Area Product Garden
Model a fraction of a fraction as overlapping parts of a unit area.
Preview the complete lesson
Eight core questions and a two-question exit ticket. Print one set per student.
c1. Calculate 2/3 × 3/4. Show a model or your reasoning.
c2. Calculate 1/2 × 2/5. Show a model or your reasoning.
c3. Calculate 3/4 × 2/3. Show a model or your reasoning.
c4. Calculate 2/5 × 4/7. Show a model or your reasoning.
c5. Calculate 3/5 × 1/2. Show a model or your reasoning.
c6. Calculate 5/6 × 3/4. Show a model or your reasoning.
c7. Calculate 2/7 × 3/5. Show a model or your reasoning.
c8. Calculate 3/8 × 4/5. Show a model or your reasoning.
e1. Calculate 2/3 × 3/4. Show a model or your reasoning.
e2. What do the product denominator and numerator count?
Before the lesson
Recognize a whole and count equal parts.
Materials
- Pencil: 1 per student
- Plain paper sheets: 2 per student
- Projector: 1 per class (optional)
Activity
- Draw one unit garden rectangle with thirds as rows and fourths as columns.
- Shade two of three rows one way and three of four columns another.
- Count the overlap and relate six of twelve cells to 2/3 × 3/4.
45-minute teaching sequence
Look at the first model or task. What is the whole? What do you notice?
Draw one unit garden rectangle with thirds as rows and fourths as columns.
Shade two of three rows one way and three of four columns another.
Count the overlap and relate six of twelve cells to 2/3 × 3/4.
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. What do the product denominator and numerator count?
Watch for this misconception
Counting all shaded cells instead of the overlap.
Mark the overlap with a third pattern, then count only those cells.
Teacher answer key. Open responses include a rubric and one valid sample.
c1. Calculate 2/3 × 3/4. Show a model or your reasoning.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
c2. Calculate 1/2 × 2/5. Show a model or your reasoning.
1/2 × 2/5 = 1/5. Multiply the numerators and denominators; interpret the groups or overlap.
c3. Calculate 3/4 × 2/3. Show a model or your reasoning.
3/4 × 2/3 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
c4. Calculate 2/5 × 4/7. Show a model or your reasoning.
2/5 × 4/7 = 8/35. Multiply the numerators and denominators; interpret the groups or overlap.
c5. Calculate 3/5 × 1/2. Show a model or your reasoning.
3/5 × 1/2 = 3/10. Multiply the numerators and denominators; interpret the groups or overlap.
c6. Calculate 5/6 × 3/4. Show a model or your reasoning.
5/6 × 3/4 = 5/8. Multiply the numerators and denominators; interpret the groups or overlap.
c7. Calculate 2/7 × 3/5. Show a model or your reasoning.
2/7 × 3/5 = 6/35. Multiply the numerators and denominators; interpret the groups or overlap.
c8. Calculate 3/8 × 4/5. Show a model or your reasoning.
3/8 × 4/5 = 3/10. Multiply the numerators and denominators; interpret the groups or overlap.
s1. Calculate 1/2 × 1/2. Show a model or your reasoning.
1/2 × 1/2 = 1/4. Multiply the numerators and denominators; interpret the groups or overlap.
s2. Calculate 1/3 × 1/2. Show a model or your reasoning.
1/3 × 1/2 = 1/6. Multiply the numerators and denominators; interpret the groups or overlap.
s3. Calculate 1/4 × 1/2. Show a model or your reasoning.
1/4 × 1/2 = 1/8. Multiply the numerators and denominators; interpret the groups or overlap.
s4. Calculate 2/3 × 1/2. Show a model or your reasoning.
2/3 × 1/2 = 1/3. Multiply the numerators and denominators; interpret the groups or overlap.
h1. Explain the overlap for 2/3 × 3/4.
Two of three rows and three of four columns overlap in six of twelve equal cells, or one half.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h2. Why does the whole garden need to be named?
The product is a fraction of that unit area, not a free-standing length.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h3. Does reversing the shading order change the area product?
No, the same six of twelve cells overlap.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
h4. Design a garden overlap with area 1/4 square unit.
Shade one half in each direction; one of four equal cells overlaps.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
e1. Calculate 2/3 × 3/4. Show a model or your reasoning.
2/3 × 3/4 = 1/2. Multiply the numerators and denominators; interpret the groups or overlap.
e2. What do the product denominator and numerator count?
The denominator counts all equal grid cells; the numerator counts overlap cells.
- Identify the whole and use equal-sized parts where needed.
- Show a correct model or numerical relationship.
- Explain why the model supports your conclusion.
Use the whole and equal-part prompts before moving to independent practice. Some items revisit the core concept with simpler models.
s1. Calculate 1/2 × 1/2. Show a model or your reasoning.
s2. Calculate 1/3 × 1/2. Show a model or your reasoning.
s3. Calculate 1/4 × 1/2. Show a model or your reasoning.
s4. Calculate 2/3 × 1/2. Show a model or your reasoning.
e1. Calculate 2/3 × 3/4. Show a model or your reasoning.
e2. What do the product denominator and numerator count?
Explain, transfer, and critique. More than one valid response may satisfy the rubric.
h1. Explain the overlap for 2/3 × 3/4.
h2. Why does the whole garden need to be named?
h3. Does reversing the shading order change the area product?
h4. Design a garden overlap with area 1/4 square unit.
e1. Calculate 2/3 × 3/4. Show a model or your reasoning.
e2. What do the product denominator and numerator count?
Teaching notes & lesson sources
Designed for Grade 5. Model a fraction of a fraction as overlapping parts of a unit area.
5.NF.B.4b curriculum reference
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.