This document outlines instructional strategies for teaching multiplication and division of whole numbers, decimals, and fractions using the concrete-representational-abstract (CRA) approach. It provides examples of using physical objects, drawings, and standard algorithms to develop conceptual understanding at each stage. The CRA approach is demonstrated for topics like multiplying large whole numbers, dividing with decimals, and solving word problems involving fractions.
3. 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. CRA – A Sequence of Instruction
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. CRA – A Sequence of Instruction
5
A = Abstract. Use previous work with
materials and drawings to make sense of
procedures with numbers and symbols.
8. CRA for
Multiplying Large Whole Numbers
8
How would you use representational?
36 x 4
36 x 24
9. CRA for
Multiplying Large Whole Numbers
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.
10. CRA for
Multiplying Large Whole Numbers
10
Abstract strategies for
36 x 4 36 x 24 436 x 24
• Partial products
• Array
• Repeated addition
• Lattice
• Standard algorithm (5th grade)
11. CRA for
Division with Whole Numbers
11
Concrete –
Use 15 chips to show 15 divided by 5
19. Strategies for Multiplying Decimals
19
How could you model 3 x 0.4
using a picture?
0.4 + 0.4 + 0.4 = 1.2
20. Strategies for Multiplying Decimals
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?
23. Strategies for Multiplying Decimals
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. Strategies for Multiplying Decimals
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. Strategies for Multiplying Decimals
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. Strategies for Multiplying Decimals
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. Strategies for Multiplying Decimals
27
What reasoning can you use to place the
decimal?
5 1 4 4
64.3 x 0.8 =
64 x 1 = 64
60 x 1 = 60
64 x ½ = 32
60 x ½ = 30
38. 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
2
1
2 of
3
40. Multiplying Fractions
40
What’s the context?
2
1
3
4
How can you use concrete objects to
model this problem?
41. Multiplying Fractions
41
How can you use a drawing to represent
this problem?
2
1
3
4
When and how do you introduce the
standard algorithm for multiplying
fractions?
42. Multiplying Fractions
42
What’s the context?
2
1 1
2 2
3
How can you use concrete objects to
model this problem?
43. Multiplying Fractions
43
How can you use a drawing to represent
this problem?
2
1 1
2 2
3
What algorithms can you use for
multiplying with mixed numbers?
44. Multiplying Fractions
44
2
1 1
2 2
3
What algorithms can
you use for multiplying
with mixed numbers?
45. Multiplying Fractions
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.
1
1
2
18 x 1 16 x 36 x ½
2
72 x ½ x
2
2
3
1
2
3
46. 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
47. Division with Fractions
47
How do these problems differ?
Write a context for each.
Draw a representation for each.
½ 5 5 ½
48. Multiply and Divide,
Big and Small
NCCTM 2014
Amanda Northrup
anorthrup@haywood.k12.nc.us
www.linkyy.com/teachandlearn
@msnorthrup
Editor's Notes
Take pics of representations – 5 groups of 3, 3 groups of 5, a 3x5 array
How is division related to multiplication? Do all of these representations have the same meaning?
Emphasize that a context can help us understand the meaning of division (and create a matching representation)
Do and discuss
Show video of my students solving this problem
Do and discuss connection between the representational strategies and the previous work with blocks. If time, demonstrate this problem on iPad app “Dare to Share” as well
Using Explain Everything demo circle division for 78 divided by 3, then challenge group with 672 / 5
Allow participants time to solve and discuss, then present their strategies.
Handouts – decimal grids, number lines
Ample discussion here to work out the meaning of multiplication in this context.
Ample discussion before clicking to reveal images
Note – model with the blocks how these are done concrete
The digits are the same. The only uncertainty is – where does the decimal go?
How can we use reasoning to determine where to place the decimal?
Step through each.
Solicit ideas.
Common possibilities included after click
Solicit ideas.
Common possibilities included after click
Before moving on, note that the next lesson would have the students solve several problems using reasoning –
then analyze them to look for patterns. Students DISCOVER the algorithm of counting decimal places this way
.3 in each of 3 groups
Circles
Decimal grids
Number line
Bar model
Notice that we are now dividing a whole number by a decimal instead of dividing a decimal by a whole number.
How does that change things in the problem?
Number line
Bar model
Why won’t circle division work here?
Number lines can help understand why one quotient is so small, and one is so large
Step through each one
Solicit ideas.
Common possibilities included after click
Step through each one
Solicit ideas.
Common possibilities included after click
This one is especially challenging because the decimal isn’t needed for the quotient.
Point out the common misconception that the algorithm of counting decimal places doesn’t work here, as in multiplication. Invite discussion on this point.
Goal end time – 3:30
Note both area model and bar model if no one else mentions these
Note both area model and bar model if no one else mentions these
Note and demonstrate the use of area model algorithm if no one else does
A good time to demonstrate Think Blocks app – Fractions Version
Note and demonstrate the use of area model algorithm if no one else does
A good time to demonstrate Think Blocks app – Fractions Version
Talk through each example End goal time (2:30 + break)
10 problems are posted around the room for groups to solve (20 mins)
Debrief with an emphasis on pictures for problems 1, 2, 4, and 5
End goal time is 4:05
Note the tendency of students to look for a rule/short cut which causes problems when ½ div by 5 is much different than 5 div by ½
Allow time to work independently, share in group, and discuss