Helping equivalent fractions make sense
Equivalent fractions are not a trick with numbers. They are different ways of naming the same amount. The most important job here is to make that sameness visible before introducing any rule.
The central idea: same amount, different pieces
The learner already knows that a whole can be split into equal pieces. Now we want them to notice something subtler: we can split those pieces again without changing the amount we had.
Take one half. Split each half into two equal pieces. The whole now has four pieces instead of two, but the selected amount has not grown or shrunk. What used to be one selected half is now two selected quarters.
This is the idea to protect throughout the lesson. If the learner can explain why the amount stayed the same, the symbolic rule later has meaning.
Use the pieces from Lesson 1
Your homemade circles, rectangles and fraction strips are ideal here. Put matching wholes side by side, because equivalent fractions only make sense when the wholes are the same size.
Place one half over two quarters. Ask whether any extra area appears. Then remove the labels and ask the learner to describe what they see.
Cover two thirds with four sixths. Make the relationship visible before writing 2/3 = 4/6.
These are especially useful because pieces line up cleanly. They make it easy to see one fifth becoming two tenths, or three fifths becoming six tenths.
A very concrete example: share, then split again
This works especially well with MathLink cubes, LEGO bricks or small sweets, because the learner can physically make the groups and then see the same share described in a new way.
Start with 12 objects and 2 people
Give each person 6 objects. Each person has one half of the whole collection.
Now make each person’s share from 2 yellow, 2 red and 2 blue objects. Do not change how many objects either person has; we are only reorganising each half.
The important point is that the learner has not been handed a rule. They have watched one share get re-described. The original half has stayed exactly the same size; it has simply been organised into three equal colour-groups.
Try the same idea with 3 people. Choose a total that divides neatly, give everyone an equal share, then subdivide every person’s share into the same smaller groups. Ask: “What fraction did each person have first? What new fraction describes exactly the same share now?”
A powerful move: split every piece in the same way
One of the clearest ways to build equivalent fractions is to take an existing partition and split every piece equally.
If a whole is divided into 3 thirds and each third is split in half, the whole now has 6 sixths. If 2 thirds were selected before, then both selected thirds also split in half, giving 4 selected sixths.
- Did we add any amount?
- Did we take any amount away?
- What changed?
- What stayed the same?
The answer we want is: the number and size of the pieces changed, but the amount selected stayed the same.
Why top and bottom must change together
The familiar rule “multiply the numerator and denominator by the same number” should come after the learner has seen the pieces split.
Suppose we start with 3/4 and split every quarter in half. The whole now has twice as many pieces: 8 instead of 4. But each of the 3 selected quarters also becomes 2 selected eighths, so there are 6 selected pieces.
The whole has been cut into twice as many equal pieces.
Each selected quarter was split too, so the number of selected pieces also doubled.
Useful phrase: “If you change the size of the pieces, you must also change how many of those pieces cover the same amount.”
Common misconceptions to watch for
Return to one whole. Sixths are smaller than thirds because the same whole has been cut into more pieces.
If 3/4 becomes 6/4, we have doubled the selected amount without changing the piece size. Build it with pieces so the change is obvious.
If 3/4 becomes 3/8, the pieces have become smaller but we kept only three of them. The amount has changed.
Half of a large pizza may be more food than three quarters of a small one. For equivalent-fraction comparisons, keep the whole fixed.
Good questions are better than telling
Equivalent fractions are an excellent place to let the learner explain what they notice.
Cross-topic links worth noticing
Equivalent fractions begin to connect fractions to several later ideas. There is no need to formalise all of these yet, but it is useful to notice them when they arise.
Half an hour is 30 minutes; two quarters of an hour is also 30 minutes.
50p is half of £1, and two 25p coins are the same amount. Likewise, 50¢ is half of $1, and two 25¢ coins make the same half-dollar amount. This is equivalent fractions in a form many children already understand.
1/2 metre = 2/4 metre = 50 cm. Different units or partitions can describe the same length.
Later, 1/2, 0.5 and 50% will become three ways of naming the same proportion.
When to move on
Do not wait for instant mental generation of every equivalent fraction. At this stage, understanding matters more than speed.
The learner is ready to move on when they can:
- show that two differently partitioned models cover the same amount;
- explain that splitting every piece changes the number and size of pieces, not the amount;
- produce simple equivalents such as 1/2 = 2/4 or 2/3 = 4/6 using a picture or prop;
- recognise that numerator and denominator must change together to keep the amount unchanged.
If the learner can apply the symbolic rule but cannot explain why it works, stay with the pictures a little longer.
Use the learner lesson alongside this guide
The learner page lets them compare the same amount represented with different partitions, repeatedly split pieces, and match equivalent fractions. Keep the homemade pieces nearby so the screen activity and physical model reinforce each other.
Open Lesson 2: Equivalent fractions →