Share one fractional amount equally

d1. One half of a sandwich is shared equally by 3 people. Does each person receive more or less than half a sandwich?

Answer: Less than half; each receives 1/6 of the original whole sandwich.

Look for:

• Predicts less than 1/2.
• Names 1/6 of the original sandwich.

Teacher response: Likely difficulty: Interpreting division as counting how many halves fit in three sandwiches. Response: Keep only one half available and separate it into three equal shares.

d2. A student writes 1/4 ÷ 2 = 2/4. Explain why this cannot describe equal sharing.

Answer: Sharing the available fourth between two people gives each 1/8, not 2/4; two fourths exceed the entire available amount.

Look for:

• Rejects 2/4 by checking the available amount.
• Finds 1/8.

Teacher response: Likely difficulty: Multiplying the unit fraction by the number of recipients. Response: Check whether two copies of the claimed share fit into the original fourth.

d3. One third of a strip is shared among 4 people. A student calls each share 1/4. Of which whole is that true, and what fraction of the original strip is each share?

Answer: Each share is 1/4 of the available third, but 1/12 of the original whole strip.

Look for:

• Distinguishes the two reference amounts.
• Names 1/12 of the original strip.

Teacher response: Likely difficulty: Changing the reference whole from the original strip to the available amount. Response: Keep two labeled outlines: the full strip and its available third.

w1. Share 1/3 meter of ribbon equally among 4 people. How much does each receive?

1. Draw one meter and divide it into three equal parts.
2. Only one third is available; split it into four equal shares.
3. Imagine the same four-way split in each third: twelve equal pieces fill one meter.
4. Each share is 1/12 meter; four shares make 4/12 = 1/3 meter.

Worked example: 1/12 meter per person.

Look for:

• Finds 1/12 meter.
• Checks four equal shares reconstruct 1/3 meter.

Teacher response: Likely difficulty: Answering one fourth without keeping the original meter as the unit. Response: Point to the one-meter outline whenever naming a fraction of a meter.

w2. One fifth of a pan of cornbread is shared equally between 2 people. Write an equation and a multiplication check.

1. Label the full pan as one whole and identify one fifth.
2. Cut that fifth into two equal pieces.
3. Each fifth could hold two such pieces, so ten fit in the original pan.
4. Give one piece to each person and check the two tenths equal one fifth.

Worked example: 1/5 ÷ 2 = 1/10 pan; 2 × 1/10 = 1/5.

Look for:

• Writes 1/10 of a pan per person.
• Checks with multiplication by 2.

Teacher response: Likely difficulty: Dividing the denominator by two or giving each person one fifth. Response: Track the total distributed amount as two copies of the proposed share.

p1. Share 1/2 liter of juice equally into 4 cups. How much goes in each cup?

Answer: 1/8 liter.

Look for:

• Finds 1/8 liter per cup.
• Checks four portions make 1/2 liter.

Teacher response: Likely difficulty: Treating four cups as four whole liters to divide by a half. Response: Draw only the half liter available and split it into four equal portions.

p2. Find 1/6 ÷ 3 and explain how many equal shares of that size fit in the original whole.

Answer: 1/18; eighteen such shares fit in one whole.

Look for:

• Finds 1/18.
• Connects 6 × 3 to pieces per whole.

Teacher response: Likely difficulty: Using 6 + 3 for the new denominator. Response: Split each of six sixths into three equal pieces.

p3. A student shares 1/4 of a pie among 3 people and gives each 1/12 of the pie. Verify the answer without repeating a division rule.

Answer: Three shares total 3 × 1/12 = 3/12 = 1/4, so the answer is correct.

Look for:

• Uses multiplication or recombination.
• Shows the total is exactly the available fourth.

Teacher response: Likely difficulty: Accepting a fraction without checking it reconstructs the starting amount. Response: Join three twelfth-sized pieces and compare them with the available fourth.

p4. Which gives a larger share: 1/3 ÷ 2 or 1/3 ÷ 5? Explain before calculating.

Answer: 1/3 ÷ 2 is larger. The same third shared by fewer people gives larger shares: 1/6 versus 1/15.

Look for:

• Chooses the two-person share.
• Explains 1/6 > 1/15 using equal available amounts.

Teacher response: Likely difficulty: Thinking a larger divisor gives a larger share. Response: Keep the available third fixed and compare two-way with five-way partitions.

p5. One eighth of a cake is given to one person. Write the division and result.

Answer: 1/8 ÷ 1 = 1/8.

Look for:

• Finds 1/8.
• Explains that dividing by 1 keeps the amount.

Teacher response: Likely difficulty: Assuming every division must make an amount strictly smaller. Response: Ask whether sharing with one recipient splits the available piece at all.

p6. Make a story for 1/5 ÷ 3. Include the available amount, number of equal shares, and amount in one share.

Answer: Example: 1/5 liter is poured equally into 3 cups, with 1/15 liter in each.

Look for:

• Matches a unit-fraction available amount and 3 recipients.
• Finds 1/15 in the same original unit; equivalent stories count.

Teacher response: Likely difficulty: Writing an integer-divided-by-unit-fraction story for the reversed operation. Response: Start the story with exactly one fifth available, not five or three whole units.

t1. One quarter kilogram of clay is shared equally among 6 students. Explain the amount per student with a diagram description and a multiplication check.

Answer: 1/24 kilogram per student. Split each quarter of a kilogram into six pieces; each piece is one twenty-fourth. Six pieces give 6/24 = 1/4 kilogram.

Look for:

• Finds 1/24 kilogram.
• Explains the whole and verifies six shares reconstruct 1/4 kilogram.

Teacher response: Likely difficulty: Calling each share one sixth of a kilogram instead of one sixth of the available quarter. Response: Keep the original kilogram outline visible around all four quarters.

t2. A group has 1/3 liter of paint. Three people will share it equally. Another group has 3 liters and fills 1/3-liter jars. Explain which situation is 1/3 ÷ 3 and solve only that one.

Answer: The equal-sharing group is 1/3 ÷ 3 = 1/9 liter per person. The jar situation asks how many thirds fit in three liters.

Look for:

• Selects the equal-sharing situation.
• Finds 1/9 liter per person and distinguishes the meanings.

Teacher response: Likely difficulty: Confusing the available amount with the size of each group. Response: Underline what is known in each story: number of recipients versus size of each jar.