Add and subtract fractions with denominators up to 12 without a calculator

What you will learn

• Add and subtract fractions with denominators up to 12 without a calculator

Models used: number sentences, stories, tables.

Before you start

• Convert between mixed numbers and improper fractions

The pieces must have the same size

We want to add one half and one quarter. Their pieces have different sizes.

1. Both strips show equal wholes. Colour shows the amount; the dividing lines show the size of each part.

   We want to add one half and one quarter. Their pieces have different sizes.

2. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Cut the half into two quarters. The amount stays the same; now the pieces match.

   1/2 = 2/4

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

3. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   Two quarters and one quarter make three quarters.

   1/2 + 1/4 = 3/4

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

4. Each strip is one whole of the same size. Count all the equal parts, then count the shaded parts.

   We can undo the joining: remove one quarter and two quarters remain.

   3/4 − 1/4 = 2/4 = 1/2

   The size of a fraction depends on both numbers: how many equal parts make the whole, and how many of those parts we take.

Worked example: A new example, together

Calculate 5/6 − 2/3.

1. The parts must have the same size. Read each value with the heading above its column.

   Check the whole, rename unequal-sized parts when necessary, then join or remove the counts.

   Adding or subtracting equal-sized fractional parts changes their count, not their size.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   10/12	8/12

2. The parts must have the same size. Read each value with the heading above its column.

   Use common denominator 12.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   10/12	8/12

   Calculate 5/6 − 2/3.

   1/6

3. The parts must have the same size. Read each value with the heading above its column.

   10/12 − 8/12 = 2/12.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   10/12	8/12

4. The parts must have the same size. Read each value with the heading above its column.

   Simplify: 1/6.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   10/12	8/12

5. The parts must have the same size. Read each value with the heading above its column.

   Check the result against the question we started with.

   2/12 + 8/12 = 10/12.

   We found 1/6. Adding or subtracting equal-sized fractional parts changes their count, not their size.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   10/12	8/12

Common mistakes to discuss

• Adding the denominators too. The denominator names the size of the piece and does not change. 2/5 + 1/5 = 3/5, not 3/10.

Check your understanding

• Add and subtract fractions with denominators up to 12 without a calculator
• explain why fraction addition and subtraction need equal-sized parts
• check the answer against the original quantities and units

Explain your reasoning to an adult, then try a fresh question in the interactive lesson. A result from one attempt is not proof of lasting mastery.