standard_description: Prove the Laws of Sines and Cosines and use them to solve problems.
problem_activity: Optional activity
In this activity, students use what they know about corresponding lengths and angles to show that one picture of a bird is
not
a scaled copy of the other. Unlike in previous tasks, minimal scaffolding is given here, so students need to decide what evidence is necessary to explain or show an absence of scaling. As students work, notice the different ways students use corresponding lengths and angles to think about scaling.
Identifying a specific measurement in one image and the corresponding measurements in the other (which show that one is not a scaled copy of the other) requires thinking about the structure of the pictures (MP7).
Monitor students for different approaches to show that one image is not a scaled copy of the other:
Identifying corresponding angles in the two images which have different measures
Identifying two pair of corresponding segments in the images with different scale factors
Select students who use these methods and ask them to present their approaches, in this sequence, during the discussion.
Digital
Launch
Arrange students in groups of 3–4. Provide access to geometry toolkits.
Give students 2 minutes of quiet work time followed by 1–2 minutes to discuss in groups how to find evidence that one picture is not a scaled copy of the other. Then, briefly discuss students’ ideas as a class. Consider listing the ideas for all to see and discussing which ones are likely to be most effective.
If students' ideas deviate from drawing corresponding points and segments and comparing distances and angles, guide them with some prompts. For example:
Pick a point that can be easily referenced (e.g., the tip of one wing) on one picture. Ask for the corresponding point on the other.
Ask if that pair of corresponding points could help them determine if one picture is scaled from the other. If not, ask what else might be needed.
Add another point and a segment connecting the two points. Ask if or how the segment could help, and so on.
Student Facing
Here are two pictures of a bird. Find evidence that one picture is not a scaled copy of the other. Be prepared to explain your reasoning.
Student Response
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Print
Launch
Arrange students in groups of 3–4. Provide access to geometry toolkits.
Give students 2 minutes of quiet work time followed by 1–2 minutes to discuss in groups how to find evidence that one picture is not a scaled copy of the other. Then, briefly discuss students’ ideas as a class. Consider listing the ideas for all to see and discussing which ones are likely to be most effective.
If students' ideas deviate from drawing corresponding points and segments and comparing distances and angles, guide them with some prompts. For example:
Pick a point that can be easily referenced (e.g., the tip of one wing) on one picture. Ask for the corresponding point on the other.
Ask if that pair of corresponding points could help them determine if one picture is scaled from the other. If not, ask what else might be needed.
Add another point and a segment connecting the two points. Ask if or how the segment could help, and so on.
Student Facing
Here are two pictures of a bird. Find evidence that one picture is not a scaled copy of the other. Be prepared to explain your reasoning.
[IMG]
Student Response
Teachers with a valid work email address can
click here to register or sign in
for free access to Student Response.
Anticipated Misconceptions
Students may draw two segments that do not share a point, or choose non-corresponding points and segments on the two figures. Refer students to earlier work involving polygons and point out pairs of distances that could be used for comparison and those that could not be. Remind them that we can only compare the corresponding parts, not just any two parts.
Activity Synthesis
Invite select students to share their reasoning. Ask how they knew that certain pairs of points in the images were corresponding points. Students could identify a unique feature such as the eyes or the tip of the wing. Make sure students understand that:
Corresponding angles on an image and a scaled copy have the same measure
Corresponding segments on an image and a scaled copy are related by the same scale factor
To conclude that an image is
not
a scaled copy of another, it is sufficient to find one pair of corresponding angle measures that are different or two pairs of corresponding segments with different scale factors.
Speaking: MLR8 Discussion Supports.
Give students additional time to make sure that everyone in their group can explain or justify why the copy of the bird is not a scaled copy. Then, vary who is called on to represent the ideas of each group. This routine will prepare students for the role of group representative and to support each other to take on that role.
Design Principle(s): Optimize output (for explanation)


aligns: no