Split every part, keep the amount

d1. A strip shows 2/3 shaded. Every third is split into two equal pieces. What fraction is now shaded?

Answer: 4/6. The amount stays fixed while both counts double.

Look for:

• Names 4/6.
• Explains that the shaded amount did not change.

Teacher response: Likely difficulty: Doubling the denominator but leaving the numerator unchanged. Response: Split both shaded thirds as well as the unshaded third, then recount.

d2. Which equals 2/5: 3/6 or 4/10? Explain the transformation.

Answer: 4/10. Both numerator and denominator are multiplied by 2.

Look for:

• Selects 4/10.
• Links the factor of 2 to splitting every fifth in half.

Teacher response: Likely difficulty: Adding the same number to both terms preserves value. Response: Compare the shaded boundaries of two same-sized strips.

d3. Complete 3/4 = __/12. What happened to each fourth?

Answer: 9/12. Each fourth was split into three equal pieces.

Look for:

• Supplies numerator 9.
• Explains the factor of 3.

Teacher response: Likely difficulty: Adding 8 to both terms because 4 + 8 = 12. Response: Draw three small pieces inside each fourth and count the shaded ones.

w1. Show why 2/3 = 8/12 with one strip.

1. Divide one strip into three equal parts and shade two.
2. Split every third into four equal pieces.
3. Count twelve equal pieces across the whole and eight shaded pieces.
4. Write 2/3 = (2 × 4)/(3 × 4) = 8/12.

Worked example: Each third becomes four twelfths; two shaded thirds become eight shaded twelfths.

Look for:

• Keeps the same whole and shading.
• Uses the same factor for both counts.

Teacher response: Likely difficulty: Changing fraction labels without connecting them to unchanged area. Response: Keep the original shaded boundary visible while adding the new lines.

w2. A student says 6/8 = 3/8 after grouping shaded pieces in pairs. Repair the reasoning.

1. Start with eight equal parts, six shaded.
2. Group every pair of neighboring eighths into one fourth.
3. The whole has four groups and the shaded region has three groups.
4. Divide both counts by 2 to write 6/8 = 3/4.

Worked example: 6/8 = 3/4. Group all eight eighths in pairs, not only the shaded six.

Look for:

• Changes both counts by division by 2.
• Explains why the new denominator is 4.

Teacher response: Likely difficulty: Regrouping the shaded pieces while retaining the old total count. Response: Circle pairs throughout the entire strip, including the unshaded region.

p1. Split each fourth in a 3/4 strip into two equal parts. Write the new fraction.

Answer: 6/8.

Look for:

• Counts six shaded eighths.
• Keeps the shaded boundary fixed.

Teacher response: Likely difficulty: Leaving the shaded count as 3. Response: Count the two smaller pieces in each of the three shaded fourths.

p2. Complete 1/3 = 4/__. Explain your factor.

Answer: 4/12, using a factor of 4.

Look for:

• Writes denominator 12.
• Uses factor 4 for both counts.

Teacher response: Likely difficulty: Using different factors for numerator and denominator. Response: Ask how many smaller pieces replace each original third.

p3. Repair the statement 1/3 = 2/4. Give a true equivalent fraction with numerator 2.

Answer: 1/3 = 2/6. Splitting each third in half creates sixths.

Look for:

• Rejects 2/4.
• Supplies 2/6 with the same factor on both counts.

Teacher response: Likely difficulty: Adding one to numerator and denominator. Response: Draw thirds and halves on equal strips to expose the changed amount.

p4. Group a strip with 8 of 12 equal parts shaded into groups of four small parts. Name the shaded fraction using the larger parts.

Answer: 2/3.

Look for:

• Forms three equal larger parts.
• Names two shaded groups out of three.

Teacher response: Likely difficulty: Dividing only the numerator when regrouping. Response: Mark three groups of four across the whole and count the two shaded groups.

p5. Are 3/6 and 2/4 equivalent? Explain using a common familiar fraction.

Answer: Yes. Both equal 1/2.

Look for:

• Identifies both as 1/2.
• Uses the same whole for the comparison.

Teacher response: Likely difficulty: Believing equivalence requires one numerator to be an integer multiple of the other. Response: Group each strip into two equal regions and compare the shaded halves.

p6. Give two fractions equivalent to 2/3 with different denominators. Describe how you made each.

Answer: Examples: 4/6 by multiplying both counts by 2, and 8/12 by multiplying both by 4.

Look for:

• Gives two valid equivalent fractions with distinct denominators.
• Explains each common factor; other valid pairs count.

Teacher response: Likely difficulty: Changing only the denominator to generate new names. Response: Ask for the same split factor in every original third.

t1. A recipe uses 3/4 cup. A cup measure is marked in twelfths. Which mark gives the same amount, and why?

Answer: 9/12 cup, because each fourth contains three twelfths.

Look for:

• Chooses 9/12 cup.
• Explains the unchanged amount and factor 3.

Teacher response: Likely difficulty: Treating a finer scale as a different amount. Response: Partition each fourth of a drawn cup scale into three intervals.

t2. Kai writes 2/3 = 4/5 because both numbers grew by 2. Explain the error and give a correct equivalent fraction with numerator 4.

Answer: Adding 2 does not preserve the fraction. 2/3 = 4/6.

Look for:

• Rejects the add-two rule.
• Gives 4/6 with a valid repartition or common-factor explanation.

Teacher response: Likely difficulty: Assuming an equal additive change preserves a ratio. Response: Split every third into two sixths and count both the shaded and total pieces.