Using precise notation to explore the structure of multiplication

Ryan Casey5th-Grade STEM TeacherBoston Public Schools

rcasey@bostonpublicschools.org

Is 3 x 5 the same as 5 x 3?

Check-in

• What’s on your mind?

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Page 3

Yesterday…

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Whole numbers are rational numbers.

Any general task that is applicable to fractions

should also work for whole numbers.

Linear measurement can help bridge this

understanding.

An area-model task

• Task 1: Answer the following 3 questions:

– Where do you see 2/3 in this diagram?

– Where do you see 1/4 in this diagram?

– Where do you see 1/8 in this diagram?

• Task 2: Could an equivalent task be created with linear measurement? If so, draw it.

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Ways to develop linear measurement

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2 problems with Cuisenaire rods

• Not everyone has them and they cost a fair amount of money

• They are measures of volume.6

Pipe-cleaner Cuisenaire rods

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2 different manipulatives

Consider the following tasks:

1.) Which is larger 1/2 or 2/3? Arrange your fraction blocks to show which is greater.

2.) Which is larger 1/2 or 2/3? Arrange your pipe cleaners to show which is greater.

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Some pipe-cleaner tasks

1. If we call the green pipe cleaner “24”, what’s the value of…

– …a red pipe cleaner?

– …a white pipe cleaner?

– …a yellow pipe cleaner?

– ….an orange pipe cleaner?

2. Now repeat #1 and call the green pipe cleaner “1”.

3. If the yellow pipe cleaner is given the value “1”, what’s the value of a white and a brown?

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What students gain

• They have to think about the unit. (This is not the case with fraction strips.)

• A common strategy is to scale down (partition) and then up (iterate), but there are many ways of engaging with the task.

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• The Lesh Model

p. 25, citing Lesh, Post, and Behr (1987)

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Definition of mathematical understanding

“A mathematical idea is understood if it is part of an internal network.”

“A mathematical idea, procedure, or fact is understood thoroughly if it is linked to exiting networks with stronger or more numerous connections.” (p. 67)

Hiebert, J., & Carpenter, T.P. (1992). Learning and teaching with understanding. Handbook of Research on Mathematics Teaching and Learning: 65-92

Is 3 x 5 the same as 5 x 3?

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3 x 5

5 x 3

5 x 3 = 3 x 5 = 7 + 8 = 21 - 6

Grade 3, CCSS-M

7+8

21 - 6

3 x 5 or 5 x 3

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3 x 5 or 5 x 3

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3 x 5 or 5 x 3

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3 x 5 or 5 x 3

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3 x 5 or 5 x 3

Unambiguously 5 x 3.

Unambiguously 3 x 5

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3 x 5 or 5 x 3

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There are 3 T of vanilla in used each recipe. After making the recipe 5 times, how much vanilla have I?

There are 3 groups of students. Each group has 5 students. How many students are there in total?

Last week, Jay’s sister ran 5 times as much as Jay did. Jay ran 3 miles. How many miles did Jay run?

Yesterday, I spent $5. I spent 3 times that much today. How much did I spend?

5 x 3

5 x 3

3 x 5

3 x 5

Some definitions

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2 x 10 10 x 2

multiplier multiplicandscale factor

Scale factor: The first factor written in a multiplication

expression, expressing the number of groups

unit rate1

Unit rate: The quantity in “one” group.

What are the units?

There are 6 doughnuts in each box. There are 5 boxes. All together there are ____ doughnuts.

Represent this in several ways.

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6 x 5 = 30

6 boxes of 55 doughnuts

30 doughnuts

2 contexts

At breakfast there were 10

muffins on each tray. trays of

muffins were eaten. In total, how

many muffins were eaten?

Every morning, Kayla runs

miles. How many miles did Kayla

run after running for 10 mornings?

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5

22

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2 equations

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1

3´ 6 = ? 6´

1

3= ?

2 diagrams

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6´1

3= ?

Multiplicative Structures

Whole number Unit fraction Non-unit fraction<1

“mixed number”

Whole number

Unit fraction

Non-unit fraction <1

“mixednumber”

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unit r

ate

scale factor

Multiplicative structures

Whole number Unit fraction Non-unit fraction “mixed number”

Divisible whole number

Non-divisiblewhole number

Unit fraction

non-unit fraction

“mixednumber”

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unit r

ate

scale factor

3´14

3 ´1

5

3 ´4

5

3´ 24

5

1

3´

1

5

1

3´

4

5

1

3´ 2

4

5

1

3´14

1

3´15

2

3´

1

5

2

3´

4

5

2

3´ 2

4

5

2

3´14

2

3´15

21

3´

1

5

21

3´

4

5

21

3´ 2

4

5

21

3´14

21

3´15

Multiplicative structures

Whole number Unit fraction Non-unit fraction “mixed number”

Divisible whole number

Non-divisiblewhole number

Unit fraction

non-unit fraction

“mixednumber”

27

unit r

ate

scale factor

3´14

3 ´1

5

3 ´4

5

3´ 24

5

1

3´

1

5

1

3´

4

5

1

3´ 2

4

5

1

3´14

1

3´15

2

3´

1

5

2

3´

4

5

2

3´ 2

4

5

2

3´14

2

3´15

21

3´

1

5

21

3´

4

5

21

3´ 2

4

5

21

3´14

21

3´15

Multiplicative structures

Whole number Unit fraction Non-unit fraction “mixed number”

Divisible whole number

Non-divisiblewhole number

Unit fraction

non-unit fraction

“mixednumber”

28

unit r

ate

scale factor

3´14

3 ´1

5

3 ´4

5

3´ 24

5

1

3´

1

5

1

3´

4

5

1

3´ 2

4

5

1

3´14

1

3´15

2

3´

1

5

2

3´

4

5

2

3´ 2

4

5

2

3´14

2

3´15

21

3´

1

5

21

3´

4

5

21

3´ 2

4

5

21

3´14

21

3´15

Disclaimer

The National Council of Teachers of Mathematics is a public voice of mathematics education, providing vision, leadership, and professional development to support teachers in ensuring equitable mathematics learning of the highest quality for all students. NCTM’s Institutes, an official professional development offering of the National Council of Teachers of Mathematics, supports the improvement of pre-K-6 mathematics education by serving as a resource for teachers so as to provide more and better mathematics for all students. It is a forum for the exchange of mathematics ideas, activities, and pedagogical strategies, and for sharing and interpreting research. The Institutes presented by the Council present a variety of viewpoints. The views expressed or implied in the Institutes, unless otherwise noted, should not be interpreted as official positions of the Council.

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