Unlike denominators

What is this skill?

Adding and subtracting fractions with different denominators means first renaming the fractions so they share one denominator, since only equal-sized pieces can be added. To add 1/2 and 1/3, rewrite both in sixths: 3/6 + 2/6 = 5/6. To subtract 1/6 from 3/4, rewrite both in twelfths: 9/12 − 2/12 = 7/12.

A common denominator can always be found by multiplying the two denominators together, although a smaller one often works. Once the pieces are the same size, only the numerators are added or subtracted, and the denominator stays put. The final step is to write the answer in simplest form. 3/4 + 5/8 becomes 6/8 + 5/8 = 11/8, which is the same as 1 3/8. When a question says “simplest form,” Tutorsaurus accepts either the mixed number or the improper fraction, as long as it is fully reduced. An answer like 22/16 is marked wrong because it can still be reduced.

Why it matters

Fractions are the biggest conceptual step in elementary math, and this is where fifth grade pulls the threads together. Equivalent fractions, factors and multiples, and number sense all come into play. Children who can do this fluently are well prepared for ratios, proportions and algebra, where fractions are part of nearly every problem.

It is also the first place that fractions are treated as numbers that can be combined, not just as pictures of shaded parts. That shift in thinking is worth taking time over.

Common mistakes

• Adding the tops and the bottoms: 1/2 + 1/3 = 2/5. It looks right to a child who has been adding across, but 2/5 is smaller than 1/2, and the sum cannot be smaller than one of the parts. Comparing the answer with the starting fractions exposes the slip.
• Changing the denominator without changing the numerator, or the other way around. Whatever is done to the bottom must be done to the top.
• Forgetting to simplify, or leaving an improper fraction unconverted when the problem asks for a mixed number.
• Subtracting the denominators as well as the numerators. After the pieces are the same size, only the numerators change.
• Picking a common denominator that is not actually a multiple of both original denominators.

How to help at home

• Cook together. Doubling or combining recipe measurements (1/2 cup plus 1/3 cup) is real fraction addition.
• Use paper strips folded into halves, thirds and sixths to show why 1/2 and 1/3 need to be renamed before they can be added.
• List the multiples of each denominator side by side until one number appears in both lists. That number is a common denominator.
• Ask your child to estimate first: is the sum more or less than 1?
• After simplifying, check by asking whether the top and bottom share any factor other than 1.

Worked examples

These are real questions from the Tutorsaurus unlike denominators practice, with the answers worked out.

1. Example 1

   1/4 + 1/12 = ? (simplest form)

   Answer: 1/3. Common denominator 12: 3/12 + 1/12 = 4/12 = 1/3.

2. Example 2

   1/2 − 1/4 = ? (simplest form)

   Answer: 1/4. Common denominator 4: 2/4 − 1/4 = 1/4.

3. Example 3

   1/2 + 6/10 = ? (simplest form)

   Answer: 1 1/10. Common denominator 10: 5/10 + 6/10 = 11/10 = 1 1/10.