1150 day 5

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1150 day 5

  1. 1. AlgebraicThinkingandAlgebra
  2. 2. Algebraic Thinking involves:•Recognizing patterns•Modeling situations withobjects, pictures or symbols•Analyzing the effects of change Understanding variables (letters that stand for an unknown) is key to algebraic thinking.
  3. 3. The evolution of algebraic thinking (an example) 3 + =3 1 + 2 = 3 + 2 = 3 x + 2 = 3
  4. 4. MAP released item, 2006, grade 3
  5. 5. DESE released item 2004 MAP test, grade 4
  6. 6. MAP released item, 2006, 5th grade
  7. 7. MAP released item, 2006, grade 7
  8. 8. What does an equal sign mean? 1 + 2 = 3 1 + 2 = ETA Cuisenaire
  9. 9. Model the equation using a balance scale. + 2 = 3
  10. 10. Model the equation using a balance scale. + 2 = 3 2 1
  11. 11. x + 2 = 3 -2 -2x = 1
  12. 12. http://illuminations.nctm.org/ActivityDetail.aspx?ID=26http://illuminations.nctm.org/ActivityDetail.aspx?ID=33
  13. 13. Properties of EquationsAddition Property of Equality If a = b, then a + c = b + c 4=4 4+2=4+2
  14. 14. Multiplication Property of Equality If a = b, then a · c = b · c 4=4 4·2=4·2
  15. 15. Distributive Property of Multiplication overAddition a(b + c) = a · b + a · c 2(3 + 4) = 2 · 3 + 2 · 4 5(14) = 5(10 + 4) = 5 · 10 + 5 · 4 = 50 + 20 = 70
  16. 16. Model the equation 2x + 1 = 7 using a balance scale. x +2x + 1 x = 7 1 6 3 3
  17. 17. If you triple a number and add 3, the result is 36. Findthe number. Algebra Algebraic methods 3x + 3 = 36 - 3 -3 3x = 33 3 11 36 11 33 11 3 3 3 x = 11
  18. 18. Kate bought a TV on sale for $160. If the sale was a“1/3 off” sale, what was the original price of the TV?AlgebraLet x = the original price original price - 1/3 of the original price = sale price 1 x - x 160 3 3 2 3 x 160 2 3 2 x = 240
  19. 19. Kate bought a TV on sale for $160. If the sale was a“1/3 off” sale, what was the original price of the TV?Algebraic methods $80 $80 $80 1 $160 Sale price off 3 3(80) = $240
  20. 20. Write down any three consecutive numbers.Multiply the first and third numbers.Square the middle number.What do you notice? First times last is one less than square of middle 5 6 7 36 n–1 n n+1 35 n2 (n – 1)(n + 1) 10 11 12 121 (n – 1)(n + 1) = n2 – 1 120 difference of squares
  21. 21. FunctionsA function from set A to set B is a correspondencefrom set A to set B in which every element of set Ais paired with exactly one element of set B. Input (domain) Output (range)
  22. 22. DESE sample MAP item, 3rd grade
  23. 23. Representations of Functions Table Ordered pairs Input Output {(1, 4), (2, 8), (3, 12), (4, 16)} 1 4 2 8 Domain: {1, 2, 3, 4} 3 12 Range: {4, 8, 12, 16} 4 16Arrow Diagram Function Notation 1 4 2 8 f(x) = 4x 3 12 4 16
  24. 24. Representations of Functions Graph Input Output 16 1 4 2 8 12 3 12 output 4 16 8 4 1 2 3 4 input
  25. 25. Use function notation to write a rule for the function {(1, 2), (2, 5), (3, 8), (4, 11)}Input Output Arithmetic sequence 1 a1 = 2 an = a1 + (n – 1)d 2 a2 = 5 a1 = 2 3 a3 = 8 d=3 4 a4 = 11 an = 2 + (n – 1)3 an = 2 + 3n – 3 an = 3n – 1 f(x) = 3x - 1
  26. 26. Composite Functions x f g f(x) g(f(x)) Notation: (g o f)(x) means g(f(x))
  27. 27. If f(x) = 2x – 1 and g(x) = 3x + 1, what is (f o g)(3) ? (f o g)(3) = f(g(3)) g(3) = 3(3) + 1 = f(10) =9+1 = 19 = 10 f(10) = 2(10) – 1 = 20 – 1 = 193 g f 10 19

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