Survey
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
* Your assessment is very important for improving the workof artificial intelligence, which forms the content of this project
Mathematical model wikipedia , lookup
List of important publications in mathematics wikipedia , lookup
Penrose tiling wikipedia , lookup
Laws of Form wikipedia , lookup
Fundamental theorem of algebra wikipedia , lookup
Factorization of polynomials over finite fields wikipedia , lookup
Mathematics of radio engineering wikipedia , lookup
Factorization wikipedia , lookup
Location arithmetic wikipedia , lookup
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 • • • • • • • 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 Geoboard Activities 2 3 4 5 6 7 8 9 10 Geoboard – TAKS Related Problems 11 12 13 14 Algebra Tiles Activities 15 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 16 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 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 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 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 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 21 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 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 = 23 Multiplying Polynomials 1. (x + 2)(x + 3) 2. (x – 1)(x +4) 3. (x + 2)(x – 3) 4. (x – 2)(x – 3) 24 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 25 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.” 26 Solving Equations Mat Left side of equation Right side of equation = 27 Algebra Tile Mat for Multiplying and Factoring 28 29 Algebra Tiles – TAKS Related Problems 30 31 32 Using Multiple Representations 33 34 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. 35 Using Multiple Representations – TAKS Related Problems 36 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 • 2 1 -10 -9 -8 -7 -6 -5 -4 -3 -2 -1 0 1 • 2 3 4 • • 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} 37 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 16 38 Graphing Calculator – TAKS Related Problems 39 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 40