Count how many unit fractions fit

d1. How many 1/2-liter bottles can be filled from 3 liters?

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

Look for:

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

Teacher response: Likely difficulty: Dividing 3 by 2 instead of counting half-unit groups. Response: Mark two half-liter groups inside each of three one-liter units.

d2. A student says 2 ÷ 1/4 must be less than 2 because it is division. Explain using quarter-sized groups.

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

Look for:

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

Teacher response: Likely difficulty: Assuming division always decreases a positive number. Response: Fit four quarters into each whole and count across both wholes.

d3. Which equation describes “4 meters cut into 1/3-meter pieces”: 4 ÷ 1/3 or 1/3 ÷ 4? What does the answer count?

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

Look for:

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

Teacher response: Likely difficulty: Reversing the total amount and the size of each group. Response: Underline 4 meters as the available total and 1/3 meter as the size of one piece.

w1. How many 1/4-meter pieces can be cut from 3 meters of ribbon?

1. Draw 0 to 3 meters with whole-meter landmarks.
2. Divide each meter into four quarter-meter intervals.
3. Count all twelve intervals as the pieces that fit.
4. Check: 12 × 1/4 meter = 3 meters.

Worked example: 12 pieces.

Look for:

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

Teacher response: Likely difficulty: Computing three quarters because the divisor denominator is mistaken for the divisor. Response: Say “How many quarter-meter pieces?” while pointing to each interval.

w2. A 2-liter jug fills cups holding 1/6 liter each. How many full cups can it fill?

1. One liter fills six cups of size 1/6 liter.
2. Two liters contain two sets of six such cups.
3. Count 2 × 6 = 12 cups.
4. Check that twelve sixth-liters make 12/6 = 2 liters.

Worked example: 12 full cups.

Look for:

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

Teacher response: Likely difficulty: Giving 1/12 liter as a share instead of a number of cups. Response: Attach a cup label to each counted group and a liter label to the total amount.

p1. How many 1/3-meter pieces fit in 2 meters? Write a multiplication check.

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

Look for:

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

Teacher response: Likely difficulty: Finding 2/3 by treating division as multiplication by the unit fraction. Response: Place three third-meter pieces within each meter.

p2. Find 5 ÷ 1/2. Explain without using an invert-and-multiply rule.

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

Look for:

• Finds 10.
• Explains two groups per whole.

Teacher response: Likely difficulty: Dividing by the denominator 2. Response: Count two half-unit hops in each of five whole units.

p3. A tray holds 4 liters of soup. Each serving is 1/8 liter. How many servings are available?

Answer: 32 servings.

Look for:

• Finds 32.
• Labels the answer servings.

Teacher response: Likely difficulty: Reporting 1/32 liter per serving rather than counting the given-size servings. Response: Label one serving 1/8 liter and count eight servings per liter.

p4. Which makes more pieces from the same 3-meter ribbon: pieces of size 1/4 meter or 1/6 meter? Give both counts.

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

Look for:

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

Teacher response: Likely difficulty: Thinking larger piece size gives more groups in a fixed amount. Response: Fit both group sizes into one meter first, then repeat for three meters.

p5. A student writes 3 ÷ 1/5 = 3/5. Check the proposed answer by multiplying it by 1/5, then repair it.

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

Look for:

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

Teacher response: Likely difficulty: Using multiplication by the divisor when counting its groups. Response: Ask how many fifths make one whole, then how many make three wholes.

p6. Write a story for 2 ÷ 1/4. State what the 2, the 1/4, and the answer mean.

Answer: 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.

Look for:

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

Teacher response: Likely difficulty: Writing a story that shares one fourth between two recipients. Response: Begin with two whole units available and a fixed quarter-unit container.

t1. A maker has 5 meters of wire and cuts 1/3-meter pieces with no waste. There must be at least 14 pieces. Is there enough wire? Explain with division and a multiplication check.

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

Look for:

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

Teacher response: Likely difficulty: Treating the divisor as a number of recipients instead of a piece length. Response: Count thirds in each meter, then compare the resulting count with 14.

t2. Compare these stories: A shares 1/4 liter among 3 people. B pours 3 liters into 1/4-liter cups. Write both expressions and explain why their answers have different meanings.

Answer: 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.

Look for:

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

Teacher response: Likely difficulty: Treating reversed division expressions as interchangeable. Response: Make a two-column sketch: known group count versus known group size.