Comparing Fractions: Which One Is Bigger?

Warm-Up

6 minutes. Write on the board: Which is bigger, 1/3 or 1/8? Take a vote before any discussion, and write both totals up.

A large part of the class will vote for 1/8, and they should not be told they are wrong yet. Ask the students who chose it to explain, and write the reason up as they give it: eight is more than three.

Say plainly that this is a good reason and that it is exactly right about whole numbers — eight cookies beat three cookies every time. The question is whether it stays right when the numbers are on the bottom of a fraction. Leave it open.

Hand out the paper strips. Everyone folds one strip into three equal parts and the other into eight, then shades one part of each. Ask which shaded piece is bigger.

Nobody needs convincing after that. Re-take the vote and let the class watch it move.

Direct Instruction

13 minutes. Name what the strips showed, in the denominator’s own terms: the bottom number says how many pieces the whole was cut into, so a bigger bottom number means more cuts and smaller pieces. Write that on the board and leave it there.

Then the condition, immediately, before the rule hardens. Hold up the small circle and the large circle. Shade half the small one and a quarter of the large one, and ask which is more.

The rule the class just learned says one half beats one quarter, and the paper in your hands says otherwise. Resolve it out loud: comparing fractions only works when both are parts of the same whole. Every comparison in this lesson and on the worksheet assumes that, and it is worth saying rather than assuming.

Now teach the three strategies, giving each one an example and a reason:

• Same denominator — compare the tops. With 3/7 and 6/7 the pieces are the same size, so whoever has more pieces has more. This is the easy one and students rarely miss it.
• Same numerator — compare the bottoms, and the smaller bottom wins. With 3/4 and 3/8 you have three pieces either way, but fourths are bigger than eighths. This is the strip demonstration again, one step up — and it is worth folding, since a strip folds into fourths and eighths easily.
• Neither one matches — compare each to one half. Is 7/12 more or less than half? Half of twelve is six, and seven is more than six, so 7/12 is over a half. Any fraction over a half beats any fraction under it.

Put the half-test on the board as something students can actually run: double the top and compare it to the bottom. More than the bottom means over a half, less means under, equal means exactly a half.

Then place four or five fractions on the number line together, so the strategies produce a picture rather than three separate tricks. A student who can see 7/12 sitting just right of the middle has something to fall back on when they forget which rule applies.

Say what the third strategy does not do. Comparing to one half settles it only when the two fractions land on opposite sides. If both are over a half, it tells you nothing and you need another approach — usually rewriting them with a common denominator, which is the next lesson rather than this one.

Guided Practice

12 minutes. Partners work Sections A, B, and C of the worksheet, which are built around the three strategies in the order they were taught.

• Require students to name the strategy before giving an answer: same bottoms, so compare the tops. Naming it is what turns three tricks into a decision, and it is also how you hear which students are guessing — two options means a coin flip is right half the time.
• Section C is the benchmark section and the one to slow down on. Do the first two items with the whole class and use the doubling test out loud each time.
• Circulate and watch for the warm-up error returning. It comes back the moment the paper strips are out of sight, especially on unit fractions like 1/5 against 1/9, and a student who has ‘learned’ the rule without believing it will revert under any pressure. Send those students back to their own strips rather than repeating the rule at them.
• Add each new fraction to the number line on the board as pairs finish it, so the picture grows through the section.

Independent Practice

9 minutes. Students work Section D independently — mixed comparisons where the strategy is not announced and has to be chosen.

That choosing is the real skill and the reason not to assign the sheet in order and stop when time runs out. Sections A to C each hand the strategy over by grouping; Section D is the first place a student decides for themselves.

Timing note. The worksheet carries 34 items across six sections, which is more than one lesson. Sections E (ordering three fractions) and F (word problems) are the natural homework or next-session material — do not try to fit all six, and do not skip Section D to reach them.

Assessment

3 minutes. Exit ticket, three items. (1) Which is bigger, 1/4 or 1/10? Explain how you know. (2) Which is bigger, 5/8 or 3/8? (3) Nina says 1/8 of a pan of brownies is more than 1/6 of the same pan, because eight is bigger than six. What would you tell her? Then: would your answer change if the two pieces came from different-sized pans?

Item 1 is the lesson’s own misconception in fresh numbers — a correct answer with the reason because ten is bigger is a wrong answer that happened to land right, and the explanation is what catches it. Item 3 asks a student to diagnose someone else’s reasoning, which is a level above producing their own. Its second half is the only place the same-whole condition is checked, and it is the one to read closely — a student who says the answer would change has kept hold of the caveat that the rest of the lesson quietly assumes.

Closure

2 minutes. Hold up the two folded strips one last time and ask why the one folded into eight has smaller pieces. Listen for an answer about the whole being cut into more parts — not for a restated rule about bigger bottoms, which a student can recite while still voting for 1/8 next week.