Multiply and Divide, Big and Small

Multiply and Divide,

Big and Small

NCCTM 2014

Amanda Northrup

[email protected]

www.linkyy.com/teachandlearn

@msnorthrup

The Need…

2

Ma & Pa Kettle Math

CRA – A Sequence of Instruction

3

C = Concrete. Use materials to focus on the

development of conceptual understanding, while

starting to make connections to procedures. During

this stage, students might work with base ten

blocks, fraction bars, red and yellow chips, tiles,

cubes, etc...


4

R = Representational. Connect the

previous work with concrete materials to

other representations, especially drawings.

Students think more deeply about both

concepts and procedures. They might use

circles, tallies, rectangles, drawings, etc...


5

A = Abstract. Use previous work with

materials and drawings to make sense of

procedures with numbers and symbols.

WHOLE NUMBERS

CRA for

Multiplying Large Whole Numbers

7

Concrete…

36 x 4

36 x 24

CRA for


8

How would you use representational?

36 x 24

36 x 4

CRA for


9

Now for abstract

36 x 4

Any abstract method should MAKE SENSE based on what we learned from

the concrete and representational strategies.

If we can’t explain the connection for the strategy, we shouldn’t be using it.

CRA for


10

Abstract strategies for

36 x 4 36 x 24 436 x 24

• Partial products

• Array

• Repeated addition

• Lattice

• Standard algorithm (5th grade)

CRA for

Division with Whole Numbers

11

Concrete –

Use 15 chips to show 15 divided by 5

CRA for


12

Concrete

78 3

CRA for


13

Concrete: Use base ten blocks to show 78 3

CRA for


14

Representational:

Show 78 3 using a drawing

Circle division: 672 5

CRA for


15

CRA for


16

Abstract:

672 5

Try it with more than one algorithm

DECIMALS

Strategies for Multiplying Decimals

18

How could you model

3 x 0.4

using concrete materials?


19

How could you model 3 x 0.4

using a picture?

0.4 + 0.4 + 0.4 = 1.2


20

0.1 x 0.1

Things to think about:

• What is 1 x 0.1? (Justify your answer)

• Will our answer be more or less?

• What is ½ of 0.1

• Will our answer be more or less?

• What could the context be?


21

How could you model 0.1 x 0.1

using concrete materials?


22

How could you model

0.1 x 0.1

using a picture?


23

What about the standard algorithm?

Solve these using only repeated addition

and/or decimal grids:

3 x 7 3 x 0.7 7 x 0.03

What do you notice stays the same?

What differences do you notice?


24

What do you notice stays the same?

What differences do you notice?

34 x 78 = 2652

3.4 x 0.78 = 2.652

34 x 7.8 = 265.2


25

Using reasoning to place the decimal –

Define a context for 7.9 x 4.6

Let’s find the digits.

7.9 x 4.6 = 3 6 3 4

0.3634

3.634

36.34

363.4


26

What reasoning can you use to place the

decimal?

7.9 x 4.6 = 3 6 3 4

8 x 5 = 40

7 x 5 = 35

7 x 4 = 28

8 x 4 = 32


27


decimal?

64.3 x 0.8 =

64 x 1 = 64

60 x 1 = 60

64 x ½ = 32

60 x ½ = 30

5 1 4 4

Strategies for Dividing Decimals

28

What’s the context?

0.9 3

What’s the concrete model?


29


0.9 3

How could you draw a pictorial model?


30


2 0.5

What’s the concrete model?

1 2 3 4


31

2 0.5

How could you draw a pictorial model?


32

What do you notice? What is tricky here?

Why do these results make sense?

0.2 4 = 0.05 4 0.2 = 20


33


33.8 13


33.8 13 = 2 6

0.26

2.6

26

260


34


decimal?

33.8 13 = 2 6

30 10 = 3

30 15 = 2

36 12 = 3


35


97.5 6.5


97.5 6.5 = 1 5

150

15

1.5

0.15


36


decimal?

97.5 6.5 = 1 5

90 10 = 9

100 10 = 10

100 5 = 20

90 5 = 18

FRACTIONS

A Context to Consider:

38

Half of Jimmy’s garden is roses. Of the

roses, two-thirds are red. What fraction of

Jimmy’s whole garden is red roses?

Model with CONCRETE materials:

Fraction bars; Paper strip

21

32 of

Multiplying Fractions

39

21

32

How can you use a drawing to represent

this problem?


40


21

43

How can you use concrete objects to

model this problem?


41

21

43


this problem?

When and how do you introduce the

standard algorithm for multiplying

fractions?


42


211

322

How can you use concrete objects to

model this problem?


43

211

322


this problem?

What algorithms can you use for

multiplying with mixed numbers?


44

211

322

What algorithms can

you use for multiplying

with mixed numbers?


45

Tasks like this are important for developing

reasoning:

Use words and numbers to explain whether the

product is larger or smaller than the underlined

factor.

18 x 1 16 x 36 x ½

72 x ½ x 2

2

2

11

3

2

3

12

Division with Fractions

46

It’s all about reasoning! Use only pictures

and words to solve these problems.

1. Choose a problem

2. Solve and discuss with your partner

3. Flag any problems you would like to

debrief with the group

4. Move to any other problem

Division with Fractions

47

How do these problems differ?

Write a context for each.

Draw a representation for each.

½ 5 5 ½

Multiply and Divide,

Big and Small

NCCTM 2014

Amanda Northrup

[email protected]

www.linkyy.com/teachandlearn

@msnorthrup

Education

Multiply and Divide, Big and Small