When Is the Same Size Not the Same Size?
5 min
Narrative
The purpose of this Warm-up is to elicit the idea that rectangles can have the same diagonal length but different areas, which will be useful when students work with aspect ratio in a later activity. While students may notice and wonder many things about these figures, the fact that all of the diagonals are the same length and that all of the rectangles have different areas are the important discussion points.
This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- They are all rectangles.
- They all have a diagonal drawn in.
- Their diagonals are all the same length.
- One of the rectangles looks like a square.
Students may wonder:
- Do the rectangles all have the same area?
- Do the rectangles all have the same perimeter?
- Are the diagonals all the same length? (if they did not measure)
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the idea that the diagonals are all the same length but the areas of the rectangles are different does not come up during the conversation, encourage students to use a ruler or the edge of a piece of paper to measure and verify these claims.
Standards
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.RP.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <em>For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</em>
- 6.RP.A.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <span>For example, “The ratio of wings to beaks in the bird house at the zoo was <span class="math">\(2:1\)</span>, because for every <span class="math">\(2\)</span> wings there was <span class="math">\(1\)</span> beak.” “For every vote candidate A received, candidate C received nearly three votes.” </span>
20 min
20 min
Knowledge Components
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When Is the Same Size Not the Same Size?
5 min
Narrative
The purpose of this Warm-up is to elicit the idea that rectangles can have the same diagonal length but different areas, which will be useful when students work with aspect ratio in a later activity. While students may notice and wonder many things about these figures, the fact that all of the diagonals are the same length and that all of the rectangles have different areas are the important discussion points.
This Warm-up prompts students to make sense of a problem before solving it by familiarizing themselves with a context and the mathematics that might be involved (MP1).
Launch
Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Students may notice:
- They are all rectangles.
- They all have a diagonal drawn in.
- Their diagonals are all the same length.
- One of the rectangles looks like a square.
Students may wonder:
- Do the rectangles all have the same area?
- Do the rectangles all have the same perimeter?
- Are the diagonals all the same length? (if they did not measure)
Activity Synthesis (Teacher Notes)
Ask students to share the things they noticed and wondered. Record and display their responses for all to see, without editing or commentary. If possible, record the relevant reasoning on or near the image. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information.
If the idea that the diagonals are all the same length but the areas of the rectangles are different does not come up during the conversation, encourage students to use a ruler or the edge of a piece of paper to measure and verify these claims.
Standards
- 6.G.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.G.A.1·Find the area of right triangles, other triangles, special quadrilaterals, and polygons by composing into rectangles or decomposing into triangles and other shapes; apply these techniques in the context of solving real-world and mathematical problems.
- 6.RP.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <em>For example, "The ratio of wings to beaks in the bird house at the zoo was 2:1, because for every 2 wings there was 1 beak." "For every vote candidate A received, candidate C received nearly three votes."</em>
- 6.RP.A.1·Understand the concept of a ratio and use ratio language to describe a ratio relationship between two quantities. <span>For example, “The ratio of wings to beaks in the bird house at the zoo was <span class="math">\(2:1\)</span>, because for every <span class="math">\(2\)</span> wings there was <span class="math">\(1\)</span> beak.” “For every vote candidate A received, candidate C received nearly three votes.” </span>