Building patterns II

Building patterns II

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Further Resources
This task is about continuing a 3-D spatial pattern.
Step 1  Step 2 Step 3
diagram of the faces of the shape
Here are the first three models of a sequence of steps. 
Each model has the faces on the front and back painted blue and all the other faces painted white.
 
a)
Use cubes to build the next 2 models in the sequence, i.e., Step 4 and Step 5.
 
b)
Use the models to complete the table for Steps 4 and 5 only in the table below.
The first three steps have been done for you.
 
 
Check with your teacher if your models and table are correctly completed for Steps 4 and 5 before continuing with parts c) to e).
 
c)
 
Without using the cubes, complete the table above for Steps 6 and 7.
 
d) i)
In Step 10 how many faces would be white? __________
  ii)
Explain in words how to work out the number of white faces in Step 20.
 
 
 
iii)
 
Complete this equation for the number of white faces (w) in Step number n.
 
w = __________
 
e) i)
How many blue faces would there be in Step 10? __________
  ii)
Complete this equation for the number of blue faces (b) in Step number n.
 
b = __________
Task administration: 
This task is completed with pencil and paper, and other equipment.
 
Equipment:
At least 25 small cubes per student.
  • This assessment activity can be carried out with one student at a time or with a group or class of students working independently.
  • Mark sections a) and b), and write in corrections necessary before allowing the student to continue with parts c) and d).
  • The first page of information could be copied onto an OHT and be displayed to the whole class.
Level:
5
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
spatial patterns, model construction, rules, functional rules, quadratic equation
Description of task: 
In this task students build the next two models of a spatial sequential pattern and then use their results to predict subsequent patterns and give general rules for these in words and in equations.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Patterns and relationships, sets 6-7
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 

a)

 Both steps correctly built.
b) All four entries in table correct for Steps 4 and 5, i.e., 16, 20 white faces, and 20, 30 blue faces.
c) 24, 28 white faces for Steps 6 and 7.
42, 56 blue faces for Steps 6 and 7.
d) i) 40
ii) Any 1 of:

  • multiply the shape number by 4, or
  • keep on adding on 4, or
  • equivalent statements.

iii) 4n or equivalent algebraic expressions.
e) i) 110
ii) Any 1 of:

  • n (n + 1), or
  • n2 + n, or
  • equivalent algebraic expressions.

NOTE: This resource spans Level 2 to Level 5.
Part a) is Level 2; Parts b) and c) are Level 3; Parts d) i) and ii) are Level 3-4.
Part d) iii) is Level 4; and Part e) is Level 4-5.