Summative Task for Quadratics – #MCF3M

My Grade 11 e-Learning math class is completing a unit on quadratic equations. I have a few things happening for their summative assessment, but the part I find most interesting is the following “experiment”. It’s heavily based on the Leaky Tower task from TIPS4RM at EduGAINS.ca. I’m going to test it out tonight with my kids before I finalize the evaluation criteria and post the task. If you have feedback, I’d love to hear it. I’ll be adding photos to help explain the setup.

Leaking Bottle – Summative Task – Part 1

You’ll be completing a short experiment and writing a report to go with it. You can get help from a classmate, family member, etc. while running the experiment, but just as an extra set of hands. No one should be helping you with the math part.

Preparation

Gather the supplies you’ll need:

  • a clear, disposable, empty, plastic bottle
  • a ruler
  • a watch, phone, or other time-keeping device OR a video-recording device.

—photo here—

Carefully poke a hole in the bottle about 3cm from the bottom. Seriously, be careful here. You might try using something sharp, like a pin or a nail, to start the hole, then widen it with a pencil. You want the final hole to have a diameter of 3-7mm. Don’t worry about being super-precise.

—photo here—

Hold a ruler next to your bottle, or tape a ruler to your bottle if you need both of your hands free. You want to be able to measure the water level, so put the “zero” end of the ruler at the bottom.

—photo here—

Cover the hole and fill the bottle with water. If your bottle has a tapered top (like the one pictured here), only fill it up in the cylindrical section (i.e. before it starts to narrow). You can cover the hole with your finger, or you might try a piece of tape (if you use tape, fold the end on itself so it’s easier to remove).

—photo here—

Data Collection

If you’re recording video (easier, I think), start recording. If you’re just using a watch or other timing device, wait for a “good” time, like a whole minute, for a starting point.

Uncover the hole, letting the water in the bottle flow out into a sink or another container. Don’t make a mess; nobody wants a mess.

—photo here—

If you’re using a watch, use the ruler to record the water level every 5 or 10 seconds or so. Pick an easy time to keep track of. Record measurements until the flow of water stops.

If you’re recording a video, let the water finish flowing out, then stop the video. Play the video back, noting the height of the water every 5 or 10 seconds or so.

Analysis

You now have a table of values: time (independent variable) and height measurements (dependent variable). If you didn’t get good data (you lost track of time, the video didn’t work, etc.), perform the experiment again. It doesn’t take long.

  1. Using Desmos, create a scatter plot for your measurements.
  2. Find an equation to fit the data as best you can.
  3. Identify the key points on the graph.
  4. How should the equation you found be restricted? i.e. what should the domain and range be?
  5. Write the equation you found in Standard Form and Vertex Form.

Leaking Bottle – Summative Task – Part 2

One small change

Repeat the above experiment, but this time put another hole about 7-10cm above the first one. Uncover them at the same time, so water will flow out of both holes.

—photo here—

Your analysis will be a little more complex, because you won’t have a single, nice equation that can accurately model the data.

  1. Using Desmos, create a scatter plot for your measurements.
  2. Find an equation (or equations!) to fit the data as best you can.
  3. Identify the key points on the graph.
  4. How should the equation(s) you found be restricted? i.e. what should the domain(s) and range(s) be?
  5. Write the equation(s) you found in Standard Form and Vertex Form.
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Using video to capture quadratic motion

In my MBF3C class today we observed quadratic motion and modelled it with http://www.desmos.com, and online graphing calculator. I’ve recreated the steps here (with some fake data).

First, I went to Desmos and projected a blank Cartesian coordinate system onto the screen at the front of the classroom.

Desmos_blank
Then, I found a black rubber stopper (I teach in a science classroom) a little more than an inch across.

I asked for two volunteers who would be willing to throw things at each other. That was easy.

They practiced lobbing the stopper back and forth to each other in front of the screen, trying to get the black rubber to crest visibly near the top of the grid. Eventually they were confident they could do it.
I held my iPhone in landscape orientation and recorded a slow-motion video. After a few attempts I felt we had a successful toss, and the students returned to their seats without injury.

We scrubbed through the video slowly and recorded the x– and y-coordinates for each major tick of the x-axis.

Then we plotted the points in Desmos (using the Table feature):

Desmos_table

We then graphed a generic quadratic using the vertex form and its parameters (y=a(x-h)^2+k). Desmos provided the sliders for each parameter:

Desmos_basic

As a group the students decided to make a negative and small, to flatten the curve, then they adjusted h and k to fit:

Desmos_transformed
Some interesting stuff to note about the process:

  • Even using 120fps there were places near the edges of the curve for which it was hard to see the coordinates (blurring and gaps).
  • The vertex wasn’t on the y-axis, which was surprising to the students.
  • The glare of the projector made the grid a little hard to see.
  • We had to have the lights out to make the grid visible at all on camera, and the dim lighting made the video a bit grainy.
  • The parabola we fit to the data worked really, really well.

What’s next

I’m going to perform some more motion tasks with them to get more quadratic data, and we’re going to do some curve-fitting to model and predict things (for example, how far can you throw a ball off a 10th-story roof?).

I’d like to have a large, physical grid on the wall or something so that I can have the lights on when we record video.

I want students to record video and analyse it. Lots of them have iPhones, and I bet some of the other phones can take good, crisp video. If not, there’s some learning there too (about interpolation if nothing else).

I’ll try some other phenomena also.

A video you can use

Here’s another video we took, if you want to use it:

Physical phenomena for quadratic relations

I’m working on a quadratics unit for my MCF3M (online) and MBF3C (F2F) classes. The Ms need to be able to do a few more things, but both groups have to be able to model quadratic “stuff” using an equation.

I’ll be using desmos.com pretty heavily, and I got some great ideas from Heather Theijsmeijer (@HTheijsmeijer).

I’m trying to find some examples of physical phenomena that I can have students in either class play with to practise/demonstrate modelling. Here are my ideas so far:

Throwing or Bouncing a Ball

This is the first thing I thought of. A ball follows a nice parabolic path in the air if it’s moving horizontally.

My plan is to have students use a phone or camera to record a video or a rapid burst of images, overlay a set of axes, and fit a curve to the path. My iPhone can record at 120fps, which is great. I also found a handy post at Stack Overflow that explains how to extract images from a video, so that might be helpful too.

Pouring Water from a Hose

Set your hose at an angle, turn on the water, and snap a picture. Parabola. Beauty. Maybe put a piece of grid paper behind it, or just import it into Desmos.

Rolling a Ball Up An Incline

This one’s messy, but I think it might work.

Dip a marble in some ink or paint. Set a piece of grid chart paper on an incline (say, a piece of plywood) and roll the ball on an angle up the paper. When it crests and rolls back down, it should have left parabolic paint. On graph paper.

Other ideas?

I’m open to suggestions. I have stuff like photos of suspensions bridges, etc., but I really want something students can generate on their own.