Fraction Products, Scaling & Two Kinds of Division

c204-pre-1. A rectangle is 2/3 meter by 3/4 meter. Sketch it inside a 1 m by 1 m square and find its area.

Answer: The area is 6/12 = 1/2 square meter. A 3-by-4 partition of the unit square has 12 equal cells; the rectangle covers 2 × 3 = 6 cells.

Look for:

• 0 points: Uses only the selected rectangle as the whole or adds the side lengths. No correct supporting model or explanation is present.
• 1 point: Finds 1/2 square meter but omits the unit-square partition, or shows six of twelve cells without the area label.
• 2 points: The area is 6/12 = 1/2 square meter. A 3-by-4 partition of the unit square has 12 equal cells; the rectangle covers 2 × 3 = 6 cells.

Teacher response: Watch for: Uses only the selected rectangle as the whole or adds the side lengths. Respond: Outline the full square meter before shading rows and columns; count overlap cells out of all twelve.

c204-pre-2. Without multiplying first, compare (4/5) × (3/7) with 3/7. Then find the product and explain your prediction.

Answer: It is smaller because 4/5 is less than 1 and 3/7 is positive. The product is 12/35, less than 3/7 = 15/35.

Look for:

• 0 points: Assumes multiplication must always increase an amount. No correct supporting model or explanation is present.
• 1 point: Gives the correct size prediction or product but does not connect the factor to the change in size.
• 2 points: It is smaller because 4/5 is less than 1 and 3/7 is positive. The product is 12/35, less than 3/7 = 15/35.

Teacher response: Watch for: Assumes multiplication must always increase an amount. Respond: Show four of five equal shares of the original 3/7 amount.

c204-pre-3. One fourth of a liter of juice is shared equally among 3 cups. How much goes in each cup? Show the original liter and the shares.

Answer: Each cup gets (1/4) ÷ 3 = 1/12 liter. The quarter-liter portion is split into three equal pieces, each one twelfth of the original liter.

Look for:

• 0 points: Names each share 1/3 liter by treating the quarter-liter portion as the original liter. No correct supporting model or explanation is present.
• 1 point: Gives 1/12 liter but does not show the original whole and three equal shares.
• 2 points: Each cup gets (1/4) ÷ 3 = 1/12 liter. The quarter-liter portion is split into three equal pieces, each one twelfth of the original liter.

Teacher response: Watch for: Names each share 1/3 liter by treating the quarter-liter portion as the original liter. Respond: Outline the liter and divide each quarter into three so the size of one share is visible.

c204-pre-4. A student claims (1/2) ÷ 4 = 2. Use a multiplication check to correct the claim and explain what the quotient means.

Answer: The quotient is 1/8, the size of each of four equal shares of 1/2. Four times 1/8 equals 1/2; four times 2 does not.

Look for:

• 0 points: Multiplies the starting half by four instead of splitting it into four shares. No correct supporting model or explanation is present.
• 1 point: Corrects to 1/8 or supplies a valid multiplication check, but does not explain both.
• 2 points: The quotient is 1/8, the size of each of four equal shares of 1/2. Four times 1/8 equals 1/2; four times 2 does not.

Teacher response: Watch for: Multiplies the starting half by four instead of splitting it into four shares. Respond: Draw four equal pieces inside one half, then repeat them to reconstruct the half.

c204-pre-5. How many 1/3-cup servings can be made from 2 cups? Draw or count the servings and state what the quotient counts.

Answer: 2 ÷ (1/3) = 6 servings. Each cup contains three third-cup servings, so two cups contain six.

Look for:

• 0 points: Divides 2 by 3 as if the question asked for the size of one shared amount. No correct supporting model or explanation is present.
• 1 point: Gives 6 but does not label servings or explain three servings per cup.
• 2 points: 2 ÷ (1/3) = 6 servings. Each cup contains three third-cup servings, so two cups contain six.

Teacher response: Watch for: Divides 2 by 3 as if the question asked for the size of one shared amount. Respond: Partition each of two cup drawings into thirds and count the portions.

c204-pre-6. A ribbon is 3 meters long and is cut into 1/4-meter pieces. Write a division equation and a multiplication equation to check the number of pieces.

Answer: 3 ÷ (1/4) = 12 pieces, and 12 × (1/4 meter) = 3 meters. The quotient counts pieces, not meters per piece.

Look for:

• 0 points: Reports 3/4 meter as the number of pieces. No correct supporting model or explanation is present.
• 1 point: Finds 12 pieces but omits one equation or its meaning.
• 2 points: 3 ÷ (1/4) = 12 pieces, and 12 × (1/4 meter) = 3 meters. The quotient counts pieces, not meters per piece.

Teacher response: Watch for: Reports 3/4 meter as the number of pieces. Respond: Make four quarter-meter pieces in each meter, then group all three meters.

c204-post-1. A garden bed is 3/5 meter wide and 1/2 meter long. Tile a unit square to show the bed’s area and label the square units.

Answer: The area is 3/10 square meter. A 5-by-2 partition creates ten equal cells in one square meter, and the bed covers three of them.

Look for:

• 0 points: Counts overlap pieces without using the original square meter as the whole. No correct supporting model or explanation is present.
• 1 point: Gives 3/10 square meter but does not relate three selected cells to the full ten-cell unit square.
• 2 points: The area is 3/10 square meter. A 5-by-2 partition creates ten equal cells in one square meter, and the bed covers three of them.

Teacher response: Watch for: Counts overlap pieces without using the original square meter as the whole. Respond: Extend the garden’s rows and columns through the full unit square before counting.

c204-post-2. A design is enlarged by a factor of 5/4. Its original width is 2/3 meter. Predict whether the new width is greater or less, then calculate and justify.

Answer: It is greater because 5/4 exceeds 1. The new width is (5/4) × (2/3) = 10/12 = 5/6 meter, greater than 2/3 = 4/6 meter.

Look for:

• 0 points: Assumes every fractional factor makes a quantity smaller. No correct supporting model or explanation is present.
• 1 point: Gives the correct prediction or 5/6 meter but omits the factor-based explanation or comparison.
• 2 points: It is greater because 5/4 exceeds 1. The new width is (5/4) × (2/3) = 10/12 = 5/6 meter, greater than 2/3 = 4/6 meter.

Teacher response: Watch for: Assumes every fractional factor makes a quantity smaller. Respond: Show one original width plus one fourth of that width.

c204-post-3. A 1/3-meter strip is divided equally into 2 tabs. Draw the tabs relative to a whole meter and find the length of each.

Answer: Each tab is (1/3) ÷ 2 = 1/6 meter. Two equal sixth-meter tabs make the original third-meter strip.

Look for:

• 0 points: Calls each tab 1/2 meter because it is half of the strip. No correct supporting model or explanation is present.
• 1 point: Gives 1/6 meter but omits the whole-meter reference or the two equal tabs.
• 2 points: Each tab is (1/3) ÷ 2 = 1/6 meter. Two equal sixth-meter tabs make the original third-meter strip.

Teacher response: Watch for: Calls each tab 1/2 meter because it is half of the strip. Respond: Keep a whole-meter bar behind the strip and split every third into halves.

c204-post-4. A baker splits 1/5 kg of dough equally among 4 rolls. Decide whether 1/20 kg per roll is correct and use multiplication to justify.

Answer: It is correct: (1/5) ÷ 4 = 1/20 kg per roll, and 4 × 1/20 = 4/20 = 1/5 kg in total.

Look for:

• 0 points: Checks by multiplying the total by four instead of recombining the four shares. No correct supporting model or explanation is present.
• 1 point: Accepts 1/20 kg but omits a valid reconstruction of the original fifth kilogram.
• 2 points: It is correct: (1/5) ÷ 4 = 1/20 kg per roll, and 4 × 1/20 = 4/20 = 1/5 kg in total.

Teacher response: Watch for: Checks by multiplying the total by four instead of recombining the four shares. Respond: Draw four equal portions inside one fifth and label their total before checking.

c204-post-5. A station has 3 liters of paint. Each sample jar holds 1/5 liter. How many full jars can it fill? Show the count per liter and in all.

Answer: 3 ÷ (1/5) = 15 jars. Five fifth-liter jars fit in each liter, so three liters fill fifteen jars.

Look for:

• 0 points: Finds a share size such as 3/5 liter instead of counting jars. No correct supporting model or explanation is present.
• 1 point: Gives 15 jars but does not explain the five-per-liter grouping.
• 2 points: 3 ÷ (1/5) = 15 jars. Five fifth-liter jars fit in each liter, so three liters fill fifteen jars.

Teacher response: Watch for: Finds a share size such as 3/5 liter instead of counting jars. Respond: Arrange five jars beside each of three one-liter containers and count the jars.

c204-post-6. A 4-meter rope is cut into 1/2-meter lengths. Someone predicts fewer than 4 pieces because this is division. Correct the prediction with two equations.

Answer: There are 8 pieces: 4 ÷ (1/2) = 8 and 8 × (1/2 meter) = 4 meters. Two half-meter pieces fit in each meter, so division here increases the numerical count.

Look for:

• 0 points: Assumes division must always reduce the numerical answer. No correct supporting model or explanation is present.
• 1 point: Gives 8 pieces or a valid equation but omits the second equation or explanation of the count.
• 2 points: There are 8 pieces: 4 ÷ (1/2) = 8 and 8 × (1/2 meter) = 4 meters. Two half-meter pieces fit in each meter, so division here increases the numerical count.

Teacher response: Watch for: Assumes division must always reduce the numerical answer. Respond: Count two half-meter pieces in every meter before writing the equations.

c204-step-1. 1. Repeated Fraction Groups: Connect repeated fractional groups to multiplication. Lesson: /lessons/repeated-fraction-groups/

Answer: Can the learner identify both the group count and each group size?

c204-step-2. 2. Fraction as Division: Interpret whole-number sharing as a fractional quotient. Lesson: /lessons/fraction-as-division/

Answer: Check that the share multiplied by the number of recipients rebuilds the total.

c204-step-3. 3. Area Product Garden: Interpret a fraction of a fraction as overlap area. Lesson: /lessons/area-product-garden/

Answer: Does the learner count cells relative to the entire unit square?

c204-step-4. 4. Scaling Stories: Predict how a factor changes the size of a quantity. Lesson: /lessons/scaling-stories/

Answer: Ask for a size prediction before any multiplication.

c204-step-5. 5. Unit Fraction Sharing: Split a unit fraction among a positive whole number of groups. Lesson: /lessons/unit-fraction-sharing/

Answer: Does the quotient name the amount per group in the original whole’s units?

c204-step-6. 6. How Many Servings: Count unit-fraction servings in a positive whole-number amount. Lesson: /lessons/how-many-servings/

Answer: Does the quotient count servings, and does the multiplication check recover the total?

c204-step-7. 7. Fraction Design Challenge: Explain a model and check a multi-step fraction design. Lesson: /lessons/fraction-design-challenge/

Answer: Ask the learner to state what each operation means and label every resulting quantity.

c204-skill-1. Products and scaling: items 1–2 on both assessments. Evidence: Interprets overlap area and predicts the effect of a factor smaller or larger than one.

Answer: 0–1: reteach. Partition one unit square in both directions and count the overlap as parts of the original square. /interventions/area-product/. 2–3: guided practice. Use Scaling Stories to predict bigger, same or smaller before calculating. /lessons/area-product-garden/. 4: extend. Design rectangles with equal area but different fractional side lengths and justify the products. /lessons/scaling-stories/. Missing evidence: collect another response before grouping.

c204-skill-2. Size of each share: items 3–4 on both assessments. Evidence: Divides a unit fraction by a positive whole number and checks by recombining shares.

Answer: 0–1: reteach. Keep the original whole outlined while splitting just its unit-fraction portion into equal shares. /interventions/unit-fraction-sharing/. 2–3: guided practice. State the size and unit of one share, then multiply by the number of shares to check. /lessons/unit-fraction-sharing/. 4: extend. Write two sharing stories with the same share size and different unit-fraction starting amounts. /lessons/fraction-as-division/. Missing evidence: collect another response before grouping.

c204-skill-3. Number of unit-fraction groups: items 5–6 on both assessments. Evidence: Counts how many unit fractions fit in a positive whole-number amount and labels the quotient as a count.

Answer: 0–1: reteach. Count unit-fraction portions in one whole, then repeat that count for every whole. /interventions/count-unit-fractions/. 2–3: guided practice. Pair each serving story with a multiplication check: number of servings × serving size = total. /lessons/how-many-servings/. 4: extend. Compare a sharing story and a counting story that use the same numbers but ask different questions. /lessons/fraction-design-challenge/. Missing evidence: collect another response before grouping.