Dot patterns

Dot patterns

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about identifying the number of dots in different sized shapes of a spatial pattern.
a)
1st 2nd 3rd
 
 
i)  

 
 
Complete the table to show how many dots are in the 3rd and 4th shapes above.
 
shape 1st 2nd 3rd 4th
number of dots 5 9    
 
ii)
 
How many dots will be in the 8th shape of this pattern? __________
 
b)

1st 2nd 3rd
 
 
i)  

   
 
Complete the table to show how many dots are in the 3rd and 4th shapes above.
 
shape 1st 2nd 3rd 4th
number of dots 8 13    
 
ii)
 
How many dots will be in the 9th shape of this pattern? __________
 
c)

1st 2nd 3rd
 
 
i)  

 
 
Complete the table to show how many dots are in the 3rd and 4th shapes above.
 
shape 1st 2nd 3rd 4th
number of dots 3 6    
 
ii)		  
How many dots will be in the 7th shape of this pattern? __________
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
spatial patterns, rules
Description of task: 
Students identify the number of dots in the <i>n</i>th shape of a series of spatial patterns.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Patterns and relationships, sets 4-5
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
 

Y7 (03/2005)
a) i)
ii)		 13, 17
33		 easy (for both correct)
difficult
b) i)
ii)		 28, 23
48		 moderate (for both correct)
difficult
c) i)
ii)		 10, 15 (non-linear pattern)
36		 moderate (for both correct)
difficult
Based on a representative sample of 220 students.
Diagnostic and formative information: 
  Common error Likely misconception
b) i)
    ii)		 16,19, ...and
34		 Only adds on the 3 dots to the top of each shape.
c) i)
c) ii)		 9,12, ...and
21		 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.