# Common Misunderstandings - Level 3.6 Proportional Reasoning Tool

## Level 3: Multiplicative Thinking

### Instructions:

Place card showing eels in front of student and say, “This card shows three eels. The blue eel is twice as long as the red eel and the orange eel is three times as long as the red eel. The eels are fed food pellets according to their length. If the red eel gets 2 food pellets, how many pellets would be fed to the other two eels?” Note the student’s response.

If done fairly easily, point to the blue eel and ask, “If the blue eel gets 14 food pellets, how many pellets would be fed to the other two eels?” Note the student’s response.

If done fairly easily, ask, “If the orange eel gets 18 food pellets, how many pellets would be fed to the other two eels?” Note the student’s response.

This classic task examines the extent to which students are able to work with proportional reasoning. In particular, it explores the extent to which students can use ‘if … then’ reasoning and multiplicative thinking to solve simple problems of the form a/b = c/d where three values are known.

Observed response Interpretation/Suggested teaching response

Irrelevant or incorrect response to initial, red eel question

May not understand task or appreciate the relationship between the length of the eels and the amount of food

• Introduce simple proportional reasoning problems involving doubling and/or tripling, model, and discuss possible solution strategies
• Use an extended range of problems to make the links to multiplication and division explicit

Able to solve the red eel question, and partially solves blue eel question (eg, halves to get 7 pellets for red eel but unable to say how many for orange eel)

Suggests that task is understand but solution strategies limited to doubling and halving

• Use an extended range of proportional reasoning problems that involve more than doubling and halving (eg, if 5 drops of nectar feed 2 butterflies, how many drops will be needed to feed 12 butterflies? How many butterflies can be feed by 25 drops?), discuss solution strategies and make the links to multiplication and division explicit
• Consider introducing problems involving non-integer ratios (eg, where red eel is 10 cm long, blue eel is 15 cm long and orange eel is 25 cm long)

Solves all parts of the task multiplicatively (ie, uses division and multiplication consistently)

Indicates a relatively sound understanding of simple proportional reasoning

• Consider introducing problems involving non-integer ratios (eg, where red eel is 10 cm long, blue eel is 15 cm long and orange eel is 25 cm long)