# Common Misunderstandings - Level 3.5 Cartesian Product Tool

## Level 3: Multiplicative Thinking

### Instructions

Place card showing T-shirt sign in front of student and say, “This sign says there are 3 different colours and 4 different sizes of T-shirt on special. If you bought one of each type, h ow many different T-shirts could you buy?” Note student’s response.

If this is done relatively easily, place the lunch order card in front of student and say, “ The school canteen offers 2 choices of rolls, 4 choices of filling and 3 choices of drink. Claire ordered a peanut-butter roll and a drink. What might she have ordered?” Note student’s response.

If this done relatively easily, say, “There are 23 students in Claire’s class. If everyone ordered a roll with a filling and a drink, could they all have a different lunch order?” Note student’s response, stop if student appears uncertain, try to find out what is making this difficult.

## 3.5 Cartesian product tool

This task examines the extent to which students are able to work with the Cartesian product or ‘for each’ idea of multiplication. In particular, it explores the strategies used by students to determine the total number of options or combinations. This idea arises in many Chance and Data problems and is needed to support later work with fractions (eg, fraction renaming) and rate (eg, 60 kilometres/hour or 3.4 kg @ \$1.29/kg) for which the groups of idea is no longer applicable.

Observed response Interpretation/Suggested teaching response

Incorrect (eg, may add) or no response to T-Shirt task

May not understand or be able to represent the task

• Introduce simple problems like the T-Shirt task (ie, two variables), discuss how this might be represented and the ‘for each’ idea (eg, for each colour there are 4 sizes, or for each size there are 3 colours)
• Consider introducing tree diagrams as a more efficient way to represent problems of this type

Solves T-Shirt task (by listing all or multiplying) but irrelevant, incorrect or no response to initial Canteen task

Suggests some capacity to model situations involving two variables but may not understand or be able to represent situations involving more than 2 variables

• Review problems involving two variables (see above) and problems like the initial Canteen task where numbers needed are not stated
• Discuss different ways to represent problems of this type (eg, tree diagrams, tables) and extend to problems involving three or more variables
• Where students are reliant on representing/counting all, make links to multiplication more explicit

Solves T-Shirt task and initial Canteen Task (6 options) by listing/counting all options (additive) or by multiplying 2 by 3 (multiplicative), may respond to last question but unable to justify thinking/reasoning

Suggests some capacity to model/work with Cartesian product situations but unable to explain or justify their thinking

• Discuss different ways to represent problems of this type and make links to multiplication more explicit (see above)
• Encourage students to construct similar problems and to justify and defend their reasoning to others

Solves all tasks multiplicatively (ie, uses multiplication directly without the need to model)

Indicates relatively sound understanding of Cartesian product idea

• Provide more complex problems (eg, from Chance and Data strand) and/or invite students to construct similar problems
• Consider applying the ‘for each’ idea to fraction models and diagrams (see Partitioning (pdf - 199.51kb) paper)