Estimate before finding equal units

d1. Without finding an exact answer, is 3/4 + 2/3 greater than 1? Explain.

Answer: Yes. Both addends exceed 1/2, so their sum exceeds 1.

Look for:

• Predicts greater than 1.
• Uses both addends being above half.

Teacher response: Likely difficulty: Adding numerators and denominators without checking magnitude. Response: Place each addend above the half mark and combine two half-sized lower bounds.

d2. A student says 1/2 + 1/3 = 2/5. Give a quick reason the result cannot be right.

Answer: 2/5 is less than 1/2, but adding a positive 1/3 to 1/2 must make the result greater than 1/2.

Look for:

• Rejects 2/5 using magnitude.
• Explains that the sum exceeds 1/2.

Teacher response: Likely difficulty: Adding the denominators along with the numerators. Response: Ask whether a positive addition can make the starting amount smaller.

d3. To subtract 5/6 − 1/4, can the numerators be subtracted directly while keeping one denominator?

Answer: No. Rename them as 10/12 and 3/12 first; the difference is 7/12.

Look for:

• Uses equal fractional units.
• Finds 7/12.

Teacher response: Likely difficulty: Subtracting counts that refer to different-sized pieces. Response: Compare a sixth-sized piece with a fourth-sized piece, then partition both into twelfths.

w1. Estimate, then calculate 3/4 + 2/3.

1. Both fractions exceed half, so the sum is greater than 1. Both are at most 3/4, so the sum is at most 1 1/2.
2. Use twelfths: 3/4 = 9/12 and 2/3 = 8/12.
3. Add the counts of twelfths: 9/12 + 8/12 = 17/12.
4. Group 12 twelfths as one whole and check that 1 5/12 fits the estimate.

Worked example: 17/12 = 1 5/12, between 1 and 1 1/2.

Look for:

• Finds 17/12 or 1 5/12.
• Connects common units and the estimated range.

Teacher response: Likely difficulty: Reporting 5/7 after adding unlike fraction symbols. Response: Keep the unit word “twelfths” in the addition sentence.

w2. Estimate, then calculate 7/8 − 1/3.

1. Compare 7/8 with 5/6: 7/8 is greater, and 5/6 − 1/3 = 1/2, so the answer is above half.
2. Rename the fractions as 21/24 and 8/24.
3. Subtract equal-sized parts: 21/24 − 8/24 = 13/24.
4. Since half is 12/24, the answer is only 1/24 above the benchmark.

Worked example: 13/24, a little more than 1/2.

Look for:

• Finds 13/24.
• Checks its distance from half.

Teacher response: Likely difficulty: Subtracting denominators or overlooking an unreasonable magnitude. Response: Show the half benchmark at 12/24 on the same line as the difference.

p1. Estimate whether 5/6 + 1/4 is below or above 1, then find the exact sum.

Answer: Above 1; 5/6 is 1/6 short of 1 and 1/4 exceeds 1/6. The sum is 13/12 = 1 1/12.

Look for:

• Predicts above 1.
• Finds 13/12 with equivalent fractions.

Teacher response: Likely difficulty: Using a whole-number denominator sum of 10. Response: Compare the missing sixth with the quarter being added before using twelfths.

p2. Find 2/3 − 1/6. Explain why renaming 2/3 is needed.

Answer: 4/6 − 1/6 = 3/6 = 1/2; both counts must refer to sixths.

Look for:

• Renames 2/3 as 4/6.
• Finds 1/2 and names the common units.

Teacher response: Likely difficulty: Subtracting 1 from 2 while ignoring different units. Response: Split every third into two sixths.

p3. A student gives 3/8 for 1/2 + 1/6. Use an estimate to reject it, then repair the calculation.

Answer: The sum must exceed 1/2, but 3/8 does not. 1/2 + 1/6 = 3/6 + 1/6 = 4/6 = 2/3.

Look for:

• Rejects 3/8 using the estimate.
• Finds 2/3 using sixths.

Teacher response: Likely difficulty: Treating a plausible-looking fraction as sufficient without a magnitude check. Response: Compare the reported sum with the starting half.

p4. Calculate 5/6 − 3/4. Is the difference greater or less than 1/4?

Answer: 1/12, less than 1/4. In twelfths, 10/12 − 9/12 = 1/12.

Look for:

• Finds 1/12.
• Explains that the quantities are close together.

Teacher response: Likely difficulty: Expecting a large difference because both fractions are large. Response: Place the two amounts near each other on a unit line and inspect the gap.

p5. Both 12 and 24 are possible common denominators for 1/3 + 1/4. Solve using one and show why the other gives the same amount.

Answer: 4/12 + 3/12 = 7/12; or 8/24 + 6/24 = 14/24, which equals 7/12.

Look for:

• Gives a correct sum.
• Explains equivalence of 7/12 and 14/24.

Teacher response: Likely difficulty: Believing only the least common denominator can produce a correct sum. Response: Split every twelfth in half to connect the two valid representations.

p6. Complete __ + 1/4 = 5/6. Estimate the missing amount relative to 1/2, then calculate.

Answer: The missing amount is above 1/2: 5/6 − 1/4 = 10/12 − 3/12 = 7/12.

Look for:

• Predicts above half.
• Finds 7/12 and checks by addition.

Teacher response: Likely difficulty: Adding to find a missing addend without considering the inverse relation. Response: Compare 1/2 + 1/4 = 3/4 with the required 5/6.

t1. A walker travels 5/8 mile, then 2/3 mile. A tracker reports a total below 1 mile. Evaluate the report and find the total.

Answer: The report is too small because both walks exceed half a mile. The total is 15/24 + 16/24 = 31/24 = 1 7/24 miles.

Look for:

• Rejects the report with a benchmark.
• Finds 1 7/24 miles using common units.

Teacher response: Likely difficulty: Accepting an exact-looking output without checking its scale. Response: Make the two half-mile lower bounds visible before computing.

t2. A container holds 3/4 liter. After 2/5 liter is poured out, someone claims 1/9 liter remains. Check the claim and explain the correct remainder.

Answer: 7/20 liter remains: 15/20 − 8/20 = 7/20. The remainder exceeds 1/4 because less than 1/2 liter was removed, so 1/9 is too small.

Look for:

• Uses a valid estimate to reject 1/9.
• Finds 7/20 liter with equivalent fractions.

Teacher response: Likely difficulty: Subtracting numerator and denominator independently. Response: First remove an imagined half liter to establish a quarter-liter lower bound.