Towards the end of last semester, we worked on a task that Tom Reardon shares as The Great Applied Problem.

What is your question? (Note that this lesson was before I read Michael Pershan’s post considering how we ask students about what we can explore mathematically.)

why are we always looking from the top and never the side 1

other than circles and rectangles what are the cross sections of shapes? 1

are the 2 bases the same size 1

what are the sizes of the different cross sections? 1

i wonder how to find the volume of a partially filled cylinder in which the base of the liquid is a sector 1

how many cubic units of water are in the shape currenty? 1

is the water filling up or flowing out. what is the length of each cord as it fills 1

I wonder if there is a relationship between the diameter of the base and the change in dimensions of the rectangle… 1

the areas of the bases created by the water 1

What is the total volume of the tank 1

what is the smallest rectangle possible to be a cross section? 1

at what rate does the volume change 1

is it filling up or draining 1

How much water is in the cylinder? 1

how much water will the cylinder hold 1

how much water would it take to fill the tank 1

I wonder what the volume is of the water in the tube. 1

how much water will it hold 1

will the water flow through the straw 1

what is the volme ofthe water 1

what is the volume of the water in the cylinder? 1

does the circumference change 1

how would the shape change as the cylinder shift 1

whats the ratio of the volume of water to the cylinder 1

how much more water do you need to fill it up 1

How much water can the tank hold? 1

how will the volume change if the water increases 1

What are the measures of its radius and horizontal height, or of either if the volume is given? 1

We settled on how much water is in the tank.

What is the least amount of information you need to answer the question?

Teams worked together to make a list of the measurements they wanted to use for their calculations. Very few teams wanted the same information. Some differences were minute, such as one wanting the radius and another wanting the diameter. Or one wanting the depth of the water and another wanting the distance between the center and the chord. Or one wanting the radius and another the length of the chord. Some differences were bigger, such as one wanting the ratio of water to total volume.

What information was I willing to give them?

Thanks to a spreadsheet included in Tom’s problem, I had plenty of measurements from which to choose to give students. But I hadn’t thought through whether I was willing to give the ratio of water to total volume instead of the length of the radius. I ended up not giving the ratio. But I did give some teams the length of the chord instead of the radius.

As I watched students work, I noticed that they had the opportunity to **look for and make use of structure**. Dylan Wiliam talks about asking students questions that push their thinking forward and probe their understanding. This task did just that.

What misconceptions would students have about this figure?

We talked about what’s there that’s not there. What do you see that isn’t pictured?

A semicircle

What else do you see that isn’t pictured?

A diameter

What else do you see that isn’t pictured?

A radius that forms a right triangle with half of the chord.

What else do you see that isn’t pictured?

An isosceles triangle formed by two radii and the chord.

What else do you see that isn’t pictured?

A sector.

What else do you see that isn’t pictured?

The region formed by decomposing the sector into a triangle and a segment of a circle.

Is that same region a semicircle?

It’s not. But that is what a lot of students thought when they first looked.

We are learning to look deeper to make sense of the structure before we jump into calculating.

One student reflected on working through this task in class. She also talks about using the practice **make sense of problems and persevere in solving them**: “trying to find what we needed to know”, “tried different ways to find the area”, “drew diagrams”, “made a plan”, “discussed different approaches”. She also talks about the math practice **construct viable arguments and critique the reasoning of others**: “We all decided what we wanted to know to figure out how much water was in the tank. And then we tried to explain our reasoning to the class. We all discussed what we wanted to know then decided together what we really needed to know.”

I’ve used this task for several years, but I’ve never introduced it like this before. Previously, I’ve asked my students only to calculate.

And so the journey to be less helpful continues …