# Constrained Construction Problems

I was motivated during my flight today to come up with physics problems that have multiple right answers, have a low barrier to entry and a high ceiling. Here’s my go at it, along with thoughts.

The idea behind these is students are supposed to come up with as many ways as possible.

1. Draw as many velocity vs time graphs that show an object moving +45m from where it started.

Extend 1: Describe each in words.

Extend 2: Pick one and draw its corresponding position vs time graph.

2. Draw pictures depicting situations where a normal force exerted on an object is different than the objects weight.

Extend 1: Pick one to draw a free-body diagram that will help you to explain your reasoning.

Extend 2: Categorize them by Fn > mg and Fn > mg.

3. Draw a picture of a situation where the initial and final states consist entirely of potential energy.

Extend 1: Draw energy pie charts for the initial and final state and at least two in between.

4. Identify the mass and initial velocities for two objects objects that when they collide, they stick together and remain motionless.

5. Draw free body diagrams for an object that will accelerate at 1 m/s/s.

6. Draw a velocity vs time graphs and categorize them into those that involve an object turning around and those that do not .

Extend 1: Come up with a rule.

Extend 2: Do the same for position vs time.

7. Draw a force that acts on an extended object such that the Torque due to that force is CCW.

Extensions: multiple forces where net torque is…

Brian’s Development Rules of Thumb:

– Situations should involve relationships with wiggle room. For example, consider a = Fnet / m. Not only can Fnet and m vary but the same Fnet can be accomplished in different ways. Torque similarly has wiggle room in location, angle, choice of pivot qualitatively and force, distance, angle quantitatively.

– Design around tasks that get close to known difficulties, but don’t over constrain things to make it narrowly about the difficulty. For example, don’t do, “Negative acceleration and speeding up”. Just do speeding up velocity graphs and see what happens. Or if you are going to go right at difficulties, don’t make it a trick or you being clever. My normal force situation I think tackles a difficulty in a straight forward manner and it may work, because there are do many ways to do this.

– I like processes where initial and end states are constrained but not the process in between. (Energy example above). This provides a large variety.

– I think you want choose representations very deliberately. Perhaps, ask students to start with or move to representations that support semi-quantification, or ask them to extend to multiple representations. I think it’s OK to start with picture, but it’s important to bridge to a representation (Normal and Energy are examples)

– When using in class, I would want to think carefully about the sequence of individual work leading to group work leading to whole class sharing and discussion.

– If I designed the task with a particular issue to come up and it didn’t spontaneously, I would just introduce it and ask students to consider it.

– I think these tasks are very amenable to the Five Practices for Orchestrating Productive Discussions framework. (Link to come on an edit)

Anyway, what do you think? I’m interested in what others would come up with.

I watched the Jo Boaler video you posted which talked about “low floor, high ceiling” tasks. What I liked about the one she showed (the growing pyramids) is that anyone with even a very basic math background could get going on it, and it easily gets you into pretty advanced math (graphing and calculus — why is the derivative of quadratic function a linear one) if you want it to. I could imagine spending weeks on this question and representations around this question. And if we think of math as being all about patterns, this is clearly “doing math.”

It also got me thinking about similar science tasks (“is every color in the rainbow” seems comparable — little kids can discuss this, and so can graduate students in physics).– And then (because I was recently at the Exploratorium) exhibit-like things that have low-floor, high-ceiling in their ability to sustain engagement around really core ideas. This one (https://www.youtube.com/watch?v=g8Us3VDO9bY) was really great — why do they go so slowly downhill? Which ones go faster? Why? How fast are they going? etc. (Have to admit that I saw someone ‘teach’ this exhibit and kind of ruin everything cool about it.) — If I think of science as being all about mechanisms (and then things that support our arguments for certain mechanisms), then the Downhill Race is good for that. (So is a heat camera, and making lifelike stop motion animations.)

These tasks you’ve written feel really different to me. Maybe more like assessments (?) – I might be selling students short, but I think you’d need to have practice with graphing, or with freebody diagrams to make sense of what you’re asking. In most cases, I want to say that the core physics idea is good practice with what is meant by a term. I’m not saying that’s not important — or that it’s not low floor/high ceiling — but just noting that it feels different from the kind of tasks Jo Boaler was talking about.

I agree with what you say. I guess I was trying to write questions for the physics class I actually teach. Not some class I wish I was teaching.