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Fractions

Grades 3–6 Fractions Number sense

A fraction is just a division that hasn't happened yet. The bottom number — the denominator — says how many equal pieces the whole was cut into. The top number — the numerator — says how many of those pieces you have. So 3/4 means "cut it into 4, take 3" — and also means exactly 3 ÷ 4.

Equivalent fractions — same amount, different cut

Cut a pizza into 2 and take 1 piece, or cut it into 8 and take 4 — you have the same pizza. 1/2 = 4/8. Multiplying (or dividing) the top and bottom by the same number never changes the amount:

1 2 4 8 --- = --- = --- = --- 2 4 8 16 ← multiply top & bottom by 2 each time — same amount

Adding fractions — you need the same cut first

You can't add pieces of different sizes directly, just like you can't add "2 slices" + "3 chunks." Make the denominators match first:

1 1 2 1 3 --- + --- = --- + --- = --- 2 4 4 4 4 ← double the halves into quarters, then add normally
  1. Find a common denominator (a number both bottoms divide into — often just their product).
  2. Rewrite each fraction in that cut.
  3. Add the tops; keep the bottom.

Comparing fractions

Two reliable ways:

Why fractions matter more than almost anything A large longitudinal study of US and UK students found that elementary knowledge of fractions and division uniquely predicted high-school algebra achievement 5–6 years later — even controlling for IQ, memory, and family income.
  • Siegler, R. S., et al. (2012). Early predictors of high school mathematics achievement. Psychological Science, 23(7), 691–697. doi:10.1177/0956797612440101
Fractions aren't a hurdle on the way to real math — they are the foundation of real math.
Watch out 1/2 + 1/3 is not 2/5. Adding tops and bottoms directly is the most common fraction mistake on Earth. If the pieces aren't the same size, you can't count them together yet — make the denominators match first.
Try it yourself:
  1. Which is bigger, 5/8 or 3/5?  (cross-multiply: 25 vs 24 → 5/8)
  2. 1/3 + 1/6  (answer: 1/2)
  3. 3/4 + 1/8  (answer: 7/8)

Next: decimals and percents are just fractions wearing different clothes — and division is what a fraction literally means.

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