VA ClassesAdding Fractions With Different Denominators
You cannot add halves to thirds any more than you can add centimetres to inches. Once the pieces are the same size it becomes ordinary counting, and everything hard about this topic happens before that point.
Why you cannot just add them
Here are a half and a third, drawn on identical bars.
Add straight across and 12 plus 13 comes out as 25, which is less than the half you began with. That is not a small error to be tidied up later, it is an impossible answer, and a child who notices that has learned something more useful than the method itself.
Recutting both bars
The fix is to recut both bars until every piece is the same size. Sixths do the job here, since 2 and 3 both go into 6.
The half has become 36 and the third has become 26. Neither amount moved, both bars just got more cuts, and now that every piece is the same size they can finally be counted together.
The method
Four steps, always the same
1. Find a number both denominators divide into. Multiplying them together always works.
2. Rewrite each fraction with that denominator, using the equivalent fraction rule.
3. Add the numerators. Keep the common denominator.
4. Simplify, and convert to a mixed number if the question asks.
Step 1 has a shortcut worth knowing. If one denominator divides into the other, use the larger one and you only have to rewrite a single fraction. For 14 + 38, use 8, not 32.
Worked examples
When one denominator divides into the other
- Check first: does 4 divide into 8? Yes, so use 8 and leave the second fraction alone.
- Rewrite 14 with a denominator of 8. Multiply top and bottom by 2, giving 28.
- Now add the tops. 2 + 3 = 5, so the answer is 58.
- Can it be simplified? 5 and 8 share no factor, so no.
When neither divides into the other
- 3 does not divide into 4, so multiply them: 3 × 4 = 12. Use twelfths.
- 23: multiply top and bottom by 4, giving 812.
- 14: multiply top and bottom by 3, giving 312.
- Add the tops. 8 + 3 = 11, giving 1112, which will not simplify.
An answer over 1
- 2 divides into 4, so use quarters.
- 12 becomes 24.
- Add the tops. 3 + 2 = 5, giving 54.
- The top is bigger than the bottom, so this is more than one whole. As a mixed number, 5 ÷ 4 = 1 remainder 1, which is 1 and a quarter.
A two-second check
Before writing the answer down, ask one question: is the answer bigger than the larger fraction I started with?
It has to be, because something was added to it. A child who answers 12 + 13 = 25 and then asks this question catches themselves in about two seconds, because two fifths is obviously less than a half.
Second check for the same price: the answer should also be less than the two fractions doubled. Between those two bounds, and you have almost certainly done it right.
Common mistakes
1. Adding straight across
12 + 13 = 25, and this is the one that costs the most marks by a wide margin. Halves and thirds are pieces of different sizes, and adding them as if they were not is the fraction equivalent of adding a length in centimetres to one in inches and writing down the total.
2. Changing the bottom but forgetting the top
The bottom of 23 gets taken up to 12 and the top is left behind, giving 212 rather than 812. It usually happens because the child is concentrating on getting the denominators to match and forgets that they were meant to be rewriting a fraction, not just changing a number.
3. Adding the denominators too, at the end
Everything goes right as far as 36 + 26 and then the last line reads 512. All the difficult work has been done correctly and the mark still goes, because the old habit resurfaces at the final step. Once the denominators match, the denominator has finished changing for good.
Practice questions
Seven questions. One attempt each, and the explanation appears once the answer is in.
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For parents
Most children can carry out the four steps perfectly well. What they skip is the half-second of judgement before and after, and that is where the marks live. Get them into the habit of saying the estimate out loud before they write anything: a half plus a third is a bit more than a half, so the answer should land somewhere close to 0.8. A child who does that almost never hands in an impossible answer, because they have already decided roughly what a possible one looks like.
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