Same Numerator, Different Size: Comparing 3/4 and 3/8
Mr. Delgado's third graders are comparing 3/4 and 3/8. On the exit ticket, 14 of 26 students wrote that 3/4 < 3/8. During the share-out, Ana says, 'It has to be 3/8, because 8 is bigger than 4.' Ben says they are equal 'because both have a 3 on top.' Cece draws two rectangles of the same size, splits one into 4 equal parts and shades 3, splits the other into 8 equal parts and shades 3, and says 3/4 is bigger. Mr. Delgado has fraction strips, number lines, and circular fraction pieces available. He wants the next lesson to target the reasoning behind the wrong answers rather than just correcting the conclusion, and he has one 45-minute period before the unit assessment.
1. Explain the flaw in Ana's reasoning and the flaw in Ben's reasoning. They are making different mistakes and need different responses.
Read the worked answer
Ana is comparing denominators as if they were whole numbers: 8 > 4, therefore eighths > fourths. But the denominator names the size of the part, and it works backwards - the more equal parts a whole is cut into, the smaller each part. Three large parts (fourths) cover more than three small parts (eighths). Her error is treating the bottom number as a magnitude rather than as a count of parts.
Ben is comparing numerators and stopping. Both fractions do have 3 parts, but they are not the same parts. Three of something big is not equal to three of something small.
Both errors share a root: reading a fraction as two independent whole numbers instead of one quantity. Once a student sees the fraction as a count of same-sized parts, the comparison is easy and runs the opposite way from Ana's answer: 3/4 > 3/8. Ben should also be asked what he would say about 3/4 and 3/100 - the structure of his error becomes obvious when the denominators are far apart.
2. Which representation would you lead with - fraction strips, a number line, or circular pieces - and why? Justify the choice against Ana's specific error.
Read the worked answer
Lead with fraction strips (identical rectangles), then move to the number line in the same lesson.
Strips work first because they hold the whole constant and vary only the size of the parts. Ana can lay the fourths strip directly under the eighths strip - same total length - and see that the three shaded fourths extend past the three shaded eighths. Her claim ('8 is bigger, so the pieces are bigger') is contradicted physically, not verbally, which is what a 45-minute reteach needs.
The number line should follow immediately because 3/4 and 3/8 can be placed on the same 0-to-1 line, where order is visible without shading: 3/8 sits left of 1/2, 3/4 sits right of it. That gives the class a general tool, not just a fact about this pair.
Circular pieces are the weakest first choice here: eighths and fourths drawn as pie slices are hard to compare by eye when the pieces have different shapes, and students often compare the shape of the slice rather than the area.
Whichever model is used, keep the wholes the same size. Comparing 3/4 of one rectangle to 3/8 of a larger one will 'confirm' Ana's answer.
3. A student marks 3/4 on a number line that has been divided into eighths by counting three tick marks from 0 and landing on 3/8. What does this reveal, and how do you move the student to 6/8 without telling them the answer?
Read the worked answer
The student is counting tick marks instead of parts of the size named by the denominator. On a line partitioned in eighths, every tick is another eighth, so three ticks from 0 is by definition 3/8 - the student's action is consistent with the line they were given and inconsistent with the fraction they were asked to locate. The numerator '3' refers to fourths; the line does not show fourths.
Move them by renaming, not by correcting the position:
- 'Where would 1/4 be on this line?' The student marks 2/8 - one fourth is two eighths. Confirm by folding the fraction strip.
- 'So where is 2/4? 3/4?' Three jumps of 2/8 each lands on 6/8.
- Ask them to say both names for the same point: 6/8 and 3/4.
The idea to leave them with: a numerator counts parts of the size the denominator names, so before counting on a line you must check that the partition matches the denominator - if the line shows eighths, rename 3/4 as 6/8 first.