In this unit we look at ratio in the context of objects, physical space, geometric quantities and mixable things such as paint.
- given a ratio find relative numbers of objects, lengths of sides
- given a ratio find the relevant fractions
- given fractions making a whole, find the relevant ratio
- find ratios between three objects
Ratios, proportion, fractions are all related items that come up in both real life and mathematical situations. Ratio is a particularly valuable concept in the context of scale drawings. Here the ratio of the drawing to the actual object gives the idea of the relative size of the drawing to the object.
In mathematics, ratios and proportions are fundamental to trigonometry, coordinate geometry and calculus.
The unit This is to That, Number, Level 4, gives an introduction to ratios and relates them to proportion and fractions. You may want to provide those experiences prior to undertaking this unit.
For more background on this topic of ratios see the Staff Tutorials Ratios.
Geometric instruments: ruler, compasses, protractor, a tape measure several metres long
In session 1 we start off with a series of questions to ask around the class. There are some sample ones below. You can probably make up more of your own. There are some more examples in Copymaster 1.
For the competition in session 1 you have the eight groups a, b, c, d, e, f, g, h. They should play each other in the following rounds:
Round 1: a & b; c & d; e & f; g & h
Round 2: a & c; b & d; e & g; f & h
Round 3: a & d; b & c; e & h; f & g
Round 4: a & e; b & f; c & g; d & h
- I have twice as many eggs as you. You have 6 eggs. How many do I have?
- Jane (use the names of people in the class) has three times as many dogs as Bill. Bill has 4 dogs. How many dogs does Jane have?
- Bob have five times as many brazil nuts as Jenny. Jenny has four brazils. How many brazil nuts has Bob?
- I have twice as many eggs as you. I have 10 eggs. How many do you have?
- Jane (use the names of people in the class) has three times as many dogs as Bill. Jane has 15 dogs. How many dogs does Bill have?
- Bob have five times as many brazil nuts as Jenny. Bob has 30 brazils. How many brazil nuts has Jenny?
- I have twice as many eggs as you. What is the ratio of the number of my eggs to the number of your eggs?
- Jane (use the names of people in the class) has three times as many dogs as Bill. What is the ratio of the number of Jane’s dogs to the number of Bill’s dogs?
- Bob have five times as many brazil nuts as Jenny. What is the ratio of the number of Bob’s brazils to the number of Jenny’s brazils?
- If the ratio of my eggs to your eggs is 2:1, what fraction of our eggs do you have? What fraction do I have?
- If the ratio of Jane’s dogs to Bill’s is 3:1, what fraction of their dogs does Jane own? What fraction does Bill own?
- If the ratio of Bob’s brazils to Jenny’s is 5:1, what fraction of the brazils does Jenny have? What fraction does Bob have?
One thing that you should note here is that ratios don’t add. So that if you have two things mixed in the ratio of 2:1 and another mixture of the same items in the ratio 3:1, then the combined mixture is NOT in the ratio 5:2. You can see this by looking at the Staff Tutorial Basic Ratios.
In this session, students are reintroduced to ratios.
- Start with a whole class quick fire quiz. Suggested questions are in the Teachers’ Notes. Go for as long as you think you need to in order for most of the students to be able to answer the type of question being asked.
- Break the class up into eight groups of no more than four in a group. Give the groups Copymaster 1. Let them make up 20 questions along the lines of the questions in 1 above. They have to answer their own questions on the Copymaster.
- Bring the groups together in pairs and get each group to ask its competition pair 5 of the questions that they have made up. Each correct answer gets 1 mark.
- Then pair up the groups again until four rounds have been played. Check the total scores to see who the winner is. (You might subtract a mark if the team that posed the question got the answer wrong.)
In this session, we consolidate ideas relating to ratio.
- Divide the session up into three parts and get the class to tackle one of Copymasters 2, 3, and 4 in each part. They can work on their own or in groups. Depending on the ability of your class you may want to revise the material from the last session. The algebraic questions might be quite hard but it will give the faster students something to puzzle over. It will also give you something to talk about in a whole class setting.
In this session we look at drawing objects involving ratios.
- Try to get hold of some scale models. Talk about the items and the way that ratio/scale is used for that object.
- Show them a metre ruler, 100 cm long. Get different students to answer the following questions
How long would a stick be it the ratio of the ruler to the stick was 2:1; 5:1; 10:1; 1:2; 1:5; 1:10?
- Get them to work in pairs on Copymaster 5. Check that everyone understands the task.
- Now get someone to measure a rectangle that you have made in cardboard. (A size of 25cm by 20 cm would be fine.)
- Get the class to draw rectangles so that your rectangle to theirs is in the ratio 5:1. (Before you do this you may need to refresh their memories of how to draw rectangles, and triangles.)
- Then give them a series of ratios to work with, say, 5:2; 5:3; and 10:1.
- Now give them the series of ratio tasks on Copymaster 6. Note that (vi) can’t be drawn. Why?
- Check the work as they go and hold a discussion at the end.
In this session we draw a physical space to scale.
- Try to get hold of maps with various scales. Talk about the items and the way that ratio/scale is used for that object.
- Talk about drawing the school campus to scale (or use the classroom or some other physical space). Have a discussion with the whole class
What would be a useful scale to use?
How would you decide that? (size of paper; size of campus)
Would it be useful to divide it up between groups of students?
What do you need to measure?
- Check that everyone understands the task
- Give the class time to go take their measurements.
- When they have completed their task get them to measure certain sections on their map (from a tree to a building, and so on).
How far apart are the objects on the ground?
- Repeat this for several distances.
- Give them the chance to measure the actual distance to see how accurate their measurements were and whether their scale is correct.
This is the session where we mix paint and other things.
- Give the class this problem to work on individually. Henry has mixed orange juice to water in a 1 litre jug in the ratio 3:1. Reina has mixed orange juice to water in a litre jug in the ratio 4:1. They put both of their mixtures in a large bowl. What is the ratio of orange to water in the bowl?
(This problem can be found in the Staff Tutorial Orange and Water. Note that the answer is not 7:2. Why?)
- Make sure that as the students finish this they are asked to (i) check their work and (ii) find another way to do it. If possible get them to do it a third way.
- At a suitable time, bring the class together and discuss the answer to the problem. (How many ways can your students find to do this problem?)
- Get them to make up and solve their own drink mixing problem.
- Discuss these problems. If you think that they need more time on this topic, get them to solve each other’s problems.
- Now consider the following paint problem. Henry mixes a litre of paint with red paint to yellow paint to base tone in the ratio of 1:2:4. Reina mixes a litre of paint with red paint to yellow paint to base tone in the ratio of 1:3:5. What is the ratio of red to yellow to base tone in the combined 2 litres of paint?
- Let them work on this in small groups. Encourage them to do it in more than one way.
- Discuss their results.