Compare unit fractions and same-numerator fractions with equal wholes to challenge whole-number denominator reasoning.
Listen for the idea
For a fixed whole, increasing the number of equal parts makes each part smaller.
Explain the order of unit fractions using piece size.
Apply the reasoning to same-numerator comparisons and name its limits.
Materials: Three equal-sized paper strips; Pencil.
A 12-minute route
2 minutes: Try the three diagnostic questions. Ask for a brief explanation.
3 minutes: Teach one worked example that addresses the difficulty.
5 minutes: Choose three practice tasks and discuss the reasoning.
2 minutes: Give one transfer task independently and decide the next step.
The second worked example, three unused practice tasks, and remaining transfer task are available for a follow-up session. Allow more time if the group needs to build the models.
Teaching move: Align equal wholes first. Partition them into halves, fourths, and eighths, then compare a single piece from each before using comparison symbols.
Use the answer notes to interpret what you hear. Choose a response for each question to get a suggested teaching step. These choices describe this brief check and are not a mastery score.
1 For the same whole, which is greater: 1/3 or 1/8? Explain without comparing only the numbers 3 and 8.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/3 is greater because splitting a whole into 3 equal parts makes larger pieces than splitting it into 8.
Listen for this difficulty: Treating a larger denominator as a larger fraction.
Try this response: Draw the same-sized whole twice and point to one part of each.
Look for:
Chooses 1/3.
Explains the sizes of parts of the same whole.
2 Compare 3/4 and 3/8 of the same whole. What stays the same, and what changes?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 3/4 > 3/8. The number of pieces is 3 in each, but fourths are larger than eighths.
Listen for this difficulty: Ignoring equal numerators and ordering by denominator size.
Try this response: Match each shaded fourth with one shaded eighth to compare three pairs.
Look for:
Uses the correct comparison.
Distinguishes piece count from piece size.
3 Does a larger denominator always mean a smaller fraction? Test the claim with 1/2 and 7/8.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. 7/8 > 1/2; the “larger denominator, smaller amount” shortcut needs the same positive numerator and same whole.
Listen for this difficulty: Overgeneralizing the unit-fraction rule to unlike numerators.
Try this response: Shade seven eighths and compare the total coverage with one half.
Look for:
Rejects the unrestricted claim.
Uses the different numerators to explain the exception.
2 · Model together
Two worked examples
Use one now. Ask the learner to explain each step before revealing the next part of your drawing.
1 Order 1/2, 1/4, and 1/8 from least to greatest.
Start with three identical whole strips.
Partition them into 2, 4, and 8 equal parts.
Compare one part from each strip.
The eighth is smallest and the half is largest.
Result: 1/8 < 1/4 < 1/2.
Teaching notes
Listen for this difficulty: Applying whole-number order directly to denominators.
Try this response: Ask how many copies of each piece would fill the same whole.
Look for:
Uses equal-sized wholes.
Orders all three fractions by part size.
2 Compare 2/3 and 2/6 without finding a common denominator.
Use two identical whole strips.
A third is larger than a sixth of the same whole.
Take two of each type of piece.
Two larger pieces cover more than two smaller pieces.
Result: 2/3 > 2/6.
Teaching notes
Listen for this difficulty: Assuming sixths are larger because 6 exceeds 3.
Try this response: Pair one third with one sixth twice rather than comparing the symbols alone.
Look for:
Compares the unit fractions correctly.
Uses the equal numerator 2 to compare totals.
3 · Guided practice
Choose three tasks
Mix a calculation or drawing with an explanation or error repair. Keep the other tasks for another short session.
1 Compare 1/4 and 1/6 of the same whole using <, >, or =. Explain with sharing.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/4 > 1/6. Four equal shares of a fixed amount are larger than six equal shares.
Listen for this difficulty: Believing more recipients means larger individual shares.
Try this response: Imagine sharing one identical strip between four people, then six.
Look for:
Writes 1/4 > 1/6.
Keeps the total amount fixed in the explanation.
2 Put 1/3, 1/8, and 1/4 in increasing order.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 1/8 < 1/4 < 1/3.
Listen for this difficulty: Sorting denominator numbers from least to greatest.
Try this response: Compare the sizes of single pieces on equal strips.
Look for:
Orders all three correctly.
Can explain why the eighth is smallest.
3 Which covers more of one whole: two sixths or two eighths? Explain.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Two sixths, because a sixth is larger than an eighth and both collections contain two pieces.
Listen for this difficulty: Ignoring unit-fraction size when the count matches.
Try this response: Compare one pair of pieces, then duplicate that comparison.
Look for:
Chooses 2/6.
Uses both equal count and different part size.
4 Choose a denominator from 3, 4, or 8 to make 1/__ smaller than 1/6.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 8.
Listen for this difficulty: Choosing the smaller denominator to make the smaller amount.
Try this response: Ask which partition creates more than six equal pieces in one whole.
Look for:
Selects 8.
Explains why eighths are smaller than sixths.
5 A student says 5/8 < 1/4 because eighths are smaller than fourths. Is the conclusion correct?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: No. 5/8 > 1/4; 1/4 is 2/8, and five eighths exceed two eighths.
Listen for this difficulty: Using a same-numerator rule when the numerators differ.
Try this response: Rename the fourth as two eighths, then compare equal-sized pieces.
Look for:
Rejects the conclusion.
Uses 2/8 or an accurate same-whole drawing.
6 Draw 3/4 and 3/6 on equal-sized strips. Circle the greater amount and describe the matching piece counts.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: 3/4 is greater. Each has three shaded pieces, but fourths are larger than sixths.
Listen for this difficulty: Making different-sized wholes to force a comparison.
Try this response: Align the strip endpoints and check equality before partitioning.
Look for:
Uses equal wholes.
Compares three fourths with three sixths correctly.
4 · Independent transfer
Try a fresh situation
Choose one task without coaching. Check the explanation against both criteria in its teaching notes; use the other task for a later check.
1 Two identical ropes are cut into 4 and 8 equal pieces. One person takes 3 pieces from each rope. Which collection is longer, and why?
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: Three fourths is longer than three eighths. The piece counts match, and each fourth is twice as long as each eighth.
Listen for this difficulty: Confusing more available pieces with larger selected pieces.
Try this response: Pair the three fourths with the three eighths and compare lengths.
Look for:
Chooses the fourths collection.
Uses identical wholes, equal selected counts, and piece size.
2 Write a careful version of this rule: “The bigger denominator makes the smaller fraction.” Include when it works and one example where it fails.
Explain with a drawing, objects, or words.
Answer & teaching response
Answer or sample response: For fractions of the same whole with the same positive numerator, the larger denominator gives the smaller fraction. It fails for 7/8 and 1/2 because 7/8 is greater.
Listen for this difficulty: Using a denominator-only comparison rule universally.
Try this response: Ask whether the same number of pieces is being compared in the example.
Look for:
States the same-whole and same-positive-numerator conditions.
Gives a valid unlike-numerator counterexample.
What to do next
If the explanation holds: Move on when the student compares equal counts of pieces and rejects the shortcut when numerators differ.
If the difficulty remains: Compare only one piece from each of two equal wholes. Add equal numbers of pieces after the unit fractions are secure.
Standards: 3.NF.A.3.d. Grade 3 alignment describes the mathematical work in this pack.
The tasks, worked examples, and teacher responses are original Fraction Classroom materials, version 2.0.0. The teaching approach draws on noticing the unit, moving between representations, and responding to student reasoning in the linked Illustrative Mathematics lesson. Its text and exercises are not reproduced here.