Multiplication boxes and triangles

Multiplication boxes and triangles

Pencil and paper
Overview
Using this Resource
Connecting to the Curriculum
Marking Student Responses
Working with Students
Further Resources
This task is about completing multiplication number sentences.
 

2      3

4      5      6

7      8      9      10
 
 
Use the numbers above to complete each equation by filling in the boxes and triangles. 
Each number can be used more than once in each equation.
 
a) × × 3 =             18
b) × 2 × =             40
c) 4 × × =             36
Task administration: 
This task is completed with pencil and paper only.
Level:
3
Curriculum info: 
Maths, Number and Algebra, Equations and expressions
Key Competencies: 
Using language, symbols, and texts
Keywords: 
equations, multiplication
Description of task: 
Students explore multiplicative relationships by completing multiplication equations using a restricted range of numbers.
Learning Progression Frameworks
This resource can provide evidence of learning associated with
Using symbols and expressions to think mathematically, set 4
within the Mathematics Learning Progressions Frameworks.
Read more about the Learning Progressions Frameworks.
Answers/responses: 
  Y6 (11/2006)
a)
b)
c)

2 & 3 or 3 & 2
4 & 5 or 5 & 4 or 2 & 10 or 10 & 2
3 & 3

easy
easy
moderate
Based on a representative sample of 163 students.

* Most students responded using whole numbers only.  A few, however, gave answers which used decimal numbers e.g. 1.5 × 16.

Teaching and learning: 
This resource is about exploring relationships between numbers.  In this case, it is the relationship between the given number and the total that goes in the square and triangle.  The question underlying the problems is: What should the numbers that go in the triangle and the square equal in order to make the number sentence true?
Diagnostic and formative information: 
  Common response Likely misconception
a)
b)
c)		 3 & 3 or 2 & 4 etc
10 & 10
3 & 6 or 5 & 4 etc		 Gives an additive rather than a multiplicative response, i.e. solves  = 6 (= 18 ¸ 3)
Next steps: 
If students are struggling to see the relationships between the numbers and are unable to come up with combinations, cover up the triangle and square with your fingers (which is equivalent to the problem  × 3 = 18) and ask them, "What number needs to go under my fingers to make the number sentence true?"  Remove your fingers and look for combinations that will multiply to that number.  For example in part a) i) cover up the square and triangle and ask, "What number needs to go under my fingers to make the number sentence true?" The answer is 6.  Lift your fingers and ask students what numbers could go in the square and triangle that multiply to 6.
If students have given an additive rather than a multiplicative response, have them look carefully at the operations in the number sentence.

Students who gave a correct answer, could explore other possible combinations of numbers that would give a correct answer without the restriction of using the numbers listed in the box.

Exploration of the number property, commutativity, can be initiated using this resource: discussion with students over whether 2 × 3 is the same as 3 × 2 can be held.

For more information about algebraic thinking and commutativity refer to the Algebraic Thinking Concept Map: Commutativity.
For further information about using discourse in the classroom refer to Assessment Strategies: Mathematical Classroom Discourse.