Grade 9-12 Tasking
There’s a bit of a revolution going on - “we’ve talked about change in the math education for a hundred years, but I think the time has come for a new kind of curriculum”. What gets the kids involved, what gets them to come in. The model is the arts, an aesthetic being, a global response to structure rather than a local one. So choose things from the curriculum based on structural artistic merit. Literature is an important model too, but perhaps English class doesn’t quite get it right either.
All of this is preliminary, with examples on the website. “If you decide to use any of these problems in a class, I’ll send you a form to sign, ethics approval, then I can use comments you make in my research.” We’re interested to hear what teachers think. Funded most recently by Mathematics Knowledge Network.
Old model: There’s a control class here, and the experimental class here, we’ll test them both before, you do your thing, then we’ll test afterwards and compare. Problem is, “there’s no test I want to give them both”. Peter said he thinks teachers know if students are learning, and have a view about whether there will be a long term effect, about if students are engaged.
We need to change the curriculum. This is more of an organic process, a different kind of philosophy; we’ll go through some types of problems and then we’ll talk. They’re set up as Grade 9-10 and Grade 11-12 instead of a single grade, a different kind of cycling and to know what the next grade would do with it. All of these are available on the website.
“Missteaks”: Something wrong, yet there’s some meat/protein here. Example: 4 4/3 = 4 times 4/3. Can you find another example where the mixed number equals the multiplication?
The mistake is the idea that there’s a plus sign missing, not multiplication. Can we get to “a + b = a x b”? Turns out there is a family of things that work. (b = a/(a-1).) Peter did this with his grandson.
Him: “Have you ever done algebra before?”
Grandsom: “Yeah, sure grandpa,” but he hadn’t.
Another “missteak”: root(a+b) = a(root(b)). Find more examples. Or 64/16, cancel the 6’s and you get 4, which is correct. Here there’s exactly 3 fractions total where this trickery works. A more high octane example: 2^4 = 4^2. Can get to (9/4)^(27/8) = (27/8)^(9/4). Many won’t get that because they don’t do fractions, they do decimals. Need the square root of one is the cube root of the other, and there is only one solution, s = 3/2. Try the trick with 3 and 4 (from 2 and 3), and of course there’s a whole family.
“It’s a lot in Grade 9...” (audience laughs) “But I do want kids to see big things, like how in English they may read books beyond their capability.”
There are two kinds of quotes: “Mom, guess what I saw today” versus “I don’t get it, what was he doing”. Hopefully the students who didn’t “get it”, they’ll get some technical stuff and saw something real about mathematics. Who knows how that will work. Had one guy saying “my brain isn’t wired for this”, he said “I want to be in an art school” and so that’s where he should be.
On to Lines and Curves: Max Profit P = R - C. In grade 9 you’re supposed to work with lines, but the ministry says they want “some non-linear too”. Often piecewise linear graphs are used, “but I think you need real curves, rate of change is a big deal”. Kids are ready to understand, they’re not good at determining information from graphs.
Consider that a microwave graph for warming an egg is a straight line, heat transfer in a straight line, while with boiling water, heat transfer to an egg is proportional, curved. “There’s a lot going on with this problem. It’s pretty sophisticated for Grade 9.” But no formulas, that’s right, we’re just working with the graphs. (In Grade 12, there is a formula you can derive, in fact the problem is in a Calculus textbook.)
For the “lineup” question, an audience member asked, “Would they use a ruler to solve that?”. Yeah! You could also use Desmos.
On to Neutrinos: Follow a straight line from the origin through a grid. At x = 3, work out the value, it’s 1.091. For radius 1/10, it’s in there, it does intersect. “The is the one thing I’ve never done in class before.” You have to make a construction and “this is hard even for grade 10.” Maybe you want to stick to the 10 by 10 grid.
Peter didn’t speak to “Transformations”, takes 2-3 weeks, it was their first one done with classes. The students like it, they thought the algebra was cool, every transformation has a matrix. (S is sine and C is cosine.) They almost always choose the algebra (over the graph) because it’s an algorithm. They don’t do square roots very much, they do decimals, even after doing Pythagorus for three years. “So I’m thinking this is more of a Grade 11.”
On to Parabolas and lines: Secants centred at x=1 are all parallel for the parabola (not for cubic). “The d’s drop out.” Then, translating a line parallel to itself, how many parabola intersections? They need to keep track of a family of lines now. Family of curves is so important in University. “My first year students can’t do this,” they have trouble with it. They should have seen it in grade 11 [U level].
On to Water tank: Bring a soda bottle, lines on the side, hole at the base. “I converted these into heights, right.” We talk a bit about why it should be a parabola. “You need a bit of physics, I can do it with a Grade 12 calculus class. If they have physics, we can argue why it’s a parabola. Because of the potential and kinetic energy, when something falls from a certain height, when it hits the bottom the energy at the top is proportional to the square. That’s where it comes in.” (If you want an exponential curve, “Tire” activity below.) Being a parabola, it actually touches down, flow rate is zero. Plot square root of z and it’s a straight line, that’s your proof.
Problem - “This point is pulling the line down.” Clearly an outlier. There should be no water in the hole, but that’s the phenomenon of surface tension. So we take that point out, because that’s what statisticians do. (What did I grade this, grade 11? Who knows about grades.)
On to Trains: In Grade 11, there’s a small discrete section. “When I’ve talked to teachers, it’s short changed, mostly financial math, but this is too good to miss.” Recursive thinking, which is a huge part of a first year university linear algebra course. How many trains of length “n” can you build with those two kinds of cars? (Train has a front and back.) 8 trains of length 5. 13 trains of length 6. Many have seen Fibonacci, is it? Can you convince me there’s 21 trains of length 7?
You have to think about the structure of trains. “What’s in the front of the train.” The ones that start with 1 car are T6, that start with 2 cars are T5. (There’s a shift of the indexing, the fifth Fibonacci number is 5, but the 4th train number is 5.) The Fibonacci Quarterly is a whole journal devoted to these things. It’s sums of Squares property (proof turns out to be double induction), but we can think about it with trains. (“If your trains have this property, drive it over there, if not, drive it over here.” T4^2 + T5^2 = T10.) “This is not well known. ... Which amazes me.”
You’ve got to think about half a train. Can you cut all the trains in half? It’s the 2 car causing the issues. (4-2-4 no, two four trains, 5-5 yes, two five trains.) Pascal’s triangle, left justified, means the diagonals are Fibonacci numbers. “That’s very hard to prove, but you can prove it with trains.” An exercise for my first year university class. A great problem, not an easy proof otherwise. [See also Gr 12 Data Management.]
Moving on to Jacqueline and the Beanstalk. (“Grade 12 IB class, I think it belongs in Grade 12”.) Done in a calculus class, so did the separation of discrete from continuous. She climbs linearly, it grows exponentially. Put them together into a recursive formula, as she’s climbing, the beanstalk not only grows but lifts her up. “Forget the growth for the time she’s climbing, then she rests.” Distance between her and the top is a little easier to work with (it’s minus 5).
This is called the “Scholarship” problem, and you did it in Gr 11 financial math! It’s an annuity problem. You get some funding that grows at 5%, keep taking out $500 for scholarships. How much to start with, to fund “n” scholarships? Present Value. That formula’s in the ministry guidelines. P = d/(1+i) + d/(1+i)^2 + ... + d/(1+i)^n . (This Grade 12 class had forgotten whether they’d seen it or not.) Jacqueline’s climb is a withdrawal of 5 metres, seen in a new context.
(It had been over an hour and a half by now, with three problems left to look at.)
GRADE 12 AND UP
On to Optimal Driving Speed: Drive how fast to minimize fuel costs over fixed distance? “A calculus problem in my book, but we’re doing it totally graphically.” Minimize gas/km, not gas/hr, which is what’s on this axis. Slope of secant, from origin to point on graph, 50 km/hr. But when you go to Toronto, you also put a value on your time... say $6/hour, don’t give yourself too much. Now minimize (z+6)/v, intersect below. “The higher wage you want to pay yourself, the faster you’re going to drive, which makes sense.” So if you pay yourself minimum wage? (“This is a cubic curve, it goes up pretty fast.”)
On to Tire Pressure: How does pressure decrease with a slow leak? Note 400 kPa (kiloPascals) is high for a car tire, but it’s what a bicycle tire would be. This IS an exponential curve. Why? “They think that pressure is somehow pushing molecules out,” they’re thinking of a balloon. But a tire has a fixed volume, there’s no pushing. Molecules are flying around, when they hit the inside of the tire, they bounce off, when some of them hit a hole, they go out. Just the imprint of molecules that happen to be moving and happen to go through the hole.
The rate at which they leave is proportional to the number that are in the tire. “Rate is proportional to number, so it’s exponential.” The water tank [above] isn’t, something else is happening, gravity. If you took the water tank, put a cap on it and took it to outer space, would water still come out? Yes, but it would be exponential. “I’m thinking of my first year students, and what they’re weak at, and what they can’t do.”
Could use this tire pressure system to develop the logarithm. “I want to get a straight line out of this graph to check.” If you write all your data as a power of ten, then it converts, your index is linear. Multiplicative change becomes additive change. “I don’t use the word logarithm yet, I use the word index.”
On to the last task, Exponential Dice. They generate their own decay process. 50 dice, roll and take out 6s, keep going, see how the population of dice decreases over time. There’s ten experiments, all start at 50. (Get a big bag of dice online.) There’s a lot of variation in this data, we plot these. Here’s the theoretical curve I’m expecting. 50(5/6)^n. Best fit line. (I bring up the discrete/continuous issue here - where is the asymptote.)
After a brief break, Peter asked how we felt about the kinds of problems, would they work in your class, what do you need, that sort of thing. Responses included:
“I like the idea, it touches on something that’s relatively new but shouldn’t be new, called spiralling. ... we’re still using textbooks, once we start using these problems there’s a lot of potential here.”
“Students can be more engaged with some of these, there’s more thinking involved.”
“Also hinted at, for a later grade where those ideas are used and developed further, they’re great opening exercises to refresh the ideas. Wiring them up for the next step.”
“The one that resonated with was the line and the curve for Grade 9, they’ll see it again later. Reinforces a continuum.”
“Fits with the idea of low floor, high ceiling.” [Easy to explain, lots of depth.]
To bring it back to the model of the arts, in the art curriculum you start with powerful works of art that are conceptually beyond the student’s understanding. Also, “There hasn’t been nearly enough work on assessment. Christine Suurtaam is probably leading in the world on this, I definitely need help in figuring this one out.” (For the Transformations exercises, there was huge variation. Some students took it seriously, others did nothing.)
But “kids love stories. We all love stories. I’m hoping there’s a story behind all of these units.”
I stuck around for over half an hour afterwards, chatting with other teachers and the like. It’s worth noting that the 9-12 website now has additional projects on it not referenced here.
Did any of this strike a chord with you? Feel free to let me know. Thanks for reading!