Simple fractions as percentages

1. Create a stack of two cubes, one yellow, one blue. 
   
   
   What fractions can you see? Look for students to say 1/2 ("one half").
   I want to use percentages as well. What percentage of the stack is yellow?
   Students often know that one half equals 50% without knowing why.
   What does the symbol % (writing) mean?
   The symbol means “in every (per) hundred (cent).”The / refers to “out of” and the two circles are the zeros from 100.
   If the stack was made of 100 parts, how many would be yellow?
    
2. Make stacks of 4 and 10 cubes that also represent 50% of each colour.
   What percentage of yellow and blue is in each stack?
   
   
   How can the percentage be 50% when there are different numbers in each stack?

• A key idea is that all three stacks are one half yellow and one half blue. The fraction is about the relationship between each part and the whole.
• Look at the stacks one by one.
  Imagine there are 100 parts in this stack. How many parts are in each cube?
• Develop the idea that one quarter equals 25% (25 out of 100) and one tenth equals 10% (10 out of 100).
   

3. Use a calculator to check that all stacks have the same percentage of yellow and blue.
   This can be done by calculating 1 ÷ 2 = 0.5 (one out of two parts), 2 ÷ 4 = 0.5 (two out of four parts), and 5 ÷ 10 = 0.5 (five out of ten parts).
   Is 0.5 the same as 50%? Why is that? 
   Note that 0.5 represents five tenths which equals 50 hundredths (Decimats could be used to show this).
    
4. Create stacks for fifths and tenths using 5 (3 yellow and 2 blue) and 10 (6 yellow and 4 blue) cubes.
   
   
   
   What fraction of each stack is yellow?
   Note that 3/5 and 6/10 are equivalent fractions.
   What percentage of each stack is yellow? What percentage is blue? How many cubes are in each stack? 
   If there are 100 parts in each stack, what percentage is a single cube worth?
   In the five stack each cube is worth 20%. The percentage for one fifth is 20%. The percentage for three fifths is 60%, 3 x 20%. Note that blue makes up the remaining percentage of 40% since both colours must combine to create 100%.
   Develop similar arguments for the ten stack where each cube is worth 10%.
    

5. Use the calculator to calculate the percentages:

   • 3 ÷ 5 = 0.6 (six tenths equals 60 hundredths)             
   • 2 ÷ 5 = 0.4 (four tenths equals 40 hundredths)
   • 6 ÷ 10 = 0.6 (six tenths equals 60 hundredths)             
   • 4 ÷ 10 = 0.4 (four tenths equals 40 hundredths)

   To avoid converting from decimals to percentages use the % key. For example:
   3 ÷ 5 % = gives 60(%).
    

6. Build other stacks to see if students can work out the percentage of each colour. Allow students to work in groupings that will encourage peer scaffolding and extension, as well as productive learning conversations. Good examples might be built on thirds, fifths, eighthts, sixths, and so on. 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).

• Thirds
  

Next steps

1. Increase the level of abstraction by moving from physical cube models to schematic diagrams that show only the parts. 
   
    
2. Move further to working with the symbols, such as 2/5 = 40/100 = 40%. 
    
3. Take a familiar whole bar and change the ratio of parts. For example, having established that 1/5 = 20/100 = 20%, find the decimals for 2/5, 3/5, 4/5.
   Likewise with 1/3 = 33.3%, what is 2/3?
    
4. Connect percentages to decimals, perhaps using decimat models.
   For example, 3/4 = 0.75, what is 3/4 as a percentage?
   2/5 = 40%, what is 2/5 as a decimal?