Proportional Reasoning Strategies progression, 6th step
The purpose of the activity
In this activity, the learners apply their understanding of ratios to rates and proportion problems. They learn that rates are examples of ratios where a comparison is made between quantities of different things and that solving proportional problems involves applying a known ratio to situations that are proportionally related and finding one of the measures when the other is given.
The teaching points
A ratio can also be a rate. Part-to-whole and part-to-part ratios compare two quantities of the same thing. Rates on the other hand are examples of ratios where a comparison is made between quantities of different things. In rates, the measuring units are different or the quantities being compared are different and the rate is expressed as one quantity per the other quantity. For example, the value of food can be expressed as price per kilogram, fuel efficiency can be expressed as litres per 100 kilometres. Finding the unit rate is one way of solving proportional problems.
Solving proportional problems involves applying a known ratio to situations that are proportionally related and finding one of the measures when the other is given. For example, if it takes 10 balls of wool to make 15 beanies, 6 beanies will take 4 balls of wool. In this example, the ratio of 2:3 (balls to beanies) can be applied to each situation.
There are three main computation approaches: using strategies to calculate a problem mentally, using traditional written methods (algorithms) and using calculators. The complexity of the problem determines the most effective and efficient computation approach.
Discuss with the learners relevant or authentic situations where rates are used (price per kilogram, kilometres per hour).
The guided teaching and learning sequence
1. Write the following problem on the board. At the supermarket, you want to buy the best value fruit yoghurt. Assuming that you are not concerned about how much yoghurt you buy, which should you purchase?
2. Discuss with the learners their understanding of the problem.
“What does ‘best value’ mean?” (The lowest price for the same amount)
“How many grams are in the packet of 4?”
3. Ask the learners to look at the list and see if there are any that are obviously not the best value.
“Which options are definitely not the best value?”
“How do you know?”
Possible explanations include:
Option 1 is close to $3 for 250 grams, so at $6 for 500 grams, it is more expensive than options 3, 4 or 5.
Option 5 is 600 grams and is more expensive than the 600 grams of option 4.
4. Ask the learners for their ideas about how to compare options 2, 3 and 4. Hopefully someone will suggest that you can work out how much each costs as a rate (price per gram or per 100 grams or per kilogram) and then do a direct comparison. This approach can be referred to as a ‘unit-rate’ method of solving proportions.
5. Add a column to the table, agree on the rate (price per 100 grams) and then ask the learners (working individually or in pairs) to calculate the unit rate for options 2, 3 and 4.
6. Remind the learners that they can choose to use a calculator, a written method (traditional algorithm) or mental strategy to work out the unit rate. Tell them you expect them to be able to explain the reasonableness of their answers irrespective of the approach they used.
7. Ask for volunteers to record one of the rates on the table and to explain to the class how they obtained it. Ask them to explain why they chose a particular approach and how they knew the answer made sense.
“Why did you choose to use [approach]?”
“Why is your answer reasonable or why does your answer make sense?”
8. Look at the table and discuss other ways that the rates could be expressed. Add these as columns to the table on the board, asking the learners to work out the new unit rates. Discuss with the learners the fact that rounding errors mean that conversions from the first rate will sometimes differ from calculations in relation to the original value. However, in relation to the context of this problem, all that was required was the ‘best value’ item rather than the exact unit rate for each item.
9. Pose the following problems for the learners to work on in pairs. As they work, discuss with them the computation approach they have selected and their explanation for the reasonableness of their answers.
Dan can run 5 kilometres in 16.3 minutes. If he keeps running at the same speed, how far can he run in 27 minutes?
Jack can run an 8-kilometre race in 37 minutes. If he runs at the same rate, how long should it take him to run a 5-kilometre race?
The cost of a box containing 48 chocolate bars is $45.00. What is the cost of 8 bars?
Ask the learners to write a proportion problem (similar to those shown above). Have them swap their problem with that of another learner, solve the new problem and then check one another’s solutions.
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