Reasoning to Find Area
5 min
Teacher Prep
Setup
Required Preparation
Narrative
This activity prompts students to use reasoning strategies from earlier lessons to compare the areas of two figures. It is also an opportunity to use (or introduce) tracing paper as a way to illustrate decomposing and rearranging a figure.
During the activity, look for students who are able to explain or show how they know that the areas are equal. Some students may simply look at the figures and say, with no justification, that they have the same area. Urge them to think of a way to show that their conclusion is true.
Launch
Give students access to their geometry toolkits, and allow for 2 minutes of quiet think time. Ask them to be ready to support their answer, and remind them to use the tools at their disposal. Have copies of the blackline master ready for students who propose cutting the figures out for comparison or as a way to differentiate the activity.
Student Task
Is the area of Figure A greater than, less than, or equal to the area of the shaded region in Figure B? Be prepared to explain your reasoning.
Sample Response
The areas are equal. Sample reasoning:
- Measuring: Measure the side lengths of the small square-shaped hole and the small shaded square on the side of Figure B. They have the same side lengths, so their areas are equal. This means the square on the side fills the hole on the inside.. Measure the side lengths of the large shaded square in Figure A and then in Figure B. They have the same side lengths, so their areas are equal.
- Using scissors: Cut off the little square on the side of Figure B, and use it to fill the hole inside Figure B. The result is a square that matches up exactly with Figure A.
- Using tracing paper: Trace the boundary of the little square on the side of Figure B and move the tracing paper over the unshaded hole. Doing this shows that the little shaded square is exactly the same size as the hole. Movingthat little shaded square to fill the unshaded hole creates a big shaded square. If the boundary of that big shaded square is traced and the drawing is placed over Figure A, it would up exactly with Figure A.
Activity Synthesis (Teacher Notes)
Start the discussion by asking students to indicate which of the three possible responses—area of Figure A is greater, area of Figure B is greater, or the areas are equal—they choose.
Select previously identified students to share their explanations. If no student mentioned using tracing paper, demonstrate the following:
- Decomposing and rearranging Figure B: Place a piece of tracing paper over Figure B. Draw the boundary of the small side square, making a dotted auxiliary line to show its separation from the large square. Move the tracing paper so that the boundary of the small square matches up exactly with the boundary of the square-shaped hole in Figure B. Draw the boundary of the large square. Explain that the small square matches up exactly with the hole, so we know the small, shaded square and the hole have equal area.
- Matching the two figures: Move the tracing paper over Figure A so that the boundary of the rearranged Figure B matches up exactly with that of Figure A. Say, “When two figures that are overlaid one on top of another match up exactly, their areas are equal.”
Highlight the strategies and principles that are central to this unit. Tell students, “We just decomposed and rearranged Figure B so that it matches up exactly with Figure A. When two figures that are overlaid one on top of another match up exactly, we can say that their areas are equal.”
Anticipated Misconceptions
Students may interpret the area of Figure B as the entire region inside the outer boundary including the unfilled square. Clarify that we want to compare the areas of only the shaded parts of Figure B and Figure A.
Standards
- 3.MD.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
- 3.MD.C.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
20 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Reasoning to Find Area
5 min
Teacher Prep
Setup
Required Preparation
Narrative
This activity prompts students to use reasoning strategies from earlier lessons to compare the areas of two figures. It is also an opportunity to use (or introduce) tracing paper as a way to illustrate decomposing and rearranging a figure.
During the activity, look for students who are able to explain or show how they know that the areas are equal. Some students may simply look at the figures and say, with no justification, that they have the same area. Urge them to think of a way to show that their conclusion is true.
Launch
Give students access to their geometry toolkits, and allow for 2 minutes of quiet think time. Ask them to be ready to support their answer, and remind them to use the tools at their disposal. Have copies of the blackline master ready for students who propose cutting the figures out for comparison or as a way to differentiate the activity.
Student Task
Is the area of Figure A greater than, less than, or equal to the area of the shaded region in Figure B? Be prepared to explain your reasoning.
Sample Response
The areas are equal. Sample reasoning:
- Measuring: Measure the side lengths of the small square-shaped hole and the small shaded square on the side of Figure B. They have the same side lengths, so their areas are equal. This means the square on the side fills the hole on the inside.. Measure the side lengths of the large shaded square in Figure A and then in Figure B. They have the same side lengths, so their areas are equal.
- Using scissors: Cut off the little square on the side of Figure B, and use it to fill the hole inside Figure B. The result is a square that matches up exactly with Figure A.
- Using tracing paper: Trace the boundary of the little square on the side of Figure B and move the tracing paper over the unshaded hole. Doing this shows that the little shaded square is exactly the same size as the hole. Movingthat little shaded square to fill the unshaded hole creates a big shaded square. If the boundary of that big shaded square is traced and the drawing is placed over Figure A, it would up exactly with Figure A.
Activity Synthesis (Teacher Notes)
Start the discussion by asking students to indicate which of the three possible responses—area of Figure A is greater, area of Figure B is greater, or the areas are equal—they choose.
Select previously identified students to share their explanations. If no student mentioned using tracing paper, demonstrate the following:
- Decomposing and rearranging Figure B: Place a piece of tracing paper over Figure B. Draw the boundary of the small side square, making a dotted auxiliary line to show its separation from the large square. Move the tracing paper so that the boundary of the small square matches up exactly with the boundary of the square-shaped hole in Figure B. Draw the boundary of the large square. Explain that the small square matches up exactly with the hole, so we know the small, shaded square and the hole have equal area.
- Matching the two figures: Move the tracing paper over Figure A so that the boundary of the rearranged Figure B matches up exactly with that of Figure A. Say, “When two figures that are overlaid one on top of another match up exactly, their areas are equal.”
Highlight the strategies and principles that are central to this unit. Tell students, “We just decomposed and rearranged Figure B so that it matches up exactly with Figure A. When two figures that are overlaid one on top of another match up exactly, we can say that their areas are equal.”
Anticipated Misconceptions
Students may interpret the area of Figure B as the entire region inside the outer boundary including the unfilled square. Clarify that we want to compare the areas of only the shaded parts of Figure B and Figure A.
Standards
- 3.MD.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.
- 3.MD.C.7.d·Recognize area as additive. Find areas of rectilinear figures by decomposing them into non-overlapping rectangles and adding the areas of the non-overlapping parts, applying this technique to solve real world problems.