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### Algebraic Patterning

Workshop presented at

National Numeracy Facilitators Conference

February 2009

Jonathan Fisher

Outline

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

Introduction

- NZ maths curriculum statement about patterns:Recognise 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.

Why Patterns?

- Patterns are everywhere; we just need to learn to notice them…and 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

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.”

“That’s amazing!” his teacher exclaimed. “How did you add them so quickly?”

“I didn’t add them,” the boy responded, “I saw the pattern.”

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)

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).

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;

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.

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 < L4
- First mention of quadratic L5+ (old C was L6)
- Explicitly mentions exponent patterns.

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.

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.

Ultimately…

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

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).

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

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

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.

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.

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.

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

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

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

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

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

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

Concept maps

Currently on the ARBs

- Algebraic patterns
- Basic facts (start of May)
- Fractional thinking
- Algebraic thinking
- Computational estimation

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