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Differentiating Math Instruction Using a Variety of Instructional
Strategies, Manipulatives and the Graphing Calculator
University of Houston – Central Campus
EatMath Workshop
October 10, 2009
Differentiating Math Instruction Using a Variety of Instructional
Strategies, Manipulatives and the Graphing Calculator
GEOBOARDS
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Explore
Coordinates
Perimeter
Area
Pick’s Formula
Functions
Slope
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TAKS RELATED PROBLEMS
ALGEBRA TILES
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Naming Algebra Tiles
Adding, Subtracting, Multiplying, Dividing Integers
Solving Equations
Distributive Property
Modeling Polynomials
Multiplying Polynomials
Factoring Polynomials
Dividing Polynomials
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TAKS RELATED PROBLEMS
Multiple Representations
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Verbal Descriptions, Graphs, Tables, Pictorial Drawing
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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 flipside) 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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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.
1. (+3) + (+1) =
2. (-2) + (-1) =
3. (+3) + (-1) =
4. (+4) + (-4) =
After students have seen many examples of addition, have them formulate rules.
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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.
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.
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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.
| 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) =
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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.
1. (+6)/(+2) =
2. (-8)/(+2) =
| A negative divisor will mean “take the opposite of.” (flip-over)
1. (+10)/(-2) =
2. (-12)/(-3) =
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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.
1.
X+2=3
2.
2X – 4 = 8
3.
2X + 3 = X – 5
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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)
1. 3(X + 2) =
2. 3(X – 4) =
3. -2(X + 2) =
4. -3(X – 2) =
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Modeling Polynomials
| Algebra tiles can be used to model expressions.
| Aid in the simplification of expressions.
| Add, subtract, multiply, divide, or factor polynomials.
1. 2x + 3
2. 4x – 2
3. 2x + 4 + x + 2 =
4. -3x + 1 + x + 3 =
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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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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.
1.
3x + 3
2.
2x – 6
3.
x2 + 6x + 8
4.
x2 – 5x + 6
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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.
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.”
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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
Draw an algebra tile model to
demonstrate how to factor
x2 – 3x + 2. What are the
factors?
Find the roots of x2 – 3x + 2 by factoring.
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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
x
y
1
2
2
4
3
6
4
8
y
10
9
8
7
6
B
5
4
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2
1
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
0
1
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2
3
4
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3
5
6
7
8
9
10
x
-1
-2
-3
-4
-5
-6
-7
-8
-9
-10
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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2. The algebraic form of a linear function is d =
1
4
l, 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.
C
l
d
0
0
2
4
1
2
1
4
D
l
d
1
1
4
1
4
4
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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%20Tile
s.ppt
www.tea.state.tx.us
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