A student claims '1/8 is bigger than 1/4 because 8 is bigger than 4.' What is wrong with this reasoning?
ANothing is wrong — 8 is bigger than 4, so 1/8 should be bigger than 1/4
BThe student confused the numerators and denominators — she should compare 1 and 1 instead
CThe denominator tells how many equal pieces the whole is divided into — more pieces means each piece is smaller, so 1/8 is actually less than 1/4
DThe student is correct for fractions greater than 1 but wrong for unit fractions
The denominator counts the number of equal shares the whole has been split into — not how much you have. The numerator (1) tells you how many shares you get. More shares (larger denominator) means each share is smaller. A pizza cut into 8 slices gives you a much smaller 1-share than a pizza cut into 4 slices. Applying whole-number intuition ('bigger number = more') to denominators is the core misconception.
Question 2 Multiple Choice
Two friends each receive one slice of the same-sized pizza. Anya's pizza was cut into 6 equal slices; Benny's was cut into 3 equal slices. Who got more pizza?
AAnya, because 6 is larger than 3
BBenny, because fewer cuts means each slice is larger
CThey got the same amount, since each received exactly one slice
DIt depends on the diameter of each pizza
Same-sized pizza, different number of cuts. Benny's pizza was divided into only 3 pieces, making each piece 1/3 of the whole. Anya's was cut into 6 pieces, making each piece only 1/6 of the whole. 1/3 > 1/6. Option C is the tempting wrong answer — 'one slice is one slice' — but it ignores that slices differ in size depending on how many there are.
Question 3 True / False
1/6 is greater than 1/3 because the pizza cut into sixths has more pieces.
TTrue
FFalse
Answer: False
More pieces means each piece is smaller, not larger. 1/6 < 1/3. The correct ordering is 1/3 > 1/6: the pizza cut into only 3 pieces gives larger individual slices than one cut into 6. Counting the total number of pieces and confusing it with the size of one piece is the central misconception for this topic.
Question 4 True / False
When comparing unit fractions, the fraction with the smallest denominator is always the greatest.
TTrue
FFalse
Answer: True
1/2 > 1/3 > 1/4 > 1/6 > 1/8. The smallest denominator (2) produces the largest pieces, so 1/2 is the greatest unit fraction. This inverse relationship holds whenever the numerators are equal — which they always are for unit fractions (numerator = 1).
Question 5 Short Answer
Why does a larger denominator make a unit fraction smaller, not bigger? Explain using the idea of equal sharing.
Think about your answer, then reveal below.
Model answer: The denominator counts how many equal pieces the whole is divided into. You always receive exactly 1 piece (numerator = 1). When the whole is cut into more pieces, each piece must be smaller to keep the pieces equal. So a larger denominator means more, smaller pieces — and 1 small piece is less than 1 large piece.
This is an inverse relationship: as the denominator increases, the size of each equal share decreases. Visualizing it with a folded strip of paper or pizza makes the relationship concrete: folding once gives halves (large), folding twice gives quarters (smaller), folding three times gives eighths (smaller still).