Fractions: equivalence, comparing, adding and subtracting

Equivalent fractions

Equivalent fractions have the same value but different numerators and denominators: ½ = 2/4 = 3/6 = 50/100. They're made by multiplying (or dividing) the numerator and denominator by the same number. Multiplying both by 3 is the same as multiplying by 3/3, which is 1, so the value doesn't change.

• ½ = 3/6Numerator and denominator both × 3.
• ½ = 2/3Adding 1 to the top and bottom changes the value.

Simplifying

Simplifying means dividing the numerator and denominator by a common factor until no common factor (other than 1) is left. Dividing by the highest common factor does it in one step: 18/24 ÷ 6 = ¾. Children who simplify only part of the way (18/24 to 9/12) lose marks in the SATs when the question says “in its simplest form”. Secure times tables make this far easier.

Comparing and ordering

To compare fractions, put them in a form where they can be compared directly. Three approaches work:

• Same denominator: convert to a common denominator, then compare numerators. ⅔ and ¾ become 8/12 and 9/12, so ¾ is larger.
• Same numerator: if the numerators match, the smaller denominator gives the larger fraction. ⅖ is larger than 2/7, because fifths are bigger than sevenths.
• Benchmarks: compare each fraction with 0, ½ and 1. 3/8 is less than ½; 5/9 is more than ½.

• 1/8 is bigger than 1/4A bigger denominator means smaller pieces.

That misconception, that a bigger denominator means a bigger fraction, is the most common in KS2 fractions. Fraction walls and number lines fix it far better than explanation.

Adding and subtracting fractions

With the same denominator, add or subtract the numerators and keep the denominator. With different denominators, convert to a common denominator first.

• ⅔ + ¼ = 8/12 + 3/12 = 11/12Common denominator 12.
• ½ + ⅓ = 2/5Adding tops and bottoms. ½ + ⅓ must be more than ½, and 2/5 isn't.

For mixed numbers, two methods work: add the whole numbers and fractions separately, or convert to improper fractions first. The second is more reliable for subtraction when the fraction part needs an exchange, as in 3⅕ − 1⅗. Converting gives 16/5 − 8/5 = 8/5 = 1⅗.

In the SATs

The arithmetic paper always includes adding and subtracting fractions with different denominators, often including mixed numbers. Answers must usually be simplified or given as mixed numbers if the question asks. The reasoning papers test comparing and ordering, equivalent fractions on number lines, and explaining which fraction is larger.

Common mistakes

• Adding or subtracting numerators and denominators separately.
• Thinking a bigger denominator means a bigger fraction.
• Making equivalent fractions by adding the same number to the top and bottom.
• Simplifying only partly.
• Forgetting the whole numbers when working with mixed numbers.

Teaching ideas

Fraction walls

Keep a fraction wall on every table during the fractions unit. Children use it to check equivalences and comparisons until they no longer need it.

Does it make sense?

Before calculating ½ + ⅓, ask whether the answer will be more or less than ½, and more or less than 1. This simple estimate catches the “add the bottoms” error every time.

Common denominator race

Give pairs of fractions and ask for the lowest common denominator only, without completing the calculation. It isolates the hardest step and builds speed.

Number line ordering

Pupils place a mixed set of fractions on a 0 to 2 number line. Mixed numbers and improper fractions in the same set show they are the same kind of number.

Try it: five quick questions

SATs-style questions to use as a starter or a quick check. Tap to see each answer.

1. Simplify 24/36.

   Show answer

   ⅔

2. Which is larger: ⅗ or ⅝?

   Show answer

   ⅝ (24/40 compared with 25/40).

3. Calculate ¾ + ⅙.

   Show answer

   11/12

4. Calculate 2⅓ − 1⅚.

   Show answer

   ½

5. Order from smallest: ½, ⅖, ⅝

   Show answer

   ⅖, ½, ⅝