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Let’s Do Algebra Tiles

Let’s Do Algebra Tiles. Roland O’Daniel, The Collaborative for Teaching and Learning October, 2009 Adapted from David McReynolds, AIMS PreK-16 Project and Noel Villarreal, South Texas Rural Systemic Initiative. Algebra Tiles.

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Let’s Do Algebra Tiles

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  1. Let’s Do Algebra Tiles Roland O’Daniel, The Collaborative for Teaching and Learning October, 2009 Adapted from David McReynolds, AIMS PreK-16 Project and Noel Villarreal, South Texas Rural Systemic Initiative

  2. Algebra Tiles • Manipulatives used to enhance student understanding of concepts traditionally taught at symbolic level. • Provide access to symbol manipulation for students with weak number sense. • Provide geometric interpretation of symbol manipulation.

  3. Algebra Tiles • Support cooperative learning, improve discourse in classroom by giving students objects to think with and talk about. • When I listen, I hear. • When I see, I remember. • But when I do, I understand.

  4. 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 opposite or flip-side) represent -1. • The yellow and red squares are additive inverses of each other.

  5. Algebra Tiles • Algebra tiles can be used to model operations involving variables. • Let the green rectangle represent +1xor x and the red rectangle (the flip-side) represent -1x or -x . • The green and red rods are additive inverses of each other.

  6. Algebra Tiles • Let the blue square represent x2. The red square (flip-side of blue) represents -x2. • As with integers, the red shapes and their corresponding flip-sides form a zero pair.

  7. Zero Pairs • Called zero pairs because they are additive inverses of each other. • When put together, they model zero. • Don’t use “cancel out” for zeroes use zero pairs or add up to zero

  8. Addition of Integers • Addition can be viewed as “combining”. • Combining involves the forming and removing of all zero pairs. • Draw pictorial diagrams which show the modeling. • Write the manipulation performed • The following problems are animated to represent how these problems can be solved.

  9. Addition of Integers 4 (+3) + (+1) = • Combine like objects to get four positives (-2) + (-1) = • Combine like objects to get three negatives -3

  10. Addition of Integers (+3) + (-1) = • Make zeroes to get two positives • Important to have students perform all three representations, to create deeper understanding • Quality not quantity in these interactions +2

  11. (+3) + (-4) = • Make three zeroes • Final answer one negative • After students have seen many examples of addition, have them formulate rules. -1

  12. Subtraction of Integers • Subtraction can be interpreted as “take-away.” • Subtraction can also be thought of as “adding the opposite.” (must be extensively scaffolded before students are asked to develop) • Draw pictorial diagrams which show the modeling process • Write a description of the actions taken

  13. Subtraction of Integers (+5) – (+2) = • Take away two positives • To get three positives (-4) – (-3) = • Take away three negatives • Final answer one negative +3 -1

  14. Subtracting Integers +8 (+3) – (-5) = Can’t take away any negatives so add five zeroes; Take away five negatives; Get eight positives

  15. Subtracting Integers (-4) – (+1)= Start with 4 negatives No positives to take away; Add one zero; Take away one positive To get five negatives -5

  16. Subtracting Integers (+3) – (-3)= • After students have seen many examples, have them formulate rules for integer subtraction. (+3) – (-3) is the same as 3 + 3 to get 6

  17. 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 • Write a description of the actions performed

  18. Multiplication of Integers • The counter indicates how many rows to make. It has this meaning if it is positive. (+2)(+3) = (+3)(-4) = +6 Two groups of three positives -12 Three groups of four negatives

  19. Multiplication of Integers • If the counter is negative it will mean “take the opposite of.” • Can indicate the motion “flip-over”, but be very careful using that terminology (-2)(+3) = (-3)(-1) = -6 • Two groups of three • ‘Opposite of’ to get six • negatives +3 • ‘Opposite of’ three groups of negative one • To get three positives

  20. Multiplication of Integers • After students have seen many examples, have them formulate rules for integer multiplication. • Have students practice applying rules abstractly with larger integers.

  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.

  22. Division of Integers (+6)/(+2) = • Divide into two equal groups • 3 in each group (-8)÷(+2) = • Divide into two equal groups 3 -4

  23. Division of Integers • A negative divisor will mean “take the opposite of” (flip-over) (+10)/(-2) = • Divide into two equal groups • Find the opposite of • To get five negatives in each group -5

  24. Division of Integers (-12)/(-3) = • After students have seen many examples, have them formulate rules. +4

  25. Polynomials “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.” David A. Reid, Acadia University

  26. 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)

  27. Distributive Property 3·x + 3·2 = 3x + 6 3(x + 2)= • Three groups of the quantity x and 2 • Three Groups of x to get three x’s • Three groups of 2 to get 6

  28. Modeling Polynomials • Algebra tiles can be used to model expressions. • Model the simplification of expressions. • Add, subtract, multiply, divide, or factor polynomials.

  29. Modeling Polynomials 2x2 4x 3 or +3

  30. More Polynomials • Represent each of the given expressions with algebra tiles. • Draw a pictorial diagram of the process. • Write the symbolic expression. x + 4

  31. More Polynomials 2x + 3 4x – 2

  32. More Polynomials • Use algebra tiles to simplify each of the given expressions. Combine like terms. Look for zero pairs. Draw a diagram to represent the process. • Write the symbolic expression that represents each step. 2x + 4 + x + 2

  33. More Polynomials = 3x + 5 2x + 4 + x + 1 Combine like terms to get three x’s and five positives

  34. More Polynomials 3x – 1 – 2x + 4 • This process can be used with problems containing x2. (2x2 + 5x – 3) + (-x2 + 2x + 5) (2x2 – 2x + 3) – (3x2 + 3x – 2)

  35. (2x2 – 2x + 3) – (3x2 + 3x – 2) Take away the second quantity or change to addition of the opposite 2x2+ -2x + 3 + -3x2+ -3x+ +2 Make zeroes and combine like terms + (-2x + -3x) (2x2 + -3x2) + (3+ 2) Simplify and write polynomial in standard form -x2 – 5x + 5

  36. Substitution • Algebra tiles can be used to model substitution. Represent original expression with tiles. Then replace each rectangle with the appropriate tile value. Combine like terms. 3 + 2x let x = 4

  37. Substitution 3 + 2x = 3 + 2(4) = 3 + 8 = 11 let x = 4

  38. Substitution 4x + 2y = let x = -2 & y = 3 4(-2) + 2(3) = -8 + 6 = -2

  39. 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. • Equivalent Equations are created if equivalent operations are performed on each side of the equation. (the additon, subtraction, mulitplication, or division properties of equality.) • Variables can be isolated by using the Additive Inverse Property (zero pairs) and the Multiplicative Inverse Property (dividing out common factors).

  40. Solving Equations x + 2 = 3 -2 -2 x = 1 x and two positives are the same as three positives add two negatives to both sides of the equation; makes zeroes one xis the same as one positive

  41. Solving Equations -5= 2x ÷2 ÷2 2½ = x • Two x’s are the same as five negatives • Divide both sides into two equal partitions • Two and a half negativesis the same as one x

  42. Solving Equations · -1 · -1 • One half is the same as one negative x • Take the opposite of both sides of the equation • One half of a negativeis the same as one x

  43. Solving Equations •3•3 x = -6 • One third of an x is the same as two negatives • Multiply both sides by three (or make both sides three times larger) • One x is the same as six negatives

  44. Solving Equations 2 x + 3 = x – 5 - x - x x + 3 = -5 + -3 + - 3 x = -8 • Two x’s and three positives are the same as one x and five negatives • Take one x from both sides of the equation; simplify to get one x and three the same as five negatives • Add three negatives to both sides; simplify to get x the same as eight negatives

  45. Solving Equations 3(x – 1) + 5 = 2x – 2 3x – 3 + 5 = 2x – 2 3x + 2 = 2x – 2 – 2 or + -2 3x = 2x – 4 -2x -2x x = -4 “x is the same as four negatives”

  46. Multiplication • Multiplication using “base ten blocks.” (12)(13) • Think of it as multiplying the two binomials (10+2)(10+3) • Multiplication using the array method allows students to see all four sub-products. • The array method reinforces the area model and the foil method

  47. Multiplication using “Area Model” (12)(13) = (10+2)(10+3) = 100 + 30 + 20 + 6 = 156 12 x 13 36 +120 156 10 x 10 = 102 = 100 10 x 3 = 30 10 x 2 = 20 10 x 2 = 20 2 x 3 = 6

  48. Multiplying Polynomials (x + 3) = x2 + (2x + 3x) + 6 = x2 + 5x + 6 (x + 2) Model the dimensions (factors) of the rectangle Fill in each section of the area model Combine like terms and write in standard form

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