Proportional reasoning progression

There is a gap at the first step of this progression because learners need to be able to count all objects before they can use the strategy of equal sharing.

Learners use the strategy of equal sharing to find fractions of a set.

There is a gap at the third step of this progression because learners need to know single-digit multiplication and division facts before they can use them to find fractions of whole numbers.

Learners use multiplication and division facts to find unit fractions of whole numbers. For example:

• 1/3 of 24 = 8 because 24 ÷ 3 = 8, which is 1/3.
• 1/5 of 35 = 7 because 35 ÷ 5 = 7.

Learners use strategies to solve problems that involve simple equivalent fractions and simple conversions between fractions, decimals and percentages. For example, a learner knows that:

• they can find 3/12 of a number by dividing by 4 because 3/12 = 1/4
• 1/4 is the same as 0.25 is the same as 25%
• 1/2 of 1/4 is 1/8 .

Learners solve problems by deriving from known fractions, decimals or percentages. For example, a learner:

• finds 20% of 80 by knowing that 10% is 8 and doubling to get 16
• finds 1/5 of 70 by knowing that 1/10 is 7 and doubling to get to 14
• finds 0.8 of 40 by knowing that 1/10 (0.1) is 4 and 4 x 8 is 32.

Learners use calculators to solve problems involving fractions, decimals and percentages and are able to explain the reasonableness of the answer. For example, “I will divide by 100 on my calculator to find 1% and then multiply by 17 to get 17%” or, alternatively, “17% of 80: 10% is 8 and 20% is 16, so my answer (13.6) is reasonable.”

Learners use strategies to solve problems that involve proportions, ratios and rates.

• Using common factors
  For example, 15:18:21 is the same as 10: ? : ? . The common factor is 3.
  10 is 2/3 of 15. 12 is 2/3 of 18. 14 is 2/3 of 21.

• Converting fractions to percentages
  For example, 70 is what percentage of 42? 70/42 = 1 28/42 = 1 4/6 = 1 2/3 = 166.66%.

• Determining the “unit”
  For example, a learner solves this problem: The tank is 4/5 full of water. Andrew will use 2/3 of that water to water his plants (200 litres). How much water does the tank hold when full? That is, 2/3 of 4/5 of ? =200; 1/3 = 100 litres; 1/3 of 4/5 is 4/15.
  If 4/15 = 100, then 1/15 is 25. 15 x 25 = 375 litres.
  1/3 of the water would be 100 litres. 1/3 of 4/5 is 4/15 . If 4/15 of the tank is 100 litres, then 1/15 is 25 litres, and 15 times 25 equals 375 litres.

Learners can calculate rates by comparing two quantities or measurements that have different units, for example:

• If it takes 2 hours to travel 130 kilometres, the average speed for the journey (the total distance travelled in one time unit) is 130 kilometres per 2 hours = 65 kilometres per hour.

Learners can calculate rates to make comparisons, for example:

• If 3 kilograms of potatoes costs $3.60, the price per kilogram is $3.60 per 3 kilograms = $1.20 per kilogram. This charge per kilogram is a rate. (At supermarkets, such rates are often expressed as cents per 100 grams so that the purchaser can compare prices of similar items.)