Head on Collisions


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Kristy Hubbs

Unit Title:

Head on Collisions




College Calculus

Estimated Duration:

Eleven - 55 minute periods                         

Unit Activities:


July 22, 2015

The Big Idea (including global relevance)
  • Frontal impacts accounted for 53% of passenger vehicle occupant deaths in 2013.
  • Another aspect in car crashes is money. The estimated cost of motor-vehicle crash-related deaths, injuries and property damage was $276.5 billion in 2012, according to the National Safety Council.
  • It is extremely important to make sure that the vehicles being built are safe.
The Essential Questions:
  • How can we measure the safety of a car in a frontal crash?
  • What aspects of design make a car safer than others in crashes?
  • What factors make one crash worse than another?
Justification for Selection of Content:
  • Misconceptions regarding this content are prevalent.
  • Content is suited well for teaching via CBL and EDP pedagogies.
  • The selected content follows the pacing guide for when this content is scheduled to be taught during the school year. (Unit 1 covers atomic structure because it is taught in October when I should be conducting my first unit.)
The Hook:

Videos of Crash Tests



Pre/Post test:

One report found that high numbers of students were unable to solve problems that did not have an obvious process to follow to get to a solution, meaning that their prior mathematics courses have left important gaps in their conceptual understanding. These students tend to look at integral calculus as a series of processes with algorithms and do not develop the grasp of concepts which gives them versatility in thinking through the problem. More specifically, the biggest misconception that students have in this area is that they confuse the integral with the derivative. In this challenge they are using the acceleration to learn about its antiderivative, which is velocity. Many students will want to take the derivative of acceleration instead of anti-differentiating it. Many times students get caught up in the solution process that they don’t grasp the overall concept of the challenge.

Unit Lessons and Activities:

Lesson 1: Introduction (2 days)

This first lesson will focus on the math that is needed to describe the motion of a car when it is accelerating. The students will discover the relationship between the acceleration and the velocity as the area under the curve. This brings in integration and Riemann sums. They will also look at creating the graph of the velocity using the data that they collect from an accelerometer.

Activity 1: Hook, Big Idea, Essential ?’s, Challenge and Guiding ?’s (2 days)

Day 1: Show video and discuss the big idea. Have students share some essential questions that they thought about throughout the video and discussion. (I will be starting this on a Friday and then on Monday come in and tell the students that I took their ideas and created the challenge from them)

Day 2: Present the challenge and then have students come up with guiding questions.

Activity 2: Velocity Investigation (1day)

Day 1: Have students send cars down track to record acceleration without modification. Then have them plot the velocity graph using the acceleration data.

Lesson 2: Crash Test (8 days)

The second lesson will focus on crash testing the cars and then using calculus to evaluate their design and redesign.

Activity 3: Brainstorm, Test Materials, Analyze Results, Present findings. (4 days)

Day 1: Have students brainstorm ideas for their design.

Day 2: Build first prototype.

Day 3: First crash test and analyze results.

Day 4: Present findings.

Activity 4: Refine, Rebuild and Present (4 days)

Day 1: Brainstorm ideas for redesign.

Day 2: Rebuild

Day 3: Retest and analyze

Day 4: Present

Description of Challenge

Students will modify a pinewood derby car to have the lowest HIC (Head Injury Criterion) as measured by . This is the formula used by the National Highway Traffic Safety Administration (NHTSA) and the Insurance Institute for Highway Safety (IIHS). In this formula, a and b represent the start and stop time of the car, a(t) is the function that models the acceleration of the car. Students will use different materials and processes to lower the HIC of a pinewood derby car crashed at a distance of 35 feet. Each group will be provided with 2 cars to test.

List of Constraints Applied

Students will only be able to use certain materials provided by teacher. Students will have one day to research the material and two days to brainstorm and build.

Anticipated Guiding Questions
  • How can we make a car safer in a frontal crash?
  • What is the best design and materials to reduce the possibility of a head injury?
  • What are the important parameters in a frontal car crash?
  • What types of injuries seem to be the most devastating?
  • What area of the body is the most important to protect?
How will students test or implement the solution? What is the evidence that the solution worked? Describe how the iterative process from the EDP applies to your Challenge.

Students will design a car with the lowest Head Injury Criterion. The HIC will be measured using an accelerometer to measure the deceleration of a pinewood derby car that is crashed 35 feet from the start. The students will use the Engineering Design Process to develop and test their cars. Students will use calculus and integration to evaluate their design. Students will refine their design to develop the safest car possible.

How will students present or defend the solution? Describe if any formal training or resource guides will be provided to the students for best practices (e.g., poster, flyer, video, advertisement, etc.) used to present work.

Students are expected to present their findings twice throughout this challenge. They will first present the results of their findings after the first test crash. This will be presented only to their classmates as a brief oral report. They will also record their findings in their engineering notebook in Google Drive. The groups will then use that information to redesign their cars and test again.

This leads to the final presentation where the groups will make a poster that shows their progression through the EDP as well as all of their findings and analyses. There are specific requirements that the poster must have:

  • sketches of their design
  • pictures of their cars
  • graphs of the acceleration and velocity and the results of their designs

The students will be graded on their overall group using a rubric that will be designed by the class at the beginning of the year. The poster will also have a rubric to grade it on its components but the groups will also be presenting their posters to an audience of resource team members, administrators and other teachers. Members of the audience will visit each group and discuss with them their poster and results and award each group a grade based on their knowledge of the project, ability to communicate the challenge and their specific results and overall poster presentation.

What academic content is being taught through this Challenge?

This challenge covers many topics from earlier courses, as listed above, but the Calculus content is dictated by the Learning Outcomes listed from Cincinnati State:

Students will be able to examine the definite integral as a Riemann or Trapezoidal sum, as an area, and as the total change in the antiderivative.

Students will be able to evaluate definite integrals using antiderivatives and the Fundamental Theorem of Calculus, including integration by substitution.

Students will be able to use definite integrals to find areas of planar regions.

Real world applications:

Safety in the construction of automobiles is important for saving lives.

What activities in this Unit apply to real world context?

The hook shows first-hand the effect of frontal crashes and also the implementation process shows what happens to their design when it hits the wall.

Societal Impact:

Vehicles are tested and rated on their safety to ensure that they are safe as possible. This leads to lower insurance costs and increased quality of life.

What activities in this Unit apply to societal impact?

The students are investigating different materials to improve the safety of their car in a frontal crash.

What careers will you introduce (and how) to the students that are related to the Challenge? (Examples: career research assignment, guest speakers, fieldtrips, Skype with a professional, etc.)

The careers that students will see that are related to this challenge are automotive engineering as well as materials engineering. These two fields are continuously studying new designs and materials to use to make cars both safe and efficient. Other career fields could include biotechnical engineering, graphic design, and actuary sciences and claims adjusters. We will discuss these careers throughout the challenge, mainly during the essential and guiding questions. The students will all choose one career connected to the industry and research it for homework. They will then need to type a summary of their findings and submit it in their engineering notebook in Google Drive.

Unit Academic Standards - NGSS

1. Asking questions (for science) and defining problems (for engineering)
2. Developing and using models
3. Planning and carrying out investigations
4. Analyzing and interpreting data
5. Using mathematics and computational thinking
6. Constructing explanations (for science) and designing solutions (for engineering)
7. Engaging in argument from evidence
8. Obtaining, evaluating, and communicating information

Unit Academic Standards - CCSS

All Standards for Mathematical Practices apply as well as:
N –Q.1 Use units as a way to understand problems and to guide the solution of multi-step problems; choose and interpret units consistently in formulas; choose and interpret the scale and the origin in graphs and data displays.
N –Q.2 Define appropriate quantities for the purpose of descriptive modeling.
N –Q.3 Choose a level of accuracy appropriate to limitations on measurement when reporting quantities.
A-CED.2 Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.


Overall, the students performed very well on the post-test compared to the pre-test. There was an increase in the number of correct answers on each question. The highest increase was 4 on several questions.


I chose this content for this unit because it had direct connections to real world applications which allows the students to see the connection between what they are learning and aspects in the real world that affect them in a daily way. The purpose for selecting this content was met because the students could see in the hook video the importance of minimizing the impact of vehicles to keep the risk of head injury low. They were also able to see this when they ran their test cars down the track and into the wall, recording the acceleration data. The students were able to find solutions that supported the content in that their second iteration came out to be lower than their first which shows that they were able to lessen the impact of the car when it hit the wall.

A few things that I would do differently for the next implementation would be to use different cars. The ones we used were too narrow on the front end to be able to securely attach the students’ modifications to. They also made it difficult to keep the accelerometer in place without effecting the roll of the wheels. Getting the whole balsa wood block instead would alleviate both of these problems. I would also make sure to run several test runs before the student runs to make sure that it is collecting enough data and you know what area of the data to focus on for the calculation of the HIC.

I would definitely teach this unit again because of its direct connection to the real world. Since the students are seniors, connecting the unit to driving is very relevant to them since most of them have recently started driving themselves and are always being told to be safe. This gives them even more insight to how manufacturers are also making it safer to drive.

Next Generation Science Standards (NGSS)

Science and Engineering Practices)

Crosscutting Concepts

Asking questions (for science) and defining problems (for engineering)

Cause and effect

Developing and using models

Scale, proportion, and quantity

Planning and carrying out investigations

Systems and system models

Analyzing and interpreting data

Stability and change

Using mathematics and computational thinking


Constructing explanations (for science) and designing solutions (for engineering)


Engaging in argument from evidence


Obtaining, evaluating, and communicating information


Common Core State Standards -- Mathematics (CCSS)

Standards for Mathematical Practice

Make sense of problems and persevere in solving them

Reason abstractly and quantitatively

Construct viable arguments and critique the reasoning of others

Model with mathematics

Use appropriate tools strategically

Attend to precision

Look for and make use of structure

Look for and express regularity in repeated reasoning