Lesson 16

The purpose of this warm-up is to elicit the idea that the triangles in the figure are similar, which will be useful when students see diagrams like this one in a subsequent activity. While students may notice and wonder many things about these images, the congruent angles that imply similar figures are the important discussion points. This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is Angle-Angle Triangle Similarity.

Display the image for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice with their partner, followed by a whole-class discussion.

Ask students to share the things they noticed and wondered. Record and display their responses for all to see. If possible, record the relevant reasoning on or near the image. After all responses have been recorded without commentary or editing, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to respectfully disagree, ask for clarification, or point out contradicting information. If similarity does not come up during the conversation, ask students to discuss this idea.

Because the angle of incidence of an object bouncing off another is equal to the angle of reflection, mirrors and pool tables make great contexts for using similar triangles. 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.

This activity works best when each student has access to devices that can run the digital applet because students will benefit from seeing the relationship in a dynamic way. If students don't have individual access, projecting the applet would be helpful during the synthesis.

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 \(x\) then the other piece is \(14-x\).)

There are many places 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 measure the height of something using indirect measurement with mirrors. 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 calculate the unknown height using similar triangles.

As shown in the diagram, students should place the mirror on the ground and then move so they can see the top of the building they are measuring in the mirror. 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 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.

Invite students to estimate the height of a building or tall object (flagpole, goalpost). Connect the image in the warm-up to the image in the task statement by informing students that in mirrors the angle of incidence is equal to the angle of reflection. Ask students for ideas about how the person in the image is using the mirror to estimate the height of the building.

Invite students to brainstorm about what measurements they could make to be able to calculate the height of the building. It may be helpful to display these two images and have students discuss why the top of the Washington Monument is visible in one reflection and not the other, and what that might tell them. (The photographer is standing too far away in one photo so the angle is too shallow to reach the top of the monument.)

Take students to the designated location with mirrors and measuring tools. Let them figure out how to place the mirrors and what to measure, given the diagram. Make sure students aren’t looking into the sun!

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 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 their calculated height? 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 the camera must be perfectly perpendicular to the ground, and not use a wide-angle lens, for the measurement to be truly accurate).
• 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.

Arrange students in groups of 2–4. Let them brainstorm freely.

After several minutes, share all the ideas and pick a few to do with the tools available. 

Take students to the designated location with the tools. Let them figure out what to measure, and how to create a diagram. Make sure students aren’t looking into the sun!

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 proved to be easiest to do. How confident are they of their calculated height? To the nearest yard? To the nearest foot? To the nearest inch?

Max, one of the authors of this unit, shared a story about using these methods in real life to measure the height of something he was curious about. Share Max’s story and note which methods of his the class came up with and which they didn’t, and which of their methods they would recommend to Max.

I wondered about the height of a wind turbine blade* (just the blade!) that was on display at the Minnesota State Fair. My first attempt to measure the height was to pace a certain distance away, and then try to measure the angle of inclination from my eye to the top of the blade using a homemade clinometer: I tied a weight to a string and tied that to a pencil and sighted off the end of the pencil up to the top of the blade, so that the angle between the pencil and the string was the same as one of the angles in the triangle I was trying measure. But I didn't have a protractor so I couldn't measure the angle.

My ​next strategy was to think about angles I could measure. I used paper folding to make a 45 degree angle and changed my strategy from using a known distance and measuring the angle, to using a known angle and finding the distance. I walked until it appeared that the top of blade was directly lined up with the tip of my pencil, and the string was hanging straight down at the 45 degree angle I had marked. Because the 45 degree angle was easy to measure, I happened to pick that, and then was happy to realize that the distance I was from the turbine blade was also the height of the turbine blade!

I shared my strategy with a volunteer who was answering math questions about the fair, who replied: “The isosceles idea is what I was thinking. I imagine something like a tree falling over - the original height and the reach along the ground have to be the same, so the two angles have to be the same, and they have to add up to 90 degrees. It's easy to make a 45-45-90 triangle out of a piece of paper. (Do you see how?) If you had one, can you think of a way to use similar triangles to figure out a good place to pace to, so you wouldn't have to measure the angle looking up?”

I ended up finally with the idea from the very beginning of this unit and basically made an eclipse! I held up my paper triangle until it lined up perfectly with the height of the turbine blade, and then made my friend Christopher stand where the third vertex of the triangle appeared to be, and measured his distance from the turbine. My homemade angle measurer was a bit more precise if I recall right, but one thing about the eclipse method is that I could have constructed other triangles, like a 30-60-90, or really any right triangle whose sides I could measure, and not had to worry about where Christopher stood. The original 45-45-90 triangle had him in the middle of a garden, but I could have had him stand in a good place and then made an eclipsing triangle, measured it, and scaled it up. Anyway the thinking I did to actually figure out what tools I could use to measure the triangle was really interesting! I wish I'd had or found a mirror to try the mirror method too.

*According to the Minnesota State Fair Foundation, the actual height is 123 feet.