L Patterns

L Patterns

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about continuing a spatial pattern and finding the rule.

Suli is playing with counters, making letter Ls of various sizes, like this:


   

a) Complete this table, giving the total number of counters in letter Ls of different sizes:


Height of letter L (counters)
 		 3 4 5 6 7 8

Total number of counters
 		 4 6 8        
 
b)
 
How many counters in total would be needed to make a letter L that is ...
 

  

i)

ii)

iii)

 
10 counters high? __________ counters
 
16 counters high? __________ counters
 
37 counters high? __________ counters
 
c)
Explain or write a rule that describes how to work out the total number of counters needed from the height of an L-shape.
 
 
 
 
 
d) If the total number of counters was 40, how many counters high would the letter L be?
 
____________ counters
Task administration: 
This task is completed with pencil and paper only.
Level:
4
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
spatial patterns, rules
Description of task: 
Students complete a table showing the number of counters used to make a series of L-shapes. They identify the number of counters needed for different situations, and describe the relationship as a rule.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Patterns and relationships, sets 5-6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
 

Y8 (11/2004)

a)

 

10, 12, 14

very easy

b)

i)
ii)
iii)

18
30
72

easy
moderate
difficult

c)

 

For any 1 of:

  • multiply the number of counters in the height of the L by 2 and subtract 2 (c = 2h – 2 or c = h + h - 2);
  • subtract 1 from the height and multiply by 2 (c = 2(h - 1));
  • other answers correctly relating height or base to number of counters, e.g., "double the number of counters in the base of the L (c = 2b)".

difficult

d)

 

21

difficult

Results are based on a trial set of 211 students.

Diagnostic and formative information: 
  Common error Likely calculation Likely misconception
b) i)
    ii)
    iii)		 19
31
73		 10 × 2 - 1 = 19
16 × 2 - 1 = 31
37 × 2 - 1 = 73		 Identifies an incorrect rule
(minus 1, instead of minus 2).
b) i)
    ii)		 16
28		 8 × 2
14 × 2		 Doubles the results from the table.
b) i)
    ii)
    iii)		 16, 20
28, 32
70, 74		   Counts up in twos and loses place.
c) +2   Explains the total number of counters needed based on the previous total not the height of any given L shape (e.g., count up in twos).