Exploring Concepts in Algebraic Patterning and in Basic Facts

Teresa Maguire, Jonathan Fisher, Alex Neill February 2008

Exploring Concepts inExploring Concepts in Algebraic PatterningAlgebraic Patterning

and inand in Basic FactsBasic Facts

Workshop presented at

National Numeracy Facilitators Conference

February 2008

Teresa Maguire and Jonathan Fisher and Alex Neill


OutlineOutline

Supporting teachers with the ARBs (10 min)

Patterns (35 min)

Basic facts (35 min)

Discussion (10 min)


Supporting teachers with the ARBsSupporting 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


Animation CD/exeAnimation CD/exe


Next steps bookletNext steps booklet


Support materialsSupport materials


Teacher information pagesTeacher 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


Concept mapsConcept 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



For example, currently on ARBs

Computational estimation

Fractional thinking

Algebraic thinking


Concept MapsConcept Maps





Under development for ARBs

Algebraic patterns

Basic facts


A bit about our research ShapesBeads Sticks

Machines

PatternsPatterns


ShapesShapes


BeadsBeads


SticksSticks

Flats

Beds

Xmas Trees


MachinesMachines


Basic FactsBasic Facts

The many faces of Basic Facts


Basic FactsBasic 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.


Memory Model from researchMemory Model from research

MEMORY

Declarative Non-declarative

SemanticFacts

EpisodicEvents

ProceduralSkills & habits

PrimingConditioning

Non-associative


Start with StrategiesStart 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


StrategiesStrategies

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


Move on to memorisationMove 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


Memory stagesMemory stages

1. Level I NoviceStep-by-step countingSlow especially as numbers get biggerEasily interfered with

2. Level II AutomatedMemory retrieval replaces algorithmic (counting)Rapid – not dependent on size of numbersHarder to interfere with

3. Level III Beyond automaticityFastestNo interference even with a costly secondary task


ExampleExample

What is 8 + 6

1. Level I 8, 9, 10, 11, 12, 13, 14

2. Level II 8 + 2 =10 10 + 4 = 14

3. Level III 14


Memory is AUTOMATICMemory 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)


Memory has RANDOM ACCESSMemory 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 II 48

What is the fifteenth letter of the alphabet?


Who wants to be a millionaire?Who wants to be a millionaire?

Put these in order from smallest to largest

A 4 × 6

B 3 × 9

C 2 × 11

D 5 × 5

C A D B



Put these in order from North to South

A Palmerston North

B New Plymouth

C Napier

D Tauranga

D B C A



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


Addition 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)


SubtractionSubtraction

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)


MultiplicationMultiplication

2× table 10× table5× tables

4× table

3× table

9× table

6× table 8× table

7× table

Doubles (from addition)Place valueFilling in the gapsDouble doubles - PATTERNS

Smaller numbers

Patterns

1 more than 5× table double 4× table

HELP!!!


PatternsPatternsInstant 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


PatternsPatternsInternal 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, 204x 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, 903x 0, 3, 6, 9, 12, 15, 18, 21, 24, 27, 30

7x 0, 7, 14, 21, 28, 35, 42, 49, 56, 63, 70


Divisibility Rules Divisibility Rules Which times table is it in?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


Multiplication – another viewMultiplication – another view

2, 10, & 5 2× table Doubling10× table Place value 10× table to 5× table Halving

2, 4, & 8 2× table to 4× table Doubling4× table to 8× table Doubling

3, 6, & 93× table to 6× table Doubling 9× table Patterns

7× table HELP!!


Other tablesOther tables11x 11 22 33 44 … 88 99

11.1x 11.1 22.2 33.3 44.4 … 88.8 100 Related 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 1000 2 x 125 = 250

3 x 125 = 375

5 x 125 = 625

7 x 125 = 875 Related to 8x table because 8 x 125 = 1000


Fractions – Decimals - PercentagesFractions – 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



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 0.2500.3750.5000.625 … 125x table



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 table 0.166 0.333, 0.500, 0.666, 0.833, 1.000


Other basic factsOther 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


Other basic factsOther 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 3

20 40 60 80 100 … Divisible by 4 (2 x 2) and 5

28 56 (84) … Divisible by 4 (2 x 2) and 7

70, 88 Divisable by 2, 5 & 7, 2,2,2 & 11

Deficient numbers (Only a few factors)

The others including 4 8 16 32 64 … 10 50 … 9 25 49 …


Powers of 10 – orders of magnitudePowers of 10 – orders of magnitudeWhole numbersWhole numbers

Extended 6 and 26 × tables

Hyper – extended 100 ×, 1000×, 10 000 etc

60 × 4 = 240

26 × 10 = 260

× 10 100 1000 10 000

10 100 1000 10 000 100 000

100 10 000 100 000 100 000 1 000 000

1000 10 000 100 000 1 000 000 10 000 000

10 000 100 000 1 000 000 10 000 000 100 000 000


Powers of 10 – orders of magnitude Powers of 10 – orders of magnitude DecimalsDecimals

Extended 6 and 26 × tables

Hyper – extended 0.1×, 0.01×, 0.001 etc

6 × 0.4 = 2.40

26 × .10 = 2.6

× 10 100 1000 10 000

0.0001 0.0010 0.01 0.1 1

0.001 0.01 0.1 1 10

0.01 0.1 1. 10 100

0.1 1 10 100 1000


Basic factsBasic facts

Start with strategies

Move on to memorisation

Strategies are in non-declarative memory

Basic facts are in declarative memory


Basic facts ResourcesBasic facts Resources

nzmathshttp://www.nzmaths.co.nz/number/numberfacts.aspx Recall column

Chandra Pinsent and Sandi Tait-McCutcheon

http://www.nzmaths.co.nz/Numeracy/Other%20material/Tutorials/BFModel.pdf

Figure it out books Basic Facts

Google"basic facts" mathematics free

ARBs Concept map coming soon


Assessment Resource BanksAssessment Resource Banks

www.arb.nzcer.org.nz

Username: arb

Password: guide


UnderstandingUnderstanding(in the style of St. Paul, I Corinthians 13)(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.


UnderstandingUnderstanding(in the style of St. Paul, I Corinthians 13)(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. CRITTENDENHouston Baptist University

Houston, TexasMarch 1975

Documents

Exploring Concepts in Algebraic Patterning and in Basic Facts