This is the conclusion of a three day lesson applying right triangles. Here is day 1 and day 2. Eric is my apprentice teacher and he initiated this discussion:

Eric: Where is the angle? Here or here? What else did we find?

student: The height of the flagpole.

student: The distance away from the flagpole

Eric: One person from each group come up to the board and write your angle and distance from the flagpole on the board.

Eric: Let’s make a plan before we go outside.

We then discussed and agreed that we should actually measure the height of the flagpole when we put barbie on the pole. We also decided to find the exact ground distance to create a 30 degree angle, but that it looks like it should be approximately 45 feet from the base of the flagpole. Students also agreed that they would like to confirm their thinking that this angle will result in a safe speed for Barbie to zipline down from the top of the flagpole.

The Wrap Up

Eric made a sheet for students to reflect on the lesson.

Students sat with their partners from Barbie zipline day 1 and we begin by reviewing the scenario and their calculated flagpole height.

Next we discussed how this zipline will work. I used a string and a binder clip to make a model zipline, using a very steep slope for the zipline and I asked students to predict what would happen if Barbie came down a zipline like this. They agreed that she would fall too fast and get hurt. Next, I held the string almost horizontal and asked how this zipline would work – and students agreed that she would get stuck or mover too slowly.

I explained that the goal for this day is to use a model in class to determine a plan for Barbie to zipline down safely from the top of the flagpole. By the end of class, students had to know what angle of elevation they planned to use and how far from the base of the flagpole they needed to place the end of the zipline.

This day felt a little chaotic, but students did end up finding errors in their measurements by verifying their calculations in a variety of ways. The worksheet belowincorporated a range of geometry topics including:

Pythagorean Theorem

right triangle trigonometry to calculate lengths

inverse trigonometric functions to determine angles

similar triangles

Tomorrow, we test our calculations outside on the the flagpole.

I have seen posts about Barbie zipline on occasion over the past few years. I’ve avoided the lesson because it seemed like a lot of advance prep work and typically I don’t allow enough time plan this far ahead and work through constructing a zip line trial run to make sure it all works in advance. To keep it completely real, I also hadn’t seen any description or resources that I thought would fit my classes well. But I found myself approaching the end of a right triangle unit in geometry with 2 full block periods mapped on my unit plan labeled “Right triangle synthesis project – need to create.”

This is my first semester with a full time apprentice teacher,Eric, so I have help and some new motivation to make this a fun project for everyone this time too. This time of the school year, with short days, cold temperatures and no end in sight, it seems a lot of students appear pretty bored with school and many of the staff here are also struggling. I really just needed to lighten things up for the students and myself. The next logical thought: Queue Barbie and high quality pulleys.

We leaned heavily on Jed’s blog post and started with the activity guides included in his post, modifying them a little to incorporate his thoughts on how the lesson could be improved and our learning goals.

I like having students select a team name because it forces them to talk to each other before they being working with content. It increases collaboration and breaks down barriers with a safe opening topic for conversation.

Given the image below, use Mr. R to estimate the height of the flagpole. This led to students getting rulers and measuring on the image and a rich discussion on whether 4 inches is the same as 0.4 feet.

Discussion

Eric: What are some ways we could find the height of the flagpole?

student: climb up the flagpole?

student: find the angle?

Eric: What angle?

student: The angle of elevation?

Eric: Where does the angle of elevation go (sketches diagram)?

student: Do we know how tall Ken is?

student: Are we Ken in this situation?

Eric: What can we measure?

student: You could measure the distance from the flagpole to the person.

student: We could use that angle tool thing that Mrs. B carries around.

student: oooohhhhhh. yeah.

Eric: How could we use that? What else would we need to measure? Would we all have the same measurements?

student: The adjacent!

student: The hypotenuse!

student: oh! So we could use tangent.

Eric: I want you to measure two different times, switching roles. Why do you think we should do it twice?

student: To see if we get the same answer?

Eric: Will we all get the same answer?

Student: No.

Eric Why not?

student: Because it is not exact, but they should be really close.

Eric: Work with your group and make a plan before we go outside.

4. Measure & Calculate

Outside measurements, then back in for calculations, using this sheet from Jed Butler’s description as a guide.

Favorite question while measuring angles outside:

student: Is it possible to get the same angle of elevation if may partner and I are different heights?

Eric: What do you think?

5. Enter both of your calculated flagpole heights into the google form (accessed using a bit.ly address from their cell phones).

6. Justify flagpole height

Project the spreadsheet from the google form as students enter their flagpole heights.

Eric: We don’t know the actual height of the flagpole. Here are all of your calculated heights. We need to determine what number to use as the height of the flagpole. What are some was to analyze data?

In the spirit of Geoff Krall’s Problem Based Curriculum Maps, I attempted to organize my geometry curriculum and learning targets along with associated activities, tasks & lessons. In order to keep this as a useful document, I tried to only include the tasks that I have actually used in my geometry classes. I am interested in adding & deleting from this document regularly to keep if useful for me (and hopefully others). I plan to have the second semester completed this summer, as I am trying to develop this as I go this school year.

[update 7/25/16: It is finally completed!]

I have shamelessly stolen from all over the MTBoS, Math Vision Project, Engage NY & the Unit Blueprints Project and tried to give credit as much as possible.

Please share any criticisms, activities that I should add, activities that are misaligned, etc… in the comments.

After determining how to write the equations of circles through the Shell Center activity: Equations of Circles 1, my students needed to practice using equations of circles. I decided to find my high school on Google Maps and make a screenshot. I then imported the screenshot into Desmos and adjusted the size of the image so that 1 unit = 1 foot in my image. I shared the link using google classroom, but it could be shared in any way that you choose.

I had students submit the link to their final design using a google form, the response spreadsheet made it simple to manage and grade.

The assignment was to design a sprinkler system that covered all of the grass, at a minimum cost. I researched commercial sprinkler systems and found that the maximum radius of a sprinkler is 15 feet. Here are some students designs (click on the image to go to Desmos):

Next year I want to provide students with a list of pricing per foot for trenching, sprinkler pipe, heads for various radii, control box and also require that they calculate the cost for their design. I also want to require that they draw in the underground piping which would allow students an opportunity to review linear equations and piece wise functions as well. Additionally, calculating costs would force students to apply distance to find lengths between the sprinkler heads too.

[update 7/10: Desmos made a formula to calculate pipe length! I could have students explain what this formula does. click on the image to go to Desmos.]

I assigned this as a final project for my geometry classes. I wanted to see (and I wanted students to see) how well they could apply their mathematical skills to real world applications. I struggle with being less helpful, while keeping students from giving up, so i really studied all of the suggested questions that go with this lesson described by the Mathematics Assessment Project. I did not follow the lesson exactly as designed, but instead adapted it for my students. I broke the project down over four 50 minute class periods:

Day 1: Introduce activity, students work in pairs: turn in all thoughts & notes for feedback at end of class. This evening I replied with next steps / questions to keep students making progress the next day.

Day 2: Time to work, select students to informally present their noticing’s so far. Introduce the cup calculator for students to study patterns and test hypotheses.

Day 3: Students analyze sample solutions, then begin revising their own work.

Day 4: Final work day & reflections, discussions, presentations

Next year I’d like to add time at the beginning for noticing & wondering & individual student work before they begin working in pairs. My fear is that many students will not even try just because they are not sure where or how to start.

I’m surprised how much students struggle with four fours. I think I should do this earlier in the school year next year so that they can practice many the standards of math practice in an environment where they are more familiar with the basic math skills required. Additionally, I should add a part 2 where students use their work to prove whether or not all numbers can be constructed using 4 fours.

Since everyone had such a hard time last class, we started by watching this from 0:23 to 0:55 over and over and over. There was productive discussion. It helped. I will start with this and a discussion next time, maybe…I am struggling finding the balance of “helping” that results in productive struggle.

In some classes I also played the video of one of Fawn’s students explaining her thought process from her post on this activity. Most students were confused by this, and focused on her nail polish. I probably will not include this next time.

Since Fawn’s similar triangle lesson was so beautifully written here, I pretty much printed out her blog post and ran the class verbatim. I quickly realized that my students are not as enthusiastic problems solvers as Fawn’s and it’s because I don’t let them struggle enough. I’m working on it. They struggle a little, but at some point I jump in, and then kick myself later. This became evident to me as I compared my students reactions to Fawn’s descriptions. I got more of the “YOU’RE NOT HELPING ME!!” “It’s your job to help, you are not doing your job” types of comments. I stopped the class, we had a good group discussion on how they could be convincing that they found the exact point to hit the ball in order for it to go in the hole. They got back to work. They need more time. Some students figured out what to do right away: Others really were a mess. I am learning a lot, at least. We will continue on Monday. I’m sure it will be a success.

I loved this project. it went better then expected. Students worked hard and learned the difference between slant height & height while reinforcing volume and spacial reasoning.