Use a benchmark, then justify

d1. Is 5/12 below or above 1/2? Name one half in twelfths.

Answer: Below. 1/2 = 6/12 and 5/12 < 6/12.

Look for:

• Names 6/12 as one half.
• Places 5/12 below it.

Teacher response: Likely difficulty: Comparing numerator and denominator without a benchmark quantity. Response: Split twelve equal parts into two groups of six.

d2. Compare 3/8 and 5/6 using one half.

Answer: 3/8 < 5/6 because 3/8 < 1/2 and 5/6 > 1/2.

Look for:

• Locates both fractions relative to half.
• Uses the separating benchmark to compare.

Teacher response: Likely difficulty: Seeking a common denominator before considering useful landmarks. Response: Show 4/8 and 3/6 as the half marks on equal wholes.

d3. Both 2/3 and 3/4 are above 1/2. Does that fact alone tell which is larger?

Answer: No. Use distance from 1 or equivalent fractions; 2/3 < 3/4.

Look for:

• Recognizes the half benchmark is insufficient.
• Gives a justified comparison by another method.

Teacher response: Likely difficulty: Treating a shared benchmark region as a complete comparison. Response: Place both in the right half of a line and ask whether their order is yet known.

w1. Compare 2/5 and 7/12 using one half.

1. Two groups of 2 fifths make 4/5, less than a whole, so 2/5 is below half.
2. Half of twelve equal parts is six parts; 7/12 exceeds 6/12.
3. One fraction is below 1/2 and the other above it.
4. Place 2/5 to the left of 7/12.

Worked example: 2/5 < 7/12.

Look for:

• Justifies the position of each fraction.
• Uses half as a separating benchmark.

Teacher response: Likely difficulty: Assuming an odd denominator prevents using one half. Response: Double the selected count to check whether it fills the whole, without needing a whole-number half of 5.

w2. Compare 5/6 and 7/8 by thinking about what is missing from 1.

1. Both fractions are one equal part short of a whole.
2. Write the missing amounts: 1/6 and 1/8.
3. A sixth of the same whole is larger than an eighth.
4. Removing the larger gap leaves the smaller fraction.

Worked example: 5/6 < 7/8 because 1/6 is a larger missing amount than 1/8.

Look for:

• Compares the gaps correctly.
• Reverses gap size to compare the remaining fractions.

Teacher response: Likely difficulty: Ordering the missing pieces in the same direction as the remaining amounts. Response: Draw both fractions ending at 1 and compare the unfilled gaps.

p1. Sort 2/5, 4/8, and 5/6 as below, equal to, or above 1/2.

Answer: 2/5 is below; 4/8 is equal; 5/6 is above.

Look for:

• Places all three correctly.
• Explains at least one placement with equal parts or doubling.

Teacher response: Likely difficulty: Using numerator size alone across different denominators. Response: Check whether doubling each numerator is less than, equal to, or greater than its denominator.

p2. Compare 3/10 and 3/5 using one half.

Answer: 3/10 < 3/5, since 3/10 < 5/10 and two copies of 3/5 exceed 1.

Look for:

• Locates each on opposite sides of half.
• Writes the correct comparison.

Teacher response: Likely difficulty: Treating 10 as evidence that 3/10 is larger. Response: Compare each fraction with its own half benchmark before comparing them.

p3. Which is closer to 1: 4/5 or 5/6? Explain with missing parts.

Answer: 5/6 is closer because its missing 1/6 is smaller than 1/5.

Look for:

• Chooses 5/6.
• Compares the gaps 1/5 and 1/6.

Teacher response: Likely difficulty: Confusing the larger missing fraction with being closer. Response: Complete both wholes with the missing piece and compare its size.

p4. A student says 1/3 = 2/5 because both are below 1/2. Explain the mistake and compare them.

Answer: Being below half does not mean being equal. 1/3 = 2/6 < 2/5 because sixths are smaller than fifths of the same whole.

Look for:

• Rejects the equality.
• Uses an accurate drawing or equivalent fractions to show 1/3 < 2/5.

Teacher response: Likely difficulty: Treating benchmark categories as exact values. Response: Draw two different points in the left half of a number line.

p5. Choose a benchmark of 0, 1/2, or 1 that separates 7/8 and 9/8. Write a comparison.

Answer: 1 separates them: 7/8 < 1 < 9/8.

Look for:

• Selects 1.
• Correctly orders both fractions around it.

Teacher response: Likely difficulty: Assuming comparison benchmarks stop at proper fractions. Response: Mark 8/8 under 1 and locate each neighboring fraction.

p6. Give one fraction below 1/2 and one above 1/2, both with denominator 5. Explain your choices.

Answer: For example, 2/5 is below and 3/5 is above because twice 2 is less than 5 and twice 3 is greater than 5.

Look for:

• Provides valid fifths on opposite sides of half.
• Justifies both; other valid choices count.

Teacher response: Likely difficulty: Expecting one half to have an integer numerator over every denominator. Response: Use two copies of each selected fraction to compare with one whole.

t1. A bottle is 5/12 full and another same-sized bottle is 3/5 full. Which has more liquid? Use a benchmark and state the needed whole-size condition.

Answer: The 3/5 bottle has more. 5/12 is below half and 3/5 is above half; the bottles must have equal capacities for this amount comparison.

Look for:

• Uses half correctly.
• States that equal capacities make the amount comparison valid.

Teacher response: Likely difficulty: Applying fraction order to amounts without checking the wholes. Response: Mark the halfway fill on both identical bottle outlines.

t2. Compare 5/6 and 3/4. Explain why “both are above half” is incomplete, then finish using a suitable method.

Answer: 5/6 > 3/4. Both are above half, but 5/6 = 10/12 and 3/4 = 9/12.

Look for:

• Explains the limitation of the half benchmark.
• Completes a correct common-denominator or distance comparison.

Teacher response: Likely difficulty: Stopping after an inconclusive benchmark check. Response: Ask for a finer scale within the upper half of the unit.