session 1: the meaning of multiplication and division jennifer suh [email protected] july 23 – 25,...

Session 1: The Meaning of Multiplication and Division

Jennifer Suh [email protected]

July 23 – 25, 2015Chicago Institute

https://www.youtube.com/watch?v=aU4pyiB-kq0

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 setsWork in your teams to find at least one interesting things that come in these sets.

Dr. Jennifer SuhJennifer M. Suh, PhD, [email protected] Professor, Mathematics EducationGeorge 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

SETS by NUMBERSWhat comes in these sets?

2 3 4 5 6 7 8 9 10

Like Scategory-add only new ideas to the category for points or to generate many examples

Revisiting…Sets by Numbers

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.

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?

Target 100 with tools

Target 1

Tenths Hundredths Total

6 .06

2 .26

7 .33

5 .38

3 .68

2 .70

5 .75

1 .85

Other versions…

Target 1 --> tenths, hundredthsTarget 10 --> tenths, ones

Target 1,000 --> hundreds, tensTarget 10,000 --> thousands, hundreds

But can we still use tools?

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 .99 1.00

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.

About the Institute…1. Explore the meaning of multiplication/division1. Examine problem solving situations of

multiplication/division2. 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

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.

Our focus this session is on:

Our focus for this section:

Standards in this Section

Problem Solving Structures of Multiplication and Division

Write a multiplication or division word problem that has

the solution 24 golf balls.

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.

Multiplication and Division StructuresUnknown Product Number of Groups

Unknown (How many groups?)

Size of Group Unknown(How many in each group?)

Equal Groups Mark 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/Arrays Mark’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?

Compare In 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?

Multiplication and Division StructuresUnknown Product Number of Groups

Unknown (How many groups?)

Size of Group Unknown(How many in each group?)

Equal Groups Mark 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/Arrays Mark’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?

Compare In 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?

Let’s look back at the problems we wrote.

Which problem solving structure does your problem represent?

What might our results tell us about problem solving

structures in our classrooms?

Thinking vs Getting Answers

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



Elliott ran 6 times as far as Andrew. Elliott ran 4 miles. How far did Andrew run?




Elliott ran 6 times as far as Andrew. Elliott ran 4 miles. How far did Andrew run?

How many legs do 6 elephants have?


• 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!

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?

KNOW WANT SOLVEWhat do I KNOW about the

problem?What do I WANT to find

out?How will I SOLVE the

problem?

More on this later…

Understanding through Context, Connection, and Children’s Literature

The Meaning of Multiplication and Division

Think to yourself…

Which representation of 4 x 6 do you think is “best”? Why?

6 + 6 + 6 + 6

A B

C

D

E

Let’s start with multiplication

• What can multiplication look like?

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?

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.


Sammy ate 6 crayons during each of his first 4 classes. How many crayons did Sammy eat?

So multiplication…

• Can also describe repeated addition.• For example, 6 x 4 would mean

4 + 4 + 4 + 4 + 4 + 4.

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 [?]

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?


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

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.

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?


Jackson caught 6 meatballs during a storm. Lea caught 6 times more than Jackson. How many meatballs did Lea catch?

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?

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.

Think to yourself…

Which representation of 4 x 6 do you think is “best”? Why?

6 + 6 + 6 + 6

A B

C

D

E

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.

• 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)

Additive or Muliplicative?

Assessing Students Understanding

Journal: What do you think?

• Many teachers tell 3rd grade students that multiplication is a shortcut for adding.

• Do you think that is an important reason for multiplying?


• The Doorbell Rang- “Fair Share”- • Partitive Division


• Divide and Ride-Measurement/QuotitiveModel with Sailor Overboard Game

And what about division?

Write a division problem for the picture.

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?


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?


Did your question ask for the size of each group? (partitive division)

Did your question ask about the size of each group? (measurement division)

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.]

Modeling Division

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

Journal: A Challenge

• What do you think?• Is it possible to draw a multiplication picture

that is not also a division picture or not?

Using Bar Diagrams to Solve Multiplication/Division

Problems

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?

8585 8585 8585 8585

??

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?

?? ?? ?? ??

5252

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$16

$80$80

??

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?

4545

?? ?? ?? ????

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?

1616

9696

??

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?

??

1515 1515 1515 1515

Comparison: Product Unknown

Alexi has 17 friend bracelets. Keisha has 3 times as many. How many bracelets does Keisha have?

Alexi 1717

Keisha 1717 1717 1717

??

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

2424

Another thought about division we need to consider…

• Remainder of One - Literature Connection (measurement model)

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?

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?

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?

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?

Promoting Math Talk

Reasoning about properties

Building Collective Knowledge

Making conjectures

Encouraging reflection

Session 2: The Properties of Multiplication and Division

Jennifer Suh [email protected]

July 23 – 25, 2015Chicago Institute

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?

To be or not to be….

• http://www.dailymotion.com/video/xhp3ac_ma-amp-pa-kettle-math_tech

• http://www.math.harvard.edu/~knill/mathmovies/swf/rainman.html

Reflections from last session

• Questioning• Engagement through Games• Assessment strategies• Tasks with Rigor• http://math4all.onmason.com/grades-3-5/

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

Product Game

• PLAY product games • Assessing from games

Our focus for this section:

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.”

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? 2x6 6x6 8x3

8x3

7x6 1x6

What about the carrots?

• How does this next picture relate to the distributive property?

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

One expression we could write for the array is (5 x 5) + (5 x 4).

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

Smal

l, M

aria

n an

d A.

Lin

. A V

isua

l App

roac

h to

Tea

chin

g M

ath

Conc

epts

Re

ston

, VA:

Nati

onal

Cou

ncil

of T

each

ers

of M

athe

mati

cs 2

013

Write different expressions to describe this array

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

X X X X X X X X X

Smal

l, M

aria

n an

d A.

Lin

. A V

isua

l App

roac

h to

Tea

chin

g M

ath

Conc

epts

Re

ston

, VA:

Nati

onal

Cou

ncil

of T

each

ers

of M

athe

mati

cs 2

013

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).

Using derived facts and distributive property to learn facts

What patterns do you notice on the chart?

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

Do you see the distributive property on the multiplication chart?

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

What do you notice about the rows?

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

The Sum of the Products of 2x4 and 3x4 equals the product of 5x4

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

• Can you find 3 other examples of the distributive property? What are they?

• We will share some of your discoveries.

Looking for Patterns

X 1 2 3 4 5 6 7 8 9 10

1 1 2 3 4 5 6 7 8 9 10

2 2 4 6 8 10 12 14 16 18 20

3 3 6 9 12 15 18 21 24 27 30

4 4 8 12 16 20 24 28 32 36 40

5 5 10 15 20 25 30 35 40 45 50

6 6 12 18 24 30 36 42 48 54 60

7 7 14 21 28 35 42 49 56 63 70

8 8 16 24 32 40 48 56 64 72 80

9 9 18 27 36 45 54 63 72 81 90

10 10 20 30 40 50 60 70 80 90 100

Did anyone notice?

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.

The Distributive Property Connecting symbolic and pictorial representations

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.

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

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.

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?

The Associative Property

(3 x 4) x 2 3 x (4 x 2)

Missing Numbers

How are multiplication and division related?

Number Talks- Strategies

Math Talks

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?

• 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

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

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

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?

Number Talks -Strings

• 4 *25• 6* 25• 12 *25

• Let’s look at 32 * 15• https://www.youtube.com/watch?v=twGipAN

cIqg• (45 min into talk)

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

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.

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

So What Does It Look Like?

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/a

ctivity.aspx?id=4197• http://www.learner.org/courses/learningmat

h/number/session4/part_a/division.html

Modeling with base tens

• 532 4• Equal groups

• 195 13• Use area model

Multiplication algorithms

from Fuson (2003b, p. 303)

Division

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?

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?

Closure: 3 of 1 or 1 of 3

• An AHA you had today• Something you’ll try • A question you have

Where did you see the Standards for

Mathematical Practices in these

ideas about multiplication and

division?

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

session 1: the meaning of multiplication and division jennifer suh [email protected] july 23 – 25,...

Documents

mathematics tools

mathematics classroom

weeks sets

introduction of multiplication

properties of multiplication

target wins

divisionjennifer suh

new ideas