Working with equivalent rates

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Purpose

The purpose of this activity is to support students in treating a given non-unit rate as a unit that can be repeated (iterated), and equally partitioned, to solve problems.

Achievement Objectives
NA4-4: Apply simple linear proportions, including ordering fractions.
Required Resource Materials
  • Copymaster (make into cards of individual foods, 12 of each food)
  • Toy money ($1 coins)
  • Calculators
Activity
  1. Use the cards of coconuts made from the Copymaster 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 able to solved by repeating (iterating) a non-unit rate. The students may still elect to calculate the unit rates and apply it 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.
    4 coconuts and 12 $1 coins. Every coconut is aligned with the middle coin in each group of 3.
     
  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 = $3.00 per coconut, eight coconuts cost 8 x 3 = $24.00.
      Four circles, each with a coconut and 3 $1 coins.
       
    • Doubling the rate of four coconuts for $12 to get eight coconuts for $24.00.
      4 coconuts and 12 $1 coins.4 coconuts and 12 $1 coins.
       
  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.
    Rate table.                     Rate table.
     
  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 befor 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 $5.00. How many mangoes will she get for $20.00?
    The initial rate, 3 for 5, makes using a unit rate strategy complicated. This may encourage students to use a "within" strategy.

    Rate table.
     

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

    Four circles, each with 3 mangoes and 5 $1 coins.
     

  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?
      Rate table.
       
    • Jianyu can buy 12 peppers for $18.00. How much will he pay for 3 peppers?
      Rate table.
       
    • Selina can buy 10 coconuts for $35.00. How many coconuts can she buy for $14.00?
      Rate table.
       

Next steps 

  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 (number they both divide by) of three. $4.00 is easily multiplied by four to get the target number of dollars, $16.00.

     
    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.
Attachments
rates-2.pdf618.64 KB
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Level Four