Proportional Reasoning Strategies progression, 4th step
The purpose of the activity
In this activity, the learners develop an understanding of simple ratios and learn to identify ratios for given quantities as well as quantities for given ratios.
The teaching points
The learners understand relative quantities, for example, ‘twice as many’ is the same as two groups of the original number.
The learners use ratio notation, for example, 2:1 is another way of writing ‘twice as many’.
Ratios, like fractions, are usually expressed in the simplest form, for example, 6:2 can be simplified to 3:1.
The learners can name ratios for given quantities. The order of the numbers in the ratio must follow the order of the quantities expressed in the ratio. For example: If I have 8 apples and 4 bananas, the ratio of apples to bananas is 8:4, which can be simplified to 2:1.
The learners can identify quantities for defined ratios. For example: If a recipe for jam requires 2 cups of fruit for every cup of sugar, possible amounts of fruit and sugar include 4 cups of fruit and 2 cups of sugar or 6 cups fruit and 3 cups of sugar.
Discuss with the learners the ways in which they have used ratios in their work or home lives.
1-centimetre grid paper.
The guided teaching and learning sequence
1. Ask the learners what the phrase “twice as many” means. Use these examples:
“If I have $1 and you have twice as much money as me, how many dollars do you have?”
“If I have 2 biscuits and you have twice as many biscuits as me, how many biscuits do you have?”
“If I have 3 pencils and you have twice as many pencils as me, how many pencils do you have?”
Check that the learners all understand that “twice as many” is a way of identifying the amount as 2 times the original or 2 groups of the original number.
2. Ask the learners what the phrases “3 times as many” or “4 times as many” mean. Uses these examples:
“If you have 3 times as many cars as me, and I have 1 car, how many cars do you have?”
“If you have 4 times as many pens as me, and I have 2 pens, how many pens do you have?”
Check the learners all understand that “3 times as many” is a way of identifying the amount as 3 times the original and “4 times as many” is a way of identifying the amount as 4 times the original.
3. Write the ratio ‘2:1’ on the board and ask:
“What does a ratio of “2 to 1” mean?”
Check the learners all understand that 2:1 is a way of representing twice as many. Ask the learners to draw some examples of simple ratios, using the grid paper and two different-coloured pens. For example:
Check all the learners understand that the order of the colours in the ratio needs to be specified. Check also that they understand that 6:2 can be simplified to 3:1.
4. Ask the learners to simplify the following ratios:
5. Tell the learners that some recipes use ratios to define quantities. Tell them a fruit salad calls for:
Bananas and apples in a ratio of 2:1
Oranges and apples in a ratio of 3:1
“How many apples and oranges are there in a fruit salad that has 2 bananas?”
(1 apple, 3 oranges)
“How many bananas and apples are there in a fruit salad that has 9 oranges?”
(6 bananas, 3 apples)
“What other fruit salads could you make with these ratios?”
“If you had the following pieces of fruit to make a fruit salad, what ratios could you use to describe the quantities?”
Check that all the learners are able to write ratios such as 3:6 bananas to apples, 3:12 bananas to oranges and 6:12 apples to oranges. Encourage the learners to simplify these ratios, for example 3:6 can be simplified to 1:2 and 3:12 can be simplified to 1:4.
Ask the learners to work in pairs to solve some of the following ratio problems:
If a photo measures 4 centimetres by 9 centimetres and it is enlarged in a ratio of 2:1, what will the measurements of the enlarged photo be? (8 centimetres by 18 centimetres.)
If Sally runs at a speed of 10 laps per minute, how far will she run in 5 minutes? If Jane runs one-and-a-half times as fast as Sally, how many laps can she do in 1 minute? How many laps can Jane run in 5 minutes? (Sally runs 50 laps in 5 minutes; Jane runs 15 laps in 1 minute and 75 laps in 5 minutes.)
A jam recipe calls for fruit and sugar in a 2:1 ratio. List three different amounts of fruit and sugar that could be used. (2 cups of fruit and 1 cup of sugar, 4 cups of fruit and 2 cups of sugar, 6 cups of fruit and 3 cups of sugar.)
Laura and Sophie are on a diet. The ratio between Laura’s weight loss to Sophie’s weight loss is 3:1. If Laura has lost 12 kilograms of weight, how much weight has Sophie lost?
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