Fraction Classroom · 2.0.0 · Teacher answers & responses
Benchmark Sort: teaching companion
Use one half as a benchmark, and know when that benchmark cannot finish a comparison. Student pages include two diagnostic probes, a reteaching check, and the original exit ticket. The teacher copy includes the complete lesson preparation, teaching sequence, diagnostic responses, all original question guides, and updated rubrics.
Complete lesson preparation
Grade 4 · 45 minutes. Use one half as a benchmark to classify and compare fractions.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Draw columns below, equal to, and above one half.
activity-2. Place fraction cards by comparing twice the numerator with the denominator.
activity-3. Partners explain a tricky card with a model rather than a decimal approximation.
phase-1. 5 min · warmup: Look at the first model or task. What is the whole? What do you notice?
Answer: Students: Sketch an initial prediction and explain it to a partner.
phase-2. 8 min · model: Draw columns below, equal to, and above one half.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Place fraction cards by comparing twice the numerator with the denominator.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners explain a tricky card with a model rather than a decimal approximation.
Answer: Students: Compare two solutions and explain the relationship.
phase-5. 10 min · practice: Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Answer: Students: Solve independently, then compare the reasoning.
phase-6. 5 min · exit: Compare 7/12 and 1/2. Choose <, =, or >. The wholes are the same size. How does twice the numerator help?
Answer: Students: Complete the two exit questions independently.
Diagnostic probes
Explain or draw your reasoning. Your teacher will use these responses to choose what to revisit.
d1. Without dividing, decide whether 3/7 is below or above one half.
Answer: Below one half: two copies give 6/7, still below one whole.
Teacher response: If the learner rounds half of seven to three and calls 3/7 exactly one half. Pair two copies of the three selected sevenths; notice that six pieces still leave one seventh unfilled.
d2. Both 3/4 and 5/6 exceed one half. Does that fact tell which is larger?
Answer: No. More evidence is needed; 9/12 < 10/12 shows 3/4 < 5/6.
Teacher response: If the learner orders them using only their shared “above half” category. Place both in the above-half region, then repartition into twelfths to locate their distinct endpoints.
Check after reteaching
Complete this check independently after discussing the model.
r1. Where does 5/11 lie relative to one half? Explain with twice the numerator.
Answer: Below half, because 2 × 5 = 10 is less than 11.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Compare 7/12 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
e2. How does twice the numerator help?
Answer: Compare two copies of the fraction with one whole; twice the numerator versus the denominator gives the relation.
Look for:
- Relate doubling the numerator to two copies of the fraction.
- Compare that doubled count with the denominator representing one whole.
- Explain the below/equal/above-half decisions, including 14 > 12 for 7/12.
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Focused reteach
Use one half as a benchmark, and know when that benchmark cannot finish a comparison.
teach-1. For 7/12, mark six twelfths as half and compare seven twelfths with it.
teach-2. For 3/7, compare two selected copies with all seven pieces instead of rounding half of seven.
teach-3. Sort examples below, at, and above half, then identify a pair needing more evidence to order.
Use the exit evidence
A wrong answer invites a follow-up probe; it does not by itself establish a particular misconception. Check e1 and e2 separately.
next-1. e1 meets its criterion: e1 gives > for 7/12 compared with 1/2. e2 also meets its criterion: e2 explains that comparing twice the numerator with the denominator compares two copies of the fraction with one whole.
Teacher response: Continue to the linked next lesson. Choose common units or another argument when fractions lie on the same side of one half. Next original lesson: /lessons/compare-and-defend/.
next-2. e1 does not yet meet this criterion: e1 gives > for 7/12 compared with 1/2.
Teacher response: Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on. Short intervention: /interventions/benchmark-comparison/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that comparing twice the numerator with the denominator compares two copies of the fraction with one whole.
Teacher response: Use two seven-twelfth selections and ask the learner to connect 14 > 12 with the original one-half comparison, not just recite “double it.” Short intervention: /interventions/benchmark-comparison/.
Original question guide and updated rubrics
Use these question-specific teaching responses alongside the original lesson. Accept other mathematically valid open responses that meet the criteria. The original question wording and mathematical answers are retained; model descriptions below make their references readable offline.
c1. Compare 7/12 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c2. Compare 3/8 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: <
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c3. Compare 4/8 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: =
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c4. Compare 5/6 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c5. Compare 2/5 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: <
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c6. Compare 6/10 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c7. Compare 3/7 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: <
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
c8. Compare 5/10 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: =
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
s1. Compare 1/4 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: <
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
s2. Compare 2/4 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: =
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
s3. Compare 3/4 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
s4. Compare 1/3 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: <
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
h1. Explain where 7/12 belongs without dividing.
Answer: Half of twelve is six; seven twelfths is above one half.
Look for:
- Identify 6/12 as one half.
- Compare seven twelfths with six twelfths without decimal division.
- Place 7/12 above one half.
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
h2. Name two fractions below half with different denominators.
Answer: 1/3 and 3/8 are both below half.
Look for:
- Name two fractions with different denominators.
- Make both fractions strictly below one half.
- Verify each using a model or 2 × numerator < denominator.
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
h3. Can the benchmark alone compare 3/4 and 5/6?
Answer: It shows both exceed half, but more reasoning is needed; 9/12 is below 10/12.
Look for:
- State that being above one half does not by itself order 3/4 and 5/6.
- Supply additional comparison evidence, such as 9/12 and 10/12.
- Conclude 3/4 < 5/6.
Teacher response: Compares unlike fractions using only the numerator, denominator, or an insufficient shared benchmark. Rename 2/3 and 3/4 as 8/12 and 9/12 on identical wholes. Compare the counts of twelfths. If both fractions lie above one half, ask for this additional evidence before ordering them.
h4. Invent a fraction between one half and one.
Answer: 3/5 is greater than half and less than one.
Look for:
- Choose a fraction strictly greater than 1/2.
- Verify that the same fraction is strictly less than 1.
- Justify both bounds with equal parts, a line, or numerical inequalities.
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
e1. Compare 7/12 and 1/2. Choose <, =, or >. The wholes are the same size.
Answer: >
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
e2. How does twice the numerator help?
Answer: Compare two copies of the fraction with one whole; twice the numerator versus the denominator gives the relation.
Look for:
- Relate doubling the numerator to two copies of the fraction.
- Compare that doubled count with the denominator representing one whole.
- Explain the below/equal/above-half decisions, including 14 > 12 for 7/12.
Teacher response: Judges position relative to one half from numerator or denominator alone. For a/b, compare two copies of a selected parts with all b parts. If 2a is less than, equal to, or greater than b, the original fraction is below, at, or above one half. Test 3/7 by comparing 6 with 7.
Teaching notes
- Teaching supplement version 2.0.0. Original lesson questions and PDF materials remain version 1.0.0. Original PDFs contain their original teaching notes and rubrics; use this supplement for the updated diagnostic guidance.
- Short intervention: /interventions/benchmark-comparison/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/compare-and-defend/. Choose common units or another argument when fractions lie on the same side of one half.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.