Different rules

Different rules

Pencil and paper
Overview
Using this Resource
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Further Resources
This task is about using different rules to continue number patterns.

 

 

 

a)
 		 i)
 
Use the rule "add three" to fill in the next three numbers in the pattern below.
 
3, 6, 9, _____, _____, _____, ...
 
  ii)

 

 Using a different rule to question a) i), fill in the next three numbers in the pattern below.
 
3, 6, _____, _____, _____, ...
 
What did you do to continue the pattern?
 
_________________________________________________________________________
 
b)

 

  
i)

 
 
Work out a rule and fill in the next three numbers in the pattern below.
 
2, 6, 10, _____, _____, _____, ...
 
What did you do to continue the pattern?
 
 
 
 
 
  ii)

 

 Using a different rule to question b) i), fill in the next three numbers in the pattern below.
 
2, 6, _____, _____, _____, ...
 
What did you do to continue the pattern?

 
 
 
 
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Key Competencies: 
Thinking
Keywords: 
number patterns, rules
Description of task: 
Students complete number sequences, and identify different rules for sequences that begin with the same numbers.
Curriculum Links: 
Key competencies
This resource involves identifying a number of different rules for a number pattern where the first few terms are given. This relates to the Key Competency: Thinking.
For more information see http://nzcurriculum.tki.org.nz/Key-competencies.
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)		 9, 12, 15
Any 1 of:

  • 12, 24, 48 and double/multiply by two (increasing multiplicative);
  • 3, 6, 3 and add 3 then subtract 3 (repeating pattern);
  • 10, 15, 21 and +3, + 4, + 5 (increasing by an incrementing amount);
  • 9, 15, 24 and 3 + 6 = 9, 6 + 9 = 15, 9 + 15 = 24 (Fibonacci pattern);
  • Other sequences with a rule stated and consistently applied.
 very easy
difficult
b) i)
    ii)		 10, 14, 18 and add four (increasing additive)
Any 1 of:

  • 18, 54, 162 and multiply by 3 (increasing multiplicative);
  • 11, 17, 24 and + 5, + 6, + 7 (increasing by an incrementing amount);
  • 8, 14, 22 and 2 + 6 = 8, 6 + 8 = 14, 8 + 14 = 22 (Fibonacci pattern);
  • Other sequences with a rule stated and consistently applied.

NOTE: the rule for a) ii) and b) ii) must be different - the most common answer for b) ii) was "plus 4".

 very easy
difficult

Based on a representative sample of 220 students.

Diagnostic and formative information: 
  • Nearly half the students in the trial gave the same rule (not a different rule) for a) ii) as already given in a) i), and a third gave the same rule for b) ii) as for b) i).
  • Many students attempted to apply a rule that changed for each number in the pattern, rather than identifying a more general rule that applies equally to each element in the pattern.

Strategies for completing the patterns

Strategy   Rule Typical answer % using
Increasing additive b) + 4 10, 14, 18 ... 85%
Increasing multiplicative a)
b)		 × 2
× 3		 12, 24, 48 ...
18, 54, 162 ...		 21%
15%
Increasing by an incrementing amount a)
b)		 e.g., + 3,+ 4,+ 5
e.g., + 4,+ 5, + 6		 10, 15, 21 ...
11, 17, 24 ...		 4%
5%
Other correct, e.g.,

  • Fibonacci (adding the two previous numbers);
  • Repeating pattern/alternating rule (add the 1st number then the 2nd and continue alternating).
 a)
b)
a)
b)
a)
b)		 3 + 6, 6 + 9, 9 + 15
2 + 6, 6 + 8, 8 + 14
e.g., + 3 and - 3
or + 3 and + 5
e.g., + 4 and - 4
or + 4 and + 2		 9, 15, 24 ...
8, 14, 22 ...
3, 6, 3 ...
11, 14, 19 ...
2, 6, 2 ...
8, 12, 14 ...		 9%
10%

 

Next steps: 

Encourage students to find a general rule that applies to all numbers in the sequence/pattern, and explore the use of operators other than addition when identifying rules for sequences