Stick patterns and rules

Stick patterns and rules

Pencil and paperOnline interactive
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about describing rules for spatial patterns.

The growing patterns in this task have been made up from ice block sticks.

Question 1Change answer

three shapes of a stick pattern

For the shapes in the pattern above, write a rule to explain how to work out how many ice block sticks are needed for any shape number (for example, someone may ask how many ice block sticks are needed to make shape 67).

Question 1Change answer

three shapes of a stick pattern

For the shapes in the pattern above, write a rule to explain how to work out how many ice block sticks are needed for any shape number (for example, someone may ask how many ice block sticks are needed to make shape 52).

Task administration: 
This task can be completed with pencil and paper or online.
Levels:
3, 4
Curriculum info: 
Maths, Number and Algebra, Patterns and relationships
Keywords: 
spatial patterns, functional rules, recursive rules
Description of task: 
Students describe a functional rule for two spatial patterns.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Patterns and relationships, sets 4-5
Patterns and relationships, sets 5-6
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
    Y8 (11/2009)
a) Accept functional and sequential descriptions of the rule.
Functional:

  • n + 3 or 4 + (n – 1) or equivalent rules (where n = shape number)
  • Add three to whatever the shape number is (this may involve recognising that the number of horizontal slats is the same as the shape number then simply add three more (implicitly n + 3).

Sequential:

  • Add 1 per shape and 3 to start off with.
  • Add 3 to the previous shape*.
 moderate
b) Accept functional and sequential descriptions of the rule.
Functional:

  • 4n + 2
  • 4(n – 1) + 6
  • Accept other equivalent rules, e.g., 6n – 2(n – 1) or 2n + 2(n + 1)
  • Start with 2 and add 4 for every shape number (this may involve recognising that it is 4 lots of the shape number).

Sequential:

  • Start with 6 and add 4 to the previous shape (reference to previous shape number indicates the sequential nature of the rule).
  • Add 4 to the previous shape*.
 moderate

Based on a representative sample of 150 Y8 students

NOTE:
If students describe a rule that is correct, they should also be able to apply it correctly to show how it works (e.g., they could show how it works for easy numbers such as 10 or 100).

*Students who answered only referring to what happens to the previous shape need to be asked to clarify their rule (see Next steps).

Teaching and learning: 

These questions have been phrased to elicit a functional (direct) rule which can be applied to find an nth term of a pattern, rather than a sequential rule which relies upon the recursive nature of the previous term/s. 
Click on the link to the Algebraic pattern concept map, to find out more about functional relationships or recursive strategies.

Diagnostic and formative information: 

Strategies
For both questions over 40% of students successfully described a functional or sequential rule. Many students answered with a correct sequential rule, and others described how the spatial pattern grew. A number of students could work out the number of sticks required for the example but not describe the rule.
The mean ability of students who described a functional rule was notably higher than students who described a sequential rule. About half of the correct strategies were functional rules for question a) and about a quarter for question b).

Examples of students' correct strategies
*These examples could be cut out and used for class or small group discussion.
**Functional (=F) and sequential (=S)

a)
  • Every time you go on to a new shape add 1 ice block stick. (S)
  • Add one per shape and 3 to start off. (F)
  • Get the shape number and add 3. (F)
  • You would already have 3 sticks -  so then the shape number is the sticks that go across,
  • e.g., shape 1   . Example   etc (50 sticks). (F)
b)
  • You start off with 6 sticks and continue adding 4 sticks.
  • So shape 4 will be 18 sticks. This explanation could be:
    - adding 4 for each shape (sequential), or 
    - 6 + 3 lots of 4 (functional, where 3 is n–1).
  • It also illustrates how the sequential and functional rules can relate to each other.
  • The number of sticks is the shape number (n) × 4 + 2. Example: shape 30 = 122 sticks. (F)
  • Every time you go on to a new shape add 4 ice block sticks. (S)


Examples of students' incorrect strategies (some of these are very close to sufficient explanations)

a)
  • You have to add one ice block stick to make the next shape number, so shape 47 would be 44 ice block sticks.
  • You start off with 4 sticks (shape 1) and add 1 more to make shape 2 and add 3 more sticks to make shape 4. [Description of what to do to the first shape, but more clarity/detail needed to develop a generalisable rule. Currently couldn't be used to find an nth term]
  • Since there are 4 sticks to make the first shape you must multiply 4 by the shape number (e.g., 47 x 4 = 188).**
b)
  • You would find how many sticks make shape 3 then multiply the answer by 3 and add a zero to the final answer.**
  • The shapes look like houses, so every tie there's a different shape in the pattern, a house is added.
  • Four sticks are added each time, so from shape 1 starting off with 6 + 4 = 10, so 4 x 30 = 120 +6 = 126.**
  • Shape number x 6 + shape number x 4 – 1 = answer.
  • You times the shape number by 4 and then add the shape number, then take 1 off and that's the answer (e.g., shape 47: 47 x 4 = 178 + 47 = 225 –1.) [This works for the first shape, but none thereafter.  It is essentially n × 5 – 1),].
  • Add to the number of the shape, that number times by itself with another 2 in the question.

**These strategies/explanations ignore that the starting point is more (Students who ignored the starting point)

Next steps: 

Students who described an incorrect/insufficient rule
Encourage the student to identify what is happening from one shape to the next to develop a sequential idea of how the pattern grows.  The resources Pyramid pattern and Stacking cans look at developing sequential rules for spatial growing patterns at Level 3. Students could share their rules for the patterns in small groups and check that each others' rules work for all members of the pattern. Once students can identify the sequential rule they could look at developing a functional rule (see Students who described a sequential rule).

Students who described or showed using examples
For students who described or showed with examples the rule for the pattern – they can be asked to find a way to write a rule as an algebraic expression.  They should test out the expression on several shape numbers for which the numbers of sticks are known, and then the larger numbers (i.e., how many sticks would be needed for shape number 100).

Students who described a sequential rule
For students who described a sequential rule, ask them how they might find a rule that would work for any given number, e.g., How many sticks for shape 100? Trying to use the sequential rule should be unwieldy and the need for a functional rule can become evident.  Additionally, as students apply their sequential rule they may see the connection between the number they are adding on to the previous shape and the coefficient of n (shape number) of the functional rule.

Students who described a functional rule with some errors
For students who described a functional rule, but made some errors, encourage them to share and explain their rule, and check that the rule works for all members of the pattern – not just the first two.   A small group discussion should provide peer support to re-check the completeness and accuracy of their rule.

Students who ignored the starting point (also students who used a simple multiplication of previous shape numbers) 
Get students to create more shapes of each pattern. Ask, "Does Shape 6 have twice as many sticks as Shape 3?" or "Does Shape 3 have three times as many as Shape 1?"  For question a) remove the 3 vertical sticks of each shape and ask how the pattern grows.  For question b) take away the first two vertical sticks of each shape and ask how the pattern grows.  This should help students see how the intercept influences the pattern. For example in question a) the first shape has 2 + 4 sticks, the second 2 + 4 + 4, so the 2 is the initial term of an arithmetic progression (as well as the intercept) and the 4 is the common difference (or increase) – which becomes the coefficient of n in the rule for the pattern.

For all above misconceptions the range of students' rules from this resource (correct and incorrect) could be printed off and shared in a whole class or small group discussion.

Numeracy resources
Thinking Ahead, Book 9: Teaching Number through Measurement, Geometry, Algebra and Statistics, 2006  

For more information about patterns, see the Algebraic Patterns Concept Map.