Systems and Submarines


Kelly DeNu

Unit Title:

Systems and Submarines 




Math and Science

Estimated Duration:

7 Days (Block Scheduling)  

Unit Activities:

Activity 1: Car Crash

Activity 2: Constant Velocity Buggies

Activity 3: Denisty Lab

Activity 4: Cartesian Data

Activity 5: Planning/Building/Testing

Background Knowledge: 

Students should have some knowledge of density beforehand, although the density lab on day three should address any gaps in this area.  Students should also know how to write an equation in slope-intercept form (y=mx+b) and also be able to graph using slope-intercept form.  


July 2013

The Big Idea (including global relevance)

Submarines are valuable to us because they are used to establish and maintain peace and they can explore the depths of the ocean not reachable by any other means. They can also be used for rescue and salvage missions in cases involving sunken or disabled vessels.

The Essential Question

How can we engineer a model submarine with a changeable density so that it can sink in the water, rise back up to the surface, and predict the time at which it will meet a scuba diver needing oxygen?

The Challenge

Students working in pairs will design, build, test, and redesign a model submarine capable of travelling to the bottom of an aquarium and back up to the surface again.  Students will be required using mathematics to predict where the submarine and the scuba diver will cross paths. It will need to meet several criteria.

The Hook

The submarine is on a mission to rescue a scuba diver who can only rise up at a slow rate.  The idea is to rescue the scuba diver and bring them back up to the surface before he runs out of oxygen! 

Teacher's Guiding Questions
  • How will we increase the 2 liter bottle’s density so that it will sink?
  • How can we decrease its density again to bring it back up?
  • What materials can we use to accomplish these tasks?
  • What methods will be best for attaching materials together and having them hold up to underwater conditions?
  • How can the submarine be equipped to rescue the scuba diver from the seafloor?
  • How can we determine when the submarine will reach the diver?
  • Can we calculate the rate of the sub descending?  How about the scuba diver ascending?  
To engage the students in conversation, the teacher must prepare a list of feasible guiding questions for The Challenge in the unit template prior to teaching, but use this as a guide only.  
ACS (Real world applications; career connections; societal impact)

Applications: students will encounter many situations in life where a knowledge of density and buoyancy will be useful to help them solve a problem.

Career connections: mechanical, structural engineers, naval personnel, salvage workers, lifeguards, and other careers would use these concepts on a regular basis.

Societal impact: Our nation relies on subs as a nuclear threat to enemy countries for the purpose of maintaining peace. They are also used to gain intelligence from other countries, salvage sunken items, exploring unknown areas, and fixing environmental problems, like oil spills from drilled wells underwater.

Engineering Design

Students will have to design and build a sub to rescue a scuba diver near the ocean floor.  

Academic Standards

Math - 8.EE.8 Analyze and solve pairs of simultaneous linear equations.

Science – Forces and Motion: 2.  Explain that motion describes the change in the position of an object (characterized by a speed and direction) as time changes. 3. Explain that an unbalanced force acting on an object changes that object's speed and/or direction.

Unit Activities

Day 1:Introduction to Systems of Equations  (Lesson 1 Activity 1)

  • Have students take the pretest and collect those (Lesson 1, Activity 0)
  • Intro to Systems of equations using a Nascar video example.
  • Pose the following question: “What causes the racing cars to catch each other?  And why do they crash?”
  • Activity 1 – Have a toy velocity “buggy” available to race to study the velocity/speed of the race car.  Students will find the constant rate of the car and derive an equation for the speed.  Use the worksheet “car crash” to guide the activity. Record and graph data. 
  • Students will then place another car (which will be slower) on the track and determine a speed for this car. 
  • Race the cars!  Give one car a head start on the track and then start the other car at the start line.  Determine what these two equations will be (one will be y=mx and the other will be y = mx+b with b being the starting point of the car and m being the car’s velocity/speed).  Conduct the race.  Determine after how many seconds the two cars will crash into each other. 
  • Graph the two equations on a piece of graph paper.  What is the point of intersection?  Does this match the “crash” time?
  • Discuss the point of intersection and what that looks like on a graph.  
Day 2: Constant Velocity Buggies (Lesson 1, Activity 2)

  • Clarify the discussion from activity 1 about the solution to a system of equations when both lines are positive, but with different slopes.
  • Show a video of a head on car crash to open discussion of:

What information would you need to determine the exact position of impact for two cars traveling at each other?

How would the equations differ from activity 1 in order to show that the two cars were traveling at each other instead of traveling in the same direction?

  • Students will then have their groups paired up with a group that they did not work with in activity 1. The paired groups should not have the same color buggy.
  • The Toy Buggy Chicken worksheet will be handed out and students will work with their groups to work through the tables and graphing in the worksheet. Students will test their theoretical solution in question number 8 with guidance from the teacher. The teacher needs to make sure they are starting both buggies at the same time and from the correct distance.
  • Students will answer the remainder of the questions in class or as homework if they are not finished. 
Day 3: Introduction to the challenge (Lesson 2)
  • Show students a video clip showing submarines in the process of rising to the surface and sinking below.
  • Construct a web of what they know about submarines; fill this in on the board with help from participants.
  • Briefly discuss some of the uses for subs: defense, exploration, rescue, etc.
  • Divide students into pairs and pass out the Challenge Sheet and have them read the unit criteria: address any questions or misconceptions
  • Display an empty 2 liter bottle and an array of possible materials for building the sub.
  • Write down five questions that they will need to answer about this project, and bring in an empty 2 liter bottle per student within the next three school days.
  • Research on the web (Lesson 3, Activity 1)
  • Collect the questions that were done as homework.
  • Distribute Guided Research handouts and take students to computer lab to find more information about their project.
  • Students should finish research handouts for homework if not able to complete them before the end of class time.
Day 4: Density Lab (Lesson 2, Activity 1)
  • Complete the Density Lab in class; conclusion questions may be done for homework if extra time is needed.
  • Investigate with Cartesian Divers (Lesson 2, Activity 2)
  • Discuss some of the useful information that was learned yesterday from the research session. Collect student papers.
  • Pass out the Cartesian Diver Activity handouts and materials.
  • Proceed through the activity together, modeling all of the steps along the way.
  • After the activity is completed, Divers should be disassembled and the parts returned to their places.
  • Conclusions should be answered and papers turned in.
  • Density Homework
Day 5: Submarine Brainstorming and Planning (Lesson 3)
  • Pass out the Submarine Brainstorming and Planning Sheets.(Worksheet a)
  • Introduce the Engineering Design Process to students
  • All materials should be displayed in an area where students can handle and inspect them.
  • Students should plan their sub designs and draw a labeled diagram on the Planning Sheet.
  • Students should bring any additional materials needed to build their subs to the next class period.

Begin building

  • Students should gather materials and build their subs. A large aquarium or other tub with water to a minimum depth of 12 inches should be available for testing their progress.
  • Any other materials needed should be brought to the next class period.
Day 6:  Finish building, test the designs, and modify (Lesson 3, Worksheet b)
  • Students should finish the building stage and modify as needed until they are satisfied with the result. Students will time and record how long it takes the submarine to get down to the bottom of the tank to judge if their designs are improving. 
  • Finished sub should be labeled with both builders’ names and safely stored until tomorrow’s launch.
  • Distribute the “Planning & Testing” worksheet (Lesson 3, Worksheet B)
  • Students will then need to test their sub and make adjustments as needed.  Three timed trials will be averaged together to get a unit rate of the submarine.
  • As a class, we will calculate the ascending rate of the scuba diver.
  • Students will write an equation for the submarine and then the scuba diver.
  • Students will then use the equations that they derived from the descending rate of the submarine and the ascending rate of the scuba diver to make a prediction at what point in time the sub will rescue the diver.  A graph will be created.
Day 7: Launch date and performance review (Lesson 3, worksheet C)
  • Groups will demonstrate their designs in the water tub. Results will be recorded on the Launch Results Sheet.
  • Different groups’ design aspects can be discussed, with advantages and disadvantages weighed.

Intersection Point

  • Students should discuss if the prediction that they made of the intersection point is actually what happened.  
Where the CBL and EDP appear in the Unit

(Lesson 3)


Students have historically been confused when trying to answer that age long question of, “when train a leaves a point at this time and train b leaves a point at this time, when will they meet?”  Using linear systems of equations, this lesson can help them “see” that the point of intersection can easily be predicted in a variety of ways

Additional Resources

Lesson 1 was adapted from Illuminations

Lesson 2 was adapted from Paul Schember

Density worksheet was found from abc exercises

Pre-Unit Assessment Instrument
Post-Unit Assessment Instrument

This unit was overall very engaging for the students.  They really struggled with the concept of being able to make the submarine sink.  They were really quick to brainstorm ways in which to make it float.  When trying to sink the submarine, students assumed that weight was going to be the trick.  However, they quickly discovered that unless their submarine was denser than water, it would never full sink and stay.  Eventually the students who used holes to release the air that was replaced by water were the successful submarines.  They learned that by adding rocks and heavy pieces, was not always the go to solution.