More complex fractions as percentages

1.

Create a stack of twelve cubes: six yellow and six blue.

• What fraction of this stack is yellow?
• What fraction is blue?
• What is each fraction as a percentage?

Students should recognise that the stack is comprised of one half of each colour. They are likely to know that one half equals 50%.

Record 6/12 = 1/2 = 50%.

• What did we do to 6/12 to create 1/2?

Highlight that both six and twelve were divided by six. Instead of considering individual cubes, as with 6/12, we considered parts of six cubes.

2.

Alter the stack of 12 cubes to create different fractions.

• What fraction of this stack is yellow?
• What fraction is blue?
• What is each fraction as a percentage?

Students may recognise that one-quarter of the stack is blue and three-quarters of the stack is blue. They may know that 1/4 = 25% (also 0.25) and 3/4 = 75% (also 0.75).

Record 3/12 = 1/4 and 9/12 = 3/4.

• What did we do to 3/12 and 9/12 to create 1/4 and 3/4?

Highlight that three, nine, and 12 are divided by three. Instead of treating single cubes as the parts, as in 3/12 and 9/12, groups of three cubes are treated as the parts.

• What fraction of this stack is yellow?
• What fraction is blue?
• What is each fraction as a percentage?

Students may recognise that one-third of the stack is yellow and two-thirds of the stack is blue. They may know that 1/3 = 33.3% (also 0.3) and 2/3 = 66.6% (also 0.6).

Record 4/12 = 1/3 and 8/12 = 2/3.

• What did we do to 4/12 and 8/12 to create 1/3 and 2/3?

Highlight that four, eight, and 12 are divided by four. Instead of treating single cubes as the parts, as in 4/12 and 8/12, groups of four cubes are treated as the parts.

3.

Develop the same ideas with a stack of 20 cubes. 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).

1.

Increase the level of abstraction with the aim of having students use symbolic form. Start with stacks of discrete numbers of cubes, then progress to schematic diagrams with only the number of cubes given:

In symbolic form, express the fractions as 6/15 and 9/15 that can be reduced to simpler forms, 2/5 and 3/5, using three as a common factor.

Ask students to demonstrate with stacks of cubes what the use of a common factor means in terms of the parts. For example, converting 9/24 and 15/24 into 3/8 and 5/8 involves using three as a common factor. Three twenty-fourths are combined to form eighths.

2.

Create a stack where the fractions are not easily expressed as percentages, such as sixths and ninths, and twelfths. Estimate the approximate percentages, then use a calculator to find the exact percentage if an exact percentage exists.

For example:

• 10/12 = 5/6 = 83.3%, and 2/12 = 1/6 = 16.6%

• 7/12 = 58.3%, and 5/12 = 41.6%