Area Product Garden: 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. Draw one unit garden rectangle with thirds as rows and fourths as columns.

activity-2. Shade two of three rows one way and three of four columns another.

activity-3. Count the overlap and relate six of twelve cells to 2/3 × 3/4.

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: Draw one unit garden rectangle with thirds as rows and fourths as columns.

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

phase-3. 10 min · partner: Shade two of three rows one way and three of four columns another.

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

phase-4. 7 min · discussion: Count the overlap and relate six of twelve cells to 2/3 × 3/4.

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. What do the product denominator and numerator count?

Answer: Students: Complete the two exit questions independently.

d1. A unit square has three equal rows and four equal columns. Two rows and three columns are selected. How many cells are in both selections?

Answer: Six of the twelve equal cells; the product is 6/12 = 1/2.

Teacher response: If the learner counts all cells selected by either color instead of the overlap. Trace one of the two selected rows and count its three doubly selected cells, then repeat in the second row.

d2. Does the denominator twelve count only the shaded cells or all cells in the whole unit square?

Answer: All twelve equal cells in the whole unit square.

Teacher response: If the learner uses the singly shaded region as the new whole. Outline the entire unit square in a third color and count all three-by-four cells before identifying the overlap.

r1. Show 1/2 × 2/3 on a unit square. What fraction overlaps?

Answer: Two of six equal cells overlap: 2/6 = 1/3.

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. What do the product denominator and numerator count?

Answer: The denominator counts all equal grid cells; the numerator counts overlap cells.

Look for:

• Identify the product denominator as all equal grid cells in the whole.
• Identify the product numerator as cells selected in both directions.
• Connect the counts to e1: 3 × 4 = 12 total and 2 × 3 = 6 overlap cells.

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.

teach-1. Name the full rectangle as one square unit and partition it in both directions.

teach-2. Use different hatch directions for the two factors and locate cells carrying both.

teach-3. Write overlap cells / all equal cells and connect both counts to the factor numerators and denominators.

next-1. e1 meets its criterion: e1 gives 1/2 for 2/3 × 3/4. e2 also meets its criterion: e2 identifies twelve total equal cells and six overlap cells as the product denominator and numerator.

Teacher response: Continue to the linked next lesson. Predict how the product changes relative to the starting amount from the factor’s size. Next original lesson: /lessons/scaling-stories/.

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 identifies twelve total equal cells and six overlap cells as the product denominator and numerator.

Teacher response: Ask the learner to point to all twelve cells and then only the six overlap cells, connecting those two counts to the written product. Short intervention: /interventions/area-product/.

c1. Calculate 2/3 × 3/4. Show a model or your reasoning. Area model: one unit rectangle has 3 equal rows and 4 equal columns. 2 rows and 3 columns are selected; look for their overlap.

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. Calculate 1/2 × 2/5. Show a model or your reasoning. Area model: one unit rectangle has 2 equal rows and 5 equal columns. 1 rows and 2 columns are selected; look for their overlap.

Answer: 1/5

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.

c3. Calculate 3/4 × 2/3. Show a model or your reasoning. Area model: one unit rectangle has 4 equal rows and 3 equal columns. 3 rows and 2 columns are selected; look for their overlap.

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.

c4. Calculate 2/5 × 4/7. Show a model or your reasoning. Area model: one unit rectangle has 5 equal rows and 7 equal columns. 2 rows and 4 columns are selected; look for their overlap.

Answer: 8/35

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.

c5. Calculate 3/5 × 1/2. Show a model or your reasoning. Area model: one unit rectangle has 5 equal rows and 2 equal columns. 3 rows and 1 columns are selected; look for their overlap.

Answer: 3/10

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.

c6. Calculate 5/6 × 3/4. Show a model or your reasoning. Area model: one unit rectangle has 6 equal rows and 4 equal columns. 5 rows and 3 columns are selected; look for their overlap.

Answer: 5/8

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.

c7. Calculate 2/7 × 3/5. Show a model or your reasoning. Area model: one unit rectangle has 7 equal rows and 5 equal columns. 2 rows and 3 columns are selected; look for their overlap.

Answer: 6/35

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.

c8. Calculate 3/8 × 4/5. Show a model or your reasoning. Area model: one unit rectangle has 8 equal rows and 5 equal columns. 3 rows and 4 columns are selected; look for their overlap.

Answer: 3/10

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.

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 1/3 × 1/2. Show a model or your reasoning.

Answer: 1/6

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.

s3. Calculate 1/4 × 1/2. Show a model or your reasoning.

Answer: 1/8

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.

s4. Calculate 2/3 × 1/2. Show a model or your reasoning.

Answer: 1/3

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.

h1. Explain the overlap for 2/3 × 3/4.

Answer: Two of three rows and three of four columns overlap in six of twelve equal cells, or one half.

Look for:

• Partition the unit garden into three rows and four columns of equal cells.
• Select two rows and three columns and count only their overlap.
• Identify six of twelve cells as 6/12 = 1/2 of the garden.

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. Why does the whole garden need to be named?

Answer: The product is a fraction of that unit area, not a free-standing length.

Look for:

• Identify the whole garden or unit square used as the reference.
• Explain that the product describes a portion of its area.
• Distinguish that area quantity from a side length or from the area of an unspecified garden.

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.

h3. Does reversing the shading order change the area product?

Answer: No, the same six of twelve cells overlap.

Look for:

• Answer that reversing the shading order leaves the product unchanged.
• Show that the same two row-shares and three column-shares are selected.
• Count the same six of twelve overlap cells in either order.

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.

h4. Design a garden overlap with area 1/4 square unit.

Answer: Shade one half in each direction; one of four equal cells overlaps.

Look for:

• Specify a unit-square garden and two fractional selections in perpendicular directions.
• Make their overlap exactly 1/4 square unit, such as 1/2 by 1/2.
• Check the overlap cell count against the total equal cells.

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.

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. What do the product denominator and numerator count?

Answer: The denominator counts all equal grid cells; the numerator counts overlap cells.

Look for:

• Identify the product denominator as all equal grid cells in the whole.
• Identify the product numerator as cells selected in both directions.
• Connect the counts to e1: 3 × 4 = 12 total and 2 × 3 = 6 overlap cells.

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.