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.

There are many places where students may have gotten stuck or had an insight into the problem. Invite students to share their thinking. Even students who did not get a complete solution might share:

• Diagrams they drew to help them understand the problem.
• How they found similar triangles.
• Equivalent ratios they set up.
• A scale factor they found.

Display the digital applet. Invite students to consider what level of accuracy is appropriate in various situations:

• Measuring segments with a ruler and angles with a protractor (Nearest mm and nearest degree are the best we can do with those tools.)
• Measuring segments and angles digitally (The computer can tell us many decimal places of accuracy, but rounding to the nearest tenth is more than enough precision.)
• Planning a bank shot while playing pool (Pool players do think about math but they won't be using rulers and protractors during the game, so they'll need to estimate.)

Note that when players are practicing, especially for trick shots, they do calculate angles and distances.

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.

If multiple students found the height of the same object, poll the class and display the results.

Invite students to reflect on possible sources of error as they compare the different measurements that students got. Are they off by a lot or a little? What might account for the differences? (The object might not make a right angle with the ground. We might not have measured the height to the person's eyeballs or the distance to the mirror exactly. We might not have been looking at the top of the object in the mirror.)

How confident are they of the height that they calculated? To the nearest yard? To the nearest foot? To the nearest inch?

Ask students to explain, using what they know about similar triangles, mirrors, and light, why this method works. Brainstorm with students what it is about mirrors that made us able to calculate the height of the object the way we did. (The angle of reflection matches the angle of incidence. So if we stood up straight to be perpendicular to the ground, there were two triangles with two pairs of congruent corresponding angles. By the Angle-Angle Triangle Similarity Theorem, the triangles must be similar.)

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.

If students are stuck, offer this image:
 

If students would like to try to measure angles but are struggling with how to use a protractor to measure the angle of inclination to the top of a tall object, show them how to attach string to a protractor, weight the end using a binder clips and attach a straw to look through.

Invite students to share their brainstorming, and record for all to see. Ask students to consider which of these methods seemed to be more accurate, and which they would prefer to do. How confident are they of the height they calculated? To the nearest yard? To the nearest foot? To the nearest inch?

If students struggle to calculate the exact location, invite them to record all the information they know. Suggest labeling the piece of information they want to know with a variable and continuing to describe the remaining information in terms of that variable. (If the total distance is 14 and one piece is , then the other piece is .)