Same Numerator, Different Denominators: Which Piece Is Larger?
In Mr. Okafor's class, students compare fractions using paper strips folded into equal parts. The teacher writes 3/4 and 3/8 on the board and asks which fraction is greater. Jada says 3/8 is greater because 8 is bigger than 4, so the numbers in the fraction are bigger. Luis says 3/4 is greater because fourths are larger pieces than eighths, so three fourths must be more than three eighths. The teacher does not say who is right. Instead she writes a new pair, 5/6 and 5/8, and asks students to predict the comparison before folding anything. Finally, she writes the sentence: if two fractions have the same numerator, the fraction with the smaller denominator is always greater, and asks whether the class will accept it as a rule. Students must keep track of which whole they are describing, since one group cut its strip from a longer sheet of paper than another group did.
1. Evaluate Jada's and Luis's claims about 3/4 and 3/8 without folding strips. Use a benchmark or a same-whole description to decide who is right.
Read the worked answer
Luis is right: 3/4 > 3/8.
Reason with the benchmark 1/2 on two strips of the same length. Cut one into 4 equal parts and shade 3; that shaded part passes the halfway point, since 2 of the 4 parts equals 1/2 and 3 parts is more than that. Cut the other into 8 equal parts and shade 3; halfway would be 4 eighths, so 3 eighths stops before halfway.
One fraction is past halfway and the other is short of halfway, so 3/4 is greater.
Jada's error is comparing the digits in the fractions rather than the sizes of the pieces: a bigger denominator means the same whole was cut into more pieces, so each piece is smaller.
2. Predict which is greater, 5/6 or 5/8, and justify without folding. Include a check using the benchmark 1/2 or the distance to 1.
Read the worked answer
Both fractions count 5 pieces, but sixths are larger pieces than eighths, so 5/6 > 5/8.
Distance-to-1 check: 5/6 is only 1/6 away from one whole, while 5/8 is 3/8 away from one whole. The fraction that is closer to 1 is larger, and 1/6 is much smaller than 3/8, so 5/6 wins.
Benchmark check: 5/6 and 5/8 are both greater than 1/2 (3/6 = 1/2 and 4/8 = 1/2), so the benchmark alone cannot separate them; that is why the size-of-pieces reasoning or the distance-to-1 reasoning is needed.
3. Should the class accept the rule that with the same numerator, the smaller denominator always gives the greater fraction? State whether it is true, give a reason, and name the condition that must be added.
Read the worked answer
Yes, the rule is true - with one condition attached: the two fractions must describe the same-size whole.
Reason: the same numerator means the same number of pieces is being counted. A smaller denominator means the whole was cut into fewer pieces and each piece is therefore larger, so the same count of larger pieces makes a larger fraction. Examples: 3/2 > 3/4; 5/6 > 5/8; 7/8 > 7/10.
The condition matters. Three fourths of a small cookie can be less than three eighths of an enormous one, because the wholes differ. So the class should accept the rule only when it is written together with the phrase same-size whole.