How many eggs?

3/7/2015

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Using Area Using Area Using Area Using Area

Models to Teach Models to Teach Models to Teach Models to Teach Math ConceptsMath ConceptsMath ConceptsMath Concepts

CANMI April 10, 2015

Connie Rivera

CREC – ATDN

How many eggs?How many eggs?How many eggs?How many eggs?

How did you know?

How many eggs?How many eggs?How many eggs?How many eggs?ObjectivesObjectivesObjectivesObjectives

Participants will…

•Learn what area models are and how they work

•Understand the value of area models as tools for building understanding

•Build area models and discuss their components

•Be equipped to use area models to teach relevant mathematical concepts

I didn’t learn area models in school. I didn’t learn area models in school. I didn’t learn area models in school. I didn’t learn area models in school. Why do my students need them?Why do my students need them?Why do my students need them?Why do my students need them?

•Begins at a concrete rather than abstract level

•Visual and kinesthetic learners will benefit

•Memorizing steps didn’t work for most of our students

•CCR Standards’ Instructional Shift: Rigor

(conceptual understanding) and Coherence

Levels of Knowing MathLevels of Knowing MathLevels of Knowing MathLevels of Knowing MathMahesh Sharma

Communication

Application

Abstract

Pictorial

Concrete

Intuitive

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What is a rectangular What is a rectangular What is a rectangular What is a rectangular array?array?array?array?How can we use them for teaching math?

How many muffins are in this pan?How many muffins are in this pan?How many muffins are in this pan?How many muffins are in this pan?

How many How many How many How many muffins?muffins?muffins?muffins?

How did you know?

Rectangular ArraysRectangular ArraysRectangular ArraysRectangular Arrays

Array Directions:Array Directions:Array Directions:Array Directions:

• Count your bag of tiles. Let me know if you do not have 25 tiles.

• Arrange the tiles in a rectangular array to demonstrate 8 x 3 = 24.

• Look around the table and help others as needed.

Discuss at your table:Discuss at your table:Discuss at your table:Discuss at your table:

• Describe your array.

• Do you see an array that is different than yours? How is it different?

• What equations can you write to describe the array that use multiplication? …division?

• What is the relationship between multiplication and division?

• What shape is your array?

• What are the properties of that shape?

www.elevatingadulteducation.com

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What are the factors of 12?What are the factors of 12?What are the factors of 12?What are the factors of 12?

Show that they are factors by using rectangular arrays.


• What is a factor?

• How would you explain a factor in the context of rectangular arrays?

• What are the factors of 12?

• What are a few examples of numbers that are not factors of 12? How would this look in a rectangular array?

• What equations can you write to describe the arrays that use multiplication? …division?

• What is the relationship between multiplication and division?

• For what other math topics is an understanding of factors essential?


Directions:Directions:Directions:Directions:

• Arrange your tiles so that the factors

are the same



• Do you see an array that is different than yours? How is it different? How is it similar?

• What are the properties of the shape of your array?

• What equations can you write to describe the array that use multiplication? …division? …something else?

• What is the relationship between the shape you see in front of you and exponents?

• How would you explain a square number in the context of rectangular arrays?

What is a measured What is a measured What is a measured What is a measured area model?area model?area model?area model?When is it useful?

A measured area A measured area A measured area A measured area mmmmodel odel odel odel is a type of rectangular arrayis a type of rectangular arrayis a type of rectangular arrayis a type of rectangular array

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Communication

Application

Abstract

Pictorial

Concrete

Intuitive

5 x 17

Area Model of MultiplicationArea Model of MultiplicationArea Model of MultiplicationArea Model of Multiplication

5

10 7

5 x 17 = +

= +

= 85

5 x 17 = 5 (10 + 7)

= 50 + 35

= 85

17

x 5

5 x 10 is

plus 30

is 80

8


26 x 34

24

100

100100100

100 100

40

40

60 60 60

10 410 10

6

10

10


34

x 26

204

+ 680

884

30 + 4

x 20 + 6

24

180

80

+ 600

884

26 x 34

Practice the area model of Practice the area model of Practice the area model of Practice the area model of multiplication with whole numbersmultiplication with whole numbersmultiplication with whole numbersmultiplication with whole numbers

•Put the tiles back in the bag and take a piece of graph paper.

•On graph paper, create a measured area model to multiply 9 x 4.

•Write 2 multiplication and 2 division equations to describe the model.

•On graph paper, create a measured area model to multiply 14 x 25.

•Show the partial products method and how it connects to your area model for 14 x 25.

DecimalsDecimalsDecimalsDecimals

one grid = one whole

0.5 x 0.4

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0.5

0.4

DecimalsDecimalsDecimalsDecimals0.5 x 0.4 = 0.20

.7 x .3

.7

.3

DecimalsDecimalsDecimalsDecimals

How would you ask your students to How would you ask your students to How would you ask your students to How would you ask your students to solve this problem?solve this problem?solve this problem?solve this problem?

Sherri lives ¾ of a mile away from the mall.

When she was half-way to the mall she stopped

to pick up her clothes at the Dry Cleaners.

What fraction of a mile did she still have to

drive to get to the mall?

So, the area model works for decimals So, the area model works for decimals So, the area model works for decimals So, the area model works for decimals –––– what about fractions?what about fractions?what about fractions?what about fractions?

1

2 ·

2

3= ?

1

2

�

x

=

�

Area Model for Area Model for Area Model for Area Model for �

x x x x

2

3

= �

B R E A KB R E A KB R E A KB R E A K

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Fraction Multiplication Fraction Multiplication Fraction Multiplication Fraction Multiplication ––––Measured Area ModelMeasured Area ModelMeasured Area ModelMeasured Area Model

Sherri lives ¾ of a mile away from the mall.

When she was half-way to the mall she stopped

to pick up her clothes at the Dry Cleaners.

What fraction of a mile did she still have to

drive to get to the mall?

What is an abstract What is an abstract What is an abstract What is an abstract area model?area model?area model?area model?How are they useful for solving problems?

As you share this with learners, As you share this with learners, As you share this with learners, As you share this with learners,

do not skip steps in conceptual do not skip steps in conceptual do not skip steps in conceptual do not skip steps in conceptual understanding!understanding!understanding!understanding!

Rectangular array

Measured area model

Abstract area model


Communication

Application

Abstract

Pictorial

Concrete

Intuitive


Long DivisionLong DivisionLong DivisionLong Division 2256 ÷ 6

6 2112

2256- 1800

300

456

456

70

- 42036

36

6

-360

Long DivisionLong DivisionLong DivisionLong Division 2256 ÷ 6

22566376

18

4542

3636

0

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Activities with Area Models will give Activities with Area Models will give Activities with Area Models will give Activities with Area Models will give opportunities for…opportunities for…opportunities for…opportunities for…

• Thinking about math

• Reasoning through problems using mathematical thinking

• Hearing math vocabulary

• Using math vocabulary

• Explaining the process

• Examining various solutions

• Asking and answering clarifying questions

• Asking and answering follow-up questions


The purpose of using Area The purpose of using Area The purpose of using Area The purpose of using Area Models…Models…Models…Models…

• Is to add a layer of conceptual understanding to the learner’s experience…

Not to replace all computation methods.

So, this is all nice, but I have to So, this is all nice, but I have to So, this is all nice, but I have to So, this is all nice, but I have to

prepare my students to be college prepare my students to be college prepare my students to be college prepare my students to be college

and career ready. I have to teach and career ready. I have to teach and career ready. I have to teach and career ready. I have to teach them algebra.them algebra.them algebra.them algebra.

Remember! Do not skip steps in Remember! Do not skip steps in Remember! Do not skip steps in Remember! Do not skip steps in conceptual understanding.conceptual understanding.conceptual understanding.conceptual understanding.

Rectangular array

Measured area model

Abstract area model


Variables Variables Variables Variables

• Take out your tiles.

• Create a rectangular array to represent 4r



• Do you see an array that is different than yours? How is it different?

• How are the other arrays similar to yours?

• What is r equal to in your array?

• On paper, sketch your array as a measured area model.

• Now draw it as an abstract area model.

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43

(x + 2)(x + 5)Imagine this

to be x.

Here is

another x

2

5

Multiplying BinomialsMultiplying BinomialsMultiplying BinomialsMultiplying Binomials

44

x · x

x

x

2

5

x · 2

5 · x 5 · 2

(x + 2)(x + 5)


45

�

x

x

2

5

2�

5� 10

Now add all the

areas together:

x2 + 2x + 5x + 10

x2 + 7x + 10

…and simplify:

(x + 2)(x + 5)


Now you try it!Now you try it!Now you try it!Now you try it!

x

+ 2

x + 6

x2+ 6x

+ 2x + 8

(x + 6)(x + 2)


What does this have to do with What does this have to do with What does this have to do with What does this have to do with math anxiety?math anxiety?math anxiety?math anxiety?

•Math has structure.

•Math fits with what I understand.

•Math is predictable.

• I can trust it.

• I don’t always need to memorize steps.

• Important ideas work at all levels.


Abstract area model practiceAbstract area model practiceAbstract area model practiceAbstract area model practice

• Evaluate 4(3 + 2) by using an area model.

• What property does this illustrate?

• Evaluate x(y + 4) by using an area model.

• What property does this illustrate?

• Find the product of (x + 3) and (x – 5).

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Multiplying Multiplying Multiplying Multiplying BinomialsBinomialsBinomialsBinomials

www.openmiddle.com

Use an abstract area model

to find the possible

combinations of factors

whose product would be

30x – 12.

Communication

Application

Abstract

Pictorial

Concrete

Intuitive


CCR Standards ConnectionsCCR Standards ConnectionsCCR Standards ConnectionsCCR Standards Connections

• Level B: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities, e.g., by using drawings and equations with a symbol for the unknown number to represent the problem.

• Level B: Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.

• Level C: Multiply a whole number of up to four digits by a one-digit whole number, and multiply two two-digit numbers, using strategies based on place value and the properties of operations. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

• Level C: Find whole-number quotients and remainders with up to four-digit dividends and one-digit divisors, using strategies based on place value, the properties of operations, and/or the relationship between multiplication and division. Illustrate and explain the calculation by using equations, rectangular arrays, and/or area models.

Explore:Explore:Explore:Explore:

Using Area Models in Using Area Models in Using Area Models in Using Area Models in Various SettingsVarious SettingsVarious SettingsVarious SettingsHow can this work for me?

Fraction MultiplicationFraction MultiplicationFraction MultiplicationFraction Multiplication

Demonstrate how to use a measured area model to

solve these problems. Choose two problems to show

a relationship to the traditional algorithm.

Materials: blank paper, graph

paper, pencils, colored pencils

Long Long Long Long DivisionDivisionDivisionDivision

Demonstrate how to use an abstract area model to

solve these problems. Choose two problems to show

a relationship to the traditional algorithm. Choose

two to write an equation to show how you can check

your answer.

Materials: blank paper, pencils,

colored pencils

For a demonstration, visit: https://www.youtube.com/watch?v=N7boECP9BAE

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Array gameArray gameArray gameArray game

1. In groups of three or two, take turns rolling the dice. Color

in an area on the grid indicated by the dice. For example,

if you roll a 2 and then a 3, color a 2 x 3 rectangle.

2. Write the number of squares in the rectangle to indicate

the product of the two sides.

3. The first player to color in all the squares in their grid wins.

4. As the grids fill up, you will roll totals that will not fit on

the grid, you can break up the factors if you choose. For

example, a you might identify that 6 x 4 is the same as

2 x 4 and 4 x 4. This implicitly reinforces the distributive

property of multiplication.

http://nzmaths.co.nz/resource/array-game

Materials: One colored pencil per person, one

sheet of graph paper per person, and one pair of

dice per group.

Right Triangles Right Triangles Right Triangles Right Triangles –––– Square TilesSquare TilesSquare TilesSquare Tiles

� + � = �

� x � + (� x �) = �

“Multiply the sides to get the square and add the legs together

it equals the square of the hypotenuse.”

Students discovered a relationship

between the squares of the legs

and the square of the hypotenuse.

Here are their own rules based on

their discoveries:

Planning the Levels of Knowing Planning the Levels of Knowing Planning the Levels of Knowing Planning the Levels of Knowing MathematicsMathematicsMathematicsMathematics

• Think of something you usually teach, possibly

something we touched on today. Suggestions:

multiplication of whole numbers, decimals, fractions,

or polynomials; long division; or distributive property

• Find “Planning the Levels of Knowing Mathematics”

• Identify where your activity belongs on the levels

listed on the planning sheet.

• Fill in the spaces before and/or after the activity

• Refer to “Questions to Develop Proficiency in the

Mathematical Practices” as needed to aid in

completing the Communication section

ObjectivesObjectivesObjectivesObjectives

Participants will…

• Learn what area models are and how they work

• Understand the value of area models as tools for building understanding

• Build area models and discuss their components

• Be equipped to use area models to teach relevant mathematical concepts

Thank you!Thank you!Thank you!Thank [email protected]

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How many eggs?