Fraction Classroom · 2.0.0 · Teacher answers & responses
Missing Number Bridges: teaching companion
Find and apply the same multiplicative factor in both directions of an equivalence. 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. Find missing numerators or denominators by using a multiplicative scale factor.
materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).
activity-1. Write 3/5 on one card and ?/10 on another. Bridge the denominators with a times-two arrow.
activity-2. Apply the same arrow to the numerators. Check with a strip.
activity-3. Partners build bridges in both directions and explain why adding is not the rule.
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: Write 3/5 on one card and ?/10 on another. Bridge the denominators with a times-two arrow.
Answer: Students: Follow the model; label the whole and the equal parts.
phase-3. 10 min · partner: Apply the same arrow to the numerators. Check with a strip.
Answer: Students: Work in pairs. One models, the other checks; switch roles.
phase-4. 7 min · discussion: Partners build bridges in both directions and explain why adding is not the rule.
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: Complete 3/5 = ?/10. Why must the same factor appear on both arrows?
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. Complete 3/5 = ?/10 and explain the two arrows.
Answer: The missing numerator is 6; both counts multiply by two.
Teacher response: If the learner writes 8 by adding five to the numerator as well as the denominator. Draw ×2 beside the 5-to-10 arrow, then apply the same action to three shaded fifths.
d2. Complete 6/9 = 2/? and explain why the denominator decreases.
Answer: The missing denominator is 3; divide both counts by three.
Teacher response: If the learner multiplies nine by three while dividing six by three. Group the nine equal pieces into threes, with two of the three groups selected; label ÷3 on both arrows.
Check after reteaching
Complete this check independently after discussing the model.
r1. Complete 2/3 = 8/? and justify the missing number.
Answer: 12, because both numerator and denominator multiply by four.
Exit ticket
Complete both original exit questions independently. Use a drawing or explanation when asked.
e1. Complete 3/5 = ?/10.
Answer: 6
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
e2. Why must the same factor appear on both arrows?
Answer: Multiplying numerator and denominator by the same nonzero factor preserves the amount.
Look for:
- State that numerator and denominator must be scaled by the same nonzero factor.
- Explain that the number and size of parts change together while the amount stays fixed.
- Connect to e1: both 3 and 5 double, producing 6/10.
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
Focused reteach
Find and apply the same multiplicative factor in both directions of an equivalence.
teach-1. Identify the known numerator or denominator pair and state its factor.
teach-2. Use that same multiplication or division on the missing pair.
teach-3. Check by drawing equal endpoints or applying the reverse factor; contrast 2/3 = 4/6 with 4/5.
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 supplies 6 in 3/5 = ?/10. e2 also meets its criterion: e2 explains that the same nonzero factor on both counts preserves the fraction’s amount.
Teacher response: Continue to the linked next lesson. Use equivalent names for one half to support benchmark comparisons. Next original lesson: /lessons/benchmark-sort/.
next-2. e1 does not yet meet this criterion: e1 supplies 6 in 3/5 = ?/10.
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/equivalent-factors/.
next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 explains that the same nonzero factor on both counts preserves the fraction’s amount.
Teacher response: Have the learner show fifths becoming tenths on a strip and use that same drawing to explain why two matching arrows are necessary. Short intervention: /interventions/equivalent-factors/.
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. 3/5 = ?/10. Enter the missing numerator.
Answer: 6
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c2. 2/3 = 8/?. Enter the missing denominator.
Answer: 12
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c3. 1/4 = ?/12. Enter the numerator.
Answer: 3
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c4. 5/6 = 10/?. Enter the denominator.
Answer: 12
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c5. 3/4 = ?/20. Enter the numerator.
Answer: 15
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c6. 4/8 = 1/?. Enter the denominator.
Answer: 2
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c7. 2/5 = ?/15. Enter the numerator.
Answer: 6
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
c8. 6/9 = 2/?. Enter the denominator.
Answer: 3
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
s1. 1/2 = ?/4. Multiply numerator by 2.
Answer: 2
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
s2. 1/3 = ?/6. Multiply numerator by 2.
Answer: 2
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
s3. 2/4 = 1/?. Divide denominator by 2.
Answer: 2
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
s4. 1/4 = ?/8. Multiply numerator by 2.
Answer: 2
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
h1. Find two possible fractions equal to 3/5 with larger denominators.
Answer: 6/10 and 9/15 use factors two and three.
Look for:
- Give two different fractions equal to 3/5 with denominators greater than five.
- Use matching numerator and denominator scale factors.
- Verify each equality, such as 3/5 = 6/10 = 9/15.
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
h2. Can 3/5 be written as a fraction with denominator 8 and an integer numerator?
Answer: No; 3/5 times 8 is 24/5, which is not an integer.
Look for:
- Answer no for an integer numerator over 8.
- Use 3 × 8 / 5 = 24/5, or 5n = 24, to test the numerator.
- Explain that 24/5 is not an integer, so no integer count of eighths is exactly 3/5.
Teacher response: Assumes every desired denominator can represent a fraction with an integer numerator. Test 3/5 = n/8 by subdividing a fixed whole into fortieths. Three fifths cover 24 fortieths, while each eighth covers five fortieths. The required count of eighths is 24/5, so no integer numerator over eight gives exactly 3/5.
h3. Repair 2/3 = 4/5 and describe the broken bridge.
Answer: Double both parts to get 4/6; adding two does not preserve value.
Look for:
- Identify 2/3 = 4/5 as false.
- Repair it to 2/3 = 4/6 or another valid equivalent relationship.
- Describe the missing matching factor: doubling 2 requires doubling 3.
Teacher response: Adds the same number to numerator and denominator and claims the fraction stays equal. Put 2/3 and 3/4 on the same twelfths strip: 8/12 and 9/12 have different endpoints. Contrast this with 2/3 = 4/6, where both counts are multiplied by two.
h4. Create a bridge where the scale factor is 4.
Answer: 2/3 = 8/12 multiplies numerator and denominator by four.
Look for:
- Write two fractions connected by a factor of four.
- Multiply both numerator and denominator by four.
- Explain that the fraction value remains unchanged, for example 2/3 = 8/12.
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
e1. Complete 3/5 = ?/10.
Answer: 6
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
e2. Why must the same factor appear on both arrows?
Answer: Multiplying numerator and denominator by the same nonzero factor preserves the amount.
Look for:
- State that numerator and denominator must be scaled by the same nonzero factor.
- Explain that the number and size of parts change together while the amount stays fixed.
- Connect to e1: both 3 and 5 double, producing 6/10.
Teacher response: Changes one part of a fraction without scaling numerator and denominator by the same factor. Shade 2/3 of a fixed strip and split each third into two equal parts. Count four shaded sixths out of six. Write ×2 on both arrows and check that the shaded endpoint did not move.
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/equivalent-factors/. Use its model for the identified difficulty, then return to this lesson’s check and exit ticket.
- Next original lesson: /lessons/benchmark-sort/. Use equivalent names for one half to support benchmark comparisons.
- Open responses are reviewed by a teacher using the specific criteria. These materials have not been field-tested in classrooms.