1. REVIEWING THE MODELS FOR SOLVING EQUATIONS Robert Yen Hurlstone Agricultural High School

2. OVERVIEW  To review and compare 3 models for teaching equations  Students often have trouble solving equations because their teachers teach only one method, the method they were taught themselves

3. OVERVIEW  To discuss and share ideas and classroom experiences on teaching equations

4. Model 1: GUESS, CHECK AND IMPROVE Alternative names Guess and check Guess and check Guess, check and refine Guess, check and refine Trial and error Trial and error ‘By inspection’ ‘By inspection’

5. Model 1: GUESS, CHECK AND IMPROVE Description The name explains the method: guess the solution, test it, make a better guess, keep testing The name explains the method: guess the solution, test it, make a better guess, keep testing It is a process that cycles, is repetitive It is a process that cycles, is repetitive

6. Model 1: GUESS, CHECK AND IMPROVE Example 1 x + 5 = 40 By inspection, x = 35 because 35 + 5 = 40.

7. Model 1: GUESS, CHECK AND IMPROVE Example 2 = 10 = 10 By inspection, d = 30 because = 10.

8. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 GuessCheckComment x = 10

9. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 GuessCheckComment x = 10 4  10 + 5 = 45 Too low x = 14

10. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 GuessCheckComment x = 10 4  10 + 5 = 45 Too low x = 14 4  14 + 5 = 61 Too high x = 12

11. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 GuessCheckComment x = 10 4  10 + 5 = 45 Too low x = 14 4  14 + 5 = 61 Too high x = 12 4  12 + 5 = 53 Too low x = 13

12. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 GuessCheckComment x = 10 4  10 + 5 = 45 Too low x = 14 4  14 + 5 = 61 Too high x = 12 4  12 + 5 = 53 Too low x = 13 4  13 + 5 = 57 Correct!

13. Model 1: GUESS, CHECK AND IMPROVE Example 3 4x + 5 = 57 x = 13. GuessCheckComment x = 10 4  10 + 5 = 45 Too low x = 14 4  14 + 5 = 61 Too high x = 12 4  12 + 5 = 53 Too low x = 13 4  13 + 5 = 57 Correct!

14. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHS x = 10 344 x = 5

15. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHS x = 10 344 x = 5 19 x = 1

16. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHS x = 10 344 x = 5 19 x = 1 7-5

17. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHSLHS-RHS x = 10 34430 x = 5 1920 x = 1 7-512 x = -2

18. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHSLHS-RHS x = 10 34430 x = 5 1920 x = 1 7-512 x = -2 -2-86 x = -4

19. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHSLHS-RHS x = 10 34430 x = 5 1920 x = 1 7-512 x = -2 -2-86 x = -4 -8-102 x = -5

20. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 GuessLHSRHSLHS-RHS x = 10 34430 x = 5 1920 x = 1 7-512 x = -2 -2-86 x = -4 -8-102 x = -5 -11-110

21. Model 1: GUESS, CHECK AND IMPROVE Example 4 3x + 4 = x – 6 x = -5 GuessLHSRHSLHS-RHS x = 10 34430 x = 5 1920 x = 1 7-512 x = -2 -2-86 x = -4 -8-102 x = -5 -11-110

22. Model 1: GUESS, CHECK AND IMPROVE What the syllabus says (p.86, PAS4.4) ‘Five models have been proposed to assist students with the solving of simple equations... Model 4 uses a substitution approach. By trial and error a value is found that produces equality for the values on either side of the equation (this highlights the variable concept).’

23. Model 1: GUESS, CHECK AND IMPROVE Advantages  ???

24. Model 1: GUESS, CHECK AND IMPROVE Advantages  Reinforces aim of solving equations and algebraic concepts of unknown, variable  Reinforces checking of solutions  Simple to understand and apply  Feedback on partial solutions, homing in on answer, unlike algebraic methods where one careless error will undermine the solution process

25. Model 1: GUESS, CHECK AND IMPROVE Advantages  Being repetitive, can be performed via technology: spreadsheet, graphics calculator  With better guesses, solution can be found quickly  Can improve students’ computation skills and number sense

26. Model 1: GUESS, CHECK AND IMPROVE Disadvantages  ???

27. Model 1: GUESS, CHECK AND IMPROVE Disadvantages  Guesswork is not an ‘elegant’ method  Harder to apply for more complex equations (such as x on both sides)  May be hard to test values that are large or negative  More time-consuming if guesses are bad

28. Model 2: BALANCING Alternative name ‘Doing the same thing to both sides’ (of the equation) ‘Doing the same thing to both sides’ (of the equation)

29. Model 2: BALANCING Description The traditional algebraic method Models the equation as balance scales, upon which the same inverse operations are performed on both sides to create equivalent equations until it simplifies to ‘x = ___’

30. Model 2: BALANCING Description Invented by Arab mathematician Invented by Arab mathematician al-Khwarizmi in AD 825, who wrote al-Khwarizmi in AD 825, who wrote Hisab al-jabr w’al-muqabalah ‘The science of restoration and cancellation’  al-jabr = restoration by balancing, from which we get the name ‘algebra’ from which we get the name ‘algebra’  muqabalah = cancellation of terms

31. Model 2: BALANCING Description From al-Khwarizmi’s name, we get the name ‘algorithm’ From al-Khwarizmi’s name, we get the name ‘algorithm’ Can be demonstrated using concrete objects such as cups, envelopes, counters, coins or coloured dots Can be demonstrated using concrete objects such as cups, envelopes, counters, coins or coloured dots Or coloured plastic bottle caps (see session by Kevin Fuller tomorrow @ 2 pm) Or coloured plastic bottle caps (see session by Kevin Fuller tomorrow @ 2 pm)

32. Model 2: BALANCING Example 1 (A concrete model) 2x + 7 = 9

33. Model 2: BALANCING Example 1 (A concrete model) 2x + 7 = 9 Subtract 7 coins from both sides

34. Model 2: BALANCING Example 1 2x + 7 = 9 Place the remaining coins into two equal rows

35. Model 2: BALANCING Example 1 2x + 7 = 9 Divide both sides by 2 Place the remaining coins into two equal rows

36. Model 2: BALANCING Example 1 2x + 7 = 9 x = 1 [Check: 2(1) + 7 = 9]

37. Model 2: BALANCING Example 1 algebraically 2x + 7 = 9 2x + 7 – 7 = 9 – 7 2x = 2 2x = 2 2x / 2 = 2 / 2 2x / 2 = 2 / 2 x = 1 x = 1

38. Model 2: BALANCING Example 2 (Another concrete model) 3x + 2 = x + 10

39. Model 2: BALANCING Example 2 (Another concrete model) 3x + 2 = x + 10 Subtract x from both sides

40. Model 2: BALANCING Example 2 3x + 2 = x + 10

41. Model 2: BALANCING Example 2 3x + 2 = x + 10 Subtract 2 from both sides

42. Model 2: BALANCING Example 2 3x + 2 = x + 10

43. Model 2: BALANCING Example 2 3x + 2 = x + 10 Divide both sides by 2

44. Model 2: BALANCING Example 2 3x + 2 = x + 10 x = 4 [Check: 3(4) + 2 = 14 4 + 10 = 14]

45. Model 2: BALANCING Example 2 algebraically 3x + 2 = x + 10 3x + 2 – x = x + 10 – x 2x + 2 = 10 2x + 2 = 10 2x + 2 – 2 = 10 – 2 2x + 2 – 2 = 10 – 2 2x = 8 2x = 8 2x / 2 = 8 / 2 2x / 2 = 8 / 2 x = 4

46. Model 2: BALANCING Note that the appropriate inverse operations must be identified and performed in the correct order. Aim to have x on its own on the LHS of the equation: ‘x = ___’

47. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘Model 1 uses a two-pan balance and objects such as coins or centicubes. A light paper wrapping can hide a ‘mystery number’ of objects without distorting the balance’s message of equality.’ ‘Model 1 uses a two-pan balance and objects such as coins or centicubes. A light paper wrapping can hide a ‘mystery number’ of objects without distorting the balance’s message of equality.’

48. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘Model 2 uses small objects (all the same) with some hidden in containers to produce the ‘unknowns’ or ‘mystery numbers’, eg place the same number of small objects in a number of paper cups and cover them with another cup. Form an equation using the cups and then remove objects in equal amounts from each side of a marked equals sign.’ ‘Model 2 uses small objects (all the same) with some hidden in containers to produce the ‘unknowns’ or ‘mystery numbers’, eg place the same number of small objects in a number of paper cups and cover them with another cup. Form an equation using the cups and then remove objects in equal amounts from each side of a marked equals sign.’

49. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling:

50. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling: x + x + x + 1 = x + x + 2 + 1

51. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling: x + x + x + 1 = x + x + 2 + 1

52. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling: x + x + x + 1 = x + x + 2 + 1

53. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling: x + x + x + 1 = x + x + 2 + 1

54. Model 2: BALANCING What the syllabus says (p.86, PAS4.4) ‘ ‘Model 3 uses one-to-one matching of terms on each side of the equation, eg 3x + 1 = 2x + 3 x + x + x + 1 = x + x + 2 + 1 By one-to-one matching and cancelling: x = 2.’

55. Model 2: BALANCING Advantages  ???

56. Model 2: BALANCING Advantages  Powerful and elegant logical method  Works for all types of equations, including those with x on both sides  If done correctly, solution emerges quickly  Reinforces algebraic concepts of balance and equivalence of expressions (by cancelling and simplifying)

57. Model 2: BALANCING Disdvantages  ???

58. Model 2: BALANCING Disdvantages  Harder to model 6 – 2x = 14, x 2 = 10 with concrete objects: how do you represent subtraction, division or squaring of objects? x 2 = 10 with concrete objects: how do you represent subtraction, division or squaring of objects?  Difficult for some students to understand, conceptualise, reason

59. Model 2: BALANCING Disdvantages  Some students have trouble knowing which inverse operation to perform first  Lines of working can appear complicated and messy  Students often don’t know what they are actually doing or why they are doing it

60. Model 2: BALANCING Example 3 (a model for negatives) 6 – 2x = 14 Subtract 6 from both sides 6 – 2x – 6 = 14 – 6

61. Model 2: BALANCING Example 3 6 – 2x = 14 -2x = 8 Divide both sides by 2

62. Model 2: BALANCING Example 3 6 – 2x = 14 -x = 4 Take the negative of both sides

63. Model 2: BALANCING Example 3 6 – 2x = 14 x = -4 [Check: 6 – 2(-4) = 14]

64. Model 3: BACKTRACKING Alternative names Un-doing or unpacking Un-doing or unpacking Reverse flowchart Reverse flowchart

65. Model 3: BACKTRACKING Description An alternative algebraic model To undo the operations performed on the variable on one side of an equation, inverse operations are performed on the other side of the equation

66. Model 3: BACKTRACKING Description Can be demonstrated using a flowchart and reverse flowchart Models the equation as a sequence of operations on a variable that are ‘undone’ by a sequence of inverse operations that ‘backtrack’ to the variable

67. Model 3: BACKTRACKING Example 1 2x + 7 = 9 Use a flowchart to go from x to 2x + 7

68. Model 3: BACKTRACKING Example 1 2x + 7 = 9 Use a flowchart to go from x to 2x + 7

69. Model 3: BACKTRACKING Example 1 2x + 7 = 9 But 2x + 7 = 9 =

70. Model 3: BACKTRACKING Example 1 2x + 7 = 9 To get back to x, undo the operations using a reverse flowchart = To undo + 7, we – 7

71. Model 3: BACKTRACKING Example 1 2x + 7 = 9 To get back to x, undo the operations using a reverse flowchart = = = = To undo  2, we  2

72. Model 3: BACKTRACKING Example 1 2x + 7 = 9 To get back to x, undo the operations using a reverse flowchart = = = = x = 1

73. Model 3: BACKTRACKING Example 1 algebraically 2x + 7 = 9 2x = 9 – 7 2x = 9 – 7 = 2 = 2 x = 2 / 2 x = 2 / 2 x = 1 x = 1 With backtracking, we (inverse) operate on one side only

74. Model 3: BACKTRACKING Inverse operations are performed in reverse order. For example, to undo putting on our socks, then our shoes, we take off our shoes first, then take off our socks.

75. Model 3: BACKTRACKING Example 2 Use a flowchart to go from y to Use a flowchart to go from y to

76. Model 3: BACKTRACKING Example 2 Use a flowchart to go from y to Use a flowchart to go from y to

77. Model 3: BACKTRACKING Example 2 Use a flowchart to go from y to Use a flowchart to go from y to =

78. Model 3: BACKTRACKING Example 2 Use a reverse flowchart to backtrack to y = To undo  5, we  5

79. Model 3: BACKTRACKING Example 2 Use a reverse flowchart to backtrack to y Use a reverse flowchart to backtrack to y = = = = To undo + 3, we – 3

80. Model 3: BACKTRACKING Example 2 Use a reverse flowchart to backtrack to y Use a reverse flowchart to backtrack to y = = = = y = 7

81. Model 3: BACKTRACKING Example 2 algebraically y + 3 = 2  5 y + 3 = 2  5 = 10 = 10 y = 10 – 3 y = 10 – 3 y = 7 y = 7

82. Model 3: BACKTRACKING What the syllabus says (p.86, PAS4.4) ‘Model 5 uses backtracking or a reverse flow chart to unpack the operations and find the solution.’ ‘Model 5 uses backtracking or a reverse flow chart to unpack the operations and find the solution.’

83. Model 3: BACKTRACKING Advantages  ???

84. Model 3: BACKTRACKING Advantages  Only un-doing one side of equation: less working  For some students, this method is more intuitive: it’s what we do when we solve an equation mentally  Conceptually easier to understand than balancing

85. Model 3: BACKTRACKING Advantages  Algebraic working consistent with balancing method  If done correctly, solution emerges quickly  Reinforces skills in algebraic notation and generalising formulas

86. Model 3: BACKTRACKING Disdvantages  ???

87. Model 3: BACKTRACKING Disdvantages  Takes longer to teach as it requires careful practice with flowcharts  Does not work if x on both sides of equation, eg 3x + 2 = x + 10  Harder to model if x is not in the first term, eg 6 – 2x = 14

88. So which is the best method to use?

89. All three methods have merit and can be used together in the classroom All three methods have merit and can be used together in the classroom Depends on the class Depends on the class Guess, check and improve is good for starting the topic, and leads to the idea of inverse operations and algebraic methods Guess, check and improve is good for starting the topic, and leads to the idea of inverse operations and algebraic methods

90. Backtracking is a useful tool for students who struggle with algebra, again convenient for introducing inverse operations Backtracking is a useful tool for students who struggle with algebra, again convenient for introducing inverse operations Once students are confident with the algebra, introduce harder equations that require balancing to simplify the equation first, for example, Once students are confident with the algebra, introduce harder equations that require balancing to simplify the equation first, for example, x on both sides x on both sides x is not in the first term x is not in the first term equations with brackets equations with brackets

91. x on both sides 3x + 2 = x + 10

92. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 Use an inverse operation to remove the x from the RHS

93. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 2x + 2 = 10 2x + 2 = 10 Simplifies to a 2-step equation: proceed by backtracking or balancing

94. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 2x + 2 = 10 2x + 2 = 10 2x = 10 – 2 2x = 10 – 2

95. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 2x + 2 = 10 2x + 2 = 10 2x = 10 – 2 2x = 10 – 2 = 8 = 8

96. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 2x + 2 = 10 2x + 2 = 10 2x = 10 – 2 2x = 10 – 2 = 8 = 8

97. x on both sides 3x + 2 = x + 10 3x + 2 – x = 10 3x + 2 – x = 10 2x + 2 = 10 2x + 2 = 10 2x = 10 – 2 2x = 10 – 2 = 8 = 8 x = 4 x = 4

98. x is not in the first term 6 – 2x = 14

99. x is not in the first term 6 – 2x = 14 -2x + 6 = 14 -2x + 6 = 14 If backtracking, rewrite so that x is in the first term

100. x is not in the first term 6 – 2x = 14 -2x + 6 = 14 -2x + 6 = 14 -2x = 14 – 6 -2x = 14 – 6 Proceed by backtracking or balancing

101. x is not in the first term 6 – 2x = 14 -2x + 6 = 14 -2x + 6 = 14 -2x = 14 – 6 -2x = 14 – 6 = 8 = 8 Proceed by backtracking or balancing

102. x is not in the first term 6 – 2x = 14 -2x + 6 = 14 -2x + 6 = 14 -2x = 14 – 6 -2x = 14 – 6 = 8 = 8

103. x is not in the first term 6 – 2x = 14 -2x + 6 = 14 -2x + 6 = 14 -2x = 14 – 6 -2x = 14 – 6 = 8 = 8 x = -4 x = -4