The purpose of this Warm-up is for students to gain some familiarity with reflection angles. In later activities of this lesson, students will use this concept along with similar triangles to find distances in real-world applications.

By estimating a location using their intuition about reflected light, students reason abstractly and quantitatively (MP2).

Invite students to share where they think the laser will hit the wall and their reasoning. It is ok if students do not have exact values at this point.

Listen for and amplify reasoning that involves the angle at which the laser hits the mirror.

Display this image and explain that angle is called the “incidence angle.” This angle is measured between the segment that is being reflected () to a segment perpendicular to the reflecting surface (). Angle is called the “reflection angle” and is measured from the segment produced by the reflected segment () to a segment perpendicular to the reflecting surface ().

A law of physics says that the angle of incidence is congruent to the angle of reflection. In this example, each of these angles measures about . 

Ask students,

• “Are triangles and similar? Explain your reasoning.” (Yes. Angles and are right angles and angle is congruent to angle , because they are complementary to congruent angles. The Angle-Angle Triangle Similarity Theorem shows that these two triangles are similar.)
• “How could you use this similarity to find the exact point on the wall where the laser will hit?” (I could use ratios of side lengths to find the length of .)

In this activity, students apply their understanding of angle of incidence and angle of reflection to determine where to bounce a cue ball off of a rail so that it knocks another pool ball into a pocket. Billiards players really do estimate and measure similar triangles when setting up their trick shots, so this is a realistic context.

Allowing students to sketch first and then check their intuition with calculations builds on the idea that math should make sense, and that math should be a tool for being more precise and accurate when needed. Students have an opportunity to reason abstractly and quantitatively (MP2) in exploring this problem.

This is the first problem in this unit in which students must find an unknown side length without knowing the length of a corresponding side. Monitor for students who use guess and check (using both visual or numeric strategies) or algebraic reasoning to find the unknown lengths.

In the digital version of the activity, students use an applet to see the relationship between the angle of incidence and angle of reflection. The applet allows students to guess where the cue ball should bounce off the side wall, to see segment distances and angles, and then to check their guesses against the actual path. The digital version helps students see the relationship in a dynamic way. If students don't have individual access, projecting the applet would be helpful during the Activity Synthesis.

In this optional activity students go outside (or to a room with high ceilings, like a gym or theater) and use indirect measurement with mirrors to measure the height of something. In the previous activity, students used the fact that equal angles of incidence and reflection create similar triangles to analyze a diagram of a bank shot on a pool table. In this activity, they have to figure out how to use the mirrors to take advantage of the equal angles, and what measurements they should take to be able to use similar triangles to calculate the unknown height.

As shown in the diagram, students should place the mirror on the ground and then move so they can see, in the mirror, the top of the building that they are measuring. Marking an X in the center of the mirror and lining up the top edge of the building with the mirror might help students understand what they are trying to see in the mirror. Students may need to move the mirror as well as their bodies to line up the building correctly. Measuring the distance from the base of the building to the mirror, and from the mirror to their toes, will allow them to set up similar triangles and to use their height to find the height of the building.

This activity emphasizes aspects of mathematical modeling (MP4), such as making assumptions, accounting for error and imprecision, and applying mathematical relationships (similar triangles) to solve problems in the real world.

In middle school, students may have used shadows to measure the height of a tall object. In this activity, students brainstorm their own methods for indirect measurement. Then they try out the methods that seem like they will be accurate and possible to do with the tools available. Some methods that students have brainstormed in the past include:

• Taking a picture of the object and a reference object and figuring out the scale factor based on the reference object’s height. (Explaining why this has to do with similar triangles requires some knowledge of optics. The explanation draws attention to the fact that, for the measurement to be truly accurate, the camera must be perfectly perpendicular to the ground and should not have a wide-angle lens.)
• Measuring the angle up to the top of the object from a certain distance away, and then constructing a similar, smaller triangle with that same angle that they can measure each side of to calculate the unknown height.
• Creating a smaller right triangle and holding it up with one eye closed so that it appears the same height as the building, and having a partner stand where the other leg appears to end, measuring that distance, and using it to compute the scale factor.
• Creating an object that appears to be the same height as the building when held at arm’s length with one eye closed, and measuring the distance from the eye to the top and bottom of the object, and from the bottom of the object to the bottom building, and finding the scale factor of the dilation.
• Measuring the building’s shadow and the shadow of a reference object.

This activity emphasizes aspects of mathematical modeling (MP4), such as making assumptions, accounting for error and imprecision, and applying mathematical relationships (similar triangles) to solve problems in the real world.