Proportional Reasoning Strategies progression, 5th step
The purpose of the activity
In this activity, the learners develop an understanding of more complex ratios, which cannot be simplified to a ratio that includes 1, for example 2:3. They explore the relationships between related ratios.
The teaching points
Ratios, like fractions, are usually expressed in the simplest form. For example, 6:2 can be simplified to 3:1.
The learners are able to 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 use ratio notation for more complex ratios. For example, the ratio 2:3 could be used to represent 4 apples and 6 bananas or 18 apples and 27 bananas.
Equal ratios result from multiplication and division of existing ratios not from addition and subtraction. For example, multiplying both numbers in the ratio 2:3 by 2 gives an equal ratio of 4:6, but adding 2 to each number in the ratio 2:3 results in an unequal ratio of 4:5.
The learners can find a ratio between two groups from two related ratios. For example, if the ratio between apples to oranges is 2:1 and the ratio between oranges and bananas is 2:1, the ratio between apples and bananas is 4:1.
Discuss with the learners authentic situations where an understanding of ratios is important.
The guided teaching and learning sequence
1. Write the ratio ‘2:3’ on the board and ask:
“What does a ratio of “2 to 3” mean?”
Check the learners all understand that 2:3 is a way of representing two groups that are made up of equal parts. One of the groups contains 2 equal parts, and the other group contains 3 equal parts.
Ask the learners to draw some examples of more complex ratios, using 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.
2. Tell the learners that some recipes use ratios to define quantities. Tell them that a fruit salad calls for:
bananas and apples in a ratio of 2:3
oranges and apples in a ratio of 5:3.
“What numbers of bananas, apples and oranges could be used to make the fruit salad?”
The simplest example would be 2 bananas, 3 apples and 5 oranges. Other possibilities include 4 bananas, 6 apples and 10 oranges or 6 bananas, 9 apples and 15 oranges.
3. Write the following recipes for fruit salad on the board:
Bananas and apples in a ratio of 4:7
Apples and oranges in a ratio of 7:10
Oranges and pears in a ratio of 2:3
Pears and kiwifruit in a ratio of 2:5
Bananas and apples in a ratio of 8:14
Apples and oranges in a ratio of 3:5
Oranges and pears in a ratio of 6:9
Pears and kiwifruit in a ratio of 5:8
“Which of these fruits have equal ratios in the 2 recipes?” (bananas and apples, oranges and pears)
“If you have 4 pears and 10 kiwifruit, which would be the best recipe to use?” (recipe 1)
“If you have 12 apples and 20 oranges, which would be the best recipe to use?” (recipe 2)
“If you have 4 oranges and 6 pears, which would be the best recipe to use?” (recipe 1)
Check all the learners understand that equal ratios are the result of multiplying or dividing both numbers in the ratio by the same number.
4. Tell the learners that Sally has apples to oranges in the ratio 2:1 and oranges to bananas in the ratio 2:1. Ask:
“If Sally has 1 banana, how many apples does she have?” (4)
“If Sally has 3 bananas how many apples does she have?” (12)
“If Sally has 20 apples, how many bananas does she have?” (5)
“What is the simplest way of expressing the ratio between apples and bananas?” (4:1)
Encourage the learners to use diagrams to answer these questions if necessary.
5. Ask the learners to solve the following problem: “Jane has apples, bananas and pears in a fruit salad. The ratio of apples to bananas is 2:1, and the ratio of bananas to pears is 3:1. What is the ratio of apples to pears?” (6:1).
6. Ask the learners to solve the following problems:
Scott says the ratio of males to females in his class is 2:3 and there are 30 people in his class.
“How many males are in Scott’s class?” (12 males)
“How many females are in Scott’s class?” (18 females)
Encourage the learners to share their ways for solving the problem with others. One solution is “I know that for every 5 learners, 2 are males and 3 are females. There are 6 lots of 5 learners in a class of 30. This means there are 6 x 2 = 12 males and 6 x 3 = 18 females.”
Ask the learners to work together in pairs to solve some of the following ratio problems:
A jam recipe calls for fruit and sugar in a 2:5 ratio. List three different amounts of fruit and sugar that could be used. (2 cups of fruit and 5 cups of sugar, 4 cups of fruit and 10 cups of sugar, 6 cups of fruit and 15 cups of sugar.)
Alice and Jess are on a diet. The ratio between Alice’s weight loss to Jess’ weight loss is 2:3. If Alice has lost 12 kilograms of weight, how much weight has Jess lost? (18 kilograms)
Find two equal ratios in each example:
3:5, 5:7, 9:15 and 13:15
2:3, 3:4, 12:16 and 7:8
2:3, 3:4, 5:6 and 6:8
3:4, 7:8, 14:16 and 6:7
3:7, 6:14, 4:8 and 3:5.
Cameron has apples, bananas and pears. The ratio of apples to bananas is 3:1 and the ratio of bananas to pears is 3:1. What is the ratio of apples to pears? (9:1)
Jack has apples, bananas and pears. The ratio of apples to bananas is 4:1 and the ratio of bananas to pears is 2:1. What is the ratio of apples to pears? (8:1)
Ann says the ratio of males to females in her class is 3:4, and there are 21 people in her class. How many males are in Ann’s class? How many females are in Ann’s class? (9 males and 12 females)
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