**ACS (Real world applications; career connections; societal impact)**

A – (Real World Application): Safety in high rise construction, can also be used in climbing gyms or transportable climbing walls.

C – (Career Connections): Construction, Mechanical engineering, climbing gyms.

S – (Societal Impact): Falls account for one-third of all fatalities in construction and with the growing popularity of rock climbing in the United States protecting people from falls is more important than ever.

**Engineering Design Process (EDP)**

The students will design a mechanical device that lowers multiple objects weighing 2.5 to 12.5 pounds from a height of 7 feet to simulate a fall.

The students will use the Engineering Design Process to develop and test their device. Students will use calculus and rates of change to describe the motion of the falling object. They will use rates of change from position to velocity, and velocity to acceleration as a way to evaluate their device. Students will then refine their design to develop the best product possible.

##### Unit Academic Standard

Topics from the AP Calculus course guide.

- Derivative presented graphically, numerically, and analytically.
- Derivative interpreted as an instantaneous rate of change.
- Interpretation of the derivative as a rate of change in varied applied contexts, including velocity, speed, and acceleration.

##### Unit Activities

**The Math Moving objects:**

Lesson 1 will focus on the math needed to describe the motion of a falling object. For this project we will introduce the relationship between position, velocity and acceleration as a rate of change (introduction to derivative). Students will develop equations to model the data that is collected.

Activity 1: Big idea (essential and guiding questions)

Activity 2: Study of gravity

**Mechanical descending device:**

Lesson 2 will focus on the requirements for a safe fall. The students will research the criteria for a safe impact with the ground. Students will design a working model that lowers a weight from a height of 8 feet. The students will use calculus to evaluate their design.

Activity 3: Research (trip to local climbing gym to research current solutions)

Activity 4: Design auto belay device.

Activity 5: Test and redesign.

Activity 6: Presentation of solution.

##### Where the CBL and EDP appear in the Unit

On the first day the students will be presented with the challenge which will start the engineering design process. The groups will be tasked with developing a working model to control the descent of an object (similar to an auto belay device found at a Rock Climbing Gym.) The students will investigate derivatives as a rate of change by graphing position over time, then creating tables and graphs for velocity and acceleration using rates of change. The groups will use Google Docs to document their steps though the engineering design process.

**Activity 1 and 2**

**Identify Problem:**

** **The students will identify the problem. How can we safely lower a person that falls during a construction accident? The students will have to define the constraints of the problem.

**Activity 3 and 4**

**Brainstorm ideas:**

At this stage the students will have to think about how they are going to slow the rate of descent. In activity 3 the class will take a trip to RockQuest Climbing Center (a local rock climbing gym) to do some research on ways to slow a person’s rate of descent. This will start the brainstorming process which will lead to the initial design phase.

**Design:**

Groups will develop their initial design during class time as well as outside of class over the course of 5 days.

**Activity 5**

**Test:**

Students will test their design by dropping 5 different weights from a height of 8 feet and videotaping the fall. Students will use the video to collect data on position vs time. For each weight the students will develop tables and graphs for the velocity and acceleration of the object. Students will use the data to evaluate the effectiveness of their device.

**Redesign:**

** **Students will have 5 days to refine their design and retest.

**Activity 6**

**Communication:**

The students will give a presentation to the class on their progress through the engineering design process high-lighting key components with pictures, sketches, tables, graphs and equations.

##### Misconceptions

Students have difficulty with the limit definition for derivative as well as the application of derivative to real world problems.

##### Additional Resources

Prevention Videos (v-Tools): Construction Hazards / Falls / Leading Edge Work

https://www.youtube.com/watch?v=l6Xu19Q5cIE

Rock Climbing Fall

https://www.youtube.com/watch?v=qJEopoD-YYI

Climber takes a big fall!

https://www.youtube.com/watch?v=fqgk59Qr8ro

Wood structures as back drop for falling objects.

##### Pre-Unit Assessment Instrument

##### Post-Unit Assessment Instrument

##### Results: EvidenPost Assessmentce of Growth in Student Learning

After analyzing the Pre-Assessment and the Post-Assessment data. I noticed that the grades increased significantly. The test was based on the Chapter 2 Unit test over derivatives. The test was comprised of 9 free response questions. The average score on the Pre-Assessment was a 1.5% The scores were low because the students had not prior knowledge of derivatives. The average score on the Post-Assessment was a 67.5%. The average student increased their score by 66 percentage points.

##### How to Make This a Hierarchical Unit

Using a similar unit students could create an auto belay with the requirement that the velocity be constant thus the position function would be linear (increasing, decreasing, negative slope). The students could simply record the motion similar to the way we are doing it in this unit. Create a table of values for position and time and graph the point’s time on the x-axis and distance on the y-axis. The student could then find the line of best fit. This will open up discussion about the meaning of slope in this case (units being distance/time. This will also open up a discussion about the meaning of the x and y intercepts in context.

##### Reflection

Overall I am pleased with this unit. The students learned the intended math. Students learned the relationship between position velocity and acceleration as it relates to the derivative. Students did real world data collection of falling objects and analyzed that data using Excel. The students used the regression tool to find the equation of best fit which was a quadratic. Students then used their data to find the slope between consecutive data points (average velocity) and compared that to the derivative (instantaneous velocity). Students created and tested their devices using the same procedures used for the falling object so they became more efficient each time.

The data collection at rock quest was the most fun for me and the students because it gave them a chance to see and use a device similar to the one we are trying to create. It also gave them the opportunity to collect and analyze real world data. As the students climbed to the top of the 35 foot wall and let go at the top the group collected data as the student fell in 5 foot increments. The students collected data using two types of belay devices. The manual belay uses a second person and a friction device called and ATC (air traffic controller). The person on the ground lowers the climber safely to the ground. The second device is an auto belay. An auto belay is a mechanical device designed to lower the climber to the ground safely. The auto belay is more consistent lower the person at about 2.5 feet per second according to our trials. The trip to the climbing gym (RockQuest) was a success the students had fun, collected and analyzed data pertinent to the Challenge. The only shortcoming for this part of the unit was the fact that there was only 1 really easy route for the students to climb (a 5.6) The other too routes on the auto belay system were 5.9’s that were a little too hard for the students as all but 1 had never climbed before. I could not get the person at the climbing gym to reset easier routes for my students. When we arrived on of the workers at Rock Quest had moved the auto belay to the beginner wall for us but I did not bring the materials to mark the wall every 5 feet. The students all collected data but many of the groups had to wait for the 5.6 route to open up so they did not have time to climb the other routes. The other short coming for this part of the unit is cost. Rock Quest gave us a pretty good deal at $8 per student with harness and shoe rental (typically $25) but I do not know if this would have been possible without funds from CEEMS.

The solution that I intended for the students to find was something similar to the auto belay. A mechanical device that worked universally with different weight objects that required not adjustment for each weight. In the end all but 1 group ended up with a solution that involved counter weights to some degree. This did not meet my intended criteria of a universal solution. My inability to see the “easy way out” limited the effectiveness of the EDP. The students came up with solutions that limited the effect of gravity but not as I had intended. This could be due to the fact that the initial solutions failed for all of the groups.

Another short coming for this unit was technology. One technology issue was the cameras used to record the falling objects. Many of the cameras we used only shot at 30 frames per second. This led to a lot of motion blur as the objects fell. This was worse during the gravity lab when the object was in free-fall, and during the test of the initial designs that failed to slow the object. If this project is done again using a camera that shoots at least 60 frames per second is recommended, 120 frames per second would be best. The other technology issue was the video editing software. Through the course of the Units that I have taught I have learned to use Windows Movie Maker. I was promised 5 PC’s in my room that run windows. That did not come to fruition and I had to use the schools MAC laptops. We found that i-Movie was not conducive to what we wanted to do. We could not get the time stamp (HH:MM:SS) to appear so that the students could collect the data of position over time. I had to bring in my 2 lap tops and have students bring in their own so the data collection could take place. On the other hand using google docs helped the students collaborate with each other outside of school hours after they learned the subtleties of the spreadsheet program.

I had initially selected this content to give students a reason and purpose for learning derivatives and show them how it relates to the real world. While the post test scores do not necessarily show that students truly learned the material the conversations that we have had in chapter 3 lead me to believe they understand how the topics relate and they can articulate how to find the velocity and acceleration of an object and what it means in terms of the derivative.

When I reteach this lesson I will get laptops or PC’s that have Windows Movie Maker, and cameras that shoot 60 frames per second minimum. I will also require the students to make a universal belay device that does not need adjustment for each individual weight (counter weight systems are not allowed and will receive a zero.)