Find missing numerators or denominators by using a multiplicative scale factor.
Teaching companion · Version 2.0.0
Find and apply the same multiplicative factor in both directions of an equivalence.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
Complete 3/5 = ?/10 and explain the two arrows.
Look for: The missing numerator is 6; both counts multiply by two.
If you observe: Writes 8 by adding five to the numerator as well as the denominator.
Respond: Draw ×2 beside the 5-to-10 arrow, then apply the same action to three shaded fifths.
Complete 6/9 = 2/? and explain why the denominator decreases.
Look for: The missing denominator is 3; divide both counts by three.
If you observe: Multiplies nine by three while dividing six by three.
Respond: Group the nine equal pieces into threes, with two of the three groups selected; label ÷3 on both arrows.
Check independently: Complete 2/3 = 8/? and justify the missing number.
12, because both numerator and denominator multiply by four.
Use the equivalent factors short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
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.
Continue to the linked next lesson. Use equivalent names for one half to support benchmark comparisons.
Next lesson: Benchmark Sorte1 does not yet meet this criterion: e1 supplies 6 in 3/5 = ?/10.
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.
Open the linked short interventione1 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.
Have the learner show fifths becoming tenths on a strip and use that same drawing to explain why two matching arrows are necessary.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Look for in the response:
Watch for: Assumes every desired denominator can represent a fraction with an integer numerator.
Try: 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.
Look for in the response:
Watch for: Adds the same number to numerator and denominator and claims the fraction stays equal.
Try: 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.
Look for in the response:
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Look for in the response:
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Watch for: Changes one part of a fraction without scaling numerator and denominator by the same factor.
Try: 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.
Look for in the response:
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Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Write 3/5 on one card and ?/10 on another. Bridge the denominators with a times-two arrow.
Apply the same arrow to the numerators. Check with a strip.
Partners build bridges in both directions and explain why adding is not the rule.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Complete 3/5 = ?/10. Why must the same factor appear on both arrows?
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.
Teacher answer key. Open responses include a rubric and one valid sample.
The denominator doubled, so the numerator doubles.
Multiply both parts by four.
Multiply by three.
Multiply by two.
Multiply by five.
Divide both parts by four.
Multiply by three.
Divide both parts by three.
One times two is two.
One times two is two.
Four divided by two is two.
One times two is two.
6/10 and 9/15 use factors two and three.
No; 3/5 times 8 is 24/5, which is not an integer.
Double both parts to get 4/6; adding two does not preserve value.
2/3 = 8/12 multiplies numerator and denominator by four.
Both numerator and denominator are doubled.
Multiplying numerator and denominator by the same nonzero factor preserves the amount.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 4. Find missing numerators or denominators by using a multiplicative scale factor.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.