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

Transcription

1 Differentiating Math Instruction Using a Variety of Instructional Strategies, Manipulatives and the Graphing Calculator University of Houston Central Campus EatMath Workshop October 10, 2009

2 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 1

3 Geoboard Activities 2

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12 Geoboard TAKS Related Problems 11

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16 Algebra Tiles Activities 15

17 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 X 2 and the red square (the flip-side represent -- -X 2 Yellow Red Green Red Blue Red 16

18 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. 17

19 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. 18

20 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) = 19

21 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) = 20

22 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 = X 4 = X + 3 = X 5 21

23 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) = 22

24 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 x x x + 2 = 4. -3x x + 3 = 23

25 Multiplying Polynomials 1. (x + 2)(x + 3) 2. (x 1)(x +4) 3. (x + 2)(x 3) 4. (x 2)(x 3) 24

26 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 x 6 3. x 2 + 6x x 2 5x

27 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. x 2 + 7x +6 x x 2 + 5x 3 x x 2 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. 26

28 Solving Equations Mat Left side of equation Right side of equation = 27

29 Algebra Tile Mat for Multiplying and Factoring 28

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31 Algebra Tiles TAKS Related Problems 30

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34 Using Multiple Representations 33

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36 Using Multiple Representations Draw an algebra tile model to demonstrate how to factor x 2 3x + 2. What are the factors? Find the roots of x 2 3x + 2 by factoring. 35

37 Using Multiple Representations TAKS Related Problems 36

38 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 y B x 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} 37

39 2. The algebraic form of a linear function is d = 1 l, where d is the distance in miles and l 4 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 D 2 4 l d

40 Graphing Calculator TAKS Related Problems 39

41 Resources Used KISD Department of Curriculum and Instruction (CSchimek). Teacher s Guide to Geosquare Activity Cards. Scott Scientific. s.ppt 40