4 bags and put the same number of boxtops into each bag. She had 52 boxtops to place in the bags. How many boxtops did she put into each bag? ? ? ? ? ? ? ? ? 52

87 Equal Groups: Number of Groups Unknown Oscar bought t-shirts that cost $16 each. He spent $80 altogether. How many t- shirts did he buy? $16 $80 ? ?

88 Equal Groups: Size of Groups A farmer has 45 pigs in 5 pens. Each pen has the same amount of pigs. How many pigs are in each pen? 45 ? ? ? ? ? ? ? ? ? ?

89 Equal Groups: Number of Groups Unknown Jenny has 96 feet of yarn. She needs 16 feet for a decoration. How many decorations can she make? 16 96 ? ?

90 Equal Groups: Unknown Product Deryn had some jellybeans. She put them in piles of 15 and was able to make 4 piles. How many jellybeans did she have to start? ? ? 15

91 Comparison: Product Unknown Alexi has 17 friend bracelets. Keisha has 3 times as many. How many bracelets does Keisha have? Alexi 17 Keisha 17 ? ?

92 Comparison: Size Unknown Ben’s dog weighs 24 pounds. This is 3 times more Stan’s dog. How much does Stan’s dog weigh? Stan’s Dog ? ? ? ? ? ? ? ? Ben’s Dog 24

93 Another thought about division we need to consider… Remainder of One - Literature Connection (measurement model)

94 How might students think about the remainders in each problem? A mother had 20 balloons. She wanted to give them to her 3 children so that each child would have the same number of balloons. How many balloons did each get? A per store owner has 14 birds and some cages. She will put 3 birds in each cage. How many cages will she need? A father has 17 cookies. He wants to give them to his 3 children so that each child has the same number of cookies. How many cookies will each child get?

95 Examining Student Thinking about Remainders A mother had 20 balloons. She wanted to give them to her 3 children so that each child would have the same number of balloons. How many balloons did each get?

96 Examine Student Thinking About Remainders A per store owner has 14 birds and some cages. She will put 3 birds in each cage. How many cages will she need?

97 Examining Student Thinking A father has 17 cookies. He wants to give them to his 3 children so that each child has the same number of cookies. How many cookies will each child get?

98

99 Promoting Math Talk

100 Reasoning about properties

101 Building Collective Knowledge

102 Making conjectures

103 Encouraging reflection

104 Session 2: The Properties of Multiplication and Division Jennifer Suh jsuh4@gmu.edu July 23 – 25, 2015 Chicago Institute

105 Think-Pair-Share If you had to choose, which of your 5 senses could you live without? If you had to choose, which of the properties of x/÷ could you live without?

106 To be or not to be…. http://www.dailymotion.com/video/xhp3ac_ ma-amp-pa-kettle-math_tech http://www.dailymotion.com/video/xhp3ac_ ma-amp-pa-kettle-math_tech http://www.math.harvard.edu/~knill/mathmo vies/swf/rainman.html http://www.math.harvard.edu/~knill/mathmo vies/swf/rainman.html

107 Reflections from last session Questioning Engagement through Games Assessment strategies Tasks with Rigor http://math4all.onmason.com/grades-3-5/

108 About this session… Examine each of the properties of multiplication. Consider how we apply these properties so that one can develop computational fluency. Multidigit computation

109 Product Game PLAY product games Assessing from games

110 Our focus for this section:

111 Impact on Teaching and Learning? “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.”

112 Deryn made a color tiles rectangle that is 8 x 6. She broke it into 2 smaller rectangles. What might be the dimensions of those rectangles? 2x66x6 8x3 7x6 1x6

113 What about the carrots? How does this next picture relate to the distributive property?

114 How does this picture help you see that there are lots of ways to think of 6 x 7? Small, Marian and A. Lin. A Visual Approach to Teaching Math Concepts Reston, VA: National Council of Teachers of Mathematics 2013 4 x 7 2 x 7 5 x 7 1 x 7 6 x 5 6 x 2 6 x 6 6 x 1

115 One expression we could write for the array is (5 x 5) + (5 x 4). XXXXXXXXX XXXXXXXXX XXXXXXXXX XXXXXXXXX XXXXXXXXX Small, Marian and A. Lin. A Visual Approach to Teaching Math Concepts Reston, VA: National Council of Teachers of Mathematics 2013

116 Write different expressions to describe this array XXXXXXXXX XXXXXXXXX XXXXXXXXX XXXXXXXXX XXXXXXXXX Small, Marian and A. Lin. A Visual Approach to Teaching Math Concepts Reston, VA: National Council of Teachers of Mathematics 2013

117 Did you notice? With the carrots, not only can you show that 7 x (4 + 2) = 7 x 4 + 7 x 2, but also that (3 + 4) x 6 = 3 x 6 + 4 x 6. Notice that either the number of groups or the size of the groups can be broken up (or distributed).

118 Using derived facts and distributive property to learn facts

119 What patterns do you notice on the chart? X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100

120 Do you see the distributive property on the multiplication chart? X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100

121 Looking for patterns What do you notice when you look at rows 2, 3 and 5 on the multiplication chart?

122 What do you notice about the rows? X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100

123 The Sum of the Products of 2x4 and 3x4 equals the product of 5x4 X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100

124 Can you find 3 other examples of the distributive property? What are they? We will share some of your discoveries. Looking for Patterns

125 X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100 Did anyone notice?

126 X12345678910 1123456789 22468 1214161820 336912151821242730 4481216202428323640 55101520253035404550 66121824303642485460 77142128354249566370 88162432404856647280 99182736455463728190 10 2030405060708090 100 Did anyone notice?

127 Would you believe? How is the number line jumps representation on the next slide also about the distributive property? Try the problem to figure it out.

128 The Distributive Property Co nnecting symbolic and pictorial representations

129 The Distributive Property on a Number Line? Assign a whole number jump size to the blue jumps. Then assign a whole number jump size to the green jumps. Does this always work? Why? 0 2 2 2 3 3 3 5 55

130 Which pictures make it easy to show 3 x 4 = 4 x 3? Which do not?

131 Or this variation The picture on the bottom right is a lot like the simpler ones on the previous slide. Do you see here both 4 sets of 3 and 3 sets of 4? 3 colors in a set, 4 sets 4 sets of 3 colors

132 Does the Commutative Property Apply to Jumps on a Number Line? Choose a multiplication fact. Show ___ jumps of ___ on the top. Show ___ jumps of ___ on the bottom.

133 Does the Commutative Property Apply to Jumps on a Number Line?

134 Journal: So how would you argue? How would you argue that no matter what two numbers I choose, the order of multiplication doesn’t matter? Practice with a partner. Be ready to share.

135 The Associative Property How would you cut this prism to see (3 x 4) x 2 ? How would you cut it to see 3 x (4 x 2)? More generally, how do you know that a x (b x c) = (a x b) x c?

136 The Associative Property (3 x 4) x 23 x (4 x 2)

137 Using Counters… How might you use counters to show that (2 x 2) x 3 = 2 x (2 x 3)? Remember it’s about seeing why the values HAVE TO be the same.

138 One possibility 1 sets of (2 sets of 3) 2 setss of (2 sets of 3) 2 x (2 x 3)

139 Another Possibility (2 sets of 2) sets of 3 (2 x 2) x 3

140 Some other properties… Choose either the principle that 1 x n = n or the principle that 0 x n = 0. How would you want a child to argue its truth?

141 Missing Numbers

142

143

144 How are multiplication and division related?

145 Agree or Disagree Your friend argues it’s a total waste of time to learn division because you you can always multiply to find the answer. Work with others to come up with an argument or arguments to support your position.

146 Utopia? On a planet called Utopia, students don’t learn to divide. All they learn is to multiply. Maria had a friend visiting from Utopia who was watching her do homework. Here are the problems Maria was given: a. 18 ÷ 6 = ☐ b. 45 ÷ 9 = ☐ c. 54 ÷ 6 = ☐ d. 63 ÷ 7 = ☐

147 How could you help Maria’s friend find the answers to the homework problems? Write four multiplication problems that give the same answers as Maria’s four division problems. a. 18 ÷ 6 = ☐ b. 45 ÷ 9 = ☐ c. 54 ÷ 6 = ☐ d. 63 ÷ 7 = ☐ Utopia?

148 Are there properties of division too? Consider the properties we associate with multiplication. Which of them hold and which do not hold for division?

149 Number Talks- Strategies

150 Math Talks

151 Purposeful Problems Write an equation for each of the problems shown in figure 4.3. What specific misconceptions about multiplication and division might these questions address?

152 Does the commutative property hold for division, i.e. Is 12 ÷ 3 = 3 ÷ 12? What sorts of convincing arguments could or should be used? Commutativity 12 ÷ 3 3 ÷ 12

153 Identity or Zero Properties of Division? Turn to a partner. Explain why the identity property or zero property does/doesn’t work. a ÷ 1 = a but 1 ÷ a ≠ a 0 ÷ a = 0 but we cannot divide by 0

154 Agree or Disagree? Why? Jackson says he used the distributive property to solve 721 ÷ 7 because it is the same as 700 ÷ 7 + 21 ÷ 7.

155 Are there partial quotients in Jackson’s problem: 721 ÷ 7? How many groups of 7 are in 700? -----> 100 (100 x 7 = 700) How many groups of 7 are in 21 (what’s left)? -----> 3 (3 x 7 = 21) So, there are (100 groups of 7) + (3 groups of 7) -or- 103 groups of 7

156 So What? (about properties) What’s Missing? How do you know? 7 x 6 is the same as (7 x ____) + (7 x 1) 9 x 25 is the same as (__x25) + (__x25) + (1x25) 18 x 6 is the same as ____ x 12 24 x 12 is the same as 6 x ____ 5 4 4 9 48

157 How Might Our Students Solve These Problems? Neha has 11 bracelets with 6 beads. Deryn has 13 bracelets with 6 beads each? Who has more beads? How do you know? A store has 7 4-wheel go-carts. Another store has 5 4-wheel go carts. How many more wheels are in the first store? How do you know? There is a tray of 12 cookies each weighing 4 ounces. There is a tray of 12 brownie bites each weighing 3.2 ounces. Which tray weighs more? How do you know?

158 Number Talks -Strings 4 *25 6* 25 12 *25 Let’s look at 32 * 15 https://www.youtube.com/watch?v=twGipAN cIqg https://www.youtube.com/watch?v=twGipAN cIqg (45 min into talk)

159 Use the area model for 12*15 http://nlvm.usu.edu/en/nav/frames_asid_192_g_1_t_1.html?open=teacher&from=vlibrary.html

160 We know that 50 x 40 = 2,000. What is an easy way to find 49 x 40? Explain your easy way. Does this always work? Create 2 other examples that prove your thinking.

161 Become A Multiplication Expert For the following questions for your strategy: What math is involved? How does it compare to traditional algorithms? What errors might students make? – Repeated Addition – Decomposing Numbers – Compensation – Partial Products – Lattice Method – Area Model 2 x 49

162 So What Does It Look Like? Context-Rich Problem Multiple Opportunities to Explore Student Strategies Demonstrate Models w/ Increasing Complexity Move Towards Traditional Algorithm Illicit Student Strategies Manipulatives Pictorial Models Decomposing Strategies Ultimate Goal Efficiency & Accuracy Ultimately Students Chose (some w/ guidance)

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170 DIVISION a. Which type of division, measurement or partitive, would be most efficient for computing 100 / 50? Why? b. Which would you use for 100/ 2? Quotient café http://illuminations.nctm.org/activity.aspx?id =4197 Quotient café http://illuminations.nctm.org/activity.aspx?id =4197 http://www.learner.org/courses/learningmath /number/session4/part_a/division.html http://www.learner.org/courses/learningmath /number/session4/part_a/division.html

171 Modeling with base tens 532 4 Equal groups 195 13 Use area model

172 Multiplication algorithms from Fuson (2003b, p. 303)

173 Division

174 Journal: Choose 1 to discuss with a partner… 1. Think of 1 word to describe the (property) property. Explain why you selected that word. 2. Which property of multiplication could you not live without? Why? 3. How would you describe (property) to someone from another planet?

175 Think-Pair-Share If you had to choose, which of your 5 senses could you live without? If you had to choose, which of the properties of x/÷ could you live without?

176 Closure: 3 of 1 or 1 of 3 An AHA you had today Something you’ll try A question you have

177 Where did you see the Standards for Mathematical Practices in these ideas about multiplication and division?

178 Disclaimer The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council. 178

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