Algebraic Patterning

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Algebraic Patterning

• Workshop presented at
• National Numeracy Facilitators Conference
• February 2009
• Jonathan Fisher

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Outline

• Why patterns?
• What were we looking for?
• Some words
• Curriculum
• Patterns Progression
• What did we do?
• Some findings
• So what?
• ARBs what else?

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Introduction

• NZ maths curriculum statement about
  patternsRecognise patterns and relationships in
  mathematics and the real world, and be able to
  generalise from these.
• The study of patterns is a key part of algebraic
  thinking. They involve relationships and
  generalisations.
• It is important that students are able to
  recognise and analyse patterns and make
  generalisations about them.

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Why Patterns?

• Patterns are everywhere we just need to learn to
  notice themand they can be quite powerful.
• The power of patterns is that they allow us to
  predict what will come next and they allow us to
  solve problems that would be very tedious to
  solve otherwise. Link

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Why Patterns?
Power of patterns The story goes that a young boy
walked into his class and read the assignment
Add up all the numbers from 1 to 100.  He
quickly calculated in his head and said,
5050.  Thats amazing! his teacher
exclaimed.  How did you add them so quickly?
I didnt add them, the boy responded, I saw
the pattern.
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Why Patterns?

• Patterning is critical to the abstraction of
  mathematical ideas and relationships, and the
  development of mathematical reasoning in young
  children. (English, 2004 Mulligan, Prescott
  Mitchelmore, 2004 Waters, 2004)
• The integration of patterning in early
  mathematics learning can promote the development
  of mathematical modelling, representation and
  abstraction of mathematical ideas. (Papic
  Mulligan, Preschoolers Mathematical Patterning)

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What were we looking for?

• How students progress from sequential rules to
  recognising a functional rule for the same
  pattern.
• What helps students and teachers to bridge the
  progressions of understandings (resources,
  questions, words, ideas, etc).
• What kind of age can we expect children start to
  deal with functional thinking in patterns (and
  using symbolic notation).

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Curriculum (1992)
Make and describe repeating and sequential
patterns Continue a repeating and sequential
pattern Continue a sequential pattern and
describe a rule for this Describe in words,
rules for continuing number and spatial
sequential patterns Make up and use a rule to
create a sequential pattern Find a rule to
describe any member of a number sequence and
express it in words Use a rule to make
predictions Generate patterns from a structured
situation, find a rule for the general term, and
express it in words and symbols Generate a
pattern from a rule Generate linear and
quadratic patterns and find and justify the rule
Generate a pattern from a rule Describe and use
arithmetic or geometric sequences or series in
common situations Use sequences and series to
model real or simulated situations and interpret
the findings Investigate and interpret
convergence of sequences and series
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Curriculum (2007)
Create and continue sequential patterns. Find
rules for the next member in a sequential
pattern. Connect members of sequential patterns
with their ordinal position and use tables,
graphs, and diagrams to find relationships
between successive elements of number and spatial
patterns. Use graphs, tables, and rules to
describe linear relationships found in number and
spatial patterns. Relate tables, graphs, and
equations to linear and simple quadratic
relationships found in number and spatial
patterns. Relate graphs, tables, and equations to
linear, quadratic, and simple exponential
relationships found in number and spatial
patterns. Use arithmetic and geometric sequences
and series.
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Curriculum (1992 to 2007)

• What about the new curriculum? What's different?
• No mention of repeating patterns
• Earlier reference to functional rules (L3 cf L4)
• Recognising the connection between graphs, table
  and functions (rules)
• Keeps linear patterns at lt L4
• First mention of quadratic L5 (old C was L6)
• Explicitly mentions exponent patterns.

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Curriculum (1992 to 2007)

• So from the curriculum we can see a progression
  from
• repeated patterns
• sequential patterns
• sequential rules
• spatial patterns
• number patterns and rules (sequential)
• rule (functional) for any member of a number
  sequence
• rule for the general term symbols
• and let's stop there.

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Patterns progression

• Copy a pattern and create the next element
• Predict relationship values by continuing the
  pattern with systematic counting
• Predict relationship values using recursive
  methods e.g. table of values, numeric expression
• Predict relationship values using direct rules
  e.g. ? x 3 1
• Express a relationship using algebraic symbols
  with structural understanding e.g. m 6f 2 or
  m 8 6(f 1)
• These relate to the first 5 levels of Algebra
  in the Maths curriculum (1992)
• Wright (1998). The learning and Teaching of
  Algebra Patterns, Problems and Possibilities.

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Ultimately
Ultimately this would suggest that we are looking
at how we can get students to a functional rule
of a pattern using symbols.
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And

• Research has indicated that many young
  adolescents experience difficulties with the
  transition to patterns as functions due to
  issues around language to describe relationships,
  predominant additive situations, and visualising.
  (Redden, 1996 Stacey Macgregor, 1995 Warren,
  2000).
• But Young children are believed to be capable
  of thinking functionally at an early age.
  (Blanton Kaput, 2004).

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What did we do?

• Numeric patterns (repeating and growing)
• Spatial repeating patterns
• Repeating patterns with beads
• Spatial growing patterns
• Spatial and number patterns
• Number Machines

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Some words
Number sequences - Number patterns
Explicit - Recursive - nth term -
Direct rules Sequential - Spatial -
Arithmetic Linear - Triangular -
Geometric Sequential rules - Functional
rules Ordinal position - Sequential
number patterns Repeating patterns -
Growing patterns
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What did we do? Spatial repeating patterns
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What did we do?Repeating patterns with beads
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What did we do? Spatial growing patterns
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What did we do? Spatial and number patterns
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What did we do? Spatial growing patterns
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What did we do? Number Machines
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Some findings
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Pre-repeating patterns (Mary)
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Post-repeating patterns (Mary)
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Pre-Spatial Number patterns (Erin)
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Post-Number patterns (Erin)
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Post-Spatial patterns (Erin)
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Pre-Number Machines (Erin)
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Post-Number Machines (Erin)
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And Post-Functions (Erin)
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Pre-Number Machines (George)
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Post-Number Machines (George)
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Some points

• Lots of hands on material based exploration
  followed by group discussion. Materials can get
  in the way and we have to move on.
• Develop understanding by decomposing spatial
  shapes in a pattern (i.e., finding what is
  different and similar)
• We found beads very helpful to elicit discussion
  leading to functional rules between the colours
• Some students preferred to work with the numbers
  than the spatial patterns (they could see
  patterns easier), therefore keep using the
  numbers and spatial patterns together. This
  supported student better than straight spatial
  patterns.
• Don't put the members of a number pattern table
  in order - it encourages sequential thinking (use
  ... Jump to other numbers).
• Take the number machines to the next level and
  then connect it (students connect it) to the
  functional rule for a number pattern.

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So what do we do with it?

• Sort out the plethora of current resources in the
  ARBs based around patterns
• Developed new ARB resources with teacher notes
• Patterns concept map with the ideas form our
  investigation linked to resource
• Add this presentation to the website.

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So what next?

• Deliberately select the spatial or number pattern
  to target learning.
• Start to use all numbers (rational, irrational,
  weird, negative) and get students to experiment
  with calculators. (Stacey and MacGregor,
  Building foundations for Algebra, 1997)
• Connecting patterns tables graphs.

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Some other Patterns (basic fact 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

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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

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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

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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 table 0.166 0.333, 0.500, 0.666, 0.833,
  1.000

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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

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ARBs what else is there?
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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

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Concept maps
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Concept maps

• Currently on the ARBs
• Algebraic patterns
• Basic facts (start of May)
• Fractional thinking
• Algebraic thinking
• Computational estimation

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Assessment Resource Banks

• www.arb.nzcer.org.nz
• Username arb
• Password guide