Levels 5–9: Proportional Reasoning
Materials:
Instructions:
Place the first two Basketball Cards in front of the student and say, “Only one of these players (Matt or Henry) can be selected for the school basketball team. Which one would you choose and why? … Note student’s response and explore his/her reasoning.
Place the remaining Basketball Cards in front of the student and say, “Which girl (Jane or Emma) improved the most over the season and why? … Note student’s response and explore his/her reasoning.
Place the first Sharing Pizza Card in front of the student and ask, “Who gets the most pizza?” … Note student’s response and explore his/her reasoning. If answered relatively easily, repeat with the second Sharing Pizza Card. Note student’s response and explore his/her reasoning.
Advice Rubric
An ability to interpret quantities relative to context appears to be strongly related to experience, that is, the opportunities students have had to consider situations that involve relative comparisons across a wide range of contexts. Indeed, the tendency of most commonly used texts to use relatively small whole numbers when introducing these types of problem may actually work against the use of more appropriate relational (ie, multiplicative) thinking.
This task examines the extent to which students are able to recognise when a relational comparison is required and what needs to be compared with what, irrespective of the situation and/or quantities involved.
Observed response 
Interpretation/Suggested teaching response 
Little/no response or incorrect response based on irrelevant or additive reasoning (eg, Matt because he misses less goals), may guess correctly but unable to explain 
May not understand how data relates to the question, interprets situation in absolute terms, may be guessing (50% probability of being correct)  Use the same context to create more disparate situations, eg, Matt scored 28 out of 78 attempts, Henry scored 24 out of 32. Discuss in terms of fairness. Invite students to create similar problems, share and discuss as appropriate
 Devise a range of situations similar to the sharing pizza problem, starting with ones that involve whole number division, eg, 6 girls share 18 apples,,,, 8 boys share 22 apples. Physically model and partition to emphasise the link to division, include examples involving very large and very small numbers

Indicates Henry, may offer additive explanation, may choose Emma on the grounds that she has a higher average, may recognise who has more pizza but unable to explain/justify, may request calculator but not sure how to interpret result 
Suggests some capacity to interpret situations relatively when comparisons fairly obvious (eg, first problem), may not feel confident about using fractions in this context  Discuss strategies for deciding ‘what needs to be compared with what’ for problems like these, (eg, for Emma and Jane, the amount of improvement compared to the starting point, or for the pizzas, focus on the amount that is being shared). Invite students to create similar problems, share and discuss as appropriate
 Review partitioning and fraction renaming strategies (see Partitioning and Partitioning (pdf  199.51kb) paper)
 Review calculator use in this situation, in particular, how to interpret/use the result of a division in relation to the problem context and goal (eg, how to read a decimal fraction directly as a percentage)

Correctly responds to most questions, generally able to explain/justify responses, may request calculator and use percentage as a basis for comparison 
Suggests reasonably well developed capacity to interpret situations relatively and access to appropriate strategies  Provide opportunities for students to work on an extended range of problems involving different contexts and quantities expressed in different forms, eg, decimals, mixed fractions, percentages, ratios, etc
 Consider relating to work on transformations, scale diagrams, variation, inverse proportion, and missing value problems such as, “If it takes 3 men to paint a house in 4 days, how long will it take 6 men to paint the house?”
