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?
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Workshop presented at
National Numeracy Facilitators Conference
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.”
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;
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.
What about the new curriculum? What's different?
**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 this would suggest that we are looking at how we can get students to a functional rule of a pattern using symbols.
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
3n + 1
4n + 2
4n + 4
3n + 1
Instant recognition of series
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
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
Instant recognition of series
Instant recognition of membership
Currently on the ARBs