Tukutuku patterns
0
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about the number of shapes used and the rule for a growing tukutuku pattern.
Task administration:
This task can be completed with pencil and paper or online (with SOME auto marking).
Students may require a reasonable length of time to work on this resource. It may be useful to get students who do not easily see the patterns to work in pairs or in groups.
Level:
5
Curriculum info:
Description of task:
This resource is about students continuing linear or quadratic spatial patterns, and giving rules for them.
Learning Progression Frameworks
This resource can provide evidence of learning associated with within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.Answers/responses:
Y10 (04/2016)  
Red squares 
20
28
40
All 3 correct
1 or more correct (more often, students get just the first one correct)
A correct functional rule (e.g., 4n ) or
a correct recursive rule (e.g., Add 4) given.

easy
moderate
moderate
moderate
easy
moderate

White squares 
25
49
100
All 3 correct 1 or more correct (most often, students get just the first one correct) A correct functional rule (e.g., n × n , Shape number squared) or
a correct recursive rule (e.g., +1, +3, +5 , ... or Add on successive odd numbers) given.

moderate
difficult
difficult
difficult
moderate
difficult

Black squares 
16
36
81
All 3 correct 1 or more correct (usually it is the first one that students get correct)
A correct functional rule (e.g., (n – 1)^{2}, previous shape number squared) or
a correct recursive rule (e.g., +3, +5, +7, ..., or Add on successive odd numbers) given.

moderate
difficult difficult
difficult
moderate
difficult

Based on a representative sample of 101 Year 10 students.
NOTE: Recursive rules were often lacking in specific details for the white and black squares either just saying "Add on successive odd numbers" or not giving the specific increments. Both of these were accepted as correct responses.
Teaching and learning:
This resource is about students continuing linear or quadratic spatial patterns, and giving rules for them. Students use either functional or recursive rules.
Diagnostic and formative information:
 More students gave recursive rules for the the red squares rather than functional rules (33 students and 24 students respectively).
 More students gave functional rules for the the white squares rather than recursive rules (22 students and 10 students respectively).
 More students gave functional rules for the the black squares rather than recursive rules (19 students and 8 students respectively).
In each case, students who gave functional rules were far more likely to correctly give all three of the number of squares for shapes 5, 7 and 10.
At intermediate and at secondary school level, students need to be moving from recursive rules to functional rules. Functional rules should increasingly be written using pronumerals rather than verbal descriptions. This is underlined by the greater success students have at extrapolating sequences when using functional rules. For more on functional and recursive rules, click on the Algebraic patterns concept map.
Students who used pronumerals in their rules tended to use either n or x, though other ones should be accepted. The usual convention is to use n.
Expressing quadratic functions using the keyboard is difficult, because there is no superscript. Students used alternative notations to get around this, for example n × n, n^2, n**2, or n squared.
Common errors  Likely misconception  
Red squares
White squares
Black squares

20, 24, 28
25, 36, 49
16, 25, 36

Incorrect reading of the table
Ignoring the pattern numbers given in the table (7 & 10) and using 6 & 7 instead.
These students may be able to state a correct rule.
Students who gave recursive rules were more prone to making this error than students who gave functional rules.

Red squares White squares Black squares 
20 correct, but other numbers incorrect
36 correct, but other numbers incorrect
16 correct, but other numbers incorrect

Unable to extrapolate more that one pattern ahead
Cannot work with more than the next shape. These students are most likely counting the squares in the next shape, but cannot visualise past that.
These students were very unlikely to state a correct rule.

White squares Black squares 
Various  Fails to see a quadratic relationship 
Red squares White squares Black squares 
Gives an expression for a functional rule rather than a full equation,
e.g., just 4n rather than t = 4n (where t = number of squares, and n = shape number).

Uses an open rather than a closed statement
Almost all students in the trial used closed statements, and these were marked as correct.

Next steps:
Incorrect reading of the table
Students need to pay close attention to the precise numbers in the table. Year 10 students could be expected to work more carefully.
Unable to extrapolate more than one pattern ahead
These students could be encouraged to draw the next few shapes, and carefully count up the number of each colour of squares. Once they have done this, they may wish to find a rule for how many red squares there are. Once they have found the recursive rule (i.e., Keep on adding on 4"), challenge them to find the number of squares in Shape 10, 20 or even 100 without drawing any more shapes.
Fails to see a quadratic relationship
These students may need to draw the next few shapes, and carefully count up the number of each colour of squares. Students who still cannot see the patterns could then work in pairs or small groups. They could firstly compare their results and come up with a consensus on the correct results. Encourage these students to persevere. If they come up with a recursive relationship (e.g., +1, +3, +5, ...) challenge them to find the number of squares in Shape 10, 20 or even 100 without drawing any more shapes. Give them plenty of time. If they are still struggling, you may give them a clue, e.g., "What do you notice about numbers like 25, 36, 49, and 100?"
Uses an open rather than a closed statement
Get these students to read their responses. They should note that "Four times the Shape number" (or 4n) is not a sentence. It is an "open" statement that lacks closure. The complete statement is "The number of red squares in a shape equals four times the Shape number" (or t = 4n — where t is the number of red squares, and n is the Shape number).
For more see the article Equations and expressions.
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