Black and white triangle patterns

Black and white triangle patterns

Pencil and paperOnline interactive
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Further Resources
This task is about continuing a spatial pattern and writing a rule.
Mani made these shapes with some black and white triangles.
Shape 1 Shape 2 Shape 3
 
This table shows the numbers of each colour of triangle Mani used when he made the first five shapes.
 
Shape number (n)
1
2
3
4
5
i)   Number of black triangles (b)
1
3
5
7
9
ii)  Number of white triangles (w)
3
5
7
9
11
iii) Total number of triangles (t)
4
8
12
16
20

Question 1Change answer

a)  Mani made bigger and bigger shapes.
     To make shape 10, how many triangles did he use that were coloured
     i)     black?
     ii)    white?

Question 1Change answer

b)  Complete this rule relating the total number of triangles (t) with the shape number (n).
     t  =

Question 2Change answer

c)  Complete this rule relating the number of white triangles (w) with the shape number (n)
     w  =

Question 3Change answer

d)  Complete this rule relating the number of black triangles (b) with the shape number (n)
     b  =

Question 4Change answer

e)  Which shape number would have 35 black triangles?
     
Task administration: 
This task is completed with pencil and paper or online.
Levels:
4, 5
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
spatial patterns, functional rules, linear equations
Description of task: 
The start of a spatial pattern of triangles is shown and described in a table. Students generate more of the pattern and describe the relationships algebraically.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Patterns and relationships, sets 5-6
Patterns and relationships, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

   

Y10 (11/1999)

a)

i)
ii)

19
21

easy
easy

b)

 

4n

moderate

c)

 

2n + 1

moderate

d)

 

2n - 1

moderate

e)

 

18

moderate


NOTE:  

  1. Accept algebraically equivalent expressions.
  2. Accept expressions such as 4 × n or n × 4 as being equivalent to 4n.
Diagnostic and formative information: 

 

Common error

Likely calculation

Likely reason

a) i)
    ii)

18
22

9 × 2
11 × 2

Doubles the number of triangles from shape 5 (n = 5).

b)
c)
d)

"add on 4"
"add on 2"
"add on 2"

 

Gives the extra number of triangles for the next shape rather than an algebraic expression.

e)

69

17

35 × 2 - 1

(35 - 1) ÷ 2 or  
n = (t - 1) ÷ 2

Calculates the number of black triangles 
in shape 35.
Shape number with 35 white triangles. 
Should have used n = (t + 1) ÷ 2.