Primer: Fraction Operations (Mixed Fractions & Improper Fractions)

Estimated Time: 10–15 minutes

Goal: Before your lesson, you should understand how each operation works so that the lesson can focus on solving problems instead of introducing the ideas for the first time.

Step 1 — READ (3–5 minutes)

Part A — Adding & Subtracting Fractions

The Vertical Rule

Whenever you are adding or subtracting fractions or mixed numbers:

✅ Rewrite the problem vertically.

If the top fraction is smaller than the bottom fraction, borrow one whole from the whole number.

Remember:

1=\frac{5}{5}

4\frac{1}{5}\space=\space3+ \frac{5}{5}+\frac{1}{5}\space=\space3\frac{6}{5}

Now subtract normally.

Part B — Multiplication & Division

Mixed Number Rule

🚫 Never multiply or divide mixed numbers.

Always change them to improper fractions first.

The Mathematical Way

A mixed number is really a whole number PLUS a fraction.

Example:

3\frac{2}{7}

↓

Rewrite the mixed number as addition.

3+\frac27

↓

Rewrite the whole number using the same denominator.

3 \;\rightarrow\; \frac{3\cdot7}{7} = \frac{21}{7}

↓

Substitute it back into the expression.

3+\frac27 \;\rightarrow\; \frac{21}{7}+\frac27

↓

Now add like fractions.

\frac{21}{7}+\frac27 = \frac{23}{7}

The Circle Method (Shortcut)

Instead of writing every intermediate step, we can combine them into one shortcut.

Example:

3\frac27

↓

Multiply the whole number by the denominator.

3\times7=21

↓

Add the numerator.

21+2=23

↓

Keep the denominator.

\frac{23}{7}

Circle Method Formula

\frac{(\text{Whole}\times\text{Denominator})+\text{Numerator}}{\text{Denominator}}

💡 Why does this work?

The Circle Method is simply a shortcut for

3+\frac27 \rightarrow \frac{21}{7}+\frac27 = \frac{23}{7}.

It combines all of those intermediate steps into one quick calculation.

The Circle Method (1 more time)

For a mixed number like

3\frac{2}{7}

Step 1.

Multiply the whole number by the denominator.

3\times7=21

Step 2.

Add the numerator.

21+2=23

Step 3.

Keep the denominator.

3\frac27=\frac{23}{7}

Cross-Simplify First

Before multiplying fractions,

look for numbers that can cancel diagonally.

Example:

\frac94\times\frac23

Notice that

9 and 3 divide by 3.

Rewrite as

\frac31\times\frac22

Now multiply.

Division Rule

Remember:

Keep • Change • Flip

Example:

1\frac12\div\frac34

Convert the mixed number.
Keep the first fraction.
Change ÷ to ×.
Flip the second fraction.
Cross-simplify.
Multiply.

Step 2 — WATCH (5–10 minutes)

Watch ONE of these videos.

Step 3 — TRY IT YOURSELF

1.

4\frac15-2\frac35=

2.

Convert to an improper fraction.

3\frac27=

3.

Cross-simplify first.

\frac94\times\frac23=

4.

Use Keep • Change • Flip

1\frac12\div\frac34=

Come Prepared

⭐ Bring any questions you have.

If something doesn't make sense yet, that's okay—that's exactly what we'll work on together during our session.