multiply and divide, big and small
TRANSCRIPT
Multiply and Divide,
Big and Small
NCCTM 2014
Amanda Northrup
www.linkyy.com/teachandlearn
@msnorthrup
The Need…
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Ma & Pa Kettle Math
CRA – A Sequence of Instruction
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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...
CRA – A Sequence of Instruction
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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...
CRA – A Sequence of Instruction
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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
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Concrete…
36 x 4
36 x 24
CRA for
Multiplying Large Whole Numbers
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How would you use representational?
36 x 24
36 x 4
CRA for
Multiplying Large Whole Numbers
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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
Multiplying Large Whole Numbers
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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
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Concrete –
Use 15 chips to show 15 divided by 5
CRA for
Division with Whole Numbers
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Concrete
78 3
CRA for
Division with Whole Numbers
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Concrete: Use base ten blocks to show 78 3
CRA for
Division with Whole Numbers
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Representational:
Show 78 3 using a drawing
Circle division: 672 5
CRA for
Division with Whole Numbers
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CRA for
Division with Whole Numbers
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Abstract:
672 5
Try it with more than one algorithm
DECIMALS
Strategies for Multiplying Decimals
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How could you model
3 x 0.4
using concrete materials?
Strategies for Multiplying Decimals
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How could you model 3 x 0.4
using a picture?
0.4 + 0.4 + 0.4 = 1.2
Strategies for Multiplying Decimals
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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?
Strategies for Multiplying Decimals
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How could you model 0.1 x 0.1
using concrete materials?
Strategies for Multiplying Decimals
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How could you model
0.1 x 0.1
using a picture?
Strategies for Multiplying Decimals
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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?
Strategies for Multiplying Decimals
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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
Strategies for Multiplying Decimals
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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
Strategies for Multiplying Decimals
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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
Strategies for Multiplying Decimals
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What reasoning can you use to place the
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
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What’s the context?
0.9 3
What’s the concrete model?
Strategies for Dividing Decimals
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What’s the context?
0.9 3
How could you draw a pictorial model?
Strategies for Dividing Decimals
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What’s the context?
2 0.5
What’s the concrete model?
1 2 3 4
Strategies for Dividing Decimals
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2 0.5
How could you draw a pictorial model?
Strategies for Dividing Decimals
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What do you notice? What is tricky here?
Why do these results make sense?
0.2 4 = 0.05 4 0.2 = 20
Strategies for Dividing Decimals
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Using reasoning to place the decimal –
33.8 13
Let’s find the digits.
33.8 13 = 2 6
0.26
2.6
26
260
Strategies for Dividing Decimals
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What reasoning can you use to place the
decimal?
33.8 13 = 2 6
30 10 = 3
30 15 = 2
36 12 = 3
Strategies for Dividing Decimals
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Using reasoning to place the decimal –
97.5 6.5
Let’s find the digits.
97.5 6.5 = 1 5
150
15
1.5
0.15
Strategies for Dividing Decimals
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What reasoning can you use to place the
decimal?
97.5 6.5 = 1 5
90 10 = 9
100 10 = 10
100 5 = 20
90 5 = 18
FRACTIONS
A Context to Consider:
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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
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32 of
Multiplying Fractions
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21
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How can you use a drawing to represent
this problem?
Multiplying Fractions
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What’s the context?
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How can you use concrete objects to
model this problem?
Multiplying Fractions
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21
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How can you use a drawing to represent
this problem?
When and how do you introduce the
standard algorithm for multiplying
fractions?
Multiplying Fractions
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What’s the context?
211
322
How can you use concrete objects to
model this problem?
Multiplying Fractions
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211
322
How can you use a drawing to represent
this problem?
What algorithms can you use for
multiplying with mixed numbers?
Multiplying Fractions
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211
322
What algorithms can
you use for multiplying
with mixed numbers?
Multiplying Fractions
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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
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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
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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
www.linkyy.com/teachandlearn
@msnorthrup