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We have a parabola and a moving line. How many times does each line intersect the parabola? Which line is tangent to the parabola?

Plot the diagram

Teacher manualWe have an alligator egg at 10 degrees which needs to be heated to 90 degrees. There are two acceptable methods–– one is to immerse the egg in boiling water and the other is to microwave it at low power. By coincidence, it turns out that the cooking time is the same for both methods––exactly 10 minutes. The graph at the right provides the temperature -time trajectories for each method.

Suppose we can also use a combination of the two––one method for a time followed by the otherand then the other––an instantaneous switch with no temperature loss. a single switch. Can I lower mycooking time by switching?

Teacher manualWe have a parabola and a rotating line. How many times does each line intersect the parabola? Which line is tangent to the parabola?

Plot the diagram

Teacher manualA circle of radius 1 with centre (0, 2) sits above the parabola 𝑦 = 𝑥2. If we let the circle move down the y-axis, it will intersect the parabola. In fact the number of intersections will change as the circle moves and at different times will take the values 0, 1, 2, 3 and 4. Your job is to report the number of intersections in terms of the centre (0, c) of the circle, and to construct an animation that illustrates this behaviour.

Teacher manualHere is a sad house with no roof. In fact the roof is lying at the side waiting to be installed. Your job is to build an animation to move it into place. To do that you’ll need more details about the roof. You know it is a member of the family 𝑥𝑦=𝑘2 and that the peak is √2 units above the ceiling of the house.

Teacher manualThe purpose of this is to give the student practice with graphing different kinds of information. At the right we have a landscape with two towns A and B at the coordinates (10, 20) and (20, 10). Any point on this graph maps into a corresponding point on the “distance graph” which plots the point’s distance from B against its distance from A. For example, note the images of the points A and B. We draw lines and circles on the landscape graph and work out their images on the distance graph. This setup turns out to be a wonderful playground.

Build a program that, given the equation of a landscape graph, will plot the corresponding distance graph.

Teacher manualYou put in x and pull the big brass handle, and you get a “payoff” A(x). The graph y = A(x) is plotted at the right. How should you play this game to maximize your profit? Well the answer depends on what sort of access you have to such an enticing machine.

Build a program that, given the equation of a landscape graph, will plot the corresponding distance graph.

Teacher manualA cylindrical bottle contains 900 ml of water. At t=0 a hole is punched in the bottom, and water begins to flow out. It takes exactly 100 seconds for the tank to empty. Draw the graph of the amount of water z in the tank against time t. Explain the shape of the graph. Now do an experiment and collect and plot the data. Basic principles of physics tell us that the graph should be a parabola. Verify that this is the case. [There is a neat way to change variable so that your points should lie in a straight line.]

Teacher manualIf possible get hold of an old tire. Pump your tire up to P=400 kPa and monitor the pressure as it goes down. Predict what you think the graph of P against time t should look like. Discuss what mathematical form it should have. [Again there is a neat way to change variable so that your points should lie in a straight line. Of course this is the logarithm that belongs in grade 12, but we work with a preliminary form that we call the “index.”]

Exponential growth. The tire model is our introduction to exponential decay. This is a huge idea at this stage as it confronts the students with the two fundamental modes of change. Just as + and × are the two basic operations of arithmetic, so additive and multiplicative change are the basic modes of growth and decay. The most powerful way to think of these is in terms of rate of change. In additive growth, the rate of change is independent of size, whereas in multiplicative growth, the rate of change is proportional to size. This distinction will be encountered in different ways in the next units.

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 manualThis is a conceptual adventure with some of the remarkable properties of the Fibonacci numbers. How many different ways are there to build a train of length 12 using cars of length either 1 or 2? The main goal of this unit is to introduce the fundamental idea of recursive thinking. However the “trains” problems gives us a surprising method for constructing proofs for some of the elusive properties of Fibonacci numbers, for example that the sum of the squares of two consecutive Fibonacci numbers is always a Fibonacci number.

Teacher manualWhen I ask my first-year students at Queen’s what they know about Pascal’s triangle, they can give the addition rule that generates it from the top down, and they also know that the entries are the binomial coefficients and some of them know that these are the combinatorial coefficients. But they seem to have never seen how these three things are all connected. For example they seem to have never encountered the wonderful argument that the combinatorial coefficients have the simple additive property.

Teacher manualA point moves counterclockwise around a circle of radius 4 at constant speed 45°/s. Its height y as a function of time is what’s called a sine curve. We build an animation of the motion of the point and that includes, in the middle, a small block going up and down that the students agree looks just like a spring. In fact the oscillation of a spring turns out to be sinusoidal (though a bit of grade 12 calculus and physics is needed for that). We get hold of a real spring (or bungee chord) and play with it.

Teacher manualFind a vertical pier at the edge of the sea, put a scale down the side, and measure the height of the water against time and you’ll get a graph something like that below. The data are taken in the Bay of Fundy––a region famous for its unusually high tides, a resonance effect caused by the funneling shape of the basin. As you can see from the graph, the amplitude of these tides is almost 6 meters whereas on the ocean, tides are a third or a quarter of that. Theoretical considerations show that tidal graphs are well modelled by sinusoidal functions and the student has to find an equation that fits the graph below. As a treat, one can take the following problem—given the period of the moon’s rotation about the earth and the earth’s rotation around its axis, calculate the period of the tides.

Teacher manualThis is an extraordinary tale and well worth playing with. The students will encounter many things they already “half know,” and they will be enchanted at the end. The technical math here is at exactly the right level (working with the exponential and the sinusoidal functions) but of much greater value to the student will be the connection of the math with a world that they are all involved with, principally music, but also, for example, the nature of perception.

Teacher manual