In plain words
Does simplifying a fraction make it smaller?
No. Simplifying changes how a fraction is written, not how big it is. To simplify 6/8, group the eighths in pairs: each pair is one fourth, so 6/8 becomes 3/4. The pieces are bigger and there are fewer of them, but the amount is exactly the same.
- The big idea
- Simplifying changes the pieces you count with, not the amount. Group six eighths in pairs and you get three fourths: bigger pieces, fewer of them, the same amount.
- The common confusion
- Simplifying makes a fraction smaller: 3/4 must be less than 6/8, because 3 and 4 are smaller numbers than 6 and 8. It often sounds like this: “If you simplify (six eighths) to (three fourths), you lose some.”
Why it works
Shade 6 of 8 equal parts of a bar. That is 6/8.
Now group the eighths in pairs. Each pair is one fourth of the bar, so the whole is 4 fourths and the shaded part is 3 of them: 6/8 = 3/4.
Nothing was taken away. The pieces got bigger and there are fewer of them, but the shaded part covers exactly the same length. Simplifying changes the units we count in, not the amount.
In symbols, simplifying divides the top and bottom by the same number. The groups have to fit evenly into both the shaded part and the whole: groups of 4 fit into 8 eighths but not into 6 shaded eighths, so 6/8 can’t be written as a whole number of halves.
A fraction is in simplest form when no grouping fits evenly any more. 3/4 is in simplest form, because no whole number bigger than 1 divides both 3 and 4.
A worked example
A pizza is cut into 8 equal slices and 6 are left. Put the leftover slices together in pairs and each pair is a quarter of the pizza, so 6/8 of the pizza is 3/4 of the pizza. It is the same leftover pizza, counted in bigger pieces.
Why this confusion is so common
Simplifying is often taught as a procedure: divide the top and bottom by the same number. Dividing makes numbers smaller, and 3/4 looks smaller than 6/8, so it is natural to think the fraction shrank. The missing piece is that the units changed too: three fourths and six eighths count pieces of different sizes. Children need to see the amount stay put while its name changes.
Try this away from the screen
Build a tower of 8 same-size bricks: 6 of one color at the bottom and 2 of another color on top. Break it into pairs. How many pairs make the whole tower, and how many pairs are the first color? You get 3 of 4, and not a single brick was lost. Ask: is 3/4 of the tower less than 6/8 of it?
Helpful things to say
| Instead of | Try |
|---|---|
| “To simplify, just make the numbers smaller.” | “If we group these pieces in pairs, how many bigger pieces do we get? Did the amount change?” |
| “Divide the top and bottom by 2, and you’re done.” | “What does dividing both numbers by 2 look like in the picture?” |
Practice questions
For grown-ups helping out: the answers are tucked away so learners can try first.
What is 4/6 in simplest form? Group the sixths to check.
Show the answer
Answer: 2/3. Group the sixths in pairs. Each pair is one third: 3 pairs make the whole and 2 pairs are shaded. So 4/6 = 2/3, with bigger pieces, fewer of them and the same amount.
Which is bigger: 10/12 or 5/6?
Show the answer
Answer: They are the same. Each sixth is 2 twelfths, so 5 sixths is 10 twelfths: 10/12 = 5/6. Simplifying 10/12 to 5/6 changes the size of the steps, not where you end up.
Six 1/10 pieces are laid end to end. How many 1/5 pieces would cover the same length?
Show the answer
Answer: 3 pieces. One fifth is as long as 2 tenths. Grouping 6 tenths in pairs makes 3 groups, so 6/10 = 3/5. The pieces got bigger; the length stayed the same.
Simplifying 8/10 gives 4/5. How does 4/5 compare with 8/10?
Show the answer
Answer: 4/5 is the same amount as 8/10. Each fifth is 2 tenths, so 4 fifths is 8 tenths. Simplifying changes the pieces you count, not the amount: 4/5 = 8/10.
Questions people ask
Does simplifying a fraction change its value?
No. 6/8 and 3/4 are the same number. Simplifying rewrites the amount using bigger pieces, so you need fewer of them.
Is 6/8 the same as 3/4?
Yes. Each fourth is made of 2 eighths, so three fourths is 3 × 2 = 6 eighths. Same amount, different-size pieces.
How do you know when a fraction is in simplest form?
When no whole number greater than 1 divides both the top and the bottom. 6/8 can be divided by 2 to give 3/4, and nothing bigger than 1 divides both 3 and 4, so 3/4 is in simplest form.
Do you always have to simplify fractions?
Teachers often ask for simplest form because it is the shortest way to name an amount. But 6/8 and 3/4 are equally true: they are the same number. Unsimplified forms are often useful, for example when adding 6/8 + 1/8 = 7/8.
Words to know
- Simplest form
- A fraction is in simplest form when its top and bottom numbers share no factor except 1, such as 3/4 (but not 6/8).
- Equivalent fractions
- Fractions that name the same amount, such as 1/2, 2/4 and 4/8. They share one point on the number line.
- Numerator
- The top number of a fraction. It counts how many of the equal parts we have.
- Denominator
- The bottom number of a fraction. It tells how many equal parts make one whole.
Curriculum connections
We don’t assume any particular curriculum. For teachers in the United States, here’s how this slice lines up with the Common Core State Standards:
- 4.NF.A.1
Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions.
- 3.NF.A.3.b
Recognize and generate simple equivalent fractions, e.g., 1/2 = 2/4, 4/6 = 2/3. Explain why the fractions are equivalent, e.g., by using a visual fraction model.
Related slices
Different Numbers, Same Amount
Is (one half) the same as (two fourths)?
Why 1/2, 2/4 and 4/8 are exactly the same amount, even though the numbers are different. · About 3 minutes
Adding Different Pieces
Is (one half) + (one fourth) equal to (two sixths)?
Why 1/2 + 1/4 is 3/4, not 2/6: make the pieces the same size before you count them. · About 3 minutes
The Bigger Denominator Trap
Is (one sixth) bigger than (one eighth)?
Why 1/8 is smaller than 1/6, even though 8 is bigger than 6. · About 3 minutes
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