Questions: Introduction to Fractions

Question 1

Maria says 'I ate 1/2 of my pizza and you ate 1/2 of yours, so we ate the same amount.' When is she wrong?

• A. She is never wrong — 1/2 always means the same amount regardless of context
• B. She is wrong when the two pizzas are different sizes, because 1/2 of a large pizza is more than 1/2 of a small pizza
• C. She is wrong because fractions cannot be used to describe pizza
• D. She is wrong only if the pizzas were cut into a different number of slices

Answer: (B)

A fraction always describes a part of a specific whole. 1/2 of a 16-inch pizza is 8 inches' worth; 1/2 of an 8-inch pizza is 4 inches' worth. Same fraction, very different amounts. This is the 'whole matters' principle: a fraction without a specified whole is incomplete. The number of slices each pizza was cut into is irrelevant — what matters is the size of the whole.

Question 2

In the fraction 3/5, what does the denominator 5 tell you?

• A. There are 5 shaded parts
• B. There are 5 unshaded parts
• C. The whole has been divided into 5 equal parts, defining one-fifth as the unit being counted
• D. The numerator 3 must be subtracted from 5 to get the remaining parts

Answer: (C)

The denominator names the unit: it tells you how many equal parts the whole has been divided into, and therefore what size each part is. Once you divide a whole into 5 equal parts, you have defined the unit 'one-fifth.' The numerator then counts how many of that unit you have — in this case, 3. Thinking of the denominator as 'total pieces' (rather than 'unit-definer') leads directly to the misconception that 3/5 means '3 shaded, 5 total' — which only works for area models and breaks down everywhere else.

Question 3

A fraction with a numerator larger than its denominator — like 7/4 — is not a valid number because fractions should usually be less than 1.

• T. True
• F. False

Answer: False

False. A fraction represents division: 7/4 means 7 ÷ 4 = 1.75. There is no rule requiring fractions to be less than 1 — that belief comes from over-relying on the 'shaded pieces of a shape' model, where you typically shade fewer pieces than the total. On a number line, 7/4 sits between 1 and 2. 'Improper fractions' (numerator ≥ denominator) are perfectly valid numbers greater than or equal to 1.

Question 4

To place the fraction 3/4 correctly on a number line from 0 to 1, you divide the segment into 4 equal parts and count 3 of them from 0.

• T. True
• F. False

Answer: True

True. The number line model directly reflects what the fraction means: the denominator tells you how many equal parts to divide the segment into (creating the unit 'one-fourth'), and the numerator tells you how many of those parts to count from 0. This model also shows that 4/4 = 1 (all four parts equals the whole), and that fractions greater than 1 extend beyond the 0-to-1 segment — for example, 5/4 is one full unit plus one more fourth.

Question 5

Your friend says '1/2 of my brownie is the same amount as 1/2 of your brownie.' What question must you ask before you can know if that's true, and why?

Model answer: You must ask: are the two brownies the same size? A fraction specifies a relationship between a part and a whole, but the actual amount depends entirely on what the whole is. If your brownie is twice as large as your friend's, then 1/2 of yours is twice the amount of 1/2 of theirs, even though both fractions are written identically. The fraction 1/2 is incomplete until you define the whole it refers to.

This question targets the single most important insight in this topic: fractions describe relationships, not fixed amounts. Two identical fractions can represent very different quantities if their wholes differ. This principle is critical for interpreting fractions in real contexts — recipes, measurements, comparisons — and it becomes even more important when comparing fractions with different wholes in later work.