About a year ago, I signed up for Samira Mian's Udemy course on Islamic Geometry. I also purchased a copy of Eric Broug's book Islamic Geometric Patterns. I wanted to learn the basics so that I could determine if this might be a good way to satisfy the grade 7 mathematics geometric constructions outcome. I designed a short unit that I described last year. Recently, I decided to try replicating some of these patterns using the online Desmos calculator and geometry tool. I think having some experience drawing these patterns with a compass and straight edge was helpful.
If you're looking for some Islamic geometric patterns to try, YouTube is a great place to get some ideas. There are some great instructional videos from Samira Mian and Nora Youssef, among others. The first pattern that I tried was a Star and Hexagon pattern that I learned from Samira's Udemy course. I learned that sticking with exact values are worth the effort. Rounding intersection points and slopes of lines to the nearest tenths or hundredths place work well at first but the errors compound and things start to get messy down the road. Interlacing the pattern gave me lots of practice with domain and range restrictions.
8 Fold Rosette
Nora Youssef has a nice video tutorial on for drawing an 8-Fold Rosette pattern. I did this pattern twice. The first time I constructed the basic pattern and the second time I added interlacing. I used the polygon function to add colour and figured out how to use trigonometry to rotate the polygons around the origin. This made it really efficient. I created a table with the vertices of the polygon and then just duplicated and rotated that polygon around the rosette. I duplicated the polygons multiple times to make the colours bold.
You can see from my notebook below that some of the math took me a few tries (this goes on for several pages). To make the weave for the 8 fold rosette, I made lines parallel to the original with a distance of 0.5 above and below. Each ribbon was then 1 unit wide. I was working with the equations in point-slope form. I'm pretty sure that there are more efficient ways to do these calculations but I haven't discovered them yet. I really like how these messy bits encourage me look for more efficient and elegant methods.
Desmos Geometry Tool
After working with the Desmos calculator for a while, I wanted to give the geometry tool a try. I decided to try a pattern that I saw on the Pattern In Islamic Art website. This site has some great resources. The pattern that I tried was from David Wade's book Pattern in Islamic Art. The geometry tool requires much less algebraic manipulation, but I find hiding the underlying grid is much more tedious than in the calculator. Everything has to be hidden individually instead of turning a whole folder on or off in the calculator. I've drawn this pattern in the past by hand and it would have been much more difficult if I didn't have that previous experience.
I've tried tiling some designs to cover the plane but I haven't come up with any good methods for this yet. I've also tried using sliders to dynamically adjust some of the relationships between the sizes of the pieces in these designs. These are great challenges and are helping me learn new features of Desmos. Dan Meyer wrote "If Math Is The Aspirin, Then How Do You Create The Headache?" I hesitate to call these graphing projects "headaches" because I enjoy the challenge. Regardless, this is a case where my need for mathematical solutions guide my learning and give me reasons to explore new graphing methods.
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.
The final chapter of the Math at Work 12 textbook deals with Trigonometry and the Law of Sines and Law of Cosines. Towards the end of the chapter there is a puzzle (p351) that asks students to create a triangle using 9 of the numbers from 1 to 10. Each side of the triangle is the sum of 4 of these numbers. I liked the construct of this puzzle but I wasn't a big fan of the questions that it asked students so I decided to give it an overhaul. An image from the textbook is below.
We followed up this warm-up with an open middle style problem using the same situation that would require students to apply the law of cosines. A challenge like the one below gives the students a reason to practice the law of cosines without feeling tedious or repetitive.
Directions: Use the numbers 1-9 (using each number no more than once) to fill in the circles. The sum of the numbers on each side of the triangle is equal to the length of that side. What is the triangle with the largest (or smallest) angle that you can make?
A triangle with the largest angle (there are several variations with the same angle):
Side A: 6+8+9+7=30 Side B: 7+4+1+3=15 Side C: 3+2+5+6=16
Angle A: 150.799 Angle B: 14.119 Angle C = 15.082
A triangle with the smallest angle (there are several variations with the same angle):
Side A: 1+2+3+4=10 Side B: 4+8+9+7=28 Side C: 1+5+6+7=19
Angle A: 10.844 Angle B: 148.212 Angle C = 20.944
Another challenging question that could be asked is how many different arrangements of the numbers 1 to 9 in the triangle diagram could you make? You have to consider that rotations of the triangle are the same. This would be a challenging combinatorics question even for Pre-calculus 12 students.
Nova Scotia Mathematics Curriculum Outcomes
Mathematics 11 - G03 Solve problems that involve the cosine law and the sine law, including the ambiguous case.
Math at Work 12 - G01 Students will be expected to solve problems by using the sine law and cosine law, excluding the ambiguous case.
Math at Work 12 - N01 Students will be expected to analyze puzzles and games that involve logical reasoning, using problem-solving strategies.
Mathematics 12 - LR01 Analyze puzzles and games that involve numerical and logical reasoning, using problem-solving strategies
Pre-calculus 12 - PC03 Determine the number of combinations of n different elements taken r at a time 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!