Exploring Circles
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to compare two figures and use the characteristics of those figures to help them sketch a possible third figure that has various characteristics of each. It invites students to hold mathematical conversations and explain their reasoning (MP3). It gives the teacher an opportunity to hear how students use terminology and talk about figures and their properties.
There are multiple reasonable answers to the question, and students should be encouraged to be creative. The grid is provided to allow students the opportunity to discuss side lengths, find the area and perimeter of Figure A, and estimate the dimensions of Figure B.
Launch
Arrange students in groups of 2–4. Display the image of the two figures for all to see. Make sure students understand that they are to draw a third Figure, C, that has features of both Figures A and B but more closely resembles Figure A. Give students 2 minutes of quiet work time to sketch Figure C, and then give them time to share their thinking with their group. After everyone has conferred in groups, ask the group to share the characteristics used in generating different versions of Figure C, and ask them to show an example of one of the sketches.
Student Task
Here are two figures.
Figure C looks more like Figure A than like Figure B. Sketch what Figure C might look like. Explain your reasoning.
Sample Response
Answers vary. Possible solutions:
- This first possibility for Figure C is almost identical to Figure A but with slightly rounded corners like Figure B. Both in size and shape, it is closer to Figure A.
- This second possibility for Figure C is a polygon (quadrilateral, parallelogram, and rhombus) like Figure A, but it is taller than it is wide like Figure B.
- This third possibility for Figure C resembles Figure A in that it is the same distance across the figure, from left to right or from top to bottom. Also like Figure A, it can be rotated, and it still look exactly the same. It looks like Figure B in that its sides are curved rather than straight line segments.
Activity Synthesis (Teacher Notes)
After students have conferred in groups, invite each group to share the characteristics that were important to them in creating their third figure. Some important points to be brought out include
- Figure A is a polygon, and Figure B is not a polygon.
- Figure A has the same width and height while the height and width of Figure B are very different.
- The area of Figure A is 36 square units while the area of Figure B is about 30 square units.
- The perimeter of Figure A is 24 units. The perimeter of Figure B is hard to determine, but it is less, maybe about 22 units.
Encourage students to be as precise as possible as they describe why they chose the figure they drew.
Display the responses for all to see. Because there is no single correct answer to the question, attend to students’ explanations and ensure that the reasons given are correct. During the discussion, prompt students to explain the meaning of any terminology that they use and to clarify their reasoning as needed.
Standards
- 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.
- 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.
20 min
5 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Exploring Circles
5 min
Teacher Prep
Setup
Narrative
This Warm-up prompts students to compare two figures and use the characteristics of those figures to help them sketch a possible third figure that has various characteristics of each. It invites students to hold mathematical conversations and explain their reasoning (MP3). It gives the teacher an opportunity to hear how students use terminology and talk about figures and their properties.
There are multiple reasonable answers to the question, and students should be encouraged to be creative. The grid is provided to allow students the opportunity to discuss side lengths, find the area and perimeter of Figure A, and estimate the dimensions of Figure B.
Launch
Arrange students in groups of 2–4. Display the image of the two figures for all to see. Make sure students understand that they are to draw a third Figure, C, that has features of both Figures A and B but more closely resembles Figure A. Give students 2 minutes of quiet work time to sketch Figure C, and then give them time to share their thinking with their group. After everyone has conferred in groups, ask the group to share the characteristics used in generating different versions of Figure C, and ask them to show an example of one of the sketches.
Student Task
Here are two figures.
Figure C looks more like Figure A than like Figure B. Sketch what Figure C might look like. Explain your reasoning.
Sample Response
Answers vary. Possible solutions:
- This first possibility for Figure C is almost identical to Figure A but with slightly rounded corners like Figure B. Both in size and shape, it is closer to Figure A.
- This second possibility for Figure C is a polygon (quadrilateral, parallelogram, and rhombus) like Figure A, but it is taller than it is wide like Figure B.
- This third possibility for Figure C resembles Figure A in that it is the same distance across the figure, from left to right or from top to bottom. Also like Figure A, it can be rotated, and it still look exactly the same. It looks like Figure B in that its sides are curved rather than straight line segments.
Activity Synthesis (Teacher Notes)
After students have conferred in groups, invite each group to share the characteristics that were important to them in creating their third figure. Some important points to be brought out include
- Figure A is a polygon, and Figure B is not a polygon.
- Figure A has the same width and height while the height and width of Figure B are very different.
- The area of Figure A is 36 square units while the area of Figure B is about 30 square units.
- The perimeter of Figure A is 24 units. The perimeter of Figure B is hard to determine, but it is less, maybe about 22 units.
Encourage students to be as precise as possible as they describe why they chose the figure they drew.
Display the responses for all to see. Because there is no single correct answer to the question, attend to students’ explanations and ensure that the reasons given are correct. During the discussion, prompt students to explain the meaning of any terminology that they use and to clarify their reasoning as needed.
Standards
- 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.
- 7.G.A·Draw, construct, and describe geometrical figures and describe the relationships between them.