Working with equivalent rates

1.

Use the cards of coconuts made from Working with equivalent rates CM and toy $1 coins to pose this problem, or pose another, similar problem that makes use of a more relevant context.

The problems should be solved by repeating (iterating) a non-unit rate. The students may still elect to calculate the unit rates and apply them to find the missing value. However, one-step strategies also work.

• Tammy pays $12 for 4 coconuts. How much should she pay for 8 coconuts?

Model the problem with food cards and $1 coins and ask students to attempt the problem with a partner.

2.

Provide time for students to share their solutions. Look for them to justify their thinking with reference to division and/or multiplication and the material model. Construct relevant diagrams and written expressions as students share their thinking. Strategies might include:

• Finding the unit rate of dollars per coconut by division. Since 12 ÷ 4 = $3 per coconut, eight coconuts cost 8 x 3 = $24.

• Doubling the rate of four coconuts for $12 to get eight coconuts for $24.

3.

Use the calculator to confirm the answer to the problem. 

• What operation can I put into the calculator to find out the cost of 8 coconuts? 

4.

Use rate tables to represent the strategies.

5.

Use the cards created from the copymaster and toy $1 coins to introduce the following problems, or similar problems involving money and more relevant contexts. Model each problem before asking students to solve it.

Allow calculators if students need to use them to make and/or check calculations. Consider grouping students to encourage peer scaffolding and extension. Look for students to apply division and justify the operation by referring to equal sharing. 

Consider introducing relevant te reo Māori kupu, such as pāpātanga (rate), whakawehe (divide, division), and whakarea (multiply, multiplication).

For each problem, students should draw a rate table, including the operational arrows.

Lexi buys 3 mangoes for $5.

• How many mangoes will she get for $20?

The initial rate, 3 for 5, makes using a unit rate strategy complicated. This may encourage students to use a within-strategy.

For some students, you may need to demonstrate with materials how the iteration to the 3 for 5 rate works.

6.

Pose other problems that develop the idea of iterating or equally partitioning a non-unit rate. A good sequence of problems might be:

• Taine can buy six taros for $21. How much will he pay for two taros?

• Jianyu can buy 12 peppers for $18. How much will he pay for 3 peppers?

• Selina can buy 10 coconuts for $35. How many coconuts can she buy for $14?

1.

Pose further contextually relevant problems that encourage students to generalise when using a unit rate is best and when using a non-unit rate is best. Much depends on the numbers: non-unit rates are easier to work with than a unit rate if there is an easy multiplicative relationship between numbers in the rates. 

For example, in this table, the numbers in the non-unit rate have a common factor (the number they both divide by) of 3. $4 is easily multiplied by 4 to get the target number of dollars: $16.

Unit rate strategies always work, irrespective of the difficulty of the relationship between the numbers. Calculators make using a unit rate even more attractive.

2.

Ask students to create their own non-unit rate problems. Let them use the cards from the copymaster and toy $1 coins to create and model the problems and solutions. Doing so will help students recognise that accessible problems require careful choices about the numbers.