Transcript Connections between multiplication and division

Connecting Division to
Multiplication
for Larger Numbers
Math Alliance Project
Tuesday, July 6. 2010
DeAnn Huinker, Beth Schefelker,
Melissa Hedges, & Chris Guthrie
Learning Intention
and Success Criteria
We are learning to…
 Explore the meaning of division with multiplication
You will be successful when you can…
 Use prior knowledge of multiplication to solve
division problems.
Nearest Answer
Ten Minute Math, Dale Seymour Publications



Study each problem and the possible
“nearest answer” choices.
Select your “nearest answer.”
Be ready to share your thinking.
Example
9 × 211 ≈
20
200
2,000
20,000
Nearest Answer
Ten Minute Math, Dale Seymour Publications
82 ÷ 4 ≈
2
10
20
40
250 ÷ 8 ≈
15
30
300
2,000
3895 ÷ 39 ≈
0.10
10
100
1,000
268 ÷ 9.9 ≈
2,500
25
250
2.5
Solving 48 ÷ 6 or 36 ÷ 9

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What thought process do you use to recall
the facts above?
Why?
Something to think about…
 When children are working on a page of
division facts, are they practicing division or
multiplication?
Tapping into multiplicative thinking…
“How close can you get?”
“What number times what number
will get me close to the target number?”
• I Don’t go over the target number
• How many will be leftover?
“Near Facts” using Missing Factors

4×
→ 23
with
left over

7×
→ 52
with
left over

9×
→ 86
with
left over

12 ×
→ 145
with
left over
Tapping into multiplicative thinking…
How close can you get?

Find the largest factor without going over
the target number.

Jot down your thinking in the recording sheet.


Write the accompanying division sentence.
Share your thinking
“groups of”
“left over”
“too high”
“too low”
Big Idea That’s Surfacing?

How is this type of multiplicative thinking
different from using the US Standard
Algorithm?
12
145
Quantity focused vs. digit-based
Applying “missing factor” thinking
to larger numbers

317 ÷ 7 =

Restate as a missing factor. “How many groups of 7?”
× 7 = 317

Where might you begin?
10 × 7 = 70 (too small)
20 × 7 = 140 (too small)
30 × 7 = 210 (too small)
40 × 7 = 280 (getting closer, I’ll keep going)
50 × 7 = 350 (too big, I’ll go back to 7 × 40)
Subtracting out “multiple same-sized
groups”
Start with → 40 × 7 = 280
317
- 280
37
- 35
2
(40 × 7 = 280) subtract out 40 groups of 7
(5 × 7 = 70) subtract out 5 groups of 7
So…how many groups of 7 are there in 317?
How do you know?
How many are leftover? How do you know?
Keeping track of thinking using the
Partial Quotient Algorithm (Ladder
Method)
Forty groups of
45
7 317
- 280
40 × 7 = 280
37
- 35 + 5 × 7 = 35
2
45
7 is equal to 280
Five groups of 7
is equal to 35
Craft a story that would “match“ this thinking.
Is it a measurement or partitive story? Why?
Partition the dividend
Solve 317 ÷ 7; Start with 40 × 7 = 280
317 ÷ 7 =
280 + 37
I know that 40 × 7 = 280.
I partition 317 into 280 and 37.
I know that there are 40 groups of 7 in 280,
so 280 ÷ 7 = 40
I know that there are 5 groups of 7 in 37 with 2
leftover, so 37 ÷ 7 = 5 with 2 left over.
I know that 40 + 5 = 45.
My answer is 45 remainder 2
Not all thinking begins this efficiently!
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7
How might a student with developing
understanding use the repeated subtraction
or ladder method?
317
-7
300
-14
286
etc...
I see the 7 in 317. I am
going to take out 1
group of 7.
1×7=7
2 × 7 =14
OK…I guess that I can
take out 2 more
groups of 7.
Scoops of lima beans…
There were 676 lima beans in a jar. I take out
18 lima beans with each scoop. How many
scoops can I make?
Using “missing factor” thinking and the Partial
Quotients/Ladder Algorithm, discuss the
continuum of possible strategies starting from
“least efficient” moving to “very efficient.”
Looking at student work
Discuss:
 The variety of approaches
 Demonstration of conceptual understanding
of division
A concept-based definition of division
Revisit the definition of division you started last week.
Share with your table.
As a table, draft a new definition of division and chart.
Goal: To develop and use a conceptually-based definition
for division.
Visualize
actions on quantities (not numbers).
General, encompass many situations and interpretations
(not limiting to just one view).
Accurate in the long term (don’t set students up for
misconceptions).
Language used attends to the conceptual meaning of the
operation.
Homework
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Beckmann
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p. 200 Class Activity 7I 1a & 1b
p. 201 Class Activity 7I #2 & #3
Also suggested but not required:
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p. 204 Class Activity 7
Numerous opportunities to practice the Partial
Quotient/Ladder Algorithm for Division