1. Algebra and the Secondary Numeracy Project

2. Find the general rule linking the number of matches needed to the number of squares built.

3. To find the number of matches needed multiply the number of squares by 3 and add 1. In algebraic shorthand m = 3s + 1

4. In this example we are doing algebra because we are making general statements in patterns that apply to any numbers and not just particular numbers. This is what distinguishes algebra from arithmetic.

5. w b A = wb Another example: A = area.

6. To be able to cope with this kind of algebra we children need to be advanced multiplicative (Stage 7) thinkers.

7. Examples of Advanced Multiplicative Thinking: Calculate 89 x 5

8. 90 x 5 is 450 then take away 5 to give 445 Or 44.5 x10 = 445 Or 80 x 5 + 9 x 5 = 445 Or ……………..

9. Work Out 72 ÷ 5

10. 72 ÷ 5 is the same as 144 ÷ 10 which is 14.4 Or …….

11. Number sense shows 73 ÷ 0.97 = 70.81 is wrong. How?

12. 73÷ 0.97 is just a little bit more than 73 since division by a number less than 1 has an answer that is bigger

13. Number Properties as Generalisation Stage 6 Part of Student Sheet Work these out. 663 + 199 =669 + 197 = 117 + 398 =698 + 127 = 272 + 296 =397 + 184 =

14. Moving to the Use of Letters Part of Student Sheet For each of the following equations write True or False in the box. Do not add up the numbers on each side 97 + 47 = 100 + 44 77 + 95 = 76 + 9 85 + 56 = 89 + 6063 + 72 = 66 + 67

15. Part of a Student Sheet For each of the following fill in the box without adding the numbers up on each side 85 + 34 = 86 + 40 + = 42 + 5 34 + 88 = 39 + 55 + = 52 + 53

16. Write three more equations using different numbers. 57 + = 54 +

17. Complete the statement: The number in the right box is always 3 than the number in the left box

18. Fill in the empty boxes. Each letter stands for any number. 75 + n = 72 + 96 + n = 99 + k + 45 = + 42m + 300 = + 40

19. Letters Only Complete down these equations. a + c + b - c = x + y + w - y = s + y + 11 - y = 18 - k + h + k =

20. Stages 5 and Earlier What algebra is there at these stages? Jean works out 57 + 8 Step 1: 57 +3 = 60 Step 2: 60 + 5 = 65 (Early Part-Whole: Stage 5)

21. Advanced Counting Stage 4 Find the ninth number in the pattern 3, 8, 13, 18 …

22. Students’ Major Misunderstandings of the Meaning of Letters in Algebra Letter Needs a Value Letter is seen as a numerical value instead of being treated as an unknown or generalised number. Sally in Peanuts says in a maths test “x equals 7, x is always 7.” Letter is an Object Letter is seen as an object. For example, 2a + 3a + a = 6a because 2 apples + 3 apples + 1 apple = 6 apples.

23. Students’ Correct Understandings of the Meaning of Letters in Algebra Letter Used in Number Generalisations For example, the correct generalisation the order of addition of two numbers does not affect the answer can be expressed in letters: a + b = b + a where a and b can be any numbers. Letter Used in Pattern Generalisations For example, the nth term in the sequence 4, 8, 12, 16,… is 4n.

24. Assessing Understanding of Letters LevelsDescription of Level 0, 1,2 At best students solve problems without realising letters represent any numbers. Students use particular numbers for letters, or regard letters as objects. Students at these levels effectively understand no algebra

25. LevelsDescription of Level 3,4 Students treat letters as specific unknowns or generalised numbers. These students have understanding of algebra

26. Percentage of English Children at Algebra Levels (Hart) Levels13 years14 years15 years 0, 1, 283%65%58% 3, 417%35%42%

27. Algebraic Thinking Is Not An Optional Extra I have a number problem. Can I work out the answer mentally? Yes No I work out the answer. I am using algebraic thinking I choose to work out the answer by pencil and paper or calculator. I am not using algebraic thinking I must estimate the answer as a check. I am using algebraic thinking

28. NCEA Level 1 = Old School Cert – Year 11 Level 2 = Old Sixth Form Cert – Year 12 Level 3 = Old Bursary – Year 13 Level 2 2003 Algebra results a disaster 500 teacher secondary pilot in 2005

29. NCEA Results 2003 (NZQA, 2003) Level 1 Mathematics (Algebra): Use straightforward algebraic methods to solve equations

30. “ Achieved” Questions Q1Simplify 3x 4.2x 3 Q2Expand and simplify 3(x + 1) + 2(x - 3) Q3Josh is estimating the area of a circle by using A = 3r 2. What is the area if r = 5?

31. Q4 Josh increases his donations to World Vision by $2 each month. Month (m)Donations (d) 1 2 3$12 4 5$16 6 7 Write a rule for the amount Josh donates after m months.

32. Q5Solve the equations: a2x(x + 3) = 0 b3x + 5 = x + 6 c2x = 5 3 7

33. Achieved Criteria 3 out of 4 of the first four questions and 2 out of 3 from Q5

34. A Merit Question Josh bought mini pizzas for his flat one night. The Supreme pizza was 50 cents more than the Hawaiian pizza. 3s + 2h = 20.75s = h + 0.5 Solve these equations to find the price of one Supreme pizza.

35. Results n = 39,067 Not AchievedAchievedMeritExcellence 50.1%32.8%12.7%4.4%

36. Summary The Numeracy Project is not about computationThe Numeracy Project is about Algebraic ThinkingThe Numeracy Project is about making generalisationsAlgebraic Thinking in Primary & Intermediate is really important

37. Purpose Looking to the future the good measure of the long-term success of the Numeracy Projects will be the improvements in algebra performance in NCEA.

38. Some Examples of Stage 8 Advanced Proportional Thinking