Estimate Before Exact: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Predict a range for 1/3 + 1/4 before doing exact work.

activity-2. Compare the proposed answer 2/7 with each addend and reject it.

activity-3. Partners estimate, solve and check an addition and a subtraction problem.

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: Predict a range for 1/3 + 1/4 before doing exact work.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Compare the proposed answer 2/7 with each addend and reject it.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners estimate, solve and check an addition and a subtraction problem.

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 1/3 + 1/4. Show a model or your reasoning. Give one reason to reject a positive sum smaller than an addend.

Answer: Students: Complete the two exit questions independently.

d1. Could 1/3 + 1/4 equal 2/7? Reject or accept it using size before finding the exact sum.

Answer: Reject it: 2/7 < 1/3, but adding positive 1/4 must increase 1/3.

Teacher response: If the learner checks only whether the numerator and denominator arithmetic looks familiar. Place the addend and proposed sum on a twenty-firsts line, then point in the direction a positive addition must move.

d2. Predict whether 5/6 + 7/8 is below or above two, and whether it is near two.

Answer: It is below and near two, since both addends are close to but below one.

Teacher response: If the learner rounds both addends to one and calls two the exact answer. Mark the missing 1/6 and 1/8 from two wholes to distinguish an estimate from the exact total.

r1. A learner says 3/4 + 2/3 = 1/5. Give a fast reason to reject it.

Answer: A positive sum must exceed 3/4, but 1/5 is smaller; both addends also exceed half, so their sum exceeds one.

e1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

e2. Give one reason to reject a positive sum smaller than an addend.

Answer: Adding a positive amount increases the starting value.

Look for:

• Specify that the added amount is positive.
• Explain that adding it moves above the starting value.
• Use this to reject a proposed sum below either positive addend, including the e1 quantities.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

teach-1. State the operation’s direction and choose a benchmark appropriate to the quantities.

teach-2. Record a justified bound or estimate before finding common units.

teach-3. Compute exactly and compare the result with the bound; if it conflicts, inspect the equivalent fractions and operation.

next-1. e1 meets its criterion: e1 gives 7/12 for 1/3 + 1/4. e2 also meets its criterion: e2 explains that adding a positive amount makes the result greater than either positive addend.

Teacher response: Continue to the linked next lesson. Use estimates and exact checks to test design quantities against material constraints. Next original lesson: /lessons/fraction-design-challenge/.

next-2. e1 does not yet meet this criterion: e1 gives 7/12 for 1/3 + 1/4.

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/unlike-estimation/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that adding a positive amount makes the result greater than either positive addend.

Teacher response: Ask the learner to compare the proposed 2/7 with 1/3 and describe the direction of positive addition, then return to the exact 7/12 result. Short intervention: /interventions/unlike-estimation/.

c1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c2. Calculate 5/6 + 7/8. Show a model or your reasoning.

Answer: 41/24

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c3. Calculate 2/5 + 3/8. Show a model or your reasoning.

Answer: 31/40

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c4. Calculate 3/4 + 2/3. Show a model or your reasoning.

Answer: 17/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c5. Calculate 7/8 - 1/3. Show a model or your reasoning.

Answer: 13/24

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c6. Calculate 5/6 - 1/2. Show a model or your reasoning.

Answer: 1/3

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c7. Calculate 9/10 - 2/5. Show a model or your reasoning.

Answer: 1/2

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c8. Calculate 4/5 - 3/8. Show a model or your reasoning.

Answer: 17/40

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

s1. Calculate 1/2 + 1/4. Show a model or your reasoning.

Answer: 3/4

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

s2. Calculate 1/3 + 1/3. Show a model or your reasoning.

Answer: 2/3

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.

s3. Calculate 3/4 - 1/4. Show a model or your reasoning.

Answer: 1/2

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.

s4. Calculate 2/3 - 1/3. Show a model or your reasoning.

Answer: 1/3

Teacher response: Adds, subtracts, or multiplies a denominator when combining counts of the same fractional unit. Place three eighth-size pieces beside five eighth-size pieces. Count eight pieces without recutting any of them. Write 8/8 and explain why 8/16 would describe smaller pieces than the ones on the table.

h1. Why is 2/7 unreasonable for 1/3 + 1/4?

Answer: A sum of positive values must exceed either addend, but 2/7 is below 1/3.

Look for:

• State that adding a positive 1/4 must make the result exceed 1/3.
• Show 2/7 < 1/3, using 6/21 < 7/21 or another valid comparison.
• Reject 2/7 on this size contradiction without relying only on a memorized addition rule.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

h2. Estimate 5/6 + 7/8 using whole benchmarks.

Answer: Both are close to one, so the sum is close to but below two.

Look for:

• Identify both 5/6 and 7/8 as close to but below one.
• Predict a sum close to but below two.
• Explain why two is an upper benchmark rather than the exact sum.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

h3. Estimate 7/8 - 1/3 using half as a benchmark.

Answer: It should be near one half; the exact result is 13/24.

Look for:

• Use one half as the target benchmark for the difference.
• Give a reason the difference is near or slightly above one half.
• Check consistency with 13/24, or use 7/8 − 1/2 = 3/8 > 1/3 to prove it is above half.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

h4. Create an incorrect answer that estimation can reject immediately.

Answer: 3/4 + 2/3 = 1/5 is impossible because the sum must exceed 3/4.

Look for:

• Invent an actual incorrect fraction-operation result.
• State a size bound or direction that a correct answer must satisfy.
• Demonstrate the proposed answer violates that bound, such as a positive sum below an addend.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.

e1. Calculate 1/3 + 1/4. Show a model or your reasoning.

Answer: 7/12

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

e2. Give one reason to reject a positive sum smaller than an addend.

Answer: Adding a positive amount increases the starting value.

Look for:

• Specify that the added amount is positive.
• Explain that adding it moves above the starting value.
• Use this to reject a proposed sum below either positive addend, including the e1 quantities.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound.