Applying multiplication and division to rates

1. Pose contextually-relevant problems that can be solved as easily using between strategies as it can using within strategies. Such problems involve equally difficult multiplicative relationships.
   For example, Tui can buy 2 peppers for $5.00. How much do 9 peppers cost?
    
2. Let students discuss their answer to the problem. They may think that the amount must be a whole number of dollars rather than $22.50, the correct amount.
   You might scaffold students by supporting or asking them to create a rate table.
    
    
3. Discuss the ways the problem might be answered. Two main strategies are likely, both involving a multiplicative operator connecting numbers.
   • Within measures: The operator connecting 2 and 9 is 4.5 since 4.5 x 2 = 9. That same operator is applied to $5.00 to get 4.5 x 5 = $22.50.
      
      
   • Between measures: The operator connecting 2 and 5 is 2.5 since 2.5 x 2 = 5. The same operator is applied to 9 to get the cost of 2.5 x 9 = $22.50.
      
      
4. Show the students how the operators 4.5 and 2.5 can be found using division on the calculator: 9 ÷ 2 = 4.5 and 5 ÷ 2 = 2.5. Note that some students may be able to find the operators mentally.
    
5. Pose further relevant problems that promote searching for operators and privilege both within and between strategies. Model the problems with materials if necessary. Ensure students create a rate table for each problem. Consider grouping students to encourage peer scaffolding and extension.
   Consider introducing relevant te reo Māori kupu, such as pāpātanga (rate), whakawehe (divide, division), and whakarea (multiply, multiplication).
   
   Examples might be:
   • Samsoni pays $3.00 for 2 mangoes. How much will he pay for 7 mangoes?
     • Within strategy
       

     • Between Strategy

       
        

   • David can buy five oranges for $8.00. How much will he pay for 12 oranges?
     • Within strategy
       
     • Between Strategy
       
        
   • Petra can buy four peppers for $5.00. How much will she pay for 14 peppers?
     • Within strategy
       
     • Between Strategy
       
        

Next steps 

1. Provide further rate problems in which the number of items, rather than their cost, is unknown. For example:
   Beau can buy 4 taros for $9.00. How many taros will she get for $40.50?

2. Explore finding the multiplication relationship between two numbers outside of rate contexts. Missing multiplier problems are often phased as “times as many”. For example:
   Moana collects 24 pinecones, and Nel collects 16 pinecones. How many times Nel’s number has Moana collected? 
   Since 24 ÷ 16 = 1.5 Moana collected “one- and one-half times as many” pinecones as Nel.
   Since 16 ÷ 24 = 0.6 we can say that Nel collected “two thirds as many” pinecones as Moana.

   The key generalisation is that a ÷ b = □ gives the operator □, such that □ x b = a. 
   Calculators can be used to easily show the generalisation. In more complex rate problems, finding the multiplicative operator between numbers is fundamental to solving the problems.