More equal pieces, smaller pieces

d1. For the same whole, which is greater: 1/3 or 1/8? Explain without comparing only the numbers 3 and 8.

Answer: 1/3 is greater because splitting a whole into 3 equal parts makes larger pieces than splitting it into 8.

Look for:

• Chooses 1/3.
• Explains the sizes of parts of the same whole.

Teacher response: Likely difficulty: Treating a larger denominator as a larger fraction. Response: Draw the same-sized whole twice and point to one part of each.

d2. Compare 3/4 and 3/8 of the same whole. What stays the same, and what changes?

Answer: 3/4 > 3/8. The number of pieces is 3 in each, but fourths are larger than eighths.

Look for:

• Uses the correct comparison.
• Distinguishes piece count from piece size.

Teacher response: Likely difficulty: Ignoring equal numerators and ordering by denominator size. Response: Match each shaded fourth with one shaded eighth to compare three pairs.

d3. Does a larger denominator always mean a smaller fraction? Test the claim with 1/2 and 7/8.

Answer: No. 7/8 > 1/2; the “larger denominator, smaller amount” shortcut needs the same positive numerator and same whole.

Look for:

• Rejects the unrestricted claim.
• Uses the different numerators to explain the exception.

Teacher response: Likely difficulty: Overgeneralizing the unit-fraction rule to unlike numerators. Response: Shade seven eighths and compare the total coverage with one half.

w1. Order 1/2, 1/4, and 1/8 from least to greatest.

1. Start with three identical whole strips.
2. Partition them into 2, 4, and 8 equal parts.
3. Compare one part from each strip.
4. The eighth is smallest and the half is largest.

Worked example: 1/8 < 1/4 < 1/2.

Look for:

• Uses equal-sized wholes.
• Orders all three fractions by part size.

Teacher response: Likely difficulty: Applying whole-number order directly to denominators. Response: Ask how many copies of each piece would fill the same whole.

w2. Compare 2/3 and 2/6 without finding a common denominator.

1. Use two identical whole strips.
2. A third is larger than a sixth of the same whole.
3. Take two of each type of piece.
4. Two larger pieces cover more than two smaller pieces.

Worked example: 2/3 > 2/6.

Look for:

• Compares the unit fractions correctly.
• Uses the equal numerator 2 to compare totals.

Teacher response: Likely difficulty: Assuming sixths are larger because 6 exceeds 3. Response: Pair one third with one sixth twice rather than comparing the symbols alone.

p1. Compare 1/4 and 1/6 of the same whole using <, >, or =. Explain with sharing.

Answer: 1/4 > 1/6. Four equal shares of a fixed amount are larger than six equal shares.

Look for:

• Writes 1/4 > 1/6.
• Keeps the total amount fixed in the explanation.

Teacher response: Likely difficulty: Believing more recipients means larger individual shares. Response: Imagine sharing one identical strip between four people, then six.

p2. Put 1/3, 1/8, and 1/4 in increasing order.

Answer: 1/8 < 1/4 < 1/3.

Look for:

• Orders all three correctly.
• Can explain why the eighth is smallest.

Teacher response: Likely difficulty: Sorting denominator numbers from least to greatest. Response: Compare the sizes of single pieces on equal strips.

p3. Which covers more of one whole: two sixths or two eighths? Explain.

Answer: Two sixths, because a sixth is larger than an eighth and both collections contain two pieces.

Look for:

• Chooses 2/6.
• Uses both equal count and different part size.

Teacher response: Likely difficulty: Ignoring unit-fraction size when the count matches. Response: Compare one pair of pieces, then duplicate that comparison.

p4. Choose a denominator from 3, 4, or 8 to make 1/__ smaller than 1/6.

Answer: 8.

Look for:

• Selects 8.
• Explains why eighths are smaller than sixths.

Teacher response: Likely difficulty: Choosing the smaller denominator to make the smaller amount. Response: Ask which partition creates more than six equal pieces in one whole.

p5. A student says 5/8 < 1/4 because eighths are smaller than fourths. Is the conclusion correct?

Answer: No. 5/8 > 1/4; 1/4 is 2/8, and five eighths exceed two eighths.

Look for:

• Rejects the conclusion.
• Uses 2/8 or an accurate same-whole drawing.

Teacher response: Likely difficulty: Using a same-numerator rule when the numerators differ. Response: Rename the fourth as two eighths, then compare equal-sized pieces.

p6. Draw 3/4 and 3/6 on equal-sized strips. Circle the greater amount and describe the matching piece counts.

Answer: 3/4 is greater. Each has three shaded pieces, but fourths are larger than sixths.

Look for:

• Uses equal wholes.
• Compares three fourths with three sixths correctly.

Teacher response: Likely difficulty: Making different-sized wholes to force a comparison. Response: Align the strip endpoints and check equality before partitioning.

t1. 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?

Answer: Three fourths is longer than three eighths. The piece counts match, and each fourth is twice as long as each eighth.

Look for:

• Chooses the fourths collection.
• Uses identical wholes, equal selected counts, and piece size.

Teacher response: Likely difficulty: Confusing more available pieces with larger selected pieces. Response: Pair the three fourths with the three eighths and compare lengths.

t2. Write a careful version of this rule: “The bigger denominator makes the smaller fraction.” Include when it works and one example where it fails.

Answer: 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.

Look for:

• States the same-whole and same-positive-numerator conditions.
• Gives a valid unlike-numerator counterexample.

Teacher response: Likely difficulty: Using a denominator-only comparison rule universally. Response: Ask whether the same number of pieces is being compared in the example.