Equal Groups, Greater Numbers
10 min
Narrative
This Warm-up prompts students to compare four area diagrams that have been decomposed into two areas, each representing a product. This comparison gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the discussion, ask students to explain the meaning of any terminology they use, such as “side lengths,” “area,” “parts,” and “decompose.”
Launch
- Groups of 2
- Display the image.
- “Pick 3 that go together. Be ready to share why they go together.”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 2–3 minutes: partner discussion
- Share and record responses.
Student Task
Which 3 go together?
Solution Steps (4)
- 1Identify each diagram's dimensions→A: 7×14 gridded, B: 7×(10+3), C: 7×(10+4), D: 7×14 with area 94
- 2Calculate area for each decomposed diagram→B: 70+21=91, C: 70+28=98
- 3Identify which show the same product→A, C, D all represent 7×14=98; B represents 7×13=91; D's label (94) is incorrect
- 4Group diagrams by shared characteristics→Multiple valid groupings based on partitioning, accurate labels, or same product
Sample Response
Sample responses:
A, B, and C go together because:- The rectangles are partitioned into smaller parts.
- The lengths of the sides and parts of sides match their labels. The labels make sense.
- They represent 7×14.
- They have labels to show the area or the value of the products.
- They are not gridded, so they do not show the square units.
Activity Synthesis (Teacher Notes)
- “Why do all 4 go together?” (They all represent multiplication. They all show a rectangle with a side length of 7 units and another side length that is a teen number of units.)
- “Why doesn’t it make sense for the rectangle in C to be split in half?” (Because one part of the rectangle should have been larger than the other because 70 is greater than 28. The part of the side labeled 10 should be longer than the part labeled 4.)
Standards
Addressing
- 3.MD.7.c·Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
- 3.MD.C.7.c·Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths <span class="math">\(a\)</span> and <span class="math">\(b + c\)</span> is the sum of <span class="math">\(a \times b\)</span> and <span class="math">\(a \times c\)</span>. Use area models to represent the distributive property in mathematical reasoning.
20 min
15 min
Knowledge Components
All skills for this lesson
Role Legend
Target
Essential
Supporting
Foundational
All lesson KCs
4 skills
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Equal Groups, Greater Numbers
10 min
Narrative
This Warm-up prompts students to compare four area diagrams that have been decomposed into two areas, each representing a product. This comparison gives students a reason to use language precisely (MP6). It gives the teacher an opportunity to hear how students use terminology and talk about characteristics of the items in comparison to one another. During the discussion, ask students to explain the meaning of any terminology they use, such as “side lengths,” “area,” “parts,” and “decompose.”
Launch
- Groups of 2
- Display the image.
- “Pick 3 that go together. Be ready to share why they go together.”
- 1 minute: quiet think time
Teacher Instructions
- “Discuss your thinking with your partner.”
- 2–3 minutes: partner discussion
- Share and record responses.
Student Task
Which 3 go together?
Solution Steps (4)
- 1Identify each diagram's dimensions→A: 7×14 gridded, B: 7×(10+3), C: 7×(10+4), D: 7×14 with area 94
- 2Calculate area for each decomposed diagram→B: 70+21=91, C: 70+28=98
- 3Identify which show the same product→A, C, D all represent 7×14=98; B represents 7×13=91; D's label (94) is incorrect
- 4Group diagrams by shared characteristics→Multiple valid groupings based on partitioning, accurate labels, or same product
Sample Response
Sample responses:
A, B, and C go together because:- The rectangles are partitioned into smaller parts.
- The lengths of the sides and parts of sides match their labels. The labels make sense.
- They represent 7×14.
- They have labels to show the area or the value of the products.
- They are not gridded, so they do not show the square units.
Activity Synthesis (Teacher Notes)
- “Why do all 4 go together?” (They all represent multiplication. They all show a rectangle with a side length of 7 units and another side length that is a teen number of units.)
- “Why doesn’t it make sense for the rectangle in C to be split in half?” (Because one part of the rectangle should have been larger than the other because 70 is greater than 28. The part of the side labeled 10 should be longer than the part labeled 4.)
Standards
Addressing
- 3.MD.7.c·Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths a and b + c is the sum of a × b and a × c. Use area models to represent the distributive property in mathematical reasoning.
- 3.MD.C.7.c·Use tiling to show in a concrete case that the area of a rectangle with whole-number side lengths <span class="math">\(a\)</span> and <span class="math">\(b + c\)</span> is the sum of <span class="math">\(a \times b\)</span> and <span class="math">\(a \times c\)</span>. Use area models to represent the distributive property in mathematical reasoning.