Uses the numbers to find the pattern, not the shapes. Adds 3 each time.
c) i)
ii)
10,14 and
26
Applies the rule "add 4" used to get the 3rd shape for the rest of the sequence.
a) ii)
b) ii)
c) ii)
29
43
28
Works out the number of dots in the shape before the nth shape
[(n-1)th shape].
a) i)
c) ii)
32
35
Calculates incorrectly – one less than the correct answer.
a) ii)
34
Doubles the 17 dots from the 4th shape to find 34 dots for the 8th shape.
Strategies to find the nth shape (ordinal position)
About a third of students trialled drew pictures of the spatial patterns to solve for the nth term.
An eighth of students extended the table to solve for the nth term.
Both strategies involving drawing and extending the table work until students are required to find the nth term. Then a more efficient strategy is needed. Under half of all students who got part i) correct for questions a)-c) correctly found the number of dots for the nth shape (part ii).
Finding a general rule and using this to calculate the number of dots for any shape is an important algebraic concept. Students often have difficulty generating rules or formulas from patterns and tables. They tend to look for rules calculating the next number in a sequence, (e.g., by "adding four" or "adding one more than before"), but have difficulty developing rules that show the relationship between each number in the sequence and its position (e.g., 3 dots for the 1st shape, 6 dots for the 2nd etc).
Next steps:
When exploring number and spatial patterns encourage students to describe the relationships they can see in words. Discuss which can be written as equations and which cannot. Help them reorganise their descriptions so they can be written algebraically. For example a written description that says "double the number each time" could be written as 2x or 2n.