More complex 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.
   The price of a carwash goes up from $8 to $10. By what percentage does the price increase? 
   How can you work out how much the percentage increase is?
   Students should either recognise that $10 is $2 more than $8 and that $2 is one quarter of $8.
   Therefore the percentage increase is 25% since 1/4 = 25%.
   You might construct a cube stack model or diagram of the problem, like this:
   
   
    
2. Draw this double number line model to represent the problem:
   
   
   
   Why is the answer equal to 125%? (The original 100% plus another 25%).
    
3. Demonstrate how calculator algorithms can be used to answer the same problem. 
   Key in 10 ÷ 8 = 1.25 and ask students what the number 1.25 represents.
   That means that $10 is 1.25 times $8 or $8 fits into $10 1.25 times.
   How does 1.25 match our answer of 25%?
   As a percentage 1.25 is 125 hundredths so equals 125%. The 125% is made up of the original 100% plus the 25% increase.
   Key in 10 ÷ 8 % = to get 125% and 2 ÷ 8 % = to get 25%.
    
4. Pose a more difficult scenario like:
   The price of a skateboard goes up from $120 to $165.
   By what percentage does the price increase?
   Draw a double number line to represent the problem.
   
   
   
   What is the increase in price? ($45)
   What percentage of $120 is $45? 
   Look for students to recognise that 45 is less than one half of 120 and then try various approximations. Use a calculator to find 45 ÷ 120 = 0.375 which is 37.5%. Some students might remember that 37.5% = 3/8. The price of the skateboard increased by 37.5%.
    
5. 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 might be:
   1. The price of a pet goes up from $80 to $104. By what percentage does the price increase?
      Students should recognise that the increase is $24. Since 10% of 80 = $8 and $24 is three times that amount, the increase is 30%.
      The calculation 24 ÷ 80 % = 30 gives the percentage increase.
      
      
       
   2. The price of a plant goes up from $35 to $42. By what percentage does the price increase?
      Students should recognise that the increase is $7 which is one fifth of $35. Since 1/5 = 20% the increase is 20%.
      The calculation 7 ÷ 35 % = 20 gives the percentage increase.
      
      
       
   3. The price of a painting goes up from $600 to $1,400. By what percentage does the price increase?
      Students should recognise that the increase is $800 which is more than 100% of $600. Since one third of $600 equals $200, the price has increased by four thirds. The percentage for 4/3 = 133.3%.
      The calculation 800 ÷ 600 % = 133.3% gives the percentage increase.
      
      
       
   4. The price of a handbag goes up from $60 to $114. By what percentage does the price increase?
      Students should recognise that the increase is $54 which is slightly less than 100% of $60. Since $6 is 10% of $60 the price has increased 100 -10 = 90%. 
      The calculation 54 ÷ 60 % = 90% gives the percentage increase.
      
      
       

Next steps

1. Introduce problems in which both the increase percentage and original price are untidy. Realistic situations of percentage increase often involve numbers that are messy. 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 percentage increase of a house that has gone up in value from $467,000 to $581,00 in two years. 
   The increase is 581,000 – 467,000 = $114,000. That is about one quarter of $467,000 so an increase of about 25% is a reasonable estimate.
   114,000 ÷ 467,000 = 0.2441 which is 24.41%. 
   That answer is close to the estimate of 25%.
    
2. Give examples in which the percentage increase is greater than 100%. For example: 
   Find the percentage increase in value for a rare coin that goes up from $4,300 to $11,350 in a single year. 
   The increase is 11,350 – 4,300 = $7, 050. That is not quite double the starting price of $4,300 so a percentage a bit smaller than 200% is expected.
   7,050 ÷ 4,300 = 1.64 which is 164%. 
   That answer is close to the estimate of below 200%.