Level 5: Proportional Reasoning
Materials:
Instructions:
Place the first ‘Then and Now’ Card in front of the student and say, “This shows Amy and Richard Smart when they were young (point to smaller images). This shows them now (point to larger images). Who grew faster between the first and second pictures? Amy or Richard?” Note student’s response and explore his/her reasoning.
Place the second ‘Then and Now’ Card in front of the student and say, “In 2003 tree A was 2 metres tall and tree B was 3 metres tall. Now, in 2006, tree A is 5 metres tall and tree B is 6 metres tall. Which tree grew more?”… Note response and ask student to explain his/her reasoning.
Place the red and yellow Rod Cards (or corresponding Cuisenaire Rods) in front of the student, point to the red rod and say, “If the red rod is 1, what would the yellow rod be?” …If no response or some hesitation, ask, “How many times longer is the yellow rod than the red rod?” … Note response and ask student to explain his/her reasoning if not obvious.
Place the purple and blue Rod Cards (or MAB tens and corresponding Cuisenaire Rods) in front of the student, point to the purple rod and say, “This purple rod is 2. What fraction name would you give to the blue rod?” …If no response or some hesitation, place the pink Rod Card (or equivalent Cuisenaire Rod) in front of the student and ask, “What fraction is the pink rod of the blue rod?” … Note response and ask student to explain his/her reasoning if not obvious.
5.1 Advice Rubric
A fundamental component of being able to think and reason proportionally is the capacity to analyse change in both absolute and relative terms and use appropriate to context. For instance, a brother and sister may each have grown 6 cm in one year, but the girl (assuming she was shorter to start with), grew ‘more’ because the 6 cm represents a greater proportion of her original height.
While most middle years’ students are able to recognise and name simple fractional parts in relation to physical models (chocolate, pizzas, sandwiches etc), many have had little or no exposure to using fractions to describe proportional relationships.
This task examines the extent to which students are able to use relational (multiplicative) as opposed to absolute thinking (additive) thinking to analyse change over time and to identify the relationship of one object to another. It assumes some prior experience with scalar multiplication (times) and a knowledge of fraction names.
| Observed response |
Interpretation/Suggested teaching response |
|---|
| May say Richard grew more but states that trees grew the same, unable to explain or accurately describe the relationship between rods |
May not understand task or interprets ‘more’ in additive (absolute) terms only, may give general description of rods (eg, “it’s bigger”), may not recognise the possibility of different units - Discuss situations involving relative comparisons (eg, 7 fish in a fish-bowl, 7 fish in a garden pond, 7 fish in a lake) in terms of what remains the same what is different
- Compare and contrast the meaning of “more” in different contexts, eg, “Harry had 17 football cards, Jamie had 23. How many more cards does Jamie have than Harry?” (additive/absolute thinking) compared to “Harry traded 9 of his cards. Jamie traded 11 of his cards. Who traded more of their cards?” (multiplicative/relational thinking)
- Use Cuisenaire Rods and/or Pattern Blocks to accurately describe relationships between rods or pieces, eg, the triangle is 1 sixth of the hexagon and half of the rhombus
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| May be able to justify why Richard grew more and recognise that Tree A grew more but unable to explain/justify, may identify yellow rod as 2½ times the red one when prompted, experiences some difficulty with remaining rods |
Suggests some capacity to recognise relational aspects of a situation but experiences difficulty describing relationships formally - Use Cuisenaire Rods and Pattern Blocks to practice identifying and describing relationships in mathematical terms irrespective of the unit, eg, If the brown rod is 1, what number name would you give to the other Rods? If the trapezium is 1, what number name would you give to the other Pattern Block pieces?
- Discuss the use of terms such as ‘more’ and ‘times’ and how to identify problems requiring relational thinking, eg, “Which basketball game was closest, the Jets (49 goals) who defeated the Swallows (46 goals), or the Amazons (34 goals) who defeated the Warriors (31 goals)?”
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| Able to explain why Richard and Tree A grew more, may justify this using per cent or fractions, able to describe rods in terms of other rods, using purple rod as a ‘measure’ |
Suggests a capacity to use relational thinking, may not be able to apply to a wider range of contexts, may need to develop more formal strategies for justifying responses - Use a wide range of problem contexts/situations that require students to discriminate between those that require an absolute response and those that require a relational response
- Compare the types of arguments that need to be used to justify responses, eg, nearly all involve a part-whole comparison which can be expressed as fractions, ratios or percentages
- Encourage students to devise their own relational problems using familiar contexts and sources such as newspapers etc
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