Fraction Classroom · 2.0.0 · Teacher answers & responses
Scaling Stories: teaching companion
Compare the product with a positive starting quantity by comparing its factor with one. 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 5 · 45 minutes. Predict how a multiplier below, equal to or above one changes a quantity.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Start with a 3/4 meter ribbon. Predict the result of scaling by 1/2, 1 and 2.
activity-2. Model each product and compare it with the original length.
activity-3. Partners write a rule based on the multiplier, then test a new ribbon length.
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: Start with a 3/4 meter ribbon. Predict the result of scaling by 1/2, 1 and 2.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Model each product and compare it with the original length.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners write a rule based on the multiplier, then test a new ribbon length.
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: Calculate 3/4 × 1/2. Show a model or your reasoning. State the scaling rule for positive amounts.
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. Before computing, predict whether 3/4 × 1/2 is below or above 3/4.
Answer: Below 3/4, because it takes half of that positive amount; the product is 3/8.
Teacher response: If the learner predicts a larger result because the operation is multiplication. Fold a 3/4-length segment in half and compare the folded share with the full starting segment.
d2. Predict how multiplying 7/8 by 1 and by 4/3 changes the starting amount.
Answer: A factor of 1 preserves 7/8; a factor of 4/3 makes it larger.
Teacher response: If the learner uses the numerator or denominator separately rather than comparing the whole factor with one. Represent 4/3 as one full copy plus another third-copy of the starting 7/8.
Check after reteaching
Complete this check independently after discussing the model.
r1. Without computing the exact product, compare 5/6 × 2/5 with 5/6.
Answer: The product is smaller because the positive factor 2/5 is below one.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
Answer: 3/8
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
e2. State the scaling rule for positive amounts.
Answer: A factor below one shrinks, one preserves, and above one enlarges the amount.
Look for:
- For a positive starting amount, state that a positive factor below one makes it smaller.
- State that a factor of one preserves it.
- State that a factor greater than one enlarges it; connect the half-factor case to e1.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
Focused reteach
Compare the product with a positive starting quantity by comparing its factor with one.
teach-1. Fix a positive starting strip and make half a copy, one copy, and two copies.
teach-2. Record whether each result is smaller, equal, or larger before calculating.
teach-3. Test a factor such as 4/3 by decomposing it into one copy plus a fraction of another.
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 3/8 for 3/4 × 1/2. e2 also meets its criterion: e2 states the positive-factor cases below one, equal to one, and above one.
Teacher response: Continue to the linked next lesson. Use size predictions when a fractional amount is divided into equal smaller shares. Next original lesson: /lessons/unit-fraction-sharing/.
next-2. e1 does not yet meet this criterion: e1 gives 3/8 for 3/4 × 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/scaling/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 states the positive-factor cases below one, equal to one, and above one.
Teacher response: Use three sketches with the same starting strip and factors 1/2, 1, and 2. Ask the learner to state all three e2 cases and connect the first to e1. Short intervention: /interventions/scaling/.
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. Calculate 3/4 × 1/2. Show a model or your reasoning.
Answer: 3/8
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c2. Calculate 3/4 × 1. Show a model or your reasoning.
Answer: 3/4
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c3. Calculate 3/4 × 2. Show a model or your reasoning.
Answer: 3/2
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c4. Calculate 2/3 × 3/4. Show a model or your reasoning.
Answer: 1/2
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c5. Calculate 2/3 × 3/2. Show a model or your reasoning.
Answer: 1
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c6. Calculate 5/6 × 2/5. Show a model or your reasoning.
Answer: 1/3
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c7. Calculate 4/5 × 5/4. Show a model or your reasoning.
Answer: 1
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
c8. Calculate 7/8 × 4/3. Show a model or your reasoning.
Answer: 7/6
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
s1. Calculate 1/2 × 1/2. Show a model or your reasoning.
Answer: 1/4
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
s2. Calculate 1/2 × 1. Show a model or your reasoning.
Answer: 1/2
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
s3. Calculate 1/2 × 2. Show a model or your reasoning.
Answer: 1
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
s4. Calculate 3/4 × 1/2. Show a model or your reasoning.
Answer: 3/8
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
h1. Why is 3/4 × 1/2 smaller than 3/4?
Answer: Taking half of a positive amount makes it smaller.
Look for:
- Interpret the factor 1/2 as taking half of the positive amount 3/4.
- Show or state that the result is 3/8.
- Explain that half of a positive amount is smaller than the starting amount.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
h2. Give a counterexample to multiplication always makes numbers bigger.
Answer: 1/2 × 1/2 = 1/4, smaller than 1/2.
Look for:
- Provide a valid multiplication example using a positive factor below one.
- Compute the product correctly.
- Compare the product with the starting positive amount and show that it is smaller.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
h3. What happens when a positive quantity is scaled by one?
Answer: Its value stays unchanged.
Look for:
- Specify a positive starting quantity.
- State that multiplying it by one preserves its value.
- Explain using one complete copy of that quantity.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
h4. Predict without computing whether 7/8 × 4/3 exceeds 7/8.
Answer: It exceeds it because the multiplier 4/3 is above one.
Look for:
- Identify the scale factor 4/3 as greater than one.
- Predict the product exceeds the starting 7/8.
- Justify the prediction from scaling rather than requiring the exact product.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
e1. Calculate 3/4 × 1/2. Show a model or your reasoning.
Answer: 3/8
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
e2. State the scaling rule for positive amounts.
Answer: A factor below one shrinks, one preserves, and above one enlarges the amount.
Look for:
- For a positive starting amount, state that a positive factor below one makes it smaller.
- State that a factor of one preserves it.
- State that a factor greater than one enlarges it; connect the half-factor case to e1.
Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.
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/scaling/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/unit-fraction-sharing/. Use size predictions when a fractional amount is divided into equal smaller shares.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.