This is the teacher guide for this lesson. A student-focused guide to assist learners as they perform the activity is available.
Teacher Guide
Ball versus Donut
Experimenting with line problem solving
Is it possible to connect three “houses” to three “utilities” without crossing the lines?
This resource was originally published in PhysicsQuest 2014: Quantum.
Is it possible to connect three “houses” to three “utilities” without crossing the lines?
- Playdough
- Six toothpicks
- Ribbon
Students start by discussing a problem. The experiment will guide students through one possible solution to the problem and then they will discuss the conclusions they can draw from their experiment.
- Total time45 - 60 minutes
- Education levelGrades 5 - 9
- Content AreaQuantum
- Educational topicConstraints, problem solving
This is the second topology experiment in this kit. This time, instead of looking at how a shape can move and flip while it is constrained in different ways, we’ll deal with what can happen as the space around the problem changes. This activity shows how a famous math problem, the “utilities problem,” can’t be solved on a flat piece of paper or on a ball, but can be solved on a donut shape, or torus. If you haven’t done Activity 3, Losing Your Shirt, you should be fine but it might help to read the “science behind…” section to get a bit of an intro. The info from that section won’t be repeated here so flip back to that section if you need to.
In this problem, the goal is to connect three houses to three utility stations (gas, water, and electric) without crossing lines. Clearly no real contractor would be worried about crossing utility lines. It seems kind of silly. But it is one of the oldest math problems in existence.
What your students will find is that it is impossible to connect the utilities to the houses without crossing lines no matter how much they try. When they are constrained to move on a surface like a sphere with no holes, there is no way they can solve the problem. It isn’t possible. They are forced to move in two dimensions. They are only allowed to move on the surface, not leave the surface into the third dimension. By trying and proving to themselves they can’t, they have done some high level math in a branch of mathematics called “graph theory.”
In graph theory, mathematicians (and your students!) prove mathematical ideas by trying to connect points with lines (usually on paper, but I’ve always had more fun with ribbon) and seeing how they can and can’t be connected. It doesn’t matter how the “houses” and “utility stations” are placed on the sphere or pancake. The fact that they can’t be connected is independent of where you put them. The fact that you can prove with some playdough and ribbons that can be applied to other surfaces like the earth or the universe is pretty darn cool.
This activity then takes things one step further and creates a space where the problem can be solved, a torus. When the same problem is done on a torus, it is easy to solve. The key thing is that a ribbon needs to be wrapped through the hole in the middle of the torus. Making it a torus instead of a sphere added an extra dimension, the third dimension. Now that the ribbons can move around into another dimension by going through the hole in the middle, the problem can be solved. What’s really neat is that this can be done on any surface that has one hole, such as a roll of tape or a coffee cup with a handle.
Key terms
These are the key terms that students should know by the END of the two lessons. They do not need to be front loaded. In fact, studies show that presenting key terms to students before the lesson may not be as effective as having students observe and witness the phenomenon the key terms illustrate beforehand and learn the formalized words afterwards. For this reason, we recommend allowing students to grapple with the experiments without knowing these words and then exposing them to the formalized definitions afterwards in the context of what they learned.
However, if these words are helpful for students on an IEP, ELL students, or anyone else that may need more support, please use at your discretion.
- Topology: A type of math and physics that studies shapes and knots.
- Constraint: A restriction of some sort that usually makes a problem harder (and more fun!) to solve.
- Perpendicular: Two things are perpendicular when they are at right angles to each other.
Objective
Students will experiment and solve problems about organizing houses and utility lines so they don’t cross. Students will analyze how different shapes affect this problem.
Before the experiment
- Ask & discuss
Pick three objects in the room. Look at their shapes, how they are put together, what is the same and what is different? Draw them.
- Turn & talk protocol
- Pair students up.
- Give them a minute to think quietly.
- Give students 2 minutes to discuss their thinking.
- Have students record their answers or share out to the whole group.
Setting up
In this activity you are going to see if it is possible to connect three utility stations to three different houses without crossing the utility lines. Toothpicks will play the part of the houses and utility stations and ribbons will act like the lines.
Take each piece of the colored ribbons and cut them into three pieces. You should now have nine pieces of ribbon, three of each color.
Tie one end of each of the three red ribbons around a toothpick. This will be your “gas” utility station and lines. Do the same for green (electric) and blue (water).
During the experiment
Make sure students are put into intentional groups. See above.
Students will complete the experiment using the Student Guide where we have outlined the experiment for students and along the way, they record results and answer questions.
In the Student Guide, they will look at their data to start to think about how other shapes could affect their solutions.
Continue to listen in on each group’s discussion; answer as few questions as possible. Even if a group is off a little, they will have a chance to work out these stuck points later.
Teacher tip
- Suggested STEP UP Everyday Actions to incorporate into activity:
- When pairing students, try to have male/female partners and invite female students to share their ideas first.
- As you put students into groups, consider having female or minority students take the leadership role.
- Take note of female participation. If they seem to be taking direction and following along, elevate their voice by asking them a question about their experiment.
- Consider using white boards so students have time to work through their ideas and brainstorms before saying them out loud.
- As students experiment, roam around the room to listen in on discussion and notice experiment techniques. If needed, stop the class and call over to a certain group that has hit on an important concept.
Consider using the RIP protocol (Research, Instruct, Plan) for lab group visits and conferring.
Consider culturally responsive tools and strategies and/or open ended reflection questions to help push student thinking, evidence tracking, and connections to their lives.
Conclusion
Follow the Write-Pass protocol to have students share and refine their thinking:
- Divide students into groups of four (different from their experimental groups).
- The teacher posts a question that students must answer with an explanation: What do all these worlds have in common?
- Students each write their own ideas on a loose piece of paper.
- Then pass the papers to the left.
- Each student silently reads the student’s response (and any of the other students’ comments, on iterating rounds of this process).
- Each student writes suggestions directly onto the original copy to help make their peers’ ideas sharper and clearer.
- Repeat the pass-read-edit until each student gets to read and comment on each other's ideas.
- The original author of each statement reads their peers' comments and writes a refined, final statement at the bottom of the paper to turn in.
After students have had a chance to discuss key ideas from the lesson and complete their student guides, you can now clarify and give concise definitions to the forces they experimented with.
- Real world connections:
- Allow students to play with the quantum logic games found here. Have them describe what the game shows and how they think this learning could be applied in their everyday lives.
- Suggestions for drawing, illustrating, presenting content in creative ways:
- Try the Wonders of Quantum Physics - Art + Polarization developed for the QuanTime project. They provide free kits and lessons.
- Engineering and design challenges connected to the content:
- Try some more advanced quantum games and lessons that explore logic gates and quantum circuits using the PhysicsQuest 2021: Introduction to the World of Quantum lessons (Mystery Tube, Particle-Wave Duality, Save Schrödinger’s Cat, Quantum Circuits).
- MS-PS4-2Develop and use a model to describe how waves are reflected, absorbed, or transmitted through various materials.
- MS-PS4-3-applicationsCCC: Influence of Science, Engineering, and Technology on Society and the Natural World. Technologies extend the measurement, exploration, modeling, and computational capacity of scientific investigations. (MS-PS4-3)
- MS-PS4-3-nature-of-scienceCCC: Science is a Human Endeavor. Advances in technology influence the progress of science and science has influenced advances in technology. (MS-PS4-3)
- MS-PS4-1-empirical-evidenceSEPs: Scientific Knowledge is Based on Empirical Evidence. Science knowledge is based upon logical and conceptual connections between evidence and explanations. (MS-PS4-1)
Credits
Written by Rebecca Thompson, PhD
Illustrations by Kerry G. Johnson
Activity illustrations and graphics by Nancy Bennett-Karasik
Updated in 2023 by Sierra Crandell, MEd, partially funded by Eucalyptus Foundation
Extension by Jenna Tempkin with Society of Physics Students (SPS)