513 Jeffery Hall, Queen's University, Kingston ON, K7L 3N6

math9-12@queensu.ca

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We start with 50 dice. They are rolled repeatedly and each time all the sixes are removed until all the dice are gone. The students repeat this experiment many times. There is lots of variation but the data ought to provide an exponential decay process with multiplier 5/6.

We take the average of 10 trials and use a log plot to check this out. We get a pretty good straight line.

In the diagram at the right, the big circle of radius 2 is fixed and the wheel of radius 1 rotates around it without slipping in a counter-clockwise direction at constant angular speed. The students must use Desmos to construct an animation of the process. When they succeed, the “video” they get is extremely satisfying.

Teacher manualThis is an excellent non-permanent vertical surfaces exercise. We ask the students to draw the graph

(in red) and then to draw its inverse graph (blue). The inverse is not a function graph but it can be broken into four functional pieces that the students must identify both graphically and algebraically.

Jacqueline, climbing 5 meters each minute, is attempting to reach the top of the beanstalk but at the same time it is also growing at the rate of 2% per minute. Will she ever reach the top and if so when? This seems to be a new problem for the students, a discrete recursive sequence that includes both additive and multiplicative change.

Imagine our surprise when, upon managing to solve it and feeling quite pleased with ourselves, we discover that this is a familiar problem in disguise, one we recently encountered in the Grade 11 financial math strand!

*The scholarship problem*. An annual scholarship of $500 is financed by a trust fund with a capital of $7500 growing at an annual rate of 5%. Can the trust
fund support the scholarship forever or will it one day run out? In the latter case, for how many years can the scholarships be awarded?

How fast should we drive on the highway if we simply want to minimize our fuel costs for a fixed distance. The students are asked to figure out what the fuel consumption graph (litres/hour vs km/hour) looks like and then use graphical means to find the optimal speed. For the graph we are using it turns out to be 50 km/h.

Now we put a value of $6/h on our driving time (and a price on gas of $1/L) and use the same graphical solution to get 90 km/h.

Finally we move into the world of algebra and build a simple model for fuel consumption of a car, giving us a formula for our graph. Armed with that, both of our problems can be solved with some simple differential calculus.

Teacher manualTake the parabola *z* = 16 - *x*^{2}, add another horizontal axis and rotate the parabola 360° about its central axis. What is the equation of the surface that you get?

Now take the contour diagram of the surface and take a walk on the hill following the path *y* = 4 – 2*x*. Describe your journey.

If player A invests a and player B invests b (0≤a, b≤1), the payoffs are P(a,b) to A and P(b,a) to B. But A goes first so that B knows A’s investment before he has to choose his. What should each player do to maximize his or her own payoff? In the first session, we work with the algebraic formula for P(x,y). In the second we work with its contour diagram seen at the right.

There is some sophisticated structure here and both activities test the students’ capacity for clear, organized thinking. There is a fun story behind this game based on the reproductive behaviour of ants.

Teacher manualStarting at x = 0, a ball moves along the curve:

at constant horizontal speed u=1. That is, the ball's x- coordinate always increases at rate 1. At some point the ball flies off the track and is then only acted on by the force of gravity. The problem is to find the point of release that maximizes the distance the ball travels before landing on the x-axis. That is, we want the landing point to be as far as possible from the origin. We take the gravitational constant to be g = 0.5

For example, in the diagram at the right, the ball is released at x=1 and it lands somewhere near x = 4.2. To increase the distance traveled, should it be released a bit later or a bit earlier?

Teacher manualYou buy a large coffee (300 ml) at 80 degrees and four milks (15 ml each) at 4 degrees, and get into your car which stays at a constant temperature of 16 degrees. It will be a 10 minute drive before you can drink the coffee, and you want it to be as hot as possible at that time. Do you put the milk into the coffee mug at the beginning or after the 10 minutes when you’re ready to drink it? Do you mix first and then drive or drive first and then mix? Does your intuition give you an answer? This is a nice example of exponential decay and the principles around heat-loss that we will encounter are ones we meet in our day-to-day lives.

Teacher manual