Add, Subtract & Check Fraction Amounts

Answer: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The addends are renamed in twelfths of the same whole.

• 0 points: Adds numerators and denominators to get 2/7. No correct supporting model or explanation is present.
• 1 point: Gets 7/12 or writes valid equivalent fractions but omits a complete common-unit explanation.
• 2 points: 1/3 + 1/4 = 4/12 + 3/12 = 7/12. The addends are renamed in twelfths of the same whole.

Watch for: Adds numerators and denominators to get 2/7.

Respond: Partition one whole into twelve parts and locate both addends within it.

Answer: 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 liter. Both amounts are compared in sixths of a liter.

• 0 points: Subtracts denominators or removes one sixth instead of one third. No correct supporting model or explanation is present.
• 1 point: Gets 1/2 liter or correctly renames 1/3 as 2/6 but leaves the explanation incomplete.
• 2 points: 5/6 − 1/3 = 5/6 − 2/6 = 3/6 = 1/2 liter. Both amounts are compared in sixths of a liter.

Watch for: Subtracts denominators or removes one sixth instead of one third.

Respond: Mark one third as two sixths on the bottle before crossing it out.

Answer: 2 1/4 = 1 5/4, so 1 5/4 − 3/4 = 1 2/4 = 1 1/2.

• 0 points: Subtracts the smaller numerator from the larger without preserving the subtraction order. No correct supporting model or explanation is present.
• 1 point: Gives 1 1/2 but omits the valid exchange, or shows 1 5/4 with a later calculation error.
• 2 points: 2 1/4 = 1 5/4, so 1 5/4 − 3/4 = 1 2/4 = 1 1/2.

Watch for: Subtracts the smaller numerator from the larger without preserving the subtraction order.

Respond: Exchange one whole for four fourths and combine those with the existing fourth.

Answer: 3 1/5 = 2 6/5. One exchanged whole supplies 5/5, and the existing 1/5 must still be included; 2 5/5 loses 1/5.

• 0 points: Drops the existing fractional part when exchanging a whole. No correct supporting model or explanation is present.
• 1 point: Corrects the rewrite to 2 6/5 but does not explain the lost fifth.
• 2 points: 3 1/5 = 2 6/5. One exchanged whole supplies 5/5, and the existing 1/5 must still be included; 2 5/5 loses 1/5.

Watch for: Drops the existing fractional part when exchanging a whole.

Respond: Keep the original fifth visible while adding the five fifths from the exchanged whole.

Answer: Reject it: both addends exceed 1/2, so the sum exceeds 1. The exact sum is 8/12 + 9/12 = 17/12 = 1 5/12, while 5/7 is below 1.

• 0 points: Accepts an added-denominators answer even though it is too small. No correct supporting model or explanation is present.
• 1 point: Gives a valid estimate or exact sum, but does not use both to evaluate the claim.
• 2 points: Reject it: both addends exceed 1/2, so the sum exceeds 1. The exact sum is 8/12 + 9/12 = 17/12 = 1 5/12, while 5/7 is below 1.

Watch for: Accepts an added-denominators answer even though it is too small.

Respond: Place both addends beyond 1/2 and compare their combined length with one whole.

Answer: No. 1 1/2 − 2/3 = 9/6 − 4/6 = 5/6 km; 5/6 + 2/3 = 9/6 = 1 1/2 km.

• 0 points: Ignores the whole-number part when finding the remaining distance. No correct supporting model or explanation is present.
• 1 point: Finds 5/6 km but omits the addition check, or rejects 1/6 with a valid partial model.
• 2 points: No. 1 1/2 − 2/3 = 9/6 − 4/6 = 5/6 km; 5/6 + 2/3 = 9/6 = 1 1/2 km.

Watch for: Ignores the whole-number part when finding the remaining distance.

Respond: Draw the entire 1 1/2 km first and show the walked and remaining parts joining to it.

Answer: 2/5 + 1/2 = 4/10 + 5/10 = 9/10 liter. Both amounts are renamed as tenths of a liter.

• 0 points: Adds unlike part counts without renaming them in a shared unit. No correct supporting model or explanation is present.
• 1 point: Gives 9/10 liter or correct tenths but leaves the common-unit explanation incomplete.
• 2 points: 2/5 + 1/2 = 4/10 + 5/10 = 9/10 liter. Both amounts are renamed as tenths of a liter.

Watch for: Adds unlike part counts without renaming them in a shared unit.

Respond: Draw fifths and halves on equal liter bars, then subdivide both into tenths.

Answer: 7/8 − 1/4 = 7/8 − 2/8 = 5/8 meter. The bar removes two eighths from seven eighths.

• 0 points: Removes one eighth for a quarter or subtracts the denominators. No correct supporting model or explanation is present.
• 1 point: Gives 5/8 meter but does not align the bar with the equivalent-fraction equation.
• 2 points: 7/8 − 1/4 = 7/8 − 2/8 = 5/8 meter. The bar removes two eighths from seven eighths.

Watch for: Removes one eighth for a quarter or subtracts the denominators.

Respond: Fold the quarter in two to show why it occupies two eighths.

Answer: 4 1/3 = 3 4/3; subtracting 1 2/3 leaves 2 2/3 cups.

• 0 points: Subtracts fractional numerators in reverse order when the first is smaller. No correct supporting model or explanation is present.
• 1 point: Finds 2 2/3 cups but omits the exchange, or correctly writes 3 4/3 before a later error.
• 2 points: 4 1/3 = 3 4/3; subtracting 1 2/3 leaves 2 2/3 cups.

Watch for: Subtracts fractional numerators in reverse order when the first is smaller.

Respond: Exchange one of the four whole cups for three thirds and add the original third.

Answer: 4 8/6 is equal. The exchanged whole adds 6/6 to the existing 2/6 and reduces the whole count from five to four.

• 0 points: Adds fractional parts from an exchange without adjusting the whole count. No correct supporting model or explanation is present.
• 1 point: Selects 4 8/6 but does not explain both the reduced whole count and the eight sixths.
• 2 points: 4 8/6 is equal. The exchanged whole adds 6/6 to the existing 2/6 and reduces the whole count from five to four.

Watch for: Adds fractional parts from an exchange without adjusting the whole count.

Respond: Draw five wholes and two sixths; replace exactly one whole with six sixths.

Answer: It is above 1 because each addend exceeds 1/2. The sum is 25/40 + 28/40 = 53/40 = 1 13/40; 12/18 is below 1.

• 0 points: Accepts a result below 1 for the sum of two amounts above 1/2. No correct supporting model or explanation is present.
• 1 point: Gives a valid estimate or 53/40, but does not connect both to rejecting 12/18.
• 2 points: It is above 1 because each addend exceeds 1/2. The sum is 25/40 + 28/40 = 53/40 = 1 13/40; 12/18 is below 1.

Watch for: Accepts a result below 1 for the sum of two amounts above 1/2.

Respond: Estimate both addends first, and keep that lower bound visible during calculation.

Answer: 2 1/4 − 5/6 = 27/12 − 10/12 = 17/12 = 1 5/12 liters used. Then 17/12 + 10/12 = 27/12 = 2 1/4.

• 0 points: Treats the remaining quantity as the amount used or drops the whole liters. No correct supporting model or explanation is present.
• 1 point: Finds 17/12 liters or a correct check equation but omits another requested part.
• 2 points: 2 1/4 − 5/6 = 27/12 − 10/12 = 17/12 = 1 5/12 liters used. Then 17/12 + 10/12 = 27/12 = 2 1/4.

Watch for: Treats the remaining quantity as the amount used or drops the whole liters.

Respond: Label start, used and remaining on a single bar, then write their part–whole equation.