Connecting Division to Multiplicationfor Larger Numbers

Math Alliance ProjectTuesday, July 6. 2010

DeAnn Huinker, Beth Schefelker, Melissa Hedges, & Chris Guthrie

Learning Intentionand 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 AnswerTen 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 AnswerTen 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 ≈ 25 250 2.5 2,500

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?

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” “too high”“left over” “too low”

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

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 (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?

Keeping track of thinking using the Partial Quotient Algorithm (Ladder Method) 45 7 317 - 280 40 × 7 = 280 37

- 35 + 5 × 7 = 35 2 45

Craft a story that would “match“ this thinking.Is it a measurement or partitive story? Why?

Forty groups of 7 is equal to

280

Five groups of 7 is equal to 35

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! How might a student with developing

understanding use the repeated subtraction or ladder method?

7 317 -7 1 × 7 = 7 300 -14 2 × 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.

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 divisionRevisit 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

Beckmann p. 200 Class Activity 7I 1a & 1b p. 201 Class Activity 7I #2 & #3 Also suggested but not required:

p. 204 Class Activity 7 Numerous opportunities to practice the Partial

Quotient/Ladder Algorithm for Division