Add and subtract mixed numbers

What you will learn

• Add and subtract mixed numbers

Models used: number sentences, stories, tables.

Before you start

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

Whole pieces and fractional pieces

Start with two and one quarter. We want to take away three quarters.

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

   Start with two and one quarter. We want to take away three quarters.

   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.

   4/4 = 4/4: the same length, so the same amount.

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

   Exchange one whole for four quarters. The amount is still the same: one whole and five quarters.

   2 1/4 = 1 + 5/4

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

   4/4 = 4/4: the same length, so the same amount.

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

   Remove three quarters from the five quarters. One whole and two quarters remain.

   2 1/4 − 3/4 = 1 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 3 2/3 + 2 1/2.

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
   22/6	15/6

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

   Use common denominator 6.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   22/6	15/6

   Calculate 3 2/3 + 2 1/2.

   6 1/6

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

   22/6 + 15/6 = 37/6.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   22/6	15/6

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

   Change back to a mixed number: 6 1/6.

   The parts must have the same size

   First amount in equal parts	Second amount in equal parts
   22/6	15/6

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.

   37/6 − 15/6 = 22/6.

   We found 6 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
   22/6	15/6

Common mistakes to discuss

• Taking the smaller fraction from the larger. The order in the question stands. For 3 1/4 − 1 3/4, exchange one whole into quarters rather than working out 3/4 − 1/4.

Check your understanding

• Add and subtract mixed numbers
• 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.