1. Find Equivalent Fractions
We are Learning to……
Find Equivalent Fractions

2. Equivalent fractions Start by showing two bars.
Set one bar to show 1/8s and the other to show 1/4s.
Ask a volunteer to shade in half of the bar showing 1/8s.
How many 1/8s make 1/2? (4)
4/8 is the same as one half.
How many sixths make one half? (3)
3/6 is also the same as one half.
Show this amount on the bar.
Turn on another bar. Set it to show fifths.
Can any number of fifths make a half?
Now set the bar to show sevenths.
What about sevenths?
Establish that to make a fraction the same as a half the bar must be divided into an even number of equal pieces.
Ask pupils to state any other fractions they can think of equal to a half.
Turn on all four bars and ask pupils to show their suggestions on the board.
Repeat the exercise for 2/3. Start by setting two of the bars to thirds and sixths. Ask a volunteer to shade in two thirds of the bar showing thirds and continue as before.
Establish that to show a fraction equal to 2/3 the bar must be divided into a number of parts equal to a multiple of three.
Ask pupils what they notice about the numerator (the top number) for all fractions equal to 2/3.
The numerator is always a multiple of 2 (an even number) and the denominator is always a multiple of 3.
Repeat the exercise for ¾.
Establish that, in this case, the denominator is always a multiple of 4 and the numerator is always a multiple of 3.
Pupils should notice that the numerator and the denominator are multiplied by the same number. Ask a volunteer to justify this using the fraction bars.

3. What does equivalent mean?
Equivalent fractions
What does equivalent mean?
Ask pupils what equivalent means.
Equal to or the same as.
Many words that start with equ- have something to do with things being equal. Can you think of any?
Some examples include equilateral, equation, equidistant.

4. 3 6 12 = = 4 8 16 Equivalent fractions Look at this diagram: ×2 ×2 ×2
Ask pupils what proportion of the diagram is shaded (3/4).
Look what happens if we cut each quarter into two equal parts.
Click to divide.
We now have 1/8s.
Exactly the same amount is shaded, but you can see how we can call this amount 6/8?
What have we done by cutting each quarter into two equal parts?
Explain that we have multiplied the number of shaded sections by two (we had three shaded sections; now we have six) and we have multiplied the number of equal parts by two (we had four; now we have eight).
Click to reveal the arrows showing the numerator and the denominator being multiplied by 2.
We’ve multiplied the numerator by 2 and the denominator by 2.
The numbers have changed but exactly the same proportion of the circle has been shaded. 3/4 and 6/8 are equivalent fractions.
We could divide each of these eights into three equal parts. Look what happens. Click to reveal.
Now, how many equal parts are there altogether? (3 x 8, 24)
How many of those equal parts are shaded? (3 x 6, 18)
So we now have 18 out of 24 parts shaded.
Click to reveal this fraction.
Explain that we have multiplied both the numerator and the denominator by three. The numbers have changed but exactly the same proportion of the circle has been shaded.
What would we multiply the numerator and the denominator of 3/4 by to get 18/24? (6)
You can se that each quarter of our original diagram has been divided into six equal parts.
3/4, 6/8 and 18/24 are equivalent fractions. Can you think of any other fractions that are equal to ¾?
How many different ways could we write ¾? (Infinitely many!) 3 6 12 = = 4 8 16 ×2 ×2

5. 2 6 24 = = 3 9 36 Equivalent fractions Look at this diagram: ×3 ×4 ×3
Explain this set of equivalent fractions as in the previous slide. 2 6 24 = = 3 9 36 ×3 ×4

6. 18 6 3 = = 30 10 5 Equivalent fractions Look at this diagram: ÷3 ÷2 ÷3
Ask pupils what proportion of the diagram is shaded. (18/30)
We could simplify this diagram by removing these horizontal lines.
Click to remove some of the horizontal divisions.
We now have ten equal parts.
Exactly the same amount is shaded, but you can see how we can call this amount 6/10. By removing those horizontal lines we have made every 3/30 into 1/10.
Explain that we have divided the number of shaded sections by 3 (we had 18 shaded sections; now we have 6) and we have divided the number of equal parts by 3 (we had 30; now we have 10).
Click to reveal the arrows showing the numerator and the denominator being divided by 3.
18/30 and 6/10 are equivalent fractions.
Tell pupils that by dividing the numerator and the denominator by the same number, we have simplified the fraction. It is simpler because the numbers are smaller.
Can we simplify this fraction any further?
Yes, 6 and 10 are both even numbers, so we could divide the numerator and the denominator by 2. Remember, if we divide the numerator and the denominator by the same number the numbers that make up the fraction change but the fraction itself has exactly the same value.
Click to show the numerator and the denominator being divided by 2.
6/10 is equivalent to 3/5.
We can see this in the diagram by grouping each 2 tenths into one fifth.
Click to reveal.
Can we simplify 3/5 any further?”
No, 3 and 5 have no common factors, there is no number which divides into both 3 and 5.”
We have expressed the fraction 18/30 in its lowest terms. This is also called cancelling the fraction down.
How could we have cancelled 18/30 to its simplest form in one step?
Establish that we could have divided the numerator and the denominator by 6.
We call 6 the highest common factor of 18 and 30. 18 6 3 = = 30 10 5 ÷3 ÷2

7. Equivalent fractions
Use this activity to generate patterns of equivalent fractions.
Stress that every fraction on the board is exactly the same fraction written in a different way. Establish that there are infinitely many ways to write the same fraction.
When all of the equivalent fractions have been revealed ask pupils how we could convert between any two given fractions on the board by multiplying and/or dividing the numerator and the denominator by the same number.
Link:
N8 Ratio and Proportion: Scale factors

8. Cancelling fractions to their lowest terms
A fraction is said to be expressed in its lowest terms if the numerator and the denominator have no common factors.
Which of these fractions are expressed in their lowest terms? 7 5 2 14 16 20 27 3 13 15 21 14 35 32 15 8 7 5
Ask pupils what we mean when we say a fraction has no common factors.
Establish that there is no number other than 1 that divides into both the numerator and the denominator.
For each fraction ask pupils whether or not they think this fraction has been shown in its lowest terms, before revealing the answer.
If pupils do not think that the fraction has been shown in its lowest terms, ask them for a number which will divide into both the numerator and the denominator.
Explain that when cancelling it is always best to divide both the numerator and the denominator by the highest number that divides into both, that is, the highest common factor. However, if you do not cancel by the highest common factor the first time round, you can always cancel again.
Go through the cancellation of each fraction asking what we are dividing by each time.
There is something different about the last fraction, what is it?
Point out that it is top-heavy. The numerator is bigger than the denominator.
This is called an improper fraction.
Ask how we could write this improper fraction as a mixed number. (2 and 2/15)
Fractions which are not shown in their lowest terms can be simplified by cancelling.

9. Drag and drop equivalent fractions
Ask pupils in turn to choose a fraction and justify where it goes by dividing the numerator and the denominator by the same number. For fraction diagrams pupils must state the fraction shaded first.
Continue until all the fractions are in the correct place.
Ask pupils to come up in turns and choose a fraction to drag and drop into the correct place.

10. Using equivalent fractions3 8 5 12
Which is bigger or ?
Another way to compare two fractions is to convert them to equivalent fractions.
First we need to find the lowest common multiple of 8 and 12.
The lowest common multiple of 8 and 12 is
24.
Now, write and as equivalent fractions over 24. 3 8 5 12
Tell pupils that another way to compare two fractions is to convert them into equivalent fractions with a common denominator.
Talk through the example on the board.
Tell pupils that the quickest way to find the lowest common multiple of two numbers is to choose the larger number and to go through multiples of this number until we find a multiple which is also a multiple of the smaller number. This method also works for a group of numbers. ×3 ×2 3 8 = 24 9 5 12 = 24 10 3 8 5 12
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and
so, ×3 ×2

11. To succeed at this lesson today you need to…
1. Be able to create equivalent fractions
2. Be able to cancel fractions to their lowest terms
3. Use equivalent fractions to order fractions
Impact 2B Page 92 Ex 6B & 6C