Workshop presented at National Numeracy Facilitators Conference February 2008

1. Unpacking Student Responses for Teacher Information Workshop presented at National Numeracy Facilitators Conference February 2008 Teresa Maguire, Jonathan Fisher and Alex Neill

2. Outline • Supporting teachers with the ARBs (10 min) • Resources with fractions (20 min) • Resources with algebra (20 min) • Other information on resources (25 min) • Discussion (15 min)

3. Supporting teachers with the ARBs How the ARBs can be used to support teachers? • Concept maps • Animation / CD • Next steps booklet • Support material • Teacher information pages

4. Concept maps Provide information about the key mathematical ideas involved Link to relevant ARB resources Suggest some ideas on the teaching and assessing of that area of mathematics Are “Living” documents

5. Concept maps

6. Animation CD/exe

7. Next steps booklet

8. Support materials

9. Teacher information pages • Task administration • Answers/responses • Calibration easy (60-79.9%) • Diagnostic and formative information (common wrong answers and misconceptions) • Strategies • Next steps • Links to other resources/information and to concept maps

10. Eating fractions of pie, pizza and cake(NM1251) • The questions • Student responses and misconceptions • Strategies • Suggested Next steps • Other resources • Discussion

11. Eating fractions of pie, pizza and cake(NM1251) 167 students Year 6 Nationwide Range of deciles

12. Eating fractions of pie, pizza and cake(NM1251) Petra ate two-fifths (2/5) of a pizza and Sarah ate one-fifth (1/5). Show how to work out how much pizza they ate altogether. 3/5 (64%, and 66% showed working)

13. Eating fractions of pie, pizza and cake(NM1251) Lima and Paul each had the same sized cake. Lima ate four-fifths (4/5) of his cake and Paul ate three-fifths (3/5) of his cake. Show how to work out how much cake they ate altogether. 7/5 or 1 2/5 (38%, and 52% showed working)

14. Eating fractions of pie, pizza and cake(NM1251) Bill ate one-fifth (1/5) of a whole apple pie. Show how to work out how much pie was left. 4/5(64%, and 70% showed working)

15. Eating fractions of pie, pizza and cake(NM1251) Andrew started with one and a half pizzas (1 1/2) and ate three-quarters (3/4) of a whole pizza. Show how to work out how much pizza is left. 3/4(46% - 53% showed working)

16. Eating fractions of pie, pizza and cake(Misconceptions) Numbering the Pieces only 3 7 4 3 Whole number (top and bottom) 3/10 7/10 Other whole number relationship/system 13 17 or 4 8 or 4/10 8/10 or 4/0 8/0 Varying referent whole 1/2 or 3/6 And the “Size of the pieces”

17. Eating fractions of pie, pizza and cake(Next steps) Partitioning Part-whole relationships Referent whole Whole class discussion Explanation and justification Diagrams

18. Eating fractions of pie, pizza and cake(Other resources) • Link to other ARB resource (keywords) • Fractional thinking concept map • NEMP • Book 7: Teaching Fractions, Decimals and Percentages, 2006 • Figure it out

19. Balance pans & Solving simple equationsAL 7111 & AL7124 = www.nzcer.org.nz/arb

20. Strategies with Zero a) i) 25 + 16 – 16 = ii) 28 + 36 – 36 + 52 – 52 = iii) 62 + 74 – 62 = iv) 78 – 44 + 44 = v) 67 + 23 + 55 – 23 – 67= b) Explain what you did Student strategies Usage and success rates using the additive identity concept In our sample, 37% of students included at least something in part b) to indicate that they were employing the concept of the additive identity. These accounted for 43% of the strategies students described. - 13% of students had an explanation including reference to the concept of the additive identity and no calculations were used. These students got an average of 94% of their answers in part a) correct, and maintained their success throughout the five parts of the question. - 19% applied the additive identity rather than a calculation but were unable to clearly explain what they had done. 85% of these students' answers in part a) were correct, but they were slightly less successful with the harder questions. - 5% had an explanation that included reference to the additive identity but they also did some calculations. Only 67% of these students' answers in part a) were correct, but they were less successful with the harder questions. The remaining 63% of students described computational methods, other methods, or gave no explanation in part b). These accounted for 57% of the strategies students described. These students had lower success rates that varied from 0% to 68%, and their success rates generally dropped off on the harder questions. For more details on the success rates of student strategies, click on the link Appendix of student strategies for AL7123

21. Strategies with Zero Based on a representative sample of 188 students

22. Car maintenance & Post a parcelAL6156 & AL 7129 • Exemplars of student responses- Graphs • Multiple representations- Graphs, tables, and equations • Other goodies

23. Car maintenance (AL6156 – Level 4) b) Draw a line graph Incorrect line graphs: Another type of graph: Step graph

24. Diagnostic and formative information Based on a representative sample of 201 students.

25. Car maintenance – Student Responses joins origin to end point starting at 1 hour bend the line

26. Histogram Relationship graph Scatterplot Bar graph

27. Car Maintenance Step graph This graph could be seen as correct if only complete hours are charged

28. Post a parcel(AL7129 – Level 5) Multiple representations Graphs Tables Equations Two courier companies This graph shows the price Peru. Show how to use this graph to find the weight of a parcel that both companies would charge the same .

29. Post a parcel-Student strategies Graphical interpretation [part a)] Indicating the x- and the y- coordinates of the intersection of the lines (5%). Indicating the x-coordinate only of the intersection of the lines (35%). Indicating the intersection of the lines only (26%). Using graphical interpolation to get a more accurate answer (19%). This was often used in conjunction with one of the other three strategies. About 80% of students who showed interpolation obtained a correct answer, while only about 50% of those whose working did not show interpolation got a correct answer. Many of the latter were satisfied with 2.5 as their answer, even though the break-even point was clearly somewhat less than that. Table interpretation [part b)] Interpolation of the table between 1 and 2 kg (4%). Using the differences between 1 and 2 kg for the two companies (9%). Averaging the costs at 1 and 2 kg (1%). Using equations [part c)] Algebraic formulation of the problem (9%). Trial and improvement methods (15%). The included a range of methods including moving sequentially towards the solution in single integers, to jumping several numbers to speed up getting to the answer. .

30. Graphical interpretaton Indicating the intersection of the lines only Indicating the x- and the y- coordinates Indicating the x-coordinate only Graphical interpolation

31. Table interpretation Interpolation of the table Using the differencesbetween 1 and 2 kg 70 - 60 = 10 100 - 90 = 10 Averaging (60 + 100) / 2 = 80 (70 + 90) / 2 = 80

32. Using equations-Algebraic formulation FastAir: Cost = 7x + 14 SafeWay: Cost = 4x + 35 Formulation only 7x + 14 = 4x + 35 Formulation and attempt to solve 7x + 14 = 4x + 35 7x = 4x + 49 3x = 49 x = 16.33 Formulation and a successful solution 7x + 14 = 4x + 35 3x = 21 x = 7

33. Using equations–Trial and improvement Initial guess only (sometimes correct) Initial guess then other guesses Starting from 1 and iterating in steps of 1 7 x 1 + 14 = 21 7 x 2 + 14 = 28 4 x 1 + 35 = 39 4 x 2 + 35 = 43 Initial guess then iterating with a step size of 1 Jumping towards the solution.

34. Future areas • Statistics • Geometry and Measurement

35. Enlargement(GM5118) Terminology of enlargementScale factor??

36. GM5118 Next steps Uses an additive model Students need to look carefully at the language clues in the question. The phrase "how many times bigger … compared with" in parts b)ii) and c)ii) needs to be interpreted as a multiplicative question. If students have misinterpreted the question this way, see if they can perform the question once this has been clarified. Unfamiliar with the term "scale factor" or of the concept of enlargement Work with students to understand that enlargement increases each dimension linearly by an amount known as the scale factor. Get the students working with enlargement on grid paper. GM5013 describes the process without using the term scale factor. Click on Level 3 and 4, enlargement AND scale factor for further resources that assess this. Also click on the Figure it out resource Enlargement Explosion (Geometry, L4+, Book 2, page 12). Unfamiliar with the effects of enlargement on area or volume Students firstly need to know that "scale factor" is a linear measure. They then need to explore the relationship between scale factor and area. The increase in area goes up by the square of the scale factor. This is because area is a square measure (m2, cm2 etc). Click on scale factors AND area OR invariant properties for further resources on this relationship. Resources GM5113, GM5038, and GM5054 are particularly useful for teaching this principle. Also click on the Figure it out resource Growing Changes (Geometry, L3, page 24). The increase in volume goes up by the cube of the scale factor. This is because area is a cubic measure (m3, cm3 etc). Click on scale factors AND volume for further resources on this relationship. Resources GM5113, and GM5038 are particularly useful for teaching this principle.

37. Perimeter (MS2178) Composite shapes

38. Next steps MS2178 The first four diagnostics above or other incorrect answers indicate that the student does not know how to calculate the circumference of a circle. These students need to have experiences in how to calculate the circumference. ARB resource MS2107 and Figure it outresourceCircle Links (Measurement, L4, Book 1, page 3) each give an activity that gets students physically measuring the circumferences and diameters of different circles and exploring the relationship between them. The classical definition of π is geometrical, and it is the ratio between the circumference and the diameter of any circle. π = circumference ÷ diameter (of any circle) So a circle with a diameter of 1 unit has a circumference of π units. Students could also be asked to explore π (pi) on the internet. The following are some historical fractional approximations to pi:

39. Assessment Resource Banks www.arb.nzcer.org.nz Username: arb Password: guide