Fractions: Wholes, Parts & Positions

c201-pre-1. A ribbon is cut into four pieces: one 1 cm piece, one 2 cm piece, one 3 cm piece and one 4 cm piece. Is each piece one fourth of the ribbon? Explain.

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

Look for:

• 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.

Teacher response: 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.

c201-pre-2. A 12 cm strip and an 8 cm strip are each cut in half. Are the two half-strip lengths equal? State each length and explain.

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

Look for:

• 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.

Teacher response: Watch for: Assumes equal fraction names always mean equal amounts. Respond: Label the original whole on both strips before comparing their halves.

c201-pre-3. From 0 to 1 there are five equally spaced tick marks, including both endpoints. What fraction is at the second tick to the right of 0? Explain how you counted.

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

Look for:

• 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.

Teacher response: 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.

c201-pre-4. One small tile is 1/6 of a strip. Draw the whole strip and show the fraction covered by four tiles.

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

Look for:

• 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.

Teacher response: 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.

c201-pre-5. Make seven equal jumps of 1/4 from 0. Name the endpoint and identify which jump reaches 1.

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

Look for:

• 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.

Teacher response: 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.

c201-pre-6. A tray has three half-liter bottles. How many liters are there altogether? Show a whole-and-parts description.

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

Look for:

• 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.

Teacher response: 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.

c201-post-1. A rectangle has three equal-width columns. One column is split in half, giving four pieces altogether. Someone labels every piece 1/4. Correct the labels and explain.

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.

Look for:

• 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.

Teacher response: 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.

c201-post-2. One student says, “Half my rope is 5 meters, so half every rope is 5 meters.” Give a counterexample and identify both whole lengths.

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.

Look for:

• 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.

Teacher response: 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.

c201-post-3. Draw a line from 0 to 1 with six equal intervals. Label the fourth endpoint after 0 and state the total number of tick marks, including 0 and 1.

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

Look for:

• 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.

Teacher response: Watch for: Equates the number of boundary marks with the number of intervals. Respond: Draw the six spaces first, then count their seven boundaries.

c201-post-4. A 3 cm card is labeled 1/6 of a whole strip. How long is the whole, and what fraction do two cards cover? Show your reconstruction.

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

Look for:

• 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.

Teacher response: 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.

c201-post-5. A walker is at 1 on a thirds number line and takes two more 1/3 steps to the right. Name the endpoint as a single fraction and a mixed number; explain.

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

Look for:

• 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.

Teacher response: 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.

c201-post-6. You have 9 quarter-meter lengths. Use as many as possible to make complete meters. State the total length and the leftover length.

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

Look for:

• 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.

Teacher response: Watch for: Uses nine pieces as nine meters or groups quarters incorrectly. Respond: Circle each set of four quarter-meters, leaving one uncircled piece.

c201-step-1. 1. Equal Parts Workshop: Distinguish an equal partition from a collection of uneven pieces. Lesson: /lessons/equal-parts-workshop/

Answer: Can the learner explain why counting pieces alone cannot name fourths?

c201-step-2. 2. Whole Size Matters: Identify the whole before comparing fractional amounts. Lesson: /lessons/whole-size-matters/

Answer: Ask whether halves of differently sized strips must have equal lengths.

c201-step-3. 3. Unit Fraction Builders: Iterate one unit fraction to reconstruct a whole. Lesson: /lessons/unit-fraction-builders/

Answer: Does the learner repeat equal units without gaps or overlaps?

c201-step-4. 4. Number Line Launch: Locate fractions by equal distances from zero. Lesson: /lessons/number-line-launch/

Answer: Look for counting intervals instead of tick marks.

c201-step-5. 5. Fraction Strip Detective: Name a shaded fraction by distinguishing selected parts from all equal parts in the whole. Lesson: /lessons/fraction-strip-detective/

Answer: Can the learner explain which count names the numerator and which count names the denominator?

c201-step-6. 6. Beyond One Walk: Continue a constant fractional unit past one. Lesson: /lessons/beyond-one-walk/

Answer: Does the step size remain unchanged on both sides of 1?

c201-step-7. 7. Fraction Match Stations: Connect words, symbols, strips and positions. Lesson: /lessons/fraction-match-stations/

Answer: Ask for the same quantity in two representations, with the whole labeled.

c201-step-8. 8. Fraction Museum: Explain and critique a complete fraction representation. Lesson: /lessons/fraction-museum/

Answer: Can a partner identify the whole, unit fraction and counted amount without help?

c201-skill-1. The whole and equal parts: items 1–2 on both assessments. Evidence: Identifies equal-sized parts and names the reference whole.

Answer: 0–1: reteach. Fold one paper strip into equal parts; compare it with an uneven partition. Say which whole each fraction names. /interventions/whole-size/. 2–3: guided practice. Use Whole Size Matters to explain when two halves can have different lengths. /lessons/equal-parts-workshop/. 4: extend. Choose different whole lengths so that one fourth of the first equals one half of the second. Justify your choices; for example, one fourth of 16 cm and one half of 8 cm each measure 4 cm. /lessons/whole-size-matters/. Missing evidence: collect another response before grouping.

c201-skill-2. Intervals and unit fractions: items 3–4 on both assessments. Evidence: Counts spaces from zero and composes a fraction from equal jumps.

Answer: 0–1: reteach. Place one counter in every interval between 0 and 1. Match each space to one unit fraction. /interventions/intervals-not-points/. 2–3: guided practice. Match a strip partition to its number-line distance, then cover the tick labels and explain. /lessons/number-line-launch/. 4: extend. Rebuild a whole from a strip labeled 3/4 and mark its unit fraction. /lessons/fraction-strip-detective/. Missing evidence: collect another response before grouping.

c201-skill-3. Composing across one: items 5–6 on both assessments. Evidence: Preserves the unit size across 1 and decomposes amounts into wholes and parts.

Answer: 0–1: reteach. Continue identical fraction steps beyond the whole; label 1 without restarting the count. /interventions/beyond-one/. 2–3: guided practice. Build the same amount with a strip and a number line and explain both names. /lessons/beyond-one-walk/. 4: extend. Create a museum card for a fraction between 1 and 2 with two equivalent descriptions. /lessons/fraction-museum/. Missing evidence: collect another response before grouping.