Exploring Concepts in Algebraic Patterning and in Basic Facts

1. Exploring Concepts in Algebraic Patterningand inBasic Facts Workshop presented at National Numeracy Facilitators Conference February 2008 Teresa Maguire and Jonathan Fisher and Alex Neill

2. Outline • Supporting teachers with the ARBs (10 min) • Patterns (35 min) • Basic facts (35 min) • Discussion (10 min)

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

4. Animation CD/exe

5. Next steps booklet

6. Support materials

7. 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

8. 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

9. Concept maps For example, currently on ARBs • Computational estimation • Fractional thinking • Algebraic thinking

10. Concept Maps

11. Concept maps

12. Concept maps Under development for ARBs • Algebraic patterns • Basic facts

13. Patterns A bit about our research Shapes Beads Sticks Machines

14. Shapes

15. Beads

16. Sticks Beds Flats Xmas Trees

17. Machines

18. Basic Facts The many faces of Basic Facts

19. Basic Facts • Times tables (and thence division) • Addition facts (and thence subtraction) • Other whole number facts • Fractional facts • Other mathematical facts (includes theorems) Definition:(a tentative go) Any number or mathematical fact that can be instantly recalled without having to resort to a strategy to derive it.

20. Memory Model from research

21. Start with Strategies There is no point in a student learning a number fact until: • They have some understanding of the domain in which the fact resides. • They have at least one strategy which they could use to construct the answer. Example: 8 × 6 = 48 • Is the student a multiplicative thinker (or at least advanced additive)? • Can they get 6 × 8 = 6 + 6 + 6 + 6 + 6 + 6 = 48 or 6× 8 = 5× 8 + 8 = 48 or some other way

22. Strategies As a student does more and more examplesusing a variety of strategies, they gradually come to instantly recall some of them. Practice with strategies Strategies are in procedural memory

23. Move on to memorisation • Practicing strategies alone is not enough to getinstant recall of basic facts. • Which is harder to do? 8 × 6 = 8 + 6 = Practice for memorisation Basic facts are in declarative memory

24. Memory stages • Level I NoviceStep-by-step countingSlow especially as numbers get biggerEasilyinterfered with • Level II AutomatedMemory retrieval replaces algorithmic (counting)Rapid – not dependent on size of numbersHarder to interfere with • Level III Beyond automaticityFastestNo interference even with a costly secondary task

25. Example What is 8 + 6 • Level I 8, 9, 10, 11, 12, 13, 14 • Level II8 + 2 =10 10 + 4 = 14 • Level III14

26. Memory is AUTOMATIC • The answer is instantly available without the need for a strategy. • The answer can be retrieved, even when the mind is occupied by other things. (Level III)

27. Memory has RANDOM ACCESS You don’t have to repeat a memorised sequence to obtain the result you want • What is 8 × 6? Level II 8 × 1 = 8; 8 × 2 = 16 ….. Level II48 • What is the fifteenth letter of the alphabet?

28. Who wants to be a millionaire? Put these in order from smallest to largest A 4 × 6 B 3 × 9 C2 × 11 D5 × 5 C A D B

29. Who wants to be a millionaire? Put these in order from North to South A Palmerston North B New Plymouth C Napier D Tauranga D B C A

30. Who wants to be a millionaire? Put these in the order of the Sound of Music Song A A name I call myself B A drink with jam and bread C A drop of golden sun D A needle pulling thread C A D B

31. Addition Facts up to 5Groupings within 5 Facts up to 10Groupings to 10 Doubles to 18 Facts to 18 Extending the facts Properties of 0 and 1 1 + 2 = 32 + 3 = 5 3 + 5 = 86 + 4 = 10 8 + 8 = 16 8 + 6 = 14 20 + 50 = 70500 + 800 = 130054 + 8 = 62 7 + 0 = 1 (Identity)6 + 1 = 7 ( Successor)

32. Subtraction Facts up to 5Groupings within 5 Facts up to 10Groupings to 10 Doubles to 18 Facts to 18 Extending the facts Properties of 0 and 1 4 – 3 = 15 – 2 = 3 7- 5 = 210 – 7 = 3 16 – 8 = 8 13 – 5 = 8 70 – 30 = 4043 – 5 = 38 4 – 0 (Identity) 8 – 1 = 7(pre-decessor)

33. Multiplication 2× table 10× table5× tables 4× table 3× table 9× table 6× table 8× table 7× table Doubles (from addition)Place valueFilling in the gaps Double doubles - PATTERNS Smaller numbers Patterns 1 more than 5× table double 4× table HELP!!!

34. Patterns Instant recognition of series • 10x 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 • 5x 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 • 2x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 • 4x 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 • 3x 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 • 9x 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 • 6x 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 • 8x 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 • 7x 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

35. Patterns Internal patterns • 10x 0, 10, 20, 30, 40, 50, 60, 70, 80, 90, 100 • 5x 0, 5, 10, 15, 20, 25, 30, 35, 40, 45, 50 • 2x 0, 2, 4, 6, 8, 10, 12, 14, 16, 18, 20 • 4x 0, 4, 8, 12, 16, 20, 24, 28, 32, 36, 40 • 6x 0, 6, 12, 18, 24, 30, 36, 42, 48, 54, 60 • 8x 0, 8, 16, 24, 32, 40, 48, 56, 64, 72, 80 • 9x 0, 9, 18, 27, 36, 45, 54, 63, 72, 81, 90 • 3x 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30 • 7x 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70

36. Divisibility Rules Which times table is it in? Instant recognition of membership • 10x It ends in a zero • 2x It ends in an even number • 5x It ends in a 0 or a 5 Iterative • 3x The sum of the digits is divisible by 3 • 9x The sum of the digits is divisible by 9 • 6x The sum of the digits is divisible by 2 and 3 684 795 487 176

37. Multiplication – another view • 2, 10, & 52× table Doubling10× table Place value10× table to 5× table Halving • 2, 4, & 82× table to 4× table Doubling4× table to 8× table Doubling • 3, 6, & 93× table to 6× table Doubling9× table Patterns • 7× table HELP!!

38. Other tables • 11x 1122 33 44 … 88 9911.1x 11.1 22.2 33.3 44.4 … 88.8 100Related to 9x table because 9 x 11.1 = 100 • 25x 25 50 75 100 2 x 25 = 50 3 x 25 = 75 Related to 4x table because 4 x 125 = 100 4 x 25 = 100 • 125x 125 250 375 500 625 750 875 10002 x 125 = 250 3 x 125 = 375 5 x 125 = 625 7 x 125 = 875 Related to 8x table because 8 x 125 = 1000

39. Fractions – Decimals - Percentages • Halves, tenths, and fifths 1/2 0.5 50% 1/10 0.1 10% 1/5 0.2 20% 1/2 x table 0.5 1.0 1.5 2.0 2.5 3.0 … 5x table 1/10 x table 0.1 0.2 0.3 0.4 0.5 0.6 … 10x table 1/5 x table 0.2 0.4 0.6 0.8 1.0 1.2 … 2x table

40. Fractions – Decimals - Percentages • Halves, quarters, and eighths 1/2 0.5 50% 1/4 0.25 25% 1/8 0.125 12.5% 1/2 x table 0.5 1.0 1.5 2.0 2.5 … 5x table 1/4 x table 0.25 0.50 0.75 1.00 1.25 … 25x table 1/8 x table 0.125 0.250 0.375 0.500 0.625 … 125x table

41. Fractions – Decimals - Percentages • Thirds, ninths, and sixths times table 1/3 0.333 33.3% 1/9 0.111 11.1% 1/6 0.166 16.6% 1/3 x table 0.333 0.666 0.999 (=1!) … 1/9 x table 0.111 0.222 0.333 0.444 …0.999 (=1) 11x table 1/6 x table0.166 0.333, 0.500, 0.666, 0.833, 1.000

42. Other basic facts Instant recognition of series Instant recognition of membership • Power series1, 2, 4, 8, 16, 32, 64, 128, 256, 512, 1024 • Square numbers1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225 • Triangular numbers1, 3, 6, 10, 15, 21, 28, 36, 45 • Cubic numbers1, 8, 27, 81, 125

43. Other basic facts • Prime numbers 2 3 5 7 11 13 17 19 23 29 31 37 41 43 47 53 59 … • Abundant numbers (Lots of factors)6 12 18 24 30 36 42 48 … 90 96 … Divisable by 2 and 320 406080 100 … Divisible by 4 (2 x 2) and 528 56(84) …Divisible by 4 (2 x 2) and 770, 88 Divisable by 2, 5 & 7, 2,2,2 & 11 • Deficient numbers (Only a few factors)The othersincluding 4 8 16 32 64 … 10 50 … 9 25 49 …

44. Powers of 10 – orders of magnitudeWhole numbers • Extended 6 and 26 × tables • Hyper – extended 100 ×, 1000×, 10 000 etc 60 × 4 = 240 26 × 10 = 260

45. Powers of 10 – orders of magnitude Decimals • Extended 6 and 26 × tables • Hyper – extended 0.1×, 0.01×, 0.001 etc 6 × 0.4 = 2.40 26 × .10 = 2.6

46. Basic facts Start withstrategies Move on tomemorisation Basic facts are in declarative memory Strategies are in non-declarative memory

47. Basic facts Resources • nzmathshttp://www.nzmaths.co.nz/number/numberfacts.aspxRecall column • Chandra Pinsent and Sandi Tait-McCutcheon http://www.nzmaths.co.nz/Numeracy/Other%20material/Tutorials/BFModel.pdf • Figure it out booksBasic Facts • Google "basic facts" mathematics free • ARBsConcept map coming soon

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

49. Understanding(in the style of St. Paul, I Corinthians 13)  Though I figure with the skills of men and computers and have not understanding, I am become as a mechanical toy, or a lifeless robot. And though I have the gift of memory, and know the multiplication tables, and all the number facts, and though I know all algorithms, so that I can grind out all answers, and have not understanding, I am not free. And though I supply right answers that please my teacher, and though I exchange my paper to be graded, and have not understanding, high marks are not enough.

50. Understanding(in the style of St. Paul, I Corinthians 13) Understanding never fails; but where there is rote memory, it shall fail; where there are skills, they shall be no longer needed; where there are algorithms and number facts, they shall vanish away. And now abideth right answers, rote memory, and understanding, these three; but the greatest of these is understanding. WILLIAM B. CRITTENDEN Houston Baptist University Houston, Texas March 1975