Compare fractions using equal wholes and useful names

Learning goal: Compare fractions with different numerators and denominators, record the relationship with a symbol and justify it using models or benchmarks. Before comparing shaded pictures, identify the whole in each. Matching outer boundaries let a strip model compare fraction values directly. A half of a large sheet can contain more paper than three fourths of a small sheet, even though 1/2 < 3/4 as numbers.

To compare 2/3 and 3/4 of the same unit, rename both in twelfths. Split each third into four equal parts: 2/3 = 8/12. Split each fourth into three: 3/4 = 9/12. Now both amounts count equal-sized pieces. Eight twelfths is less than nine twelfths, so 2/3 < 3/4. Draw aligned strips to show the one-twelfth difference.

A common denominator supplies a common piece size. Once that size matches, compare the numerators. Comparing two and three in the original fractions without considering thirds and fourths would not be a sufficient reason. Sometimes the larger numerator belongs to the smaller fraction, as in 4/10 < 3/4.

A common numerator offers another strategy. Compare 2/3 and 4/8 by renaming 2/3 as 4/6. Both fractions now count four pieces. A sixth is larger than an eighth of the same whole, so four sixths are larger than four eighths: 2/3 > 4/8. This reverses a common mistaken rule that a larger denominator must mean a larger fraction.

The benchmark one half can make a comparison quick and meaningful. Compare 3/8 and 2/3. Half of eight eighths is four eighths, so 3/8 is below one half. Rename 2/3 as 4/6; half of six sixths is three sixths, so 2/3 is above one half. Therefore 3/8 < 2/3. The fractions lie on opposite sides of the same benchmark.

A benchmark does not always settle the question. Both 3/4 and 5/6 exceed one half. That fact alone does not tell which is greater. Rename them as 9/12 and 10/12, or compare their distances from one: a fourth is more missing than a sixth. Either way, 3/4 < 5/6. Explain why the chosen method resolves this particular pair.

Use the equality symbol when different names describe the same amount. Comparing 3/5 and 6/10 gives 3/5 = 6/10, because splitting every fifth in two leaves the shaded length unchanged. The symbols < and > describe order; the open side faces the greater amount. Read the completed sentence aloud to check its direction.

A number line also uses one consistent whole: the distance from zero to one. Locate both fractions using that same distance. The point farther right represents the greater number. Do not stretch one fraction's unit interval to make its placement easier; doing so would invalidate the comparison.

Worked decision: Which is greater, 5/8 or 2/4? Rename 2/4 as 4/8. Five eighths exceeds four eighths by one eighth, so 5/8 > 2/4. Both fractions describe the same-sized unit; the comparison is supported by counting equal eighths.

Worked critique: Jo says 3/8 > 2/3 because three is greater than two. Draw both on equal strips or use the half benchmark. Three eighths is below half and two thirds is above half, so Jo's conclusion is reversed. The numerators count different-sized pieces.

Guided practice: Sort fraction pairs by a useful strategy: common denominator, common numerator or one-half benchmark. Solve each pair and explain why the method works. Use one pair on the same side of half so learners must notice the benchmark's limitation.

Independent evidence: Compare 2/3 with 3/4, 3/8 with 2/3 and 3/5 with 6/10. Supply symbols and evidence. Finally explain why drawings of different-sized pizzas cannot directly justify a comparison of portions as physical amounts. Accept more than one valid method, but require the whole and piece sizes to be clear.