Fraction Design Challenge: teaching companion

prepare. Before starting: Recognize a whole and count equal parts.

materials. Materials: Pencil: 1 per student; Plain paper sheets: 2 per student; Projector: 1 per class (optional).

activity-1. Plan a one-square-unit mosaic with 2/3 of the rows and 3/4 of the columns highlighted.

activity-2. Compute the overlap and represent it with both a grid and fraction strips.

activity-3. Partners critique whether the two models show the same whole and final amount.

phase-1. 5 min · warmup: Look at the first model or task. What is the whole? What do you notice?

Answer: Students: Sketch an initial prediction and explain it to a partner.

phase-2. 8 min · model: Plan a one-square-unit mosaic with 2/3 of the rows and 3/4 of the columns highlighted.

Answer: Students: Follow the model; label the whole and the equal parts.

phase-3. 10 min · partner: Compute the overlap and represent it with both a grid and fraction strips.

Answer: Students: Work in pairs. One models, the other checks; switch roles.

phase-4. 7 min · discussion: Partners critique whether the two models show the same whole and final amount.

Answer: Students: Compare two solutions and explain the relationship.

phase-5. 10 min · practice: Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.

Answer: Students: Solve independently, then compare the reasoning.

phase-6. 5 min · exit: Calculate 2/3 × 3/4. Show a model or your reasoning. Give two checks a partner can use on your design.

Answer: Students: Complete the two exit questions independently.

d1. A board selects 2/3 of its rows and 3/4 of its columns. Is the overlap found by adding the two fractions?

Answer: No. The shared selection is an area product: 6/12 = 1/2 of the board.

Teacher response: If the learner adds the two selections or counts singly shaded cells as overlap. Draw a three-by-four grid and hatch the selections in different directions; count only double-hatched cells.

d2. Three banners need 2/5 meter each. A 2-meter supply is available. Does the design fit, and how much remains?

Answer: Yes. It uses 6/5 meters and leaves 4/5 meter.

Teacher response: If the learner compares the group count three with the supply two without considering meters per banner. Label the three 2/5-meter pieces, combine their lengths, then compare the total length with the 2-meter supply.

r1. A board is 1/2 green and 1/4 blue without overlap. What remains blank, and how can a partner check?

Answer: 1/4 remains blank. Check that the three non-overlapping shares total one board and that the drawing uses the same whole.

e1. Calculate 2/3 × 3/4. Show a model or your reasoning.

Answer: 1/2

Teacher response: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product. Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.

e2. Give two checks a partner can use on your design.

Answer: Check equal parts and common wholes; check computed quantities against the stated material limits.

Look for:

• Give one concrete model check, such as equal parts and a stated common whole.
• Give a second independent check, such as verifying a calculation against a material limit.
• Describe what evidence a partner should examine for each check; merely saying “check the answer” is insufficient.

Teacher response: Counts pieces without checking equal area. Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition. Confuses a fraction of an identified whole with a length, area, or count. Label the whole and the answer unit before counting. For a unit-square garden use square units; for a ribbon use meters; for cut pieces use the number of pieces. Check that the drawing and equation describe the same quantity.

teach-1. For one design claim, write the known quantities and the desired answer with units.

teach-2. Choose the model: overlap for a product, disjoint parts for a sum, equal recipients for sharing, or repeated serving sizes for a count.

teach-3. Compute and check against the stated whole or material limit; have a partner inspect a different property independently.

next-1. e1 meets its criterion: e1 gives 1/2 for 2/3 × 3/4. e2 also meets its criterion: e2 supplies two concrete, independent design checks, such as equal partitions/common wholes and arithmetic against a stated material limit.

Teacher response: Continue to the linked next lesson. Use the overlap result as a starting point for a new garden design with an explicitly labeled unit area. Next original lesson: /lessons/area-product-garden/.

next-2. e1 does not yet meet this criterion: e1 gives 1/2 for 2/3 × 3/4.

Teacher response: Ask the learner to explain the attempt, then use the matching diagnostic probe to identify the difficulty. Work through the focused reteach below and its check. Use the linked short intervention if the difficulty persists; retry e1 and e2 before moving on. Short intervention: /interventions/area-product/.

next-3. e1 meets its criterion, but e2 does not yet show this evidence: e2 supplies two concrete, independent design checks, such as equal partitions/common wholes and arithmetic against a stated material limit.

Teacher response: Ask the learner to apply each proposed check to an actual model or h3 material claim. These two exit items check overlap and checking practices; use c2–c8 separately before judging the other operations. Short intervention: /interventions/area-product/.

c1. A unit mosaic highlights 2/3 of its rows and 3/4 of its columns. What fraction overlaps?

Answer: 1/2

Teacher response: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product. Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.

c2. Half the display is green and one quarter is blue, without overlap. What share is colored?

Answer: 3/4

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

c3. If 3/4 of the board is colored, what share stays blank?

Answer: 1/4

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

c4. Three equal banners each use 2/5 meter of ribbon. Find total meters.

Answer: 6/5

Teacher response: Confuses the number of equal groups with the size of each group. Draw three separate bags, each holding 2/5 cup. Label “3 groups” and “2/5 cup per group.” Count six fifth-cup pieces in total and connect the drawing to 2/5 + 2/5 + 2/5.

c5. A 5/6 meter strip loses 1/4 meter. What length remains?

Answer: 7/12

Teacher response: Subtracts unlike numerator or denominator counts without preserving equivalent values. Show 5/6 as 10/12 and 1/4 as 3/12 on a shared unit. Remove three of the ten twelfths. Check 7/12 + 3/12 = 10/12 before returning to the original fractions.

c6. Two thirds of a 3/4 meter border is painted. How many meters are painted?

Answer: 1/2

Teacher response: Assumes multiplication always enlarges a positive amount. Keep a 3/4-meter ribbon as the starting amount. Compare half a copy, one copy, and two copies. Record 3/8, 3/4, and 3/2 meters, then relate each change to a factor below, equal to, or above one.

c7. Half a sheet is shared equally among three designs. What share of the original sheet is used by each?

Answer: 1/6

Teacher response: Names a share relative to the shaded fraction instead of the original whole. Shade half of a whole strip and divide that half into three equal shares. Extend the same cuts across the unshaded half: six shares fill the original whole, so each is 1/6. Check 3 × 1/6 = 1/2.

c8. Two meters are cut into quarter-meter borders. How many borders can be made?

Answer: 8

Teacher response: Treats a serving-count quotient as the size of a shared amount. Draw two one-liter containers and mark quarter-liter servings in each. Count 4 + 4 = 8 servings. Label the result “servings” and check 8 servings × 1/4 liter per serving = 2 liters.

s1. Calculate 1/2 × 1/2. Show a model or your reasoning.

Answer: 1/4

Teacher response: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product. Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.

s2. Calculate 1/2 + 1/4. Show a model or your reasoning.

Answer: 3/4

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced.

s3. Calculate 1 - 1/4. Show a model or your reasoning.

Answer: 3/4

Teacher response: Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

s4. Calculate 2 × 1/4. Show a model or your reasoning.

Answer: 1/2

Teacher response: Confuses the number of equal groups with the size of each group. Draw three separate bags, each holding 2/5 cup. Label “3 groups” and “2/5 cup per group.” Count six fifth-cup pieces in total and connect the drawing to 2/5 + 2/5 + 2/5.

h1. Represent the 2/3 by 3/4 overlap with both grid and strip models.

Answer: Six of twelve grid cells and six twelfth-size strip sections both show one half.

Look for:

• Show the 2/3 by 3/4 selection as six overlap cells out of twelve in a grid.
• Show six selected twelfth-sized sections in a strip representing the same unit amount.
• Explain why both models represent 6/12 = 1/2, while the grid makes the multiplication structure visible.

Teacher response: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product. Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.

h2. Design a colored board with exactly 3/4 colored and two non-overlapping colors.

Answer: Color one half green and one quarter blue; one quarter remains blank.

Look for:

• Choose two positive colored shares with no overlap.
• Verify their sum is exactly 3/4 of the same board.
• Show or state that 1/4 remains blank.

Teacher response: Combines numerator counts before making the fractional units alike. Overlay thirds and fourths on the same whole to make twelfths. Rename 1/3 as 4/12 and 1/4 as 3/12; combine seven twelfths. Ask which original amount would change if only its denominator were replaced. Finds the missing count to one without retaining the fractional unit. Build one as eight eighths, shade three eighths, and count the five unshaded eighths. Write 3/8 + 5/8 = 8/8 = 1 and ask the learner to name the size of every missing piece.

h3. State and check a material limit for your ribbon design.

Answer: A 2-meter supply covers three 2/5-meter banners and leaves 4/5 meter.

Look for:

• State a material supply and a design demand in the same measurement unit.
• Calculate demand from the number and size of pieces.
• Compare demand with supply and compute any remainder; for three 2/5-meter banners from 2 meters, leave 4/5 meter.

Teacher response: Accepts an exact answer that contradicts the size or direction of the operation. Before calculating, mark the starting amount and predict the direction of change. A positive sum must exceed either addend; a positive subtraction result is below the starting amount. Use 0, 1/2, 1, or 2 to bound the result, then compare the exact answer with that bound. Confuses a fraction of an identified whole with a length, area, or count. Label the whole and the answer unit before counting. For a unit-square garden use square units; for a ribbon use meters; for cut pieces use the number of pieces. Check that the drawing and equation describe the same quantity.

h4. Explain why matching numbers alone is not enough to match models.

Answer: Both models must refer to the same whole and quantity, such as area rather than length.

Look for:

• Identify the whole represented by each model.
• Require matching quantities and units, such as area with area or length with length.
• Explain why equal numerical labels alone do not establish that the physical models represent the same amount.

Teacher response: Confuses a fraction of an identified whole with a length, area, or count. Label the whole and the answer unit before counting. For a unit-square garden use square units; for a ribbon use meters; for cut pieces use the number of pieces. Check that the drawing and equation describe the same quantity. Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ. Compare half of a 12 cm strip with half of a 6 cm strip. Measure 6 cm and 3 cm. Keep the fraction label visible and ask what changed; then repeat the comparison using two equal-length strips.

e1. Calculate 2/3 × 3/4. Show a model or your reasoning.

Answer: 1/2

Teacher response: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product. Partition one unit square into 3 equal rows and 4 equal columns. Shade 2 rows one way and 3 columns another. Count only the 6 cells shaded both ways, out of all 12 equal cells.

e2. Give two checks a partner can use on your design.

Answer: Check equal parts and common wholes; check computed quantities against the stated material limits.

Look for:

• Give one concrete model check, such as equal parts and a stated common whole.
• Give a second independent check, such as verifying a calculation against a material limit.
• Describe what evidence a partner should examine for each check; merely saying “check the answer” is insufficient.

Teacher response: Counts pieces without checking equal area. Keep one rectangle fixed. Draw four unequal pieces beside four equal-area pieces. Ask which set lets every piece be called one fourth, then have the learner repair the unequal partition. Confuses a fraction of an identified whole with a length, area, or count. Label the whole and the answer unit before counting. For a unit-square garden use square units; for a ribbon use meters; for cut pieces use the number of pieces. Check that the drawing and equation describe the same quantity.