Multiplication and Division

Math Content – Part 3March 4, 2013

Afternoon Meeting

Welcome Outstanding Coaches,Please greet at least three other people in the room and share why you what you want to know about multiplication and division.

Activity: Form a circle. “I Have… Who Has…..

From, Erny

The Devil’s Library

Learning

We are learning to…Understand how the CCSSM views the

development of multiplicative thinking.Apply strategies that promoting fluency with

single-digit multiplication and division.We will be successful when…

We can help students apply properties of operations as strategies to multiply.

What is Multiplication?Respond in writing in your notebook…

What is multiplication?

Check with a neighbor to see: How is your thinking similar? How is your choice of language similar

or different?

Repeated Addition

Repeated Addition

Begin multiplication instruction with problems involving repeated addition and introduce symbolic multiplication as a shortcut for representing repeated addition problems. Children can be encouraged to first represent repeated addition as an addition expression and then devise a shorthand for such symbolic expressions.

- Baroody, 1998

Strategies- Creatures

Standards…

Read and Reflect on Standards 3.OA: 1 – 6

Share with your shoulder partner a few ideas that struck you as critical to developing a sound understanding of multiplication and division.

From Counting by Ones to Thinking In Groups

Place a large amount of counters in the middle of the table.

On the word “go” grab as many groups of two that you can before we say stop.

Don’t count the total number of sets, just concentrate on making groups of two.

Return the counters to the middle of the table.

On the word “go” grab as many sets of ___3____ that you can before we say stop.

Making groups: What did you notice?

What kind of thinking were you doing as you were making groups?

What would this activity tell you about students’ thinking?

You were just unitizing!(To make or transform into a single unit)

Dot Images

How many dots so you see?

How do you see It?

Draw what you see in your notebook.

Image 1

Image 2

Image 3

Grounding thinking in CCSSM

3.OA.5: Apply properties of operations as strategies to multiply and divide.CommutativeAssociativeDistributive

With your shoulder partner, use an

example, remind each other how these “rules of numbers work.”

Reflect back on 3.OA.5.

Think about how the images were described. Where do the properties show up in the reasoning?

Individually:Quickly glance at the dot image and

determine the number of dots.Jot down using “language” how you saw it.Write an equation that matches your image

and your description.Identify the property or properties you used.

Turn and share.

Image 4

Image 5

Image 6

Revisit Standard 3.OA.5 and 3.OA.7 Take turns with a Shoulder Partner to summarize:

Reflecting on our “dot image” work, what are the main messages of these standards?

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Examine your word problem:What does the 4 mean?What does the 7 mean?

4 x 7 = ☐

Pose a word problem for:

Number of Groups

Group Size

The total number of objects in 4 groups of 7 objects each.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Examine your word problem:What is unknown in your problem:

Number of groups?Group size?

28 ÷ 4 = ☐

Pose a word problem for:

Tell a different story for 28 ÷ 4 = ☐.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

28 ÷ 4 = ☐

TotalNumber of shares

or Number of objectsin each shareDivision means to partition

to find the number of shares or find the amount in each share.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

4OA1Interpret a multiplication equation as a comparison, e.g., interpret 35 = 5 x 7 as a statement that 35 is 5 times as many as 7 and 7 times as many as 5. Represent verbal statements of multiplicative comparisons as multiplication equations.

4OA2 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.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

What’s the new thinking being developed in 40A1 & 2?

“This one is 32 feet higher than that one” 40 feet high, the other 8 feet high

Students in earlier grades learned to compare quantities additively….

Students in Grade 4 learn to compare these quantities multiplicatively…..

“This one is 5 times as high as that one.”

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Experiences with Compare Word Problems

Individually pose compare word problems for:

6 x 9 = ☐54 ÷ 6 = ☐ with Group Size Unknown54 ÷ 6 = ☐ with Number of Groups

Unknown

When you have a draft of each, turn and check-in with a partner.

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Connie has 8 times as much money as Melissa. Connie has $56. How much money does Melissa have?

7 x 8 = ☐ Total Unknown

56 ÷ 8 = ☐ Group Size Unknown

56 ÷ 8 = ☐ Number of Groups Unknown

Unknown: Group Size

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Building Blocks of Algebra

Understand problem situationsRepresent the situation with

objects or diagramsRepresent quantitative relationships

with equationsUse properties of operations as the

basis for strategies

p.13 OA Progressions

Common Core Leadership in Mathematics Project, University of Wisconsin-Milwaukee, Summer Institute 2011

Expanding View of Multiplication

Grade 3: Equal groups--discrete objects, arrays….

Grade 4: Comparison situations--continuous quantities

Grade 5: Stretches or Shrinks (scale factor)

--Context for reasoning multiplicatively with continuous quantities

Oldest Multiplication Chart

Napier Bones

The lattice algorithm for multiplication has been traced to India, where it was in use before A.D.1100. Many students find this particular multiplication algorithm to be one of their favorites. It helps them keep track of all the partial products without having to write extra zeros – and it helps them practice their multiplication facts

1. Create a grid. Write one factor along the top, one digit per cell.

2. Draw diagonals across the cells.

3.Multiply each digit in the top factor by each digit in the side factor. Record each answer in its own cell, placing the tens digit in the upper half of the cell and the ones digit in the bottom half of the cell.

4. Add along each diagonal and record any regroupings in the next diagonal

06

24

1

8

08

32

24

Write the other factor along the outer right side, one digit per cell.

06

24

1

8

08

32

24

35

15

10

49

21

1

4

Partial Products Algorithm for Multiplication

Calculate 50 X 60

6 7X 5 3

Calculate 50 X 7

3,000 350 180 21

Calculate 3 X 60

Calculate 3 X 7 +Add the results 3,551

To find 67 x 53, think of 67 as 60 + 7 and 53 as 50 + 3. Then multiply each part of one sum by each part of the other, and add the results

Calculate 10 X 20

1 4X 2 3

Calculate 20 X 4

200 80 30 12

Calculate 3 X 10

Calculate 3 X 4 +Add the results 322

Let’s try another one.

Calculate 30 X 70

3 8X 7 9

Calculate 70 X 8

2, 100 560 270 72

Calculate 9 X 30

Calculate 9 X 8 +Add the results

Do this one on your own.

3002

Let’s see if you’re right.

Partial QuotientsA Division Algorithm

The Partial Quotients Algorithm uses a series of “at least, but less than” estimates of how many b’s in a. You might begin with multiples of 10 – they’re easiest.

12 158There are at least ten 12’s in 158 (10 x 12=120), but fewer than twenty. (20 x 12 = 240)

10 – 1st guess

- 12038

Subtract

There are more than three (3 x 12 = 36), but fewer than four (4 x 12 = 48). Record 3 as the next guess

3 – 2nd guess- 362 13

Sum of guesses

Subtract

Since 2 is less than 12, you can stop estimating. The final result is the sum of the guesses (10 + 3 = 13) plus what is left over (remainder of 2 )

Let’s try another one

36 7,891100 – 1st guess

- 3,6004,291

Subtract

100 – 2nd guess

- 3,600

7 219 R7 Sum of guesses

Subtract

69110 – 3rd guess- 360

3319 – 4th guess - 324

Now do this one on your own.

43 8,572100 – 1st guess

- 4,3004272

Subtract

90 – 2nd guess-3870

15199 R 15

Sum of guesses

Subtract

4027 – 3rd guess- 301

1012 – 4th guess - 86