Common Misunderstandings - Levels 5–9 Working With Rate Tool

Levels 5–9: Proportional Reasoning

Instructions:

Place Harry’s Snail Card in front of the student and say, “What would you need to do to work that out? … Could you do that please?” Indicate pen and paper.Note response and ask student to explain his/her reasoning if appropriate.

Place Ron’s Snail Card in front of the student and say, “What would you need to do to work that out? … Could you do that please?” Indicate pen and paper.Note response and ask student to explain his/her reasoning if appropriate.

Place large Snail Card in front of the student and say, “Could you do that please?” Indicate pen and paper.Note response and ask student to explain his/her reasoning if appropriate.

Rate is a particular type of ratio that involves a comparison of different measures. It also implicitly involves an understanding of the ‘for each’ idea for multiplication, for example, speed is the measure of distance travelled ‘per unit of time. While some problems involving rate can be solved in terms of the repeated addition or ‘lots of’ idea (eg, the problem, At a constant speed of 80 km/hr, how far will a car travel in 4 hours? can be interpreted as ‘4 lots of 80’), but this only applies for unit rates and relatively simple numbers.

This task examines the extent to which students are able to interpret and use rates to solve problems involving speed. It assumes some prior experience with measures involved and the concept of speed as distance divided by time.

Observed response Interpretation/Suggested teaching response
Little/no response, may recognise that 21 × 7 is needed for the first question May understand the concept of speed but unable to deal with situations involving 2 or more steps and/or less ‘conducive’ numbers
• Explore similar problems, discuss what is required and why, review solution strategies as numbers change, eg, What if the speed of Harry’s Snail was given as 28 cm in 2 minutes? What if the time involved was 5&frac12; minutes?
• Review the area idea for multiplication using mm graph paper to model what happens when a decimal is multiplied by a whole number, or a decimal is multiplied by another decimal. Focus on language involved, eg, ones by ones are ones, ones by tenths and tenths by ones are tenths, tenths by tenths are hundredths and so on
• Use a broader range of problems to identify what is required and review calculator use
• Review strategies for solving multiple-step problems, ie, identify steps, make a plan, carry it out, review at the end of each step
Able to say how far in 7 minutes, may make a start on remaining problems but loses track of what is required and/or makes arithmetic error(s), may recognise relationships, eg, common factor of 7 for speed and distance on Card 2, but unable to use these effectively Suggests that the concept of rate is reasonably well understood, at least in relation to speed, but needs to consolidate knowledge of rational numbers and strategies for dealing with these effectively
• Review partitioning strategies to ensure students are able to make and name their own fraction diagrams and number line representations (see Partitioning (pdf - 199.51kb) paper)
• Review renaming strategies for fractions (see Partitioning) and associated language, eg, factors, common factors, divisors. Provide experience in renaming and simplifying fractions. Discuss other ways of representing numbers and any issues involved (eg, 4 minutes and 30 seconds can be written as 4.5 minutes but 2 minutes and 45 seconds cannot be written as 2.45 minutes)
• Introduce a range of similar problems with easier numbers to identify what needs to be done and to set up appropriate equations, have students rewrite problems using ‘tricky’ numbers and measures. Discuss.
Multiplies to solve the first problem but tends to use an additive approach for remaining problems, eg, for Ron’s Snail, adds/doubles 14 to get 28, realises that 7 cm more would take 30 seconds or half a minute, may or may not use relationships inherent in last question Appears to understand concept of rate and how it applies but tends to rely on additive strategies to solve missing value problems
• Use a range of similar problems (ie, where there is an obvious relationship between the numbers involved as in questions 2 and 3) to compare and contrast different solution strategies, in particular, those involving the use of a common factor/composite unit as in the case of Samantha’s Snail (possible to work with a unit of 3 minutes rather than find for 1 minute). Invite students to decide which strategies are better (more efficient, generalisable), apply to solve class-generated problems where there is no obvious relationship between the numbers involved
• Extend to missing value problems more generally involving different rates and quantities, reviewing the use of equations to solve problems if necessary
Able to solve all problems fairly efficiently using relationships and/or systematic multiplicative approach Suggests a sound understanding of rate and how such problems can be solved by multiplication and division
• Extend to a broader range of problems, discuss solution strategies and use of relationships where they exist
• Discuss the use of tables and graphs to solve rate problems more generally, provide experience constructing and interpreting graphical ‘stories’, such as filling up and emptying a bath at different rates.