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.
I like the investigation above because they share several common features.
Do you have a favourite activity or resource for activities? Please let me know what it is.
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 check 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 student 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 more close 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.
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 Fountain
I 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 Tank
We 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.
Swimming in the hotel pool I saw these depth markers. As a math teacher, they made me a bit uneasy. What do you notice in the photos below? What do you wonder?
Just look at those significant digits. They look so precise. I first thought... going from the shallow end to the deep end, it gets 1 foot / 0.2 metres deeper. That must mean that 1 ft = 0.2 m right? But then if 1 ft is 0.2 m then shouldn't 3 ft in the shallow end be 0.6 m instead of 1.0 m?
So I looked at it another way... 1 m is the same as 3 ft... So 1 ft must be about 0.33 m. Which would make 4 ft equal to about 1.33 m not the 1.2 m as shown. But I know that a meter stick is shorter than a yard stick so this is just an approximation. No problem, they just rounded off both values.
Then I had a moment of doubt... in the shallow end the values are in a ratio of 1/3 and in the deep end the values are in a ratio of 4/12 which is also 1/3 so shouldn't this work out? Then I realized the errors and misconceptions in this line of thinking.
Other Linear Conversions
Today, as I was driving around, I looked more closely at the clearance signs that I passed under. There doesn't seem to be much consistency in the units used or precision. Do people with tall cars know the height of their car? I just know that I'm about 6 ft tall and my car is shorter than I am. Of course you can always just wing it. If you clear the warning bar, you're good to go. Anyway, I know for sure that my car is less than 11 foot 8 if I ever end up in North Carolina.
I saw this relatively accurate sign at a parking garage today so I took a photo. 6'0" is approximately 1.8288 metres so these values are the closest I've seen.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 10 M02 - Students will be expected to apply proportional reasoning to problems that involve conversions between SI and imperial units of measure.
Mathematics at Work 10 M01 - Students will be expected to demonstrate an understanding of the International System of Units (SI) by describing the relationships of the units for length, area, volume, capacity, mass, and temperature and applying strategies to convert SI units to imperial units.
Mathematics Essentials 10 D1 - Demonstrate a working knowledge of the metric system and imperial system.
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.
Next I showed students two different paper airplane designs; the Suzanne and the Classic Dart. I asked students to predict which would fly the farthest. I also asked how much difference, if any, they expected to see between the two designs. Most students predicted that the Suzanne would fly farthest. The next step was to create an experiment in order to test our predictions.
We split the class in half. Each half followed a specific set of instructions to fold one or the other of these planes (I had a handout with instructions for each design). We used different coloured paper for each design. Each student threw their plane three times and recorded each flight distance. We measured in feet since the floor tiles in the hallway were one square foot. The students then calculated their mean distance and shared this mean with their team. Each team then calculated a five number summary and sketched a box plot for their data.
Students declared the Suzanne to be the clear winner. The low ceiling height in the hallway seems to have favoured the glider design. We conjectured that the Dart may have performed better than Suzanne if they were thrown outdoors where students could throw at a higher launch angle. We also conjectured that the greater variation in the data for Suzanne was a result of the more complex folding required. Some planes were folded very well and others were a bit of a mess.
We finished class by watching a video of the world record throw for distance (we just watched the first 3 minutes of the video). The Suzanne, designed by John Collins and thrown by football quarterback Joe Ayoob holds the Guinness World Record for the farthest flight by a paper aircraft. The record throw was 226 feet, 10 inches (approx. 69.14 m). Our longest flight was just over 40 feet. Students seemed to really enjoy this activity. It allowed them to incorporate some movement in class and asked them to use mathematics and statistics in an authentic way to answer a real question.
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.
I prepared a lesson plan to work with a student. I carefully considered how I would introduce the topic, the path that the lesson might take and the questions that I would ask to prompt our discussion. I thought about the manipulatives that we might use to visualize and physically interact with the problem. I had a course carefully laid out.
I started by drawing an irregular, kidney shaped area on the desk and asked the student how he would estimate the area of the shape. I was prepared for a number of different responses that I thought I might hear... but the student didn't follow my carefully plotted course for our lesson. Instead he replied, "I'd use Pick's Theorem."
I grew up sailing on the Columbia River. When changing course on a sailboat, you can either turn the bow (the front of the boat) through the wind (i.e. tacking) or you can turn the stern (the back of the boat) through the wind (i.e. jibing). When tacking, the boom gently moves from from one side of the boat to the other. Jibing on the other hand can be dangerous as the boom suddenly jumps to the other side of the boat. When the student suggested Pick's Theorem, it felt like changing course by jibing instead of by tacking.
After our excursion through Pick's Theorem we found our way back to estimating the area with some manipulatives. First we covered the shape with square tiles and then we covered the shape with pennies.
We found that we could cover the shape with 66 square tiles. I asked the student how the area we found with Pick's Theorem and the area we found with square tiles compared. Through our discussion we decided that we needed a common way to talk about these areas so we converted both to square centimeters. We found that the area from Pick's Theorem was 382.5 cm^2 and the area using square tiles was 412.5 cm^2. Next, we looked at our penny solution. We looked up the diameter of a penny online and found that 135 pennies at 2.85 cm^2 each gave us a total area of 384.75 cm^2. While discussing how this estimate compared to our others, the student started talking about Alex Thue and his theorem on circle packing (this student has a really good memory). The student remembered that the efficiency of hexagonal packed pennies was about 91%. So we used this efficiency to correct our penny estimate to make it even better. This led to another discussion that I hadn't planned on about tesselations and polygons that tile the plane. The student said he had read in a book that there were 14 irregular pentagons that tile the plane. His book was a few years old however so he didn't know that a 15th pentagon had been discovered in 2015 or other recent work in this area.
While the lesson didn't go quite as I had planned, I was really happy to be able to take the student's contributions to the discussion and weave them into the overall narrative of our work. Being flexible, listening to students and incorporating their contributions into a discussion can sometimes throw you off course and you might end up someplace unexpected. The journey along these altered courses however can be incredible.
Over the Christmas holiday, the number of LEGO bricks in my house increased significantly. My son received LEGO sets as gifts from numerous grandparents, aunts and uncles. I was a LEGO fan when I was a child and now I have an excuse to play with them again as an adult. We've had lots of fun recently building sets and designing our own creations. At some point I became inspired to create a scale model of our home.
Planning and Building
I started this small project by building a test model to try out the proportions and to see what kinds of bricks I would need. The sizes of the door and window established the overall size. I continued revising the structure it until it looked right and then started collecting the bricks I needed.
Building this model reminded me of working on an OpenMiddle.com math problem. In an "open middle" problem, there is a one starting point and one solution but many different paths to get to the solution. With LEGO, there are many different ways to create, revise and improve your model. There are lots of different building techniques that will all result in a well designed scale model.
After I created my initial rough model I did some reading up on LEGO scale. It turns out that it is a fairly complex topic that lots of different people have investigated. I found the Brick Architect web site to be very helpful. For "classic minifigure" scale a ratio of 1:42 can be used. One major difficulty in discussing scale is that the proportions of a LEGO minifigure are not even close to the proportions of an actual person. A LEGO minifigure is about 4 cm tall and 1.6 cm wide. An average male human is about 175 cm tall and 40 cm wide... about half as wide as a minifigure would be at that height. Another challenge is converting units. The architectural drawings of my house are in feet, which I converted to metric (cm), then a scale factor is applied and finally the metric units are converted into LEGO bricks. I found an awesome tool that does this all for you, the LEGO Unit Converter.
I used a lot of estimation to determine how many bricks of each type I would need. LEGO bricks are not cheap so you don't want to order more than you need (Check out Jon Orr's activity involving cost, Is LEGO Gender Biased?). I purchased the bricks I needed on BrickLink.com, a large online LEGO marketplace. BrickLink provides a detailed price guide for every brick available which makes it really easy to know if you're getting a good deal or not.
I needed lots of 45 degree angle slope bricks for the roof of my house. These price stats let me know what a reasonable price is to pay for new or used bricks of this type. It is amazing to see how many bricks are sold on this site. I think that the stats from this site could make for an interesting grade 12 math research project.
The Finished Project
Constructing Rectangular and Triangular Prisms
Determining the surface area of a prism can get a bit stale. Textbooks contain lots of pictures of various right rectangular and triangular prisms. These prisms are carefully labeled with the exact information that a student needs. Students are given the task of inserting these numbers into a formula and doing some basic calculations. These types of problems often don't require much thought. I've recently had the pleasure of working in some junior high classrooms. We were looking for a more hands-on and thought provoking activity for surface area. We were also looking for an activity in which students could be creative. This is what we came up with.
Students, working in pairs, are given either a yellow or blue piece of coverstock. Students with a yellow piece are asked to design and draw the net of a right rectangular prism. Students with a blue piece are asked to design and draw the net of a right triangular prism. Students can draw whatever size or shape prism they wish as long as it covers the majority of the paper (at least half). Students use a ruler to carefully draw and measure the net. They measure and label the length and width of each face and calculate the area of each face on the net they have drawn. Once students have accurately drawn their nets and labeled the area of each side, a teacher will review their work. If it is an accurate net, the teacher will give the students a pair of scissors to cut it out. Make sure students do their calculations inside the net so that it is not lost when they cut it out. Once cut out, students can fold and tape their prism.
Students found this activity to be more challenging than they expected. Several had to start over after realizing that the prism they started wouldn't fit on the page or their net wouldn't fold into a proper prism. You could extend this activity by having students tape their nets inside out (with the calculations on the inside) and then challenging them to order the prisms from least surface area to greatest surface area.
Why I Like This Task
Double the Surface Area
Nova Scotia Mathematics Curriculum Outcomes
Grade 8 M02 - Students will be expected to draw and construct nets for 3-D objects.
Grade 8 M03 - Students will be expected to determine the surface area of right rectangular prisms, right triangular prisms, and right cylinders to solve problems.
Grade 9 G01 - Students will be expected to determine the surface area of composite 3-D objects to solve problems
Math at Work 11 M01 - Students will be expected to solve problems that involve SI and imperial units in surface area measurements and verify the solutions.
There are some really big doors around Halifax. The door on Irving Shipbuilding's Halifax Shipyard Assembly and Ultra Hall facility is big enough for large "mega-blocks" of ships under construction to pass through. The doors at IMP Aerospace's Hangar #9 at the Stanfield International airport is big enough for large aircraft to pass through. Which door do you think is the largest?
What Do You Mean by Largest?
The first thing you might want to do is settle on what you mean by "largest". Do you mean width, height, area, mass or some other measurement? Each of these doors might be the largest for a specific measurement. For example, the aircraft hangar door is made of metal and quite probably has more mass than the shipyard door which is constructed of a polyester fabric.
The shipyard door is really tall but the aircraft hangar door is really wide. Below are pictures of the two facilities from Google earth with the same scale so that you can compare the buildings that these doors are on.
Door Dimensions and Surface Area
So the shipyard door has the largest height and the largest area but the hangar door has the largest width and the largest mass. Would you call this a tie? How would you determine the winner?
More Big Doors
Do you know of other big doors around Halifax? Have you seen bigger doors in other parts of Nova Scotia or the rest of Canada? What is your definition of door?
Note of Thanks: I want to say thank you to the people at both Irving Shipyard and IMP Aerospace who were very helpful providing information for this post.
The sign reads "No Vehicles Over 3200 kg." I realized that I have no idea what the mass of my car is nor what type of vehicles would have masses greater than 3200 kg. I have personal referents for smaller masses such as 1 gram (such as a jellybean or a paperclip) and 1 kilogram (like a bag of sugar). I also know that my son weighs about 20 kg but I have no similar reference for 1000 kg. 1000 kg is also known as 1 megagram (Mg) and 1 metric ton (t). I know that a cubic metre of water has a mass of 1000 kg but that doesn't really seem to be helpful to me because I have no experience with an actual cubic metre of water.
This made me realize the importance of personal referents when teaching students about SI units of measurement. Students are introduced to grams and kilograms in Grade 3. The curriculum guides states that, "as with all measurement units, it is important that students have a personal referent for a gram and a kilogram. Students should recognize which mass unit (gram or kilogram) is appropriate for measuring the mass of a specific item." The curriculum guide also states that as they begin to estimate and measure masses using the gram (g) and kilogram (kg), they should develop a sense of what a kilogram feels like, "by lifting and holding a variety of objects that have a mass of 1 kg." One activity that the curriculum guide suggests is for students to make a kilogram mass of their own. "Provide students with materials such as sand, flour, sugar, and small cubes from base-ten materials to fill a container until it exactly balances with a 1 kg mass on a balance scale. Using this kilogram container they can now compare its mass to items in the classroom to help them find a personal referent for 1 kg."
Unfortunately, it is not feasible for me to fill a container with 3 200 000 jellybeans in order to compare its mass to a car ( ...but maybe you could use Skittles). I'm going to have to find my own reference for this mass. I found that a subcompact car, such as a Toyota Yaris hatchback has a curb weight of approximately 1000 kg. On the opposite end of the passenger vehicle size spectrum, a full size luxury SUV, such as a Cadillac Escalade, has a curb weight of about 2600 kg (I had to read up on the difference between curb weight and gross vehicle weight). Once students have an general sense of vehicle weight, it might be help solidify this understanding (and be a bit of fun) to challenge them to estimate the mass of a number of different vehicles... from motorcycles to truck cranes (similar to Dan Meyer's "How Old is Tiger Woods?" activity but with mass instead of age). I made a public Google Sides document with some photos and weights of various vehicles (still a work in progress).
Now, when I'm crossing the bridge, I'm constantly estimating the size of the different vehicles around me. I keep my distance from really big vehicles on the bridge... just in case.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 3 - M04 Students will be expected to demonstrate an understanding of measuring mass (g, kg) by: selecting and justifying referents for the units gram and kilogram (g, kg); modelling and describing the relationship between the units gram and kilogram (g, kg); estimating mass using referents; and measuring and recording mass.
Mathematics at Work 10 - M01 Students will be expected to demonstrate an understanding of the International System of Units (SI) by: describing the relationships of the units for length, area, volume, capacity, mass, and temperature; and applying strategies to convert SI units to imperial units.