## How high is that wall?Elizabeth Daugherty

New Pit Wall

Students are told that the teacher is curious about the height of the wall in the New Pit (an area with a two story tall wall with skylights above it). Unfortunately, since the wall is so tall, it isn’t possible to measure the height with a measuring tape. So, geometry will be needed to complete this task. More specifically, students will need to learn trigonometry first.

#### Know the Learners

• Low-level geometry
• Twelve high school sophomore students, ten with IEPs
• Co-taught with special education and mathematics teachers
• Alternating block schedule (90 minutes every other day)

#### Standards/I Can Statements

G.SRT.6 Understand that by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles.

• I can identify which trigonometric ratio should be used to find a missing side in a right triangle.
• I can use trigonometric ratios to find a missing side length when given a side and an angle of a right triangle.

G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.

• I can accurately measure an angle of elevation using an altimeter.
• I can accurately measure lengths using measuring tape.
• I can use trigonometric ratios and measurements to find missing lengths in real world situations.

#### Lessons Prior to the Performance Task

Day 1:

The lesson will start with students learning how to determine which side in a right triangle is the opposite, adjacent, and hypotenuse when given a specific acute angle. They will then be shown a Sketchpad demonstration that introduces the idea of sine, cosine, and tangent. The focus will be on the idea that trigonometry is the relationship between the sides and angles in a right triangle and that the trigonometric value of a specific angle stays the same no matter the length of the sides of the triangle. After this demonstration, students will be told that they are going to become experts on one of the trigonometric ratios. The students will be split into three groups. One group will learn about the sine ratio with a teacher, one group will learn about the cosine ratio with the other teacher, and one group will learn about the tangent ratio with a video. Students will complete a worksheet while learning about their ratio. At the end of class, an online trig “mini golf” game will be used to summarize what the students learned in their separate groups for the entire class. Students will have homework on their specific ratio.

Day 2:

There will be a brief class discussion of what was learned last class. Afterwards, the students will be split into new groups in which each group has a sine expert, cosine expert, and tangent expert. They will all be given the same worksheet that will need to be completed in the group. Each expert will be expected to teach the other students about their ratio as it is used. The teachers will walk around and help as necessary. In order to prevent students from simply splitting up the problems and just working on their own ratio, the numbers for the sides will be given to students as they complete the previous problem. Students will have homework on all three trig ratios.

Day 3:

Class will begin with a Kahoot in which students need to determine which trigonometric ratio that should be used to solve for a missing side. They will then have time to practice finding the side lengths of right triangles through a relay activity. Each student will be given a question and when they have the answer correct, they will receive the next question. Students will be reminded that the reason they learned all of these ratios is because we wanted to measure the height of the wall in the New Pit. They will then complete the performance task described in detail below. Homework will be a review of all the trig ratios in preparation for the quiz the next class.

#### Learning Community

Both in the lessons leading up to the performance task as well as during the learning task, students are immersed in a learning community. During the lessons, they become experts on their own specific trig ratio then have to share their knowledge with others. During the class discussion prior to the performance task, they work together to come up with a plan to find the height of the wall. This uses all of the students’ knowledge and perspective on the situation. Then they work together in pairs to complete the task. Throughout these lessons students work together to help support each other’s learning.

Overview:

Students will work in pairs to complete a worksheet that guides them through determining the height of the wall in the New Pit. In order to do this, they will need to make accurate measurements of their distance from the wall and their eye level. They will then have to determine the angle of elevation to the top of the wall using an altimeter. They will then need to use their measurements and knowledge of trigonometry to find the height of the wall.

Materials Needed:

• How high is that wall? worksheet
• Measuring tape
• Altimeter
• Calculator with trig functions
Altimeter

Class discussion before activity:

The teacher will tell the students that the goal is to determine the height of the wall in the New Pit. However, we do not have a ladder or a measuring tape that is long enough. The students will then be asked how they can use trigonometry to find the height of the wall. The teacher will lead the students through a discussion in which they come up with the plan of which measurements they will need to make in order to accurately find the height of the wall.

Activity:

The class will be split into two groups and each student will work with a partner. The class is split in half so that the Pit is not too crowded when the students are making their measurements. Both groups will have tasks to complete in the classroom and in the Pit. In the classroom, the students will need to measure the eye height of the partner that is using the altimeter. In the Pit, students will need to measure their distance from the wall and the angle of elevation to the top of the wall. The measuring tape will be taped to the ground so that the students can easily and accurately determine their distance from the wall. One partner will use the altimeter to spot the top of the wall while the other student reads the angle measurement. Once these measurements are completed, students will return to the classroom and send a new pair of students to the Pit to take their measurements. When the students return to the classroom, they will need to determine which trigonometric ratio they should use to find the measurement of the wall. They will then need to complete the appropriate calculations to find the height. Because everyone’s eye height is different and the distance from the wall may be different, each pair will be working with different values. However, they should all have about the same measurement for the height of the wall. Throughout the activity, the teachers are there to answer questions from the students and to double check measurements if the students desire.

Example of student work

Class Discussion after activity:

Students will share their answers with the class. Some questions that will need to be addressed are:

• Why did you chose to use the trigonometric ratio that you chose?
• Does anyone have the exact same answer? Why might everyone have a slightly different answer? Should it be the same?
• Do you need to consider your eye height in the final answer?
• As a class, can you agree on the measurement of the wall?

#### Rubric

There are a few changes that I would make to the rubric next year: Combine the two “I Can” statements that refer to measuring. Add an “I Can” statement that refers to the students participation in the discussion and the activity.

#### Results

• Exceeds: 60%
• Approaches: 40%

All students exceeded in the first two “I Can” statements in which they needed to identify the correct trig ratio and correctly find the missing side of the triangle using that ratio. They have had a lot of practice doing this so it was great to see that they were successful in performing their new skill.

Some students struggled with measuring the distance and angle of elevation. My co-teacher and I expect measuring to be troublesome for them, as we have had this issue while measuring lengths throughout the year. We were both positioned by the students while they were measuring to help them obtain accurate results. However, some students chose not to take the measuring of the angle seriously which resulted in inaccurate results. Students that did not exceed in the “I Can” statements regarding measuring did so because of a lack of effort, which can be prevalent in this class.

The last “I Can” statement referred to the students ability to transfer their knowledge to a real world situation. In this case, students needed to think through this problem (which we did through a whole class discussion) and remember to consider the eye height of the person who measured the angle of elevation. Only one of the five groups successfully did this without prompting from the teacher. However, once it was pointed out to students that they did not find the height of the entire wall, all considered the diagram we drew during the class discussion and promptly changed their answer.

#### Reflection

Overall, this was a good task for my students. They were able to see the application of trigonometric ratios in real life. During the class discussion the students were excited about finding the actual height of the wall. They were all very good at sharing ideas of how we could go about finding the height given what we know were able to measure. There was a lot of scaffolding within the discussion and the actual performance task which is definitely needed in this class.

I wish that there had been more time available to also go outside and measure the height of the flagpole or building. I think that after finding the height of the wall, the students would have been more confident in finding other heights. They seemed to excited to be using a math concept in a real world situation and I think that they genuinely enjoyed the experience.

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