Investigations where students discover the relationship between variables can help students build a deeper understanding of functions. Often these explorations are hands on and engaging lessons. They typically start with some sort of interesting video or question prompt such as "What makes for an exciting bungee jump?" or "Which cup will keep my coffee warm the longest?". I was prompted to think about my favourite investigations after seeing a post from Nat Banting on Twitter. Below are a collection of links and descriptions of my favourite secondary mathematics investigations. ## Linear Relationships- Vroom Vroom from Fawn Nguyen and Super Racers - Students use a toy pull back car to investigate how far it travels depending on how far it is pulled back.
- Knot Again! from Jon Orr and Ropes of Different Thickness from Alex Overwijk - Students investigate the relationship between the length that a rope can be stretched and how many knots are tied in the rope.
- Paper Clip Chain - Students investigate the relationship between the number of paper clips and the time it take to make a chain from them.
- Spaghetti Bridges from Mary Bourassa and Spaghetti Bridges from Andrew Busch - Students build a "bridge" using more and more strands of spaghetti and see how many coins/washers it can hold before breaking.
- Barbie Bungee from Fawn Nguyen and Barbie Bungee from Matt Vaudrey - Students investigate the number of rubber bands used in a bungee cord and how far the bungee stretches when attached to a toy doll.
## Quadratic Relationships- Down the Drain - Students measure how quickly water drains from a cylindrical tank.
- Speedy Squares from Mary Bourassa - Students time how long it takes them to make squares of different sizes using multi-link cubes.
- Pendulum Investigation - Ask students to measure the time it takes a pendulum to swing 10 times. Change the length of the pendulum and measure again. Take several more data points and a quadratic relationship will emerge.
- Penny Circle from Dan Meyer - Students count how many pennies can fit into circles of different diameters and then make predictions using this information.
## Exponential Relationships- Bouncing a Ball from Andrew Busch - Students investigate how high a ball rebounds after each successive bounce. David Wees has a great suggestion of using audio to take more accurate measurements.
- Coin Flipping and Pennies and Dice from Andrew Busch - Students use coins, dice, candy (e.g. Skittles) or two coloured counters to investigate an exponential relationship based on the probability of landing on a specified side when thrown. I've seen variations of this activity in numerous textbooks.
- Too Hot to Handle from Andrew Busch - Students investigate how quickly a cup of how liquid cools using a thermometer.
I like the investigation above because they share several common features. - They use inexpensive or easy to find materials
- They are hands-on and engaging.
- They don't require extensive preparations or difficult cleanup (Spaghetti Bridges probably requires the most cleanup as it will leave bits of broken dried spaghetti all over the floor, plan accordingly)
- The instructions are easy to explain. The majority of class time is spent carefully collecting and analyzing data.
Do you have a favourite activity or resource for activities? Please let me know what it is. EL
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I was recently looking for an activity to explore a linear relationship, preferable one that included some practice with decimals. I found a few examples but none of them really seemed to satisfy what I was looking for. Here are a few of my criteria for good experiments to explore function relationships: - Uses easy to find and inexpensive materials or materials that are commonly found in a mathematics classroom.
- Hands on activity for students to investigate in small groups
- The instructions are easy to explain and all students have an entry point into the activity.
## Notice and WonderThere were some great questions about volume and surface area, weight, and size of the paper clips (what is a #4 sized paper clip?). The questions the we went on to investigate was how long would it take to make a paper clip chain from all 100 paper clips. I was inspired by Dan Meyer's Guinness World Record for the longest paperclip chain in 24 hours. Dan blogged about breaking the record as well as asking student to see how many paperclips they could chain in one minute. ## EstimationI asked students to estimate how long they thought it would take to create a chain of 100 paper clips. I also asked them to think about an estimate that they know was too low (that creating a chain this fast was not possible) and too high (that they would have no problem creating a chain in this time even going slowly). Most students thought that a time between 5 and 6 minutes was a good "just right" estimate. ## Gathering Data
## Revising EstimatesAfter collecting and analyzing some data, I ask students if they'd like to revise their estimate for 100 paper clips. Then we test their revised estimate using a plot of the values they collected and extrapolating. Below is one student's data plotted in Desmos. They estimated 300 seconds (5 minutes) to chain all 100 paperclips. This lesson could be modified to include outcomes from a number of different grade levels. I closed the lesson by showing students the record for the most paper clips linked together in one minute and asked students how they would compare. Nova Scotia Mathematics Curriculum Outcomes Mathematics 6 SP01 - Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02 - Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media.Mathematics 6 SP03 - Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 7 PR02 - Students will be expected to create a table of values from a linear relation, graph the table of values, and analyze the graph to draw conclusions and solve problems.Mathematics 7 N02 - Students will be expected to demonstrate an understanding of the addition, subtraction, multiplication and division of decimals to solve problems (for more than one-digit divisors or more than two-digit multipliers, the use of technology is expected). Mathematics 7 SP01 - Students will be expected to demonstrate an understanding of central tendency and range by: determining the measures of central tendency (mean, median, mode) and range; determining the most appropriate measures of central tendency to report findings. Mathematics 7 SP02 - Students will be expected to determine the effect on the mean, median, and mode when an outlier is included in a data set.EL
I was recently invited by a class to work with them on collecting and analysing data. After brainstorming some ideas with the classroom teacher, we settled on collecting data from pull back cars. I checked out Fawn Nguyen's Vroom Vroom lesson and Simon Job's Car Racing lesson to get some ideas on how to organize this lesson. We started the lesson by sharing the first half of Simon's video of cars racing across the floor. We had the students do some notice and wonder about the action taking place in the video and then introduced activity. We showed students the recording sheet that we would be using and how we would be taking measurements (A link to the record sheet Google Doc is here). Then we brainstormed some ways to make sure that we all collected good data and avoided errors: we would all use the same units (centimeters), all measure our distances the same way (from the front bumper), not use data if the car bumped into a wall or a desk, etc. We split up into racing teams of three students each. Each group got a measuring tape, a pull back car and a recording sheet on a clip board. The classroom teacher and I circulated the room (and a bit of the hallway) to help students and answer questions. After students finished collecting their data and plotting their values we came back together as a class. We asked several groups to plot their data on the whiteboard at the front of the room. We then had a discussion about general trends as well as why each car had a slightly different graph. Cars might have different wind up springs, different tire grip, dusty floors, aerodynamics, etc. We finished the class with a bit of excitement... the 150 Challenge. Each team had to use the data for their car to predict how much they would need to pull back to make the car travel as close to 150 cm as possible. Teams huddled to interpret their data and select a pull back distance. Each team brought their car to the front of the class to give it their best shot. There was lots of cheering and excitement as some teams got very close. The winning distance was only 2.5 cm! Much closer than I had expected. This activity could be easily extended for higher grade levels by incorporating linear relationships, linear equations and linear regression. Nova Scotia Mathematics Curriculum Outcomes Mathematics 6 SP01 - Students will be expected to create, label, and interpret line graphs to draw conclusions. Mathematics 6 SP02 - Students will be expected to select, justify, and use appropriate methods of collecting data, including questionnaires, experiments, databases, and electronic media.Mathematics 6 SP03 - Students will be expected to graph collected data and analyze the graph to solve problems. Mathematics 9 PR02 - Students will be expected to graph a linear relation, analyze the graph, and interpolate or extrapolate to solve problems.Mathematics 10 RF07 - Determine the equation of a linear relation, given: a graph, a point and the slope, two points and a point and the equation of a parallel or perpendicular line to solve problems. (including RF07.06 Determine the equation of the line of best fit from a scatterplot using technology and determine the correlation)Mathematics Extended 11 S01 - Analyze, interpret, and draw conclusions from one-variable data using numerical and graphical summaries.EL
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 (A similar experiment is described in Canavan-McGrath, Foundations of Mathematics 12, 429). 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
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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