4 ÷ (1/2) asks: 'How many half-sized pieces fit into 4 wholes?' Each whole contains 2 halves, so 4 wholes contain 4 × 2 = 8 halves. The answer is 8 — LARGER than what we started with (4). This surprises students who expect division to always make numbers smaller, but that assumption only holds when dividing by numbers greater than 1. Dividing by a fraction less than 1 always produces a result larger than the dividend.
Question 2 Multiple Choice
To solve (2/3) ÷ (3/4) using invert-and-multiply, a student writes (3/2) × (4/3). What error did the student make?
ANo error — the student correctly inverted the divisor
BThe student inverted the dividend (2/3) instead of the divisor (3/4)
CThe student should have added the fractions first, then inverted
DInvert-and-multiply only works for unit fractions
The invert-and-multiply rule requires flipping the DIVISOR — the fraction you are dividing BY (the one after the ÷ sign). The divisor here is 3/4, so you flip it to 4/3. The dividend (2/3) stays unchanged. The correct setup is (2/3) × (4/3). The student flipped 2/3 instead of 3/4 — inverting the wrong fraction. Always ask: 'What am I dividing by?' — that is the fraction to flip.
Question 3 True / False
Dividing a whole number by a fraction less than 1 always produces a result larger than the original whole number.
TTrue
FFalse
Answer: True
When you divide by a fraction less than 1 (like 1/4 or 1/3), you are asking how many tiny pieces fit into the whole. More small pieces fit than the original count — so the result grows. For example, 5 ÷ (1/4) = 20, because 20 quarter-pieces fit in 5 wholes. The assumption that division always makes numbers smaller is only valid for divisors greater than 1.
Question 4 True / False
The expression (1/3) ÷ 4 gives the same answer as (1/3) ÷ (1/4), because both involve the numbers 3 and 4.
TTrue
FFalse
Answer: False
(1/3) ÷ 4 asks: 'If 1/3 of a pizza is shared equally among 4 people, how much does each get?' The answer is 1/12. (1/3) ÷ (1/4) asks: 'How many quarter-pieces fit into 1/3?' The answer is 4/3. These are completely different questions with different answers. The numbers 3 and 4 appear in both, but their role (divisor vs. part of a fraction) changes the entire meaning of the problem.
Question 5 Short Answer
Explain in your own words why dividing 3 by (1/4) gives 12, not a number smaller than 3.
Think about your answer, then reveal below.
Model answer: 3 ÷ (1/4) asks: 'How many quarter-sized pieces fit into 3 wholes?' Each whole contains 4 quarter-pieces, so 3 wholes contain 3 × 4 = 12 quarter-pieces. The result is 12 — larger than 3 — because dividing by a small number means fitting many small pieces into the total. Division only makes numbers smaller when the divisor is greater than 1. When the divisor is a fraction less than 1, the result is larger than the original.
Visualizing this with a picture — 3 rectangles, each cut into 4 pieces = 12 pieces total — makes the counterintuitive result concrete. The confusion comes from overgeneralizing 'division makes things smaller' beyond its actual scope. That pattern only holds for divisors greater than 1; it breaks down as soon as the divisor is a proper fraction.