Lesson 15

This warm-up prompts students to compare four images. It 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.

Ask each group to share one reason why a particular item does not belong. Record and display the responses for all to see. After each response, ask the class if they agree or disagree. Since there is no single correct answer to the question of which one does not belong, attend to students’ explanations and ensure the reasons given are correct.

During the discussion, ask students to explain the meaning of any terminology they use. Also, press students on unsubstantiated claims.

This info gap activity gives students an opportunity to determine and request the information needed to calculate side lengths of similar right triangles. This activity previews the trigonometry concepts students will use in the next lesson. Students have enough information about one triangle to determine the ratios of any pair of sides within it, and to use the ratios to find missing side lengths in the other triangles. This is the work students will do in trigonometry, where they will learn to look up the information about a triangle from a trigonometry table or using calculating technology rather than being given it in a diagram.

The info gap structure requires students to make sense of problems by determining what information is necessary, and then to ask for information they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and ask increasingly more precise questions until they get the information they need (MP6).

Here is the text of the cards for reference and planning:

Monitor for students using:

• a scale factor between triangles (\(XY = \frac{2}{3} PQ\))
• equivalent ratios between triangles (\(\frac{XY}{PQ} = \frac{YZ}{QR}\))
• equivalent ratios within triangles (\(\frac{XZ}{ZY} = \frac{PR}{RQ}\))

Tell students they will be calculating side lengths of similar triangles. Explain the info gap structure, and consider demonstrating the protocol if students are unfamiliar with it. 

Arrange students in groups of 2. In each group, distribute a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give them the cards for a second problem and instruct them to switch roles. 

If students are stuck at first, suggest that asking for the similarity statement would be a good place to start. Then they can draw a diagram and figure out what other information they might want to ask for.
 

If students are stuck after asking for the available side lengths, ask what type of triangle would allow them to calculate the third side. (A right triangle.)

Ask students when the Pythagorean Theorem works. (Only with right triangles.) Remind students to always check that a triangle has a right angle before using the Pythagorean Theorem.

Invite a pair to share who:

• used the scale factor between triangles (either by asking for it or figuring it out).
• used the ratio within one triangle to determine the sides of the other triangle (either by creating a scale factor or using equivalent ratios)

Both methods work and one might be easier than the other depending on given information. In a subsequent unit students will learn trigonometry which relies on within triangle ratios.

This activity gives students a chance to recall all of the relationships within similar right triangles they know to be true, and attempt to use those relationships creatively to best estimate the side lengths of the unknown triangle. Monitor for students:

• estimating the length of one side
• using tracing paper and other materials to help with the estimation
• estimating a scale factor or establishing a scale factor based on an estimated side
• using the Pythagorean theorem

Demonstrate how not to do this activity: Compare each side to its corresponding side using some gesturing and squinting, and announce after about 3 seconds that you’re done and that the sides are 5, 6, 7. Call on students to explain why that approach is not sufficient, what other clues they could use to check their thinking, and why 5, 6, 7 definitely can’t be the correct lengths of the small triangle. (5, 6, 7 doesn't work because \(\frac56 \neq \frac58\) so the sides aren't proportional.)

Invite students to share all the different tools they used in their estimation. If possible, surface each of these tools:

• estimating the length of one side
• using tracing paper and other materials to help with the estimation
• estimating a scale factor or establishing a scale factor based on an estimated side
• using the Pythagorean theorem