Differentiating Math Instruction Using a Variety of Instructional Strategies, Manipulatives and the Graphing Calculator

University of Houston – Central Campus EatMath Workshop October 10, 2009

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Differentiating Math Instruction Using a Variety of Instructional Strategies, Manipulatives and the Graphing Calculator

GEOBOARDS • Explore • Coordinates • Perimeter • Area • Pick’s Formula • Functions • Slope • TAKS RELATED PROBLEMS

ALGEBRA TILES

• Naming Algebra Tiles • Adding, Subtracting, Multiplying, Dividing Integers • Solving Equations • Distributive Property • Modeling Polynomials • Multiplying Polynomials • Factoring Polynomials • Dividing Polynomials • TAKS RELATED PROBLEMS

Multiple Representations

• Verbal Descriptions, Graphs, Tables, Pictorial Drawing • TAKS RELATED PROBLEMS

Graphing Calculator

• TAKS RELATED PROBLEMS

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Geoboard Activities

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Geoboard – TAKS Related Problems

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Algebra Tiles Activities

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Algebra Tiles

Algebra tiles can be used to model operations involving integers. Let the small yellow square represent +1 and the small red square (the flip-

side) represent -1. Let the green rectangle represent X and the red rectangle (the flip-side

represent –X Let the blue square represent X2 and the red square (the flip-side represent --

-X2

Yellow Red

Green Red

Blue Red

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1. (+3) + (+1) =

2. (-2) + (-1) =

3. (+3) + (-1) =

4. (+4) + (-4) = After students have seen many examples of addition, have them formulate rules.

Addition of Integers

Addition can be viewed as “combining”. Combining involves the forming and removing of all

zero pairs. For each of the given examples, use algebra tiles to

model the addition. Draw pictorial diagrams which show the modeling.

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1. (+5) – (+2) =

2. (-4) – (-3) =

3. (+3) – (-5) =

4. (-4) – (+1) =

5. (+3) – (-3) =

After students have seen many examples, have them formulate rules for integer subtraction.

Subtraction of Integers

Subtraction can be interpreted as “take-away.” Subtraction can also be thought of as “adding the

opposite.” For each of the given examples, use algebra tiles to

model the subtraction. Draw pictorial diagrams which show the modeling

process.

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The counter indicates how many rows to make. It has this meaning if it is positive.

1. (+2)(+3) = 2. (+3)(-4) =

If the counter is negative it will mean “take the opposite of.” (flip-over)

1. (-2)(+3) =

2. (-3)(-1) =

Multiplication of Integers

Integer multiplication builds on whole number multiplication. Use concept that the multiplier serves as the “counter” of sets needed. For the given examples, use the algebra tiles to model the multiplication. Identify the multiplier or counter. Draw pictorial diagrams which model the multiplication process.

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1. (+6)/(+2) =

2. (-8)/(+2) =

A negative divisor will mean “take the opposite of.” (flip-over)

1. (+10)/(-2) =

2. (-12)/(-3) =

Division of Integers

Like multiplication, division relies on the concept of a counter.

Divisor serves as counter since it indicates the number of rows to create.

For the given examples, use algebra tiles to model the division. Identify the divisor or counter. Draw pictorial diagrams which model the process.

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1. X + 2 = 3

2. 2X – 4 = 8

3. 2X + 3 = X – 5

Solving Equations

Algebra tiles can be used to explain and justify the equation solving process. The development of the equation solving model is based on two ideas. Variables can be isolated by using zero pairs. Equations are unchanged if equivalent amounts are added to each side of the equation.

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1. 3(X + 2) =

2. 3(X – 4) =

3. -2(X + 2) =

4. -3(X – 2) =

Distributive Property

Use the same concept that was applied with multiplication of integers, think of the first factor as the counter.

The same rules apply. 3(X+2)

Three is the counter, so we need three rows of (X+2)

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1. 2x + 3

2. 4x – 2

3. 2x + 4 + x + 2 =

4. -3x + 1 + x + 3 =

Modeling Polynomials

Algebra tiles can be used to model expressions. Aid in the simplification of expressions. Add, subtract, multiply, divide, or factor polynomials.

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Multiplying Polynomials

1. (x + 2)(x + 3)

2. (x – 1)(x +4)

3. (x + 2)(x – 3)

4. (x – 2)(x – 3)

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1. 3x + 3

2. 2x – 6

3. x2 + 6x + 8

4. x2 – 5x + 6

Factoring Polynomials

Algebra tiles can be used to factor polynomials. Use tiles and the frame to represent the problem.

Use the tiles to fill in the array so as to form a rectangle inside the frame.

Be prepared to use zero pairs to fill in the array. Draw a picture.

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1. x2 + 7x +6 x + 1 2. 2x2 + 5x – 3 x + 3 3. x2 – x – 2 x – 2 “Polynomials are unlike the other “numbers” students learn how to add, subtract, multiply, and divide. They are not “counting” numbers. Giving polynomials a concrete reference (tiles) makes them real.”

Dividing Polynomials

Algebra tiles can be used to divide polynomials. Use tiles and frame to represent problem. Dividend

should form array inside frame. Divisor will form one of the dimensions (one side) of the frame.

Be prepared to use zero pairs in the dividend.

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Solving Equations Mat

=

Left side of equation Right side of equation

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Algebra Tile Mat for Multiplying and Factoring

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Algebra Tiles – TAKS Related Problems

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Using Multiple Representations

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Using Multiple Representations

Find the roots of x2 – 3x + 2 by factoring.

Draw an algebra tile model to demonstrate how to factor x2 – 3x + 2. What are the factors?

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Using Multiple Representations – TAKS Related Problems

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1. The function f(x) = {(1, 2), (2, 4), (3, 6), (4, 8)} can be represented in several other ways. Which is not a correct representation of the function f(x)? A B C x is a natural number less than 5 and y is twice x D y = 2x and the domain is {1, 2, 3, 4}

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x y

x0 1 2 3 4 5 8 9 1076-1-2-3-4-5-6-7-8-9-10

10 9 8 7 6 5 4 3 2 1

-10

-9

-8

-7

-6

-5

-4

-3

-2

-1

• • • •

y

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2. The algebraic form of a linear function is d = 14l, where d is the distance in miles and l

is the number of laps. Which of the following choices identifies the same linear function?

A For every 4 laps on the track, an athlete runs 1 mile. B For every lap on the track, an athlete runs 1

8 mile.

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l d

0 0

2 12

4 14

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l d 14

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1 4

4 16

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Graphing Calculator – TAKS Related Problems

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Resources Used

KISD – Department of Curriculum and Instruction (CSchimek). “Teacher’s Guide to Geosquare Activity Cards”. Scott Scientific. http://www.delmar.edu/aims/Files/Presentations/David_Let's%20Do%20Algebra%20Tiles.ppt www.tea.state.tx.us