Fractions: Wholes, Parts & Positions

Answer: No. Fourths require four equal lengths; these pieces have different lengths and do not each represent 1/4.

• 0 points: Treats any four pieces as fourths. No correct supporting model or explanation is present.
• 1 point: Rejects the fourths claim but does not explain equal-sized parts.
• 2 points: No. Fourths require four equal lengths; these pieces have different lengths and do not each represent 1/4.

Watch for: Treats any four pieces as fourths.

Respond: Align the four pieces against a common start, then fold an uncut copy of the ribbon into four equal lengths.

Answer: No. Their halves are 6 cm and 4 cm; the same fraction of different wholes can have different lengths.

• 0 points: Assumes equal fraction names always mean equal amounts. No correct supporting model or explanation is present.
• 1 point: Finds 6 cm and 4 cm or identifies different wholes, but does not connect the two ideas.
• 2 points: No. Their halves are 6 cm and 4 cm; the same fraction of different wholes can have different lengths.

Watch for: Assumes equal fraction names always mean equal amounts.

Respond: Label the original whole on both strips before comparing their halves.

Answer: 2/4 or 1/2. Five ticks make four equal intervals; the position is two intervals from zero.

• 0 points: Uses five tick marks as a denominator of five. No correct supporting model or explanation is present.
• 1 point: Locates the midpoint or writes 2/4 without distinguishing intervals from ticks.
• 2 points: 2/4 or 1/2. Five ticks make four equal intervals; the position is two intervals from zero.

Watch for: Uses five tick marks as a denominator of five.

Respond: Put a finger in each of the four spaces, then trace two spaces from zero.

Answer: Six equal tiles form the whole; four cover 4/6, also 2/3, of that whole.

• 0 points: Uses the number of selected tiles to define the whole. No correct supporting model or explanation is present.
• 1 point: Writes 4/6 but omits the six-tile whole, or draws the whole without labeling the amount.
• 2 points: Six equal tiles form the whole; four cover 4/6, also 2/3, of that whole.

Watch for: Uses the number of selected tiles to define the whole.

Respond: Build all six tiles first and outline the whole before selecting four.

Answer: The endpoint is 7/4 or 1 3/4. The fourth jump reaches 1; all seven jumps have the same size.

• 0 points: Restarts the fraction count or changes step size after 1. No correct supporting model or explanation is present.
• 1 point: Gives the endpoint or the fourth jump correctly, but does not establish both.
• 2 points: The endpoint is 7/4 or 1 3/4. The fourth jump reaches 1; all seven jumps have the same size.

Watch for: Restarts the fraction count or changes step size after 1.

Respond: Lay two identical fourths strips end to end and count every interval from zero.

Answer: 3/2 liters, or 1 1/2 liters: two half-liters make 1 liter and one half-liter remains.

• 0 points: Counts containers as whole liters instead of half-liter units. No correct supporting model or explanation is present.
• 1 point: Gives 3/2 or 1 1/2 but does not show how two halves form a whole.
• 2 points: 3/2 liters, or 1 1/2 liters: two half-liters make 1 liter and one half-liter remains.

Watch for: Counts containers as whole liters instead of half-liter units.

Respond: Pair two half-liter drawings into one liter and label the remaining half.

Answer: The two unsplit columns are each 1/3; the two smaller pieces are each 1/6. Four pieces need not be fourths because these pieces are unequal.

• 0 points: Names parts from the total piece count despite unequal areas. No correct supporting model or explanation is present.
• 1 point: Rejects fourths and identifies unequal areas but does not correctly label both piece sizes.
• 2 points: The two unsplit columns are each 1/3; the two smaller pieces are each 1/6. Four pieces need not be fourths because these pieces are unequal.

Watch for: Names parts from the total piece count despite unequal areas.

Respond: Overlay the small pieces on an unsplit column to show two sixths equal one third.

Answer: The first whole is 10 meters. For example, an 8-meter rope has a 4-meter half; the fraction 1/2 depends on the whole.

• 0 points: Treats a fractional amount as fixed independently of its whole. No correct supporting model or explanation is present.
• 1 point: Supplies a valid different half-length but omits one whole length or the explanation.
• 2 points: The first whole is 10 meters. For example, an 8-meter rope has a 4-meter half; the fraction 1/2 depends on the whole.

Watch for: Treats a fractional amount as fixed independently of its whole.

Respond: Write each whole length above a bar and each half-length below it.

Answer: The point is 4/6 or 2/3. There are seven tick marks enclosing six equal intervals.

• 0 points: Equates the number of boundary marks with the number of intervals. No correct supporting model or explanation is present.
• 1 point: Labels 4/6 correctly or gives seven tick marks, but does not show a consistent partition.
• 2 points: The point is 4/6 or 2/3. There are seven tick marks enclosing six equal intervals.

Watch for: Equates the number of boundary marks with the number of intervals.

Respond: Draw the six spaces first, then count their seven boundaries.

Answer: The whole is 18 cm: six 3 cm cards. Two cards cover 2/6, or 1/3, of that same whole.

• 0 points: Makes the two selected cards into a new whole. No correct supporting model or explanation is present.
• 1 point: Finds 18 cm or 2/6 correctly but does not connect both to six equal cards.
• 2 points: The whole is 18 cm: six 3 cm cards. Two cards cover 2/6, or 1/3, of that same whole.

Watch for: Makes the two selected cards into a new whole.

Respond: Keep the label 1/6 on one card while placing six equal copies in a row.

Answer: 5/3 = 1 2/3. One whole contains three thirds, then two more thirds make five thirds.

• 0 points: Changes the unit or resets the numerator after passing 1. No correct supporting model or explanation is present.
• 1 point: Gives either correct name but does not connect the five identical thirds to the whole.
• 2 points: 5/3 = 1 2/3. One whole contains three thirds, then two more thirds make five thirds.

Watch for: Changes the unit or resets the numerator after passing 1.

Respond: Mark 1 as 3/3, then continue 4/3 and 5/3 with equal spaces.

Answer: 9/4 meters = 2 1/4 meters. Eight quarters make two whole meters and one quarter-meter remains.

• 0 points: Uses nine pieces as nine meters or groups quarters incorrectly. No correct supporting model or explanation is present.
• 1 point: Finds the total or the remainder but does not explain four quarters per meter.
• 2 points: 9/4 meters = 2 1/4 meters. Eight quarters make two whole meters and one quarter-meter remains.

Watch for: Uses nine pieces as nine meters or groups quarters incorrectly.

Respond: Circle each set of four quarter-meters, leaving one uncircled piece.