This is to That

Purpose

In this unit we look at ratio, what it means for numbers or objects to be in a given ratio, and how ratios and fractions are related.

Specific Learning Outcomes
  • find twice as many, three times as many, etc., objects
  • use ratio notation for twice as many etc.
  • given a ratio find relative numbers of objects
  • given a ratio find the relevant fractions
  • given fractions making a whole, find the relevant ratio
Description of Mathematics

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.

This unit gives an introduction to ratios and relates them to proportion and fractions.

For more background on this topic of ratios see the Staff tutorials Basic Ratio, Exchange Rates, and Dinner in London.

Required Resource Materials
Copymaster 1

Various objects to manipulate (what you have handy is fine)

A4 paper for tables

Activity

In session 1 we look at the idea of ‘twice as many’ and its variations. There are several ways to demonstrate this. Here are a few. You can probably think of more situations that fit the resources in your classroom.

Method 1: Here I have 1 ball. I want to put twice as many balls in this box. How many balls should I put in the box?

Method 2: Mary has two pencils. There are twice as many pencils in this desk. How many pencils are there in the desk?

Method 3: Petra has coloured three squares yellow. If we want to have twice as many red squares, how many do we have colour in?

11 squares, 3 coloured.

 

Method 4: Laslo has four blocks. If Martine has twice as many blocks as Laslo how many blocks does she have?

These ideas can be elaborated on by changing the information that you ask for. For instance in Method 1, tell the students how many balls are in the box and ask how many balls you have. The other situations can be changed similarly.

In session 2 we look at the concept of ratio. More details can be found on this in the Staff Tutorial Basic Ratios. By now though, your class may be asking what use are ratios. Where do people use them? Start to collect examples. You could make up a ratio poster with these examples on them. You should be able to find examples of maps (a school atlas should show ratios), exchange rates (though usually they are not usually expressed using the ratio notation), and models (often the scale – a ratio – is given on the box). What other examples can be found?

Teaching Sequence

Session 1

In this session, students consider comparison problems involving multiplication.

  1. Introduce the several representations of ‘twice as many’ from the Teachers’ Notes.
  2. Get the students to think about ‘three times as many’ situations.
  3. Now you can vary the methods of the Teachers’ Notes by giving the students the number of yellow squares and the number of red squares and asking what the ratio of red to yellow is.
  4. Divide the class into pairs to play the ‘twice as many’ game. Give each pair 40 cubes. Roll a dice. The first person takes as many cubes as the number that comes up. Then the second player takes twice as many cubes as this. The first player takes twice as many cubes as the second player. The winner is the last person who is able to take all of the cubes needed to double the other player’s number of cubes.
  5. Let the pairs play the game several times each time changing the person who goes first.

Session 2

In this session, introduce the idea of ratio.

  1. Now explain the notion of ratio. (If I have 10 blocks and you have 5 blocks I have twice as many as you and we say that the ratio of the number of blocks I have to the number you have is 2:1 – two to one.)
  2. Give several examples of situations where objects are in the ratio 2:1 and 3:1.
  3. Once you’ve introduced the idea make sure that they are involved in making up situations where the ratio is 2:1 or 3:1.
  4. Ask them how they can tell whether things are in the ratio 2:1 or 3:1.
  5. Let them work in their pairs again. First get them to draw a couple of examples where the ratio of yellow squares to red squares is 2:1 and 3:1. Second ask them to make up their own ratios and draw yellow and red squares in that ratio.
  6. Get the class together to discuss this second task. Try to extend their concept of ratio from the simple cases of 2:1 and 3:1.
    Is it possible to have ratios 3:2?
    What does this mean?
    Give me some examples.
  7. Suppose that Hare had apples to oranges in the ratio of 2:1 and oranges to bananas in the ratio of 2:1.
    If Hare has 1 banana, how many apples does he have? 4
    If Hare has 3 bananas how many apples does he have? 12
    If hare has 20 apples, how many bananas does he have? 5
    How can you explain this?
  8. Do similar problems with other ratios.
  9. Ask where can you find examples in everyday life where ratios are used?
    Can you try to find some?

    Over the next few days collect examples.

Session 3

In this session we look at more situations with ratios.

  1. Suppose that Hare had apples to oranges in the ratio of 2:1 and oranges to bananas in the ratio of 2:1.
    What is the ratio of apples to bananas?
    How can you convince me of this?
    Is the ratio the same no matter how many apples Hare has? Yes. Show this by example. If Hare only has 1 apple, how many bananas does he have? .
    How can you explain this?
  2. Do similar problems with other ratios.
  3. Let the students work together on the problems of copymaster 1. 
  4. Then bring together the examples of ratios that you have found and discuss how they are used. Use an example in a practical situation (perhaps draw a scale diagram in the ratio of 10:1).

Session 4

In this session we see the link between ratios and fractions.

  1. Get the students to work in pairs to draw and colour squares in two colours in various ratios of their choosing. (2:1, 3:1 and 5:2 would be fine.) Three different ratios should be enough.
  2. Bring the class together with their drawings.
    Choose an example of a simple ratio (2:1 would be ideal).
    How many squares are red?
    How many squares are blue?
    How many coloured squares are there altogether?
    What fraction of the squares is red?
    What fraction of the squares is blue?
  3. Start to draw up a table like the one below.
ratioNo. red squaresNo. blue squaresNo. of squaresFraction of redFraction of blue
2:11051510/15 = 2/35/15 = 1/3
2:184128/12 = 2/34/12 = 1/3
      
  1. Repeat the last set of questions with the same ratio. Put the numbers into the table.
  2. Repeat the last ratio again.
  3. What patterns do you see? (always twice as many red as blue; always the same fractions, etc.)
  4. Now take another ratio and enter at least three values into the table above.
    What patterns do you see? (if the ratio is a:b then the fractions are a/(a+b) and b/(a+b))
    If you have the ratio 3:2 can you fill the table?
    Would you get the same fractions no matter how many squares you had?

Session 5

In this session we look at the problem the other way round and show how fractions give us ratios.

  1. Remind the class of what has happened so far in this unit.
  2. Let them work in pairs to complete the table below. (You don’t have to use squares if you would like to use something else.)
No. of squaresFraction of redFraction of blueNo. red squaresNo. blue squaresRatio of red to blue
92/3    
152/3    
212/3    
  1. Discuss the entries that they have added. Check that the students understand how to get each unknown value by letting them explain how they got them.
  2. Let them fill in the next table in groups.
No. of squaresFraction of redFraction of blueNo. red squaresNo. blue squaresRatio of red to blue
154/5    
25 1/5   
27  18  
  1. Again discuss the answers as a class. Let different students explain how they got the different answers.
  2. Get the students to make up a table of their own like the ones that we have been working with. They should put in any two entries in a row and be able to find the other entries. Ask them to draw their table on an A4 sheet, fill in two entries and work out the others (but do not enter them on the sheet).
  3. In a whole class setting, get different groups to come to the front and challenge the rest of the class to find the numbers that go into their blank spaces.
  4. Ask different students if they can see the link between the fractions and the ratios. If you give them different fractions can they give you the corresponding ratios? (fractions like 5/6 and 1/6 go into the ratio 5:1 – it’s just a matter of taking the numerators as the ratio).
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