a/b · Fractions
Counting the equal pieces of a whole — the fraction.
Share one pizza equally among 4 people — how do you write one person's share?
Explain why dividing the same whole into more equal parts makes each part smaller.
Start with a picture of equal sharing.
Give a name to a share that does not divide into whole numbers.
Open the immersive version →A reading path containing this lesson: Parts of one whole · 1/3
Why is the part smaller when 8 is bigger?
Imagine sharing one chocolate bar equally among four people. Each receives one of four equal parts. We write that amount as 1/4 of a bar.
Now eight people share a bar of the same size. Twice as many people does not create more chocolate. Each share must be smaller.
This may feel surprising: 8 is larger than 4, so why is 1/8 smaller than 1/4? The bottom number, the denominator, tells us how many equal parts make one whole. It is not the number of cuts or the size of a part.
Split a quarter in half again. That quarter now consists of two eighths, so 1/4=2/8. The name has changed, but the amount of chocolate has not.
Does a larger denominator always mean a smaller fraction? Compare 3/8 with 1/4. Since 1/4=2/8, three eighths is larger. The numerator, the top number, also tells us how many parts were taken.
To compare actual amounts of chocolate, the wholes must also be the same size. One eighth of a large bar may weigh more than one quarter of a small bar. First check what counts as one whole.
Read these symbols together
- b
- In a/b, the denominator b counts the equal parts in one whole. In this picture, b is a positive integer.
- a
- The numerator a counts how many of those parts are taken.
Pause and try it yourself
Make a prediction first. Change one input at a time and compare what happens with what you expected.
Predict first — one slice of a 4-way cut (1/4) vs one slice of an 8-way cut (1/8). Which is bigger?
Slice the pizza
Choose the number of slices. The red slice is 'one person's share'.
one slice = 1/4
The size of a part and how many parts we take
A fraction is one quantity, not two numbers.
Reading 3/4 as a 3 and a 4 can confuse. See it as one amount — three of four equal parts — and comparison and arithmetic become visible in pictures.
Common denominators are a translation.
A half and a third have different piece sizes. Translate them into sixths: 3/6 and 2/6. Now compare or add counts of equally sized pieces. A common denominator provides a common language.
Write both the division and the amount taken
Read the pizza picture in mathematical language.
What a fraction means
In this picture, b is a positive integer counting equal parts, and a counts the parts taken. More generally, a/b represents a divided by b when b is nonzero.
Equivalent fractions
Subdivide each piece into k equal parts. The piece count and the parts taken grow together while the amount stays the same: 1/2 = 2/4 = 4/8. Here k is a positive integer; the general identity requires b and k to be nonzero.
Adding with different denominators
For the same whole, match the piece size and then add their counts. Fractions with different denominators can be added; simply adding the two original numerators is not the correct rule.
Make one more distinction
1/8 is bigger than 1/4 — because 8 is bigger than 4.
For equally sized wholes, the opposite is true. The denominator counts equal parts of the whole. With numerator 1 in both fractions, a larger denominator means a smaller part.
To add fractions, add tops and bottoms (1/2 + 1/4 = 2/6).
Adding counts of differently sized pieces does not preserve the amount. Rewrite 1/2 as 2/4 so the pieces match, then 2/4 + 1/4 = 3/4. A common denominator matches the piece size.
Where else can this idea help?
Cooking measures
A recipe may call for 1/2 cup of sugar or 3/4 of a stick of butter. Scaling the same recipe from two servings to three uses fraction multiplication.
Reading time
15 minutes is 1/4 hour; 30 minutes is 1/2 hour. An hour and a half also uses a fraction.
Note lengths
At the same tempo, a quarter note is one quarter of a whole note's duration, while an eighth note is one eighth. A time signature also describes how beats are grouped.
Screens and chances
A display ratio of 16:9 compares width with height as 16/9. The probability of a specified face on a fair die is also expressed as a fraction, 1/6.
Explain it in your own words
Compare with one possible explanation
Halving each quarter makes eight equal pieces. The original quarter becomes two pieces without changing its amount.
Try another question
Q1Which is bigger: 3/5 or 4/7?Show answer ▾
Express both in thirty-fifths: 3/5 = 21/35 and 4/7 = 20/35, so 3/5 is larger. Comparing the original numerators or denominators alone can mislead.
Q2Share two equally sized pizzas among three people. What is one share?Show answer ▾
2 ÷ 3 = 2/3 of a pizza. A fraction also represents the result of division.
Q3Why is 1/2 + 1/3 not 2/5?Show answer ▾
2/5 is 0.4, less than 1/2. Adding the positive amount 1/3 cannot reduce the total. In sixths, 3/6 + 2/6 = 5/6.
Try answering yourself before revealing it — getting it wrong is where learning starts.
The idea to take with you
Read both how the whole is divided and how many parts are taken.
What question comes next?
A reading path containing this lesson: Parts of one whole · 1/3
Leads to next
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Ratios
Connect fraction and ratio notation, keeping the conditions on equivalent expressions in view.
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When outcomes are equally likely, favorable outcomes divided by total outcomes gives a probability as a fraction.
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