I recently had the opportunity to work with a student to investigate parabolas and quadratic functions. We used one activity to investigate two different quadratic relationships. First we observed the shape of the stream of water coming out the side of a water bottle and then we observed the rate the water drains. ## The Water FountainI set up a cylindrical bottle of water on a crate. The bottle had a whole in it covered with a piece of tape. I asked students for some predictions. What will the shape of the water coming out of the side of the water bottle look like. What will happens to the stream of water as the water level goes down? I noticed that the student drew the water stream coming out of the bottle like it comes out of a water fountain (where we had just filled the bottle). We took the tape off the hole and then watched the water come out while making some observations and taking some photos. We selected a good photo (the black bulletin board in the background really helped) and loaded into Desmos. Then we used a table to record some points along the steam of water. After that we did a linear and then a quadratic regression on the point to see that the parabola was a much better fit than a line. We then had a chat about parabolas and projectile motion. ## Draining the TankWe set up the water bottle again but this time instead of looking at the shape of the stream of water, we focused on how fast the water level fell. I asked the student to predict what this might look like. You might ask students to predict what a graph of the water level might look like over time for the two situations below. How would the graph look when filling the tank compared to emptying the tank? The water flowing into a tank should rise at a linear rate. Students should expect that when the water drains from an open tank, the flow will be greatest at first and then gradually decrease as the water level decreases. (This is an application of Torricelli's Law). Next we taped a measuring tape to the side of the bottle and collected some data as the water flowed out of the bottle. We used the stopwatch on my cell phone to record the time at each centimeter of height. This wasn't as accurate as I had hoped due to some distractions in the room. We set up the experiment again and the second time I recorded the water falling using a video (I used the CoachMyVideo app). We were able to get much more accurate values this way. We entered the data in a table on Desmos and then did a quadratic regression to fit a curve to our points. I was a bit surprised at how well the data from the video analysis on our second attempt fit to a quadratic curve (R^2 = 0.9999. I really liked how we could use the exact same setup to investigate two different quadratic relationships. Nova Scotia Mathematics Curriculum Outcomes Mathematics 11 RF02 - Demonstrate an understanding of the characteristics of quadratic functions, including: vertex, intercepts, domain and range and the axis of symmetry.Pre-calculus 11 RF04 - Students will be expected to analyze quadratic functions of the form y = ax^2 + bx + c to identify characteristics of the corresponding graph, including vertex, domain and range, direction of opening, axis of symmetry, x-intercept and y-intercept, and to solve problems.EL
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I recently did an activity with students to answer a question by collecting and analysing data. I was inspired by similar activities from Bruno Reddy, Mean Paper Aeroplanes, and Julie Reulbach, Paper Airplanes for Measures of Central Tendencies. We started class by watching a video of the Paper Airplane World Championship - Red Bull Paper Wings 2015. This short video (about 3 minutes) shows the highlights of three paper airplane competitions; Distance, Airtime, and Aerobatics. After watching the video I let students know that we would be making paper airplanes for a distance competition. |

We started by brainstorming with students the characteristics of a good paper airplane that will fly a long distance. Most students have some experience in making paper airplanes. Several features that we discussed were: the shape (a glider or dart shape, wing angles), design features (symmetry, vertical flaps or a heavy nose), construction methods (sharp creases and accurate folds), materials (type of paper) and how it is thrown (launch angle, strength and accuracy of the thrower). |

## Predictions

## The Experiment

## The Results

## Reflection

Students had a handout where they were asked to reflect on how they could make this experiment more accurate and reliable. We thought that a few practice throws would help before we started collecting data (most student's last throw was their farthest distance). Another option would be to have more than three trials for each plane to increase the reliability of the data collected. There were lots of other really good suggestions as well. |

**Nova Scotia Mathematics Curriculum Outcomes**

**Extended Mathematics 11**

**S01**- Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.

**Mathematics 9 SP03**- Students will be expected to develop and implement a project plan for the collection, display, and analysis of data by: formulating a question for investigation; choosing a data collection method that includes social considerations; selecting a population or a sample; collecting the data; displaying the collected data in an appropriate manner; drawing conclusions to answer the question.

- The x-axis has an inconsistent scale (sometimes 3 months, sometimes 4 months) that was automatically generated by Excel.
- Is a y-axis necessary or would a simple timeline do? Perhaps with a count below the timeline for successful land/drone ship vs failed land/drone ship landings.
- The graph doesn't indicate the two periods of investigation following launch failures.
- The choice of a line graph is a controversial one since these are discrete data points and not continuous data.
- The label on the y-axis should say "Number of successful landings to date". Otherwise it it looks like there were 9 successful landing in March 17 when there were actually 9 total successful landings by March 17.
- Using red and green dots will make this graph meaningless to people who are colorblind. You could use different symbols as well as different colors to remedy this.
- The success rate is more important than the total number of successful launches. Also, some successes are more ambitious than others (e.g. ocean/land/drone ship or the first re-used booster).
- Why not differentiate between land and drone ship landing attempts?.
- Not all launch data is included in this graph. How do you decide which launches to include (e.g. ocean landings where there was no drone ship).

## In the Classroom

**Nova Scotia Mathematics Curriculum Outcomes**

**Grade 8 SP01**- Students will be expected to critique ways in which data is presented.

**Mathematics Essentials 11 F2**- Select an effective data display for a given set of data and explain the reasons for the choice.

**Mathematics at Work 11 S01**- Students will be expected to solve problems that involve creating and interpreting graphs, including bar graphs, histograms, line graphs, and circle graphs.

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