Count how many unit fractions fit

Answer or sample response: 6 bottles, because two half-liter bottles fit in each liter.

Listen for this difficulty: Dividing 3 by 2 instead of counting half-unit groups.

Try this response: Mark two half-liter groups inside each of three one-liter units.

Look for:

• Counts 6 groups.
• Labels the result bottles rather than liters per bottle.

Answer or sample response: There are 8 quarter-sized groups in 2 wholes. Counting groups smaller than a whole can give a number greater than the starting number.

Listen for this difficulty: Assuming division always decreases a positive number.

Try this response: Fit four quarters into each whole and count across both wholes.

Look for:

• Finds 8.
• Explains that the result counts small groups.

Answer or sample response: 4 ÷ 1/3; the answer is 12 pieces.

Listen for this difficulty: Reversing the total amount and the size of each group.

Try this response: Underline 4 meters as the available total and 1/3 meter as the size of one piece.

Look for:

• Selects 4 ÷ 1/3.
• Names the output as 12 pieces.

Listen for this difficulty: Computing three quarters because the divisor denominator is mistaken for the divisor.

Try this response: Say “How many quarter-meter pieces?” while pointing to each interval.

Look for:

• Counts four groups per meter and twelve total.
• Checks total length with multiplication.

Listen for this difficulty: Giving 1/12 liter as a share instead of a number of cups.

Try this response: Attach a cup label to each counted group and a liter label to the total amount.

Look for:

• Finds 12 cups.
• Keeps the group count and volume units distinct.

Answer or sample response: 6 pieces; 6 × 1/3 meter = 2 meters.

Listen for this difficulty: Finding 2/3 by treating division as multiplication by the unit fraction.

Try this response: Place three third-meter pieces within each meter.

Look for:

• Finds 6 pieces.
• Checks the total length is 2 meters.

Answer or sample response: 10. Each whole contains two halves, so five wholes contain ten halves.

Listen for this difficulty: Dividing by the denominator 2.

Try this response: Count two half-unit hops in each of five whole units.

Look for:

• Finds 10.
• Explains two groups per whole.

Answer or sample response: 32 servings.

Listen for this difficulty: Reporting 1/32 liter per serving rather than counting the given-size servings.

Try this response: Label one serving 1/8 liter and count eight servings per liter.

Look for:

• Finds 32.
• Labels the answer servings.

Answer or sample response: Sixth-meter pieces make more: 18 pieces versus 12 quarter-meter pieces.

Listen for this difficulty: Thinking larger piece size gives more groups in a fixed amount.

Try this response: Fit both group sizes into one meter first, then repeat for three meters.

Look for:

• Gives 12 and 18 correctly.
• Explains smaller groups produce a larger count.

Answer or sample response: The check gives 3/25, not 3. The correct answer is 15 because 15 × 1/5 = 3.

Listen for this difficulty: Using multiplication by the divisor when counting its groups.

Try this response: Ask how many fifths make one whole, then how many make three wholes.

Look for:

• Shows the proposed check does not return 3.
• Finds and verifies 15.

Answer or sample response: Example: 2 liters fill 1/4-liter cups, so 8 cups are filled. The 2 is the total volume and 1/4 is each cup size.

Listen for this difficulty: Writing a story that shares one fourth between two recipients.

Try this response: Begin with two whole units available and a fixed quarter-unit container.

Look for:

• Matches total amount and group size to the expression.
• Finds 8 groups and labels each quantity; equivalent stories count.

Answer or sample response: Yes. 5 ÷ 1/3 = 15 pieces, which meets the requirement. Fifteen pieces use 15 × 1/3 = 5 meters.

Listen for this difficulty: Treating the divisor as a number of recipients instead of a piece length.

Try this response: Count thirds in each meter, then compare the resulting count with 14.

Look for:

• Finds 15 pieces and answers yes.
• Checks total wire length and compares with the required count.

Answer or sample response: A: 1/4 ÷ 3 = 1/12 liter per person. B: 3 ÷ 1/4 = 12 cups. A finds the size of a share; B counts known-size groups.

Listen for this difficulty: Treating reversed division expressions as interchangeable.

Try this response: Make a two-column sketch: known group count versus known group size.

Look for:

• Writes and solves both allowed division types.
• Distinguishes amount per person from number of cups.