Plan a fraction-based design and justify its quantities with two models.
Teaching companion · Version 2.0.0
Match each design action with its quantity, whole, operation, and practical check.
The original lesson PDFs remain version 1.0.0. This supplement adds diagnostic probes, teaching responses, and question-specific rubrics. Use the supplement below for the updated guidance.
Ask for an explanation or drawing before choosing a response. A wrong answer alone does not identify its cause.
A board selects 2/3 of its rows and 3/4 of its columns. Is the overlap found by adding the two fractions?
Look for: No. The shared selection is an area product: 6/12 = 1/2 of the board.
If you observe: Adds the two selections or counts singly shaded cells as overlap.
Respond: Draw a three-by-four grid and hatch the selections in different directions; count only double-hatched cells.
Three banners need 2/5 meter each. A 2-meter supply is available. Does the design fit, and how much remains?
Look for: Yes. It uses 6/5 meters and leaves 4/5 meter.
If you observe: Compares the group count three with the supply two without considering meters per banner.
Respond: Label the three 2/5-meter pieces, combine their lengths, then compare the total length with the 2-meter supply.
Check independently: A board is 1/2 green and 1/4 blue without overlap. What remains blank, and how can a partner check?
1/4 remains blank. Check that the three non-overlapping shares total one board and that the drawing uses the same whole.
Use the area product short intervention if the difficulty persists. Return to this check before repeating the exit ticket.
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.
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 lesson: Area Product Gardene1 does not yet meet this criterion: e1 gives 1/2 for 2/3 × 3/4.
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.
Open the linked short interventione1 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.
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.
Open the linked short interventionUse these as follow-up prompts, then examine the learner’s model or explanation. Open responses may differ from the sample while satisfying the criteria.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: 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.
Watch for: Combines numerator counts before making the fractional units alike.
Try: 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.
Watch for: Finds the missing count to one without retaining the fractional unit.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Subtracts unlike numerator or denominator counts without preserving equivalent values.
Try: 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.
Watch for: Assumes multiplication always enlarges a positive amount.
Try: 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.
Watch for: Names a share relative to the shaded fraction instead of the original whole.
Try: 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.
Watch for: Treats a serving-count quotient as the size of a shared amount.
Try: 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.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: 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.
Watch for: Combines numerator counts before making the fractional units alike.
Try: 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.
Watch for: Finds the missing count to one without retaining the fractional unit.
Try: 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.
Watch for: Confuses the number of equal groups with the size of each group.
Try: 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.
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: 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.
Look for in the response:
Watch for: Combines numerator counts before making the fractional units alike.
Try: 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.
Watch for: Finds the missing count to one without retaining the fractional unit.
Try: 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.
Look for in the response:
Watch for: Accepts an exact answer that contradicts the size or direction of the operation.
Try: 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.
Watch for: Confuses a fraction of an identified whole with a length, area, or count.
Try: 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.
Look for in the response:
Watch for: Confuses a fraction of an identified whole with a length, area, or count.
Try: 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.
Watch for: Assumes equal fraction labels guarantee equal physical amounts even when the wholes differ.
Try: 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.
Look for in the response:
Watch for: Counts all singly shaded cells, or only one direction, instead of the overlap in an area product.
Try: 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.
Watch for: Counts pieces without checking equal area.
Try: 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.
Watch for: Confuses a fraction of an identified whole with a length, area, or count.
Try: 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.
Look for in the response:
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Eight core questions and a two-question exit ticket. Print one set per student.
Recognize a whole and count equal parts.
Look at the first model or task. What is the whole? What do you notice?
Plan a one-square-unit mosaic with 2/3 of the rows and 3/4 of the columns highlighted.
Compute the overlap and represent it with both a grid and fraction strips.
Partners critique whether the two models show the same whole and final amount.
Choose core practice, or use the support prompts before moving to core. Use challenge prompts for explanation and transfer.
Calculate 2/3 × 3/4. Show a model or your reasoning. Give two checks a partner can use on your design.
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.
Teacher answer key. Open responses include a rubric and one valid sample.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
1/2 + 1/4 = 3/4. Use equal-sized parts before combining or removing them.
1 - 3/4 = 1/4. Use equal-sized parts before combining or removing them.
3 × 2/5 = 6/5. Multiply the numerators and denominators; interpret the groups or overlap.
5/6 - 1/4 = 7/12. Use equal-sized parts before combining or removing them.
3/4 × 2/3 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
1/2 ÷ 3 = 1/6. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
2 ÷ 1/4 = 8. Divide by multiplying by the reciprocal of the divisor. Check by multiplying back.
1/2 × 1/2 = 1/4. Multiply the numerators and denominators; interpret the groups or overlap.
1/2 + 1/4 = 3/4. Use equal-sized parts before combining or removing them.
1 - 1/4 = 3/4. Use equal-sized parts before combining or removing them.
2 × 1/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
Six of twelve grid cells and six twelfth-size strip sections both show one half.
Color one half green and one quarter blue; one quarter remains blank.
A 2-meter supply covers three 2/5-meter banners and leaves 4/5 meter.
Both models must refer to the same whole and quantity, such as area rather than length.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
Check equal parts and common wholes; check computed quantities against the stated material limits.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
Designed for Grade 5. Plan a fraction-based design and justify its quantities with two models.
Version 1.0.0. Original, AI-assisted lesson content. Free local classroom use; public reuse terms have not yet been assigned.
These lessons have not yet been field-tested in classrooms. Use the exit ticket and student explanations to decide what to revisit.