All units / Unit 4
Fraction Products, Scaling & Two Kinds of Division
Interpret products and scaling, share a unit fraction among whole-number groups, and count unit-fraction servings.
Learning goal: Explain fraction products and distinguish the size of a share from the number of unit-fraction groups.
Prerequisite: Generate equivalent fractions, multiply whole numbers, interpret area, and compose unit fractions into a whole.
This unit’s fraction-divisor work uses positive whole numbers divided by unit fractions, and unit fractions divided by positive whole numbers. The fraction-as-division lesson also connects whole-number sharing to fractional answers.
Materials for your group
Student pages and teacher answers are separate. Download an HTML file to keep or use your browser to print or save as PDF.
Teach in a connected sequence
Each complete lesson is 45 minutes. Use the pre-assessment to select a starting point; revisit a prerequisite whenever its observation is not yet secure. The short interventions in the response guide provide focused practice between lessons.
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Repeated Fraction Groups
Goal: Connect repeated fractional groups to multiplication.
Observe: Can the learner identify both the group count and each group size?
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Fraction as Division
Goal: Interpret whole-number sharing as a fractional quotient.
Observe: Check that the share multiplied by the number of recipients rebuilds the total.
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Area Product Garden
Goal: Interpret a fraction of a fraction as overlap area.
Observe: Does the learner count cells relative to the entire unit square?
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Scaling Stories
Goal: Predict how a factor changes the size of a quantity.
Observe: Ask for a size prediction before any multiplication.
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Unit Fraction Sharing
Goal: Split a unit fraction among a positive whole number of groups.
Observe: Does the quotient name the amount per group in the original whole’s units?
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How Many Servings
Goal: Count unit-fraction servings in a positive whole-number amount.
Observe: Does the quotient count servings, and does the multiplication check recover the total?
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Fraction Design Challenge
Goal: Explain a model and check a multi-step fraction design.
Observe: Ask the learner to state what each operation means and label every resulting quantity.
Before teaching: pre-assessment
Allow about 15–20 minutes. Ask for a model, explanation or equation. Keep access supports consistent between assessments. Items in the same position assess the same skill through a different task.
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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, scoring & response
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.
- 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.
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.
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Without multiplying first, compare (4/5) × (3/7) with 3/7. Then find the product and explain your prediction.
Answer, scoring & response
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.
- 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.
Watch for: Assumes multiplication must always increase an amount.
Respond: Show four of five equal shares of the original 3/7 amount.
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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, scoring & response
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.
- 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.
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.
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A student claims (1/2) ÷ 4 = 2. Use a multiplication check to correct the claim and explain what the quotient means.
Answer, scoring & response
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.
- 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.
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.
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How many 1/3-cup servings can be made from 2 cups? Draw or count the servings and state what the quotient counts.
Answer, scoring & response
Answer: 2 ÷ (1/3) = 6 servings. Each cup contains three third-cup servings, so two cups contain six.
- 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.
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.
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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, scoring & response
Answer: 3 ÷ (1/4) = 12 pieces, and 12 × (1/4 meter) = 3 meters. The quotient counts pieces, not meters per piece.
- 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.
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.
After teaching: post-assessment
Allow about 15–20 minutes. Ask for a model, explanation or equation. Keep access supports consistent between assessments. Items in the same position assess the same skill through a different task.
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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, scoring & response
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.
- 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.
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.
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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, scoring & response
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.
- 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.
Watch for: Assumes every fractional factor makes a quantity smaller.
Respond: Show one original width plus one fourth of that width.
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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, scoring & response
Answer: Each tab is (1/3) ÷ 2 = 1/6 meter. Two equal sixth-meter tabs make the original third-meter strip.
- 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.
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.
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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, scoring & response
Answer: It is correct: (1/5) ÷ 4 = 1/20 kg per roll, and 4 × 1/20 = 4/20 = 1/5 kg in total.
- 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.
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.
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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, scoring & response
Answer: 3 ÷ (1/5) = 15 jars. Five fifth-liter jars fit in each liter, so three liters fill fifteen jars.
- 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.
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.
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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, scoring & response
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.
- 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.
Watch for: Assumes division must always reduce the numerical answer.
Respond: Count two half-meter pieces in every meter before writing the equations.
Read the three skill areas separately
Items 1–2, 3–4 and 5–6 form matching pre/post pairs by skill. Each item earns 0, 1 or 2 points using its own rubric. A blank or inaccessible response stays “not observed” until you collect evidence. Use a drawing, oral explanation or manipulatives consistently if that gives the learner a fair way to respond.
Products and scaling
Evidence: Interprets overlap area and predicts the effect of a factor smaller or larger than one.
0–1 points — reteach: Partition one unit square in both directions and count the overlap as parts of the original square. Open the intervention.
2–3 points — guided practice: Use Scaling Stories to predict bigger, same or smaller before calculating. Open the next lesson.
4 points — extend: Design rectangles with equal area but different fractional side lengths and justify the products. Open the extension lesson.
Size of each share
Evidence: Divides a unit fraction by a positive whole number and checks by recombining shares.
0–1 points — reteach: Keep the original whole outlined while splitting just its unit-fraction portion into equal shares. Open the intervention.
2–3 points — guided practice: State the size and unit of one share, then multiply by the number of shares to check. Open the next lesson.
4 points — extend: Write two sharing stories with the same share size and different unit-fraction starting amounts. Open the extension lesson.
Number of unit-fraction groups
Evidence: Counts how many unit fractions fit in a positive whole-number amount and labels the quotient as a count.
0–1 points — reteach: Count unit-fraction portions in one whole, then repeat that count for every whole. Open the intervention.
2–3 points — guided practice: Pair each serving story with a multiplication check: number of servings × serving size = total. Open the next lesson.
4 points — extend: Compare a sharing story and a counting story that use the same numbers but ask different questions. Open the extension lesson.
Record evidence & plan groups
Use codes S01–S40 and keep the same code for the pre- and post-assessment. Enter rubric scores only. Records stay in this browser; names and written responses are not collected. Download a CSV before clearing browser data.
Each skill has two items and a maximum of 4 points. Use 0–1: reteach, 2–3: guided practice, and 4: extend as planning prompts. An unobserved item keeps that skill out of a group until you collect evidence. These are instructional decisions, not standardized proficiency levels.
Groups for the selected assessment
No records yet. Score an assessment to see the next teaching steps.
Alignment & use
These original tasks follow the concepts in the linked standards. The pre- and post-assessments pair the same constructs with different contexts or representations; they are classroom planning tools, not a standardized test. Discuss unexpected score differences before changing a group.
Use the number line, fraction strips or area model to make reasoning visible.