1. Connecting Division to Multiplication for Larger Numbers
Math Alliance Project
Tuesday, July
DeAnn Huinker, Beth Schefelker, Melissa Hedges, & Chris Guthrie

2. 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.

3. 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 ≈ , ,000

4. Nearest Answer Ten Minute Math, Dale Seymour Publications
82 ÷ 4 ≈
250 ÷ 8 ≈ ,000
3895 ÷ 39 ≈ ,000
268 ÷ 9.9 ≈ ,500
Chart thinking 4

5. Solving 48 ÷ 6 or 36 ÷ 9
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?

6. 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?
“What number times what number will get me close to the target number?”
I can’t go over the target number
How many will be leftover? “

7. “Near Facts” using Missing Factors
4 × → with left over
7 × → with left over
9 × → with left over
12 × → with left over

8. 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” “too high”
“left over” “too low”

9. Big Idea That’s Surfacing?
How is this type of multiplicative thinking different from using the US Standard Algorithm?
Quantity focused vs. digit-based 12
145

10. 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)‏

11. Subtracting out “multiple same-sized groups”
Start with → 40 × 7 = 280
317
- 280 (40 × 7 = 280) subtract out 40 groups of 7 37
- 35 (5 × 7 = 70) subtract out 5 groups of 7 2
So…how many groups of 7 are there in 317? How do you know?
How many are leftover? How do you know?

12. Keeping track of thinking using the Partial Quotient Algorithm (Ladder Method)
Forty groups of 7 is equal to 280 45
7 317
× 7 = 280 37
× 7 = 35
Craft a story that would “match“ this thinking.
Is it a measurement or partitive story? Why?
Five groups of 7 is equal to 35

13. Partition the dividend Solve 317 ÷ 7; Start with 40 × 7 = 280
317 ÷ 7 =
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 = 45.
My answer is 45 remainder 2

14. Not all thinking begins this efficiently!
How might a student with developing understanding use the repeated subtraction or ladder method?
× 7 = 7
300
× 7 =14
286
etc...
I see the 7 in 317. I am going to take out 1 group of 7.
OK…I guess that I can take out 2 more groups of 7.

15. 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.”

16. Looking at student work
Discuss:
The variety of approaches
Demonstration of conceptual understanding of division

17. 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.

18. Homework Beckmann p. 200 Class Activity 7I 1a & 1b
Also suggested but not required:
p. 204 Class Activity 7
Numerous opportunities to practice the Partial Quotient/Ladder Algorithm for Division