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.
I recently read an article on Wired about the Solar Voyager. A pair of engineers, Isaac Penny and Christopher Sam Soon, designed and built an autonomous, solar powered vessel. On June 1st, 2016 the 18 foot vessel, named Solar Voyager set off on its trans-Atlantic adventure from Gloucester, Massachusetts to Portugal, a journey of more than 4800 kilometres. They are predicting that this trip will take 4 months, assuming that there are no catastrophic events mid-Atlantic. One cool thing about this trip is that the Solar Voyager reports it position and other data online every 15 minutes at http://www.solar-voyager.com/trackatlantic.html. Currently, about two-weeks into its journey, Solar Voyager is just South of Halifax, Nova Scotia where I live.
The image below shows how far the Solar Voyager has traveled during its first two weeks. That is 1/8 of the time estimated for the crossing. Based on the information below, do you think that it will reach its destination in 4 months? What factors did you consider when making your estimation?
Some factors you might consider are currents, weather, equipment malfunction, obstacles/collisions, wear and tear, etc. There are so many variables at play that it must be very hard to make an accurate estimation.
Some Questions/Estimates for Students:
One of the coolest things about this project is that these young engineers "built Solar Voyager in their free time, undertaking this voyage simply for the challenge." How can I commandeer this type of intrinsic motivation for students in math class? What about this project made them want to work so hard "just for the challenge" and not for some extrinsic reward. Was it because they were the ones who selected and designed the task? Did they have just the right skills so that they felt confident that they would be successful? What is something that was relevant to their lives? How did this project captivate their curiosity?
Update: Solar Voyager ran into some trouble south of Nova Scotia. It appears it got tangled in some fishing gear and the props and rudders were fouled. After drifting for over a week, the vessel was picked up by HMCS St. John's, a Canadian Navy offshore patrol vessel.
I get so excited when my kids tell me stories of what is happening in their math classes. This is a favourite.
My youngest son (age 7, grade 2) began his story as soon as I picked him up from school.
"Mommy, did you know that if you wanted to buy, let's say some....fabric, you couldn't just like go the fabric store and say 'I'll have 20 pencils of fabric'"
I was curious where this was heading; being a mathematics consultant, I knew what grade 2's were working on at this time of year (measurement). I didn't want to steal his thunder, so I just went with it.
"Really, Michael?" I turned into teacher mode: "Can you tell me some more about that?"
He went on to explain in great detail and with loads of enthusiasm about all the trouble he would run into if he wanted to measure fabric with random objects. He actually had a lot of fun naming all of the things that would be silly to use to measure fabric. He went on for a while and wrapped up the conversation telling me there was this "thing" called a "centimeter" that we could all use and understand. You would swear he discovered the metric system himself; he took such ownership of the concept.
Keep in mind, I can't think of a time he has ever been in a fabric store (I am not the crafty type) and I am almost certain that before this math lesson, he would never have used the word fabric (cloth or material, maybe?).
So he had no previous experience with the concept but he was still engaged? Yes.
When I was at Dan Meyer's NCTM presentation (Beyond Relevance & Real World: Stronger Strategies for Student Engagement) last week, I couldn't help but think of this story from my son. I can imagine the kind of "teacher moves" my son's teacher used. She is a natural story teller, her enthusiasm is contagious and she loves to laugh. I can imagine her telling a story to the class, strategically leaving out important parts, having them experience her fabric store dilemma for themselves and brainstorming ideas with the class on how they can fix this problem!
Even if he didn't really discover the metric system, he certainly thought he did. And his teacher created those conditions. And I think that's pretty cool.
Revisiting the Classic Ferris Wheel Problem
This type of pseudo-context word problem robs students of the opportunity to explore and analyze real-world problems in much depth. Dan Meyer has written quite quite a lot about pseudocontext. My concern with the Ferris wheel problem is not that you can't model the height of a seat on a Ferris wheel with a sine function, it is why would you do it?
Instead of doing a textbook problem with a fictional Ferris wheel, I decided to use a real Ferris wheel from a nearby amusement park that some of my students would be familiar with. I visited the park to take a video of the Ferris wheel in action. Below is a 30 second clip of the "Big Ellie" Ferris Wheel at Atlantic Playland.
Notice and Wonder
I started by asking students what they noticed in the video. After brainstorming and recording the students observations I asked students what they wondered about in the video. They asked questions like "how fast is the ride going?", "how tall is this Ferris wheel?", "how far can you see from the top of the ride?", "how long does the ride last?". In order to investigate these questions further we needed to estimate some values such as the radius of the wheel, how long it takes to make one revolution, and the height of the central axis about the ground. I asked students to estimate these values using the clues in the video we watched. We watched it several times in order to get some good estimates.
I also talked about some of the mental math required to operate a ride like this. Because it is belt driven, you have to load the Ferris wheel so that it is equally balanced around the wheel. Otherwise, one side of the wheel would become too heavy and the drive cable would slip in the rim and the wheel wouldn't be able to turn! This requires a lot of on the fly estimates of weights of the riders as it is being loaded.
In order to get a see how good we did with our estimations we turned to the internet in order to try to hunt down some of these values with a Google search. This lead to a discussion about what keywords we could use to hunt down this information. A search of "height of the central axis of the Ferris wheel at Atlantic Playland" was not very fruitful... an essential skill to solve a problem like this is to translate mathematical language into common terms that you can use for a Google search. Ve Anusic has a great blog post where he discusses a similar problem and the discussion with his students about the information you need and the information you might find online. First we did a search to find Atlantic Playland's website and found that they called their ride "Big Ellie". A search for this name lead us to believe that this Ferris wheel is a No. 5 Big Eli wheel made by Eli Bridge (I later emailed the park and confirmed that this is indeed the model of their Ferris wheel). Eli Bridge's website gave us some interesting information but not exactly what we were looking for. A bit more searching and we were able to find a pdf of the owner's manual for this ride that included a helpful diagram.
It is only after we were able to answer some of the students' questions regarding the video of the Ferris wheel did we start to talk how we might mathematically modeling the height of a person riding the wheel over time and the periodic nature of this function. Students were much better able to make sense of this visual model once they had a good grasp of the context of the problem.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 12 - RF03 Represent data, using sinusoidal functions, to solve problems.
Pre-calculus 12 - T04 Graph and analyze the trigonometric functions sine, cosine and tangent to solve problems.
What is the tallest man-made structure around Halifax? Ask students to brainstorm a few ideas. Students might suggest a tall building. Some of the tallest buildings around Halifax are the Maritime Centre 78 m (256 ft), Purdy's Wharf 88 m (289 ft) and Fenwick Tower, the tallest building in Halifax at 98 m (322 ft) tall. Students might also suggest one of the two harbour bridges. The towers on the MacDonald Bridge are 103 m (338 ft) high and the towers on the MacKay Bridge are 96 m (315 ft). The towers on the MacDonald bridge are taller than the tallest buildings in Halifax. An observant student might even suggest the red and white painted smokestacks at the Tufts Cove Generating station. The smokestacks are tall indeed. From the picture below, you can see that the smokestacks are taller than the bridge towers of the MacKay Bridge.
You might ask students how you know by looking at the picture that the smokestacks are taller than the bridge towers. This would be a fun opportunity to talk about perspective.
We can check out the height of the smokestacks by using a little trigonometry.
I found a spot across the harbour from Tuft's Cove to measure the angle to the top of the smokestacks using a clinometer. (My favourite school/education clinometer is the Invicta MK1 Clinometer... not only does it have a cool name, it looks really cool as well!) It was an angle of elevation of 8 degrees. Next I used Google Earth to see that my distance to the centre stack is approx. 1160 metres. So that means tan(8) = x/1160. Solving for x gives us x = 1160*tan(8) = 163 metres. I emailed Emera an they said that the stacks are actually 500 ft. (152.4 m) tall. So I'm about 11 metres off. Not bad considering that at this distance, a variation of 1 degree is about 20 meters. The actual angle should have been about 7.5 degrees vice 8 degrees.
So, how could I minimize the amount of error? As I get farther away, the tangent value gets smaller but the distance that I'm multiplying by gets bigger. At what angle does 1 degree of error create the least amount of difference in the height being measured?
But is this the tallest man-made structure in Halifax? I used to think so, but I was only considering free standing structures. There is a radio transmission tower that is taller but not free standing... the tower has guy wires to hold it up. The CBC radio tower on Geizer’s Hill is even taller than the smokestacks at Tuft's Cove.
So how tall is this tower? I drove up to the top of Geizer's Hill to find out. I found a spot level to the base of the tower at a distance of 475 meters along Washmill Lake Dr. From this spot, my clinometer measured an angle of inclination to the top of the tower of about 23 degrees. 475 x tan(23) = 201 meters (about 659 ft).
A bit of digging led me to a website that stated the antenna height above ground level for the CBC radio tower is 192 m (629 ft). My measurement was only 9 meters different from this height... pretty close. This is so far the tallest thing I've found around Halifax. Let me know if you find something taller!
Have you ever been on a road trip with a math teacher? They're always calculating distances and estimating how much farther they can go on a tank of gas and noticing things...
This was just what happened during a trip with my wife. We were headed towards the New Brunswick boarder from Halifax and we saw a highway mileage sign. It said it that Amherst was 42 km away and New Brunswick was 47 km away.
"So how far is it from Amherst to New Brunswick?" I asked. 5 km? But is it exactly 5 km? If we assume that these distances are rounded to the nearest kilometer then it might be as far as 6 km (41.5 km to Amherst and 47.4 km to NB) or as short as 4 km (42.4 km to Amherst and 46.5 to NB)?
(46.5 - 42.4) <= distance <= (47.4 - 41.5)
A few minutes driving brought us to the next mileage sign and some additional information.
So the next mileage sign down the road seems to indicate that there are only 4 km between Amherst and the NB boarder. Using our previous logic however we could say that it is anywhere between 3 km and 5 km right? (39.5 - 36.4) <= distance <= (40.4 - 35.5). Now we are getting somewhere. We know that it is between 4 and 6 km and between 3 and 5 km.
Now we can narrow it down and say with some certainty that the distance between Amherst and NB was between 4 and 5 km. I'm feeling confident now. A few more kilometers down the road we pass the next mileage sign.
What the what! What is going on here? Is this a conspiracy of cartographers? What strange and inscrutable measurement system is at work here? Time for some Google Earth mapping. I used Google Street View to find the second road sign along the Trans-Canada Hwy 104 where the sign says there is 36 km to Amherst. A straight line from this spot on the road to the geographical centre of Amherst is only about 27 km so I think we can assume that the road signs are measured as the distance along the road.
So lets use Google Maps to measure the distance along the road. From the mileage sign to the city centre, Google Maps says that it is still only about 32 km away. Still about 4 km off what the sign says. I wonder how accurate these signs are? How long ago and using what technology were these signs created?
Do you know how road distances are measured in Nova Scotia? Have you encountered any inconsistent mileage signs? Stay tuned for updates. I've sent an email to the Nova Scotia Department of Transportation and Infrastructure Renewal to ask for some clarification in the method of measurement. I'll keep you posted of any responses.