Finding the whole

1. Pose this word problem.
   You score 20 goals. Your success rate is 80%.
   How many shots do you take? 
   Let students attempt the problem, possibly in pairs. Trial and error might be a commonly used strategy. For example, since the base is more than 20 shots students may try various numbers of shots, like 30, 24 or 25.
   An alternative is to consider:
   If 20 shots are worth 80%, what percentage is each shot worth? How do you know? (4% since 80% divided by 20 equals 4%).
   How many shots make up 100%?  (25 since 25 x 4 = 100%)
   Support students to use a double number line to organise the information from the problem and record their strategies.
   
   
   
    
2. Pose a more difficult problem that allows for less reliance on trial and error.
   You score 12 goals. Your success rate is 30%. How many shots do you take? 
   If 12 shots are worth 30%, what percentage is each shot worth? How do you know? (2.5% since 30% divided by 12 equals 2.5%)
   How many shots make up 100%? (40 since 100 ÷ 2.5 = 40 shots)
   Ask students to draw a double  number line that represents the important information.
   Use a calculator to check: 12 ÷ 40 % = 30%
   
    
3. Pose increasingly difficult problems so students rely more on generalisation methods and less on trial and error. Allow students to work in groupings that will encourage peer scaffolding and extension, as well as productive learning conversations. Consider your students' fraction and multiplication basic facts knowledge when setting these problems. You might also introduce relevant te reo Māori kupu, such as ōrau (percent). 
   Examples might include:

• You score 45 goals. Your success rate is 75%.
  How many shots do you take? 
  If 45 shots are worth 75%, what percentage is each shot worth? How do you know? (1.6666…% since 75% divided by 45 equals 1.6%)
  How many shots make up 100%?  (60 since 100 ÷ 1.6= 60 shots)
  Use a calculator to check: 45 ÷ 60 % = 75%.
  
   
• You score 28 goals. Your success rate is 80% .
  How many shots do you take?
  Each shot is worth about 2.857% since 80 ÷ 28 = 2.85714…
  100 ÷ 2.857 = 35.001… so 35 shots were taken.
  Check: 28 ÷ 35 % = 80%. 
  
   
• You score 90 goals. Your success rate is 60%.
  How many shots do you take? 
  Each shot is worth 0.6% since 60 ÷ 90 = 0.6
  100 ÷ 0.6 = 150 so 150 shots were taken.
  Check: 90 ÷ 150 % = 60%.
  
   

Next steps 

1. Increase the level of abstraction with the aim of students using symbolic form. Use empty number lines to organise the information then encourage students to use equations to record their thinking.
    
2. Use examples in which the division of the percentage given by the number of goals is a decimal. For example:
   You score 36 goals. Your success rate is 66.6%.
   How many shots do you take?  
   Each shot is worth 1.85185% since 66.6 ÷ 36 = 1.85185
   100 ÷ 1.85185= 54, so 54 shots were taken.
   Check: 36 ÷ 54 % = 66.6%