Trade one whole before subtracting

d1. Complete the trade: 2 1/4 = 1 __/4. Explain what happened to one whole.

Answer: 1 5/4. One whole became 4/4, which joins the existing 1/4.

Look for:

• Writes 5 in the numerator.
• Reduces the whole-number part from 2 to 1.

Teacher response: Likely difficulty: Adding a whole to the fractional part without decreasing the whole-number count. Response: Exchange one complete strip for four fourth-sized pieces.

d2. A student computes 2 1/5 − 3/5 as 2 2/5 by reversing 1 − 3. Explain the problem.

Answer: Subtraction order cannot be reversed to avoid regrouping. 2 1/5 = 1 6/5, so the difference is 1 3/5.

Look for:

• Rejects the reversed subtraction.
• Gets 1 3/5 by a valid trade or count back.

Teacher response: Likely difficulty: Taking the positive difference of fractional numerators regardless of order. Response: Start at 2 1/5 on a line and move left three fifths; the result must be below 2 1/5.

d3. Which is equal to 3: 2 4/4 or 2 3/4? Explain.

Answer: 2 4/4, because 4/4 is exactly one whole.

Look for:

• Chooses 2 4/4.
• States that 4/4 = 1.

Teacher response: Likely difficulty: Trading a whole for one fewer part than the denominator. Response: Reassemble four fourths into one full strip.

w1. Find 3 1/4 − 1 3/4 by trading one whole.

1. Represent 3 1/4 as three wholes and one fourth.
2. Trade one whole for 4/4: 3 1/4 = 2 5/4.
3. Remove one whole and three fourths: 2 5/4 − 1 3/4 = 1 2/4.
4. Check by adding 1 2/4 and 1 3/4 to recover 3 1/4.

Worked example: 1 1/2, or 1 2/4.

Look for:

• Preserves the value in 2 5/4.
• Finds 1 2/4 and checks by addition.

Teacher response: Likely difficulty: Keeping three wholes after adding four fourths. Response: Record the trade as an equality before doing any subtraction.

w2. Find 2 − 5/6 using a whole-number starting amount.

1. Write 2 as two whole strips.
2. Trade one whole for six sixths: 2 = 1 6/6.
3. Remove five of the six sixths.
4. One whole and one sixth remain; 1 1/6 + 5/6 = 2.

Worked example: 1 1/6.

Look for:

• Writes the correct regrouped form.
• Finds 1 1/6 with a valid check.

Teacher response: Likely difficulty: Treating the absent fraction in 2 as 0/6 and subtracting in reverse. Response: Physically replace a whole with six equal parts before removing any.

p1. Rewrite 4 2/5 using exactly three wholes.

Answer: 3 7/5.

Look for:

• Uses three wholes.
• Writes seven fifths.

Teacher response: Likely difficulty: Changing the whole count without preserving total value. Response: Trade one of the four wholes for five fifths.

p2. Calculate 2 1/3 − 2/3. Show the trade.

Answer: 1 2/3; 2 1/3 = 1 4/3.

Look for:

• Writes 1 4/3 or another equivalent decomposition.
• Finds 1 2/3.

Teacher response: Likely difficulty: Subtracting 1 from 2 in the fractional part to get a positive result. Response: Show why taking away two thirds crosses below 2.

p3. Find 3 − 7/8 and check by addition.

Answer: 2 1/8; 2 1/8 + 7/8 = 3.

Look for:

• Finds 2 1/8.
• Checks by recombining eight eighths.

Teacher response: Likely difficulty: Trading one whole for seven eighths instead of eight eighths. Response: Use eight equal eighths for the traded whole, then remove seven.

p4. Does 4 5/6 − 2 1/6 need a trade? Find the difference.

Answer: No. Five sixths can supply one sixth directly; the difference is 2 4/6 = 2 2/3.

Look for:

• Explains why no trade is needed.
• Finds 2 4/6 or 2 2/3.

Teacher response: Likely difficulty: Regrouping automatically without inspecting the fractional parts. Response: Compare the available and required sixths before touching the wholes.

p5. A student writes 5 1/4 = 4 4/4. Locate the lost amount and repair the equality.

Answer: The original 1/4 was lost. The correct trade is 5 1/4 = 4 5/4.

Look for:

• Identifies the missing 1/4.
• Repairs the form to 4 5/4.

Teacher response: Likely difficulty: Replacing rather than adding to the existing fractional part. Response: Place the old fourth beside the four fourths received in the trade.

p6. Complete __ + 1 5/8 = 3 1/8. Use subtraction, then check.

Answer: 1 1/2, or 1 4/8. Regroup 3 1/8 as 2 9/8 and subtract 1 5/8.

Look for:

• Finds 1 4/8.
• Checks the sum returns to 3 1/8.

Teacher response: Likely difficulty: Ignoring the whole consumed when fractional parts cross one. Response: Use the missing addend as a distance from 1 5/8 to 3 1/8.

t1. A rope is 4 1/6 meters long. You cut off 2 5/6 meters. How much remains? Explain why the answer must be between 1 and 2 meters.

Answer: 1 1/3 meters remain. 4 1/6 = 3 7/6, so subtracting 2 5/6 leaves 1 2/6. The cut is between 2 1/6 and 3 1/6 meters, so the remainder is between 1 and 2.

Look for:

• Finds 1 1/3 meters.
• Preserves the traded value and supports the 1-to-2 estimate.

Teacher response: Likely difficulty: Reversing the fractional subtraction and keeping too many wholes. Response: Compare the cut with the lengths that would leave exactly 1 or exactly 2 meters.

t2. Write a subtraction problem starting with 2 1/4 that needs a trade and has a positive answer. Solve it and explain the trade.

Answer: Example: 2 1/4 − 3/4 = 1 1/2. Trade to 1 5/4 before removing 3/4.

Look for:

• Creates a valid same-denominator problem that requires a trade.
• Shows an equivalent regrouping and correct positive result; other valid examples count.

Teacher response: Likely difficulty: Choosing a subtrahend that does not actually require a trade. Response: Ask whether the fractional amount being removed exceeds the available one fourth.