Reasoning
Algebra - Stepped Pyramids
Element
Reasoning
f
The learner
Applies the rules that govern growing and shrinking patterns to extend to next terms, and to predict future values.
g
The learner
Generates expressions (expressions can be verbal or symbolic) for a variety of patterns, structures and rules and uses to make predictions about future terms.
h
The learner
Justifies the structure and rules that describe patterns, comparing perspectives with others.
i
The learner
Extends and generalises patterns within the sequences of square numbers.
J
The learner
Uses verbal, symbolic, graphical or diagrammatical representations of relationships between quantities to predict future values and to solve for specific events.
What learners can typically do
Element
Reasoning
f
The learner
Uses cubes (concrete materials or visual representation) to create a pyramid with a 5 x 5 base.
Counts cubes individually to check response.
g
The learner
Generates expression
Layer one = 1
Layer two = Layer one + (4 x 2)
1 + 8 = 9
Layer three = Layer two + (4 x 4)
9 + 16 = 25
Layer four = Layer three + (4 x 6)
25 + 24 = 49
So Layer five = Layer four + (4 x 8)
Sums products of layers for answer.
h
The learner
Justifies the structure that the pattern skips in odd numbers multiplied by themselves.
​
Layer one = 1 x 1
​
Layer two = 3 x 3
Layer three = 5 x 5
Layer four = 7 x 7
Applies reasoning to identify layer five = 9 x 9.
​
Calculates 9 x 9 = 81 etc.
i
The learner
Extends patterns within the sequences of square numbers.
​
The layers follow odd squares: 1 , 3 , 5 , 7 , 9 , 11 etc.
​
Generalises pattern within the sequences of square numbers.
​
The difference between consecutive odd square numbers increases by 8.
9 - 1 = 8,
25 - 9 = 16 (which is 8 + 8)
49 - 25 = 24 (which is 16 + 8)
81 - 49 = 32 (which is 24 + 8) etc
​​
2
2
2
2
2
2
J
The learner
Predicts future layers.
​
(2 x Layer Number - 1) x (2 x Layer Number - 1)
​​
Uses order of operations to solve
​
( 2x Layer Number - 1)
​
Checks to justify.
​
Layer one:
(2 x 1 - 1) = 1
​
Layer two:
(2 x 2 - 1) = 9
​
Layer three
(2 x 3 - 1) = 25
​
Layer ten
(2 x 10 - 1) = 361
2
2
2
2
2