Percentage increases

1. Pose this problem, or another that is adapted to a more specific context that is more relevant to your students' interests, cultural backgrounds, and learning from other curriculum areas.
   You read that the price of a video game has gone up by 20%.
   If the game used to cost $18, what does it cost now? 
   How can you work out how much the 20% increase is?
   Students should either find 10% equals 18 ÷ 10 = $1.80 and double it to get $3.60, or remember that 20% equals one fifth and calculate 1/5 x 18 = $3.60.
   How much is the new price? (18 + 3.6 = $21.60)
   You might construct a diagram of the problem, like this:
   
   
    
2. Why is the answer equal to 120%? (The original 100% plus another 20%).
   Draw a double number line model to represent the problem:
   
   
    
3. Demonstrate how calculator algorithms can be used to answer the same problem. Begin with a semi-written method: 120/100 x 18 and develop a set of steps, 18 ÷ 100 x 120 = $21.60.
   What does 18 ÷ 100 = 0.18 tell you? (1% of 18 = $0.18)
   Why do you multiply by 120? (The increased price equals 120% of the normal price)
   A simpler method is to key 18 x 120% = into the calculator. In this case the calculator knows to enact the calculations above. Explore what happens if only 18 x 120 = is keyed in. The answer 2160 is one hundred times greater than the actual answer because no division by 100 has occurred.
    
4. Pose similar problems and support students to create double number line models, and use calculator algorithms, to check the reasonableness of their answers. Adapt the problems as necessary and consider grouping students to encourage tuakana-teina. You might introduce relevant te reo Māori kupu such as whakahekenga ōrau (percentage discount) and rārangi tau matarua (double number line). 
   Examples of problems might be:
   1. You read that the price of a mobile phone has gone up by 25%.
      If the phone used to cost $240, what does it cost now? 
      Students should recognise that 25% equals one quarter so the increase is 1/4 x 240 = $60. Another method is to use 10% as a benchmark. 10% of 240 equals $24, so 5% of 240 equals $12. Putting 2 x 10% and 1 x 5% together gives 24 + 24 + 12 = $60.
      The calculation 240 ÷ 100 x 125 = $300 gives the increased price. Keying in 240 x 125% = gives the same answer.
      
       
   2. You read that the price of a bicycle has gone up by 30%.
      If the bicycle used to cost $540, what does it cost now?
      Students should recognise that 30% equals three tenths so the increase is 3/10 x 540 = $162. 
      The calculation 540 ÷ 100 x 130 = $702 gives the increased price. Keying in 540 x 130% = gives the same answer. Since 130% = 1.3 you can also calculate 1.3 x 80 =.
      
       
   3. You read that the price of a kettle has gone up by 40%.
      If the kettle used to cost $80, what does it cost now? 
      Students should recognise that 40% equals four tenths so the increase is 4/10 x 80 = $32. 
      The calculation 80 ÷ 100 x 140 = $112 gives the increased price. Keying in 80 x 140% = gives the same answer. Since 140% = 1.4 you can also calculate 1.4 x 80 =.
      
       
   4. You read that the price of a toaster has gone up by 56%.
      If the toaster used to cost $46, what does it cost now? 
      Students should recognise that 56% equals slightly more than one half so the increase is about 1/2 x 46 = $23. 
      The calculation 46 ÷ 100 x 156 = $71.76 gives the increased price. Keying in 46 x 156% = gives the same answer. Since 156% = 1.56 you can also calculate 1.56 x 46 =.
      
       

Next steps 

1. Introduce problems in which both the increase percentage and original price are untidy. Look to see if students can generalise an algorithm that works for any percentage increase problem, and use estimation strategies to check if an answer is reasonable.
   For example, find the price of an item that usually costs $453 and is increased by 26%.
   26% is close to 25% or one quarter. 
   $453 is close to $440 which is easy to quarter. 1/4 x 440 = $110. Note that the estimate is likely to be low, since both the percentage increase and original price were both rounded down.
   Calculating that the discounted price is 126% of the original price is important (100 + 26 = 126%). 453 x 126 % = gives $570.78 which is close to the estimate of 453 + 110 = $563.