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Grade 11

Grade 11 Curriculum Outline

The Grade 11 schedule that appears below is out of date. It will be corrected by Aug 26. Here you will find an updated copy of the Grade 11 Curriculum Outline

Mirrors

You are in the middle of two mirrors set at an angle of 50° and your eyes are 1 meter from the vertex. You can see 6 images of yourself. Here’s the problem:

Plot, for each image, the path of the light ray as it travels from you back to your eye and calculate the apparent distance from you to each of the six images.

The configuration is symmetric so we only need solve the problem for the three reflection in the right-hand mirror. Just to get you started, I have drawn the simplest case. You are looking orthogonally at the mirror and your image is as far behind the mirror as you are in front of it. Of course the light ray does not go through the mirror––it goes from your face to the mirror and then right back to your eye––a distance of 2d where d=sin⁡(25).

Teacher manual
Mirrors

Trains

This 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 manual
Parabola and rotating line graph

Pascal’s triangle

When 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 manual
Parabola and rotating line graph

Tow Towns

The 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 manual
Parabola and rotating line graph

Parabola and rotating line

We 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 manual
Parabola and rotating line graph

Parabola and circle

A 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 manual
Parabola and rotating line graph

Roofing

Here 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 manual
Parabola and rotating line graph

Money machine

You 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.

Teacher manual
Parabola and rotating line graph

Tire

If possible get hold of an old tire. Pump your tire up to P=400 kPa, drill a small hole, 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.

The idea we develop is that the rate at which molecules escape from the tire is always proportional to the number in the tire. We can verify this by measuring the pressure at regular intervals and checking whether the change over the interval is proportional to the pressure at the start of the interval. From this we can get an exponential equation for P against time t.

The answer: Exponential decay. As seen in the above animation, the rate that air is escaping is proportional to the amount of air currently in the tire--resulting in an exponential decay function.

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 manual
Parabola and rotating line graph

Exponential Growth&Decay

In this unit we study animations of exponential decay and exponential growth and use these to understand these processes better and find equations for the graphs. We study the mechanisms behind the process and try to extract from these as much information about the mathematics as we can.

Teacher manual

Arithmetic and Geometric growth

Our grade 11 journey into financial math really might be called multiplicative growth meets additive change. Indeed we are always trying to make our enterprises prosper and their rates of growth and decline are a typically a mixture or processes that are fundamentally exponential (change proportional to size, e.g. interest) and linear (constant change, e.g. salary, weekly expenses). In this section we take an abstract look at these two basic modes of change.

Teacher manual

The scholarship problem

Here we study additive and multiplicative change in the context of compound interest and annuities. In the scholarship problem, an endowment fund grows as it gains interest but capital each year for an annual scholarship. This is a mixed arithmetic and geometric process and the concept of present value gives us an unexpected way to analyze the process.

Teacher manual

Circle and spring

A 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 manual
Parabola and rotating line graph Parabola and rotating line graph

Tides

Find 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. But that’s just the beginning and we follow it with a beautiful structural problem. Given the period of the moon’s revolution about the earth and the earth’s rotation around its axis, calculate the period of the tides.

Teacher manual
Tides

Music—the magic of 12

This 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
Music