RabbitMath Grade 11 August 2019

The RabbitMath Curriculum Grade 11

Curriculum Outline

Peter Taylor Queen’s University

Chris Suurtamm

University of Ottawa

and the RabbitMath Team For frequent updates and technical resources visit rabbitmath.ca

RabbitMath Grade 11 August 2019

RabbitMath is a mathematics curriculum. That means that the problems and the activities are all true to

the nature of mathematics. Indeed, a potential activity is not accepted for this collection, unless

mathematicians find it interesting and engaging.

There is an idea that such problems might not be accessible to grade 11 students. Certainly there are lots

of mathematical results that high school students would indeed not be ready for. But there are also

wonderful mathematical “stories” that interest and even delight mathematicians and are a good fit to the

standard high school curriculum. Of course mathematics works and plays with structures of considerable

sophistication, and the projects in this curriculum are indeed sophisticated. In the past, teachers have

been intrigued with many of the problems in this collections and have wondered whether their students

are “ready” for them. I have always believed that mathematics students are ready to engage at a higher

level of sophistication than we give them credit for.

In many ways our model for student readiness comes from our experience in disciplines such as drama,

English and the creative arts. In these subjects teachers bring resources into the school classroom that

would typically appear in the university classroom as well, and that indeed would be the subject of

focused work among professionals. In short they bring to their students sophisticated works of art that can

be experienced and enjoyed on more than one level. We believe that this can and should also be the case

in the school mathematics curriculum.

This is important. We aim for a curriculum structure that will allow and even encourage students with

different interests and abilities to work together on projects that they can all find, perhaps in different

ways, meaningful.

The purpose of the RabbitMath project is to provide a model for such a curriculum. We hope that

teachers will bring these units into their classroom and work with them together with their students.

Certainly we will all come away from that experience with ideas about what works well and what might

need to be redrafted. Such contributions, shared among teachers and mathematics educators, will allow

these curriculum materials to effectively meet the needs of our students.

Main Features of the RabbitMath curriculum Focus on hands-on student engagement—manipulatives and animations. Significant objective is university preparation—focus on the analysis of complex structures. Emphasis on collaboration and communication. Students will work on Jupyter Notebooks using laptops or Chrome books and Python. Mathematical modeling—math interacting with the world. Resources: documents, videos, animations and online interactive support. Coverage of all Overall Expectations and most Specific Expectations of the Ontario curriculum. A recent article provides some background ideas. Find it at: The Conversation RabbitMath.

RabbitMath Grade 11 August 2019

5025

1

d

RabbitMath Grade 11

Curriculum Outline

1. Mirrors Page 8

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 these images.

The configuration is symmetric so we only need solve the

problem for the three reflections 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).

2. Trains Page 17

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” problem 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.

Art by Meg

RabbitMath Grade 11 August 2019

row0 1

row1 1 1

row2 1 2 1

row3 1 3 3 1

row4 1 4 6 4 1

row5 1 5 10 10 5 1

row6 1 6 15 20 15 6 1

row7 1 7 21 35 35 21 7 1

row8 1 8 28 56 70 56 28 8 1

3. Recursive thinking Page 21

The trains problem introduced us to the

powerful idea of recursive thinking. Here

we extend this approach looking at the

well-known “sum” property of Pascal’s

triangle. Then we look at a number of

interesting recursive equations connected

with the Fibonacci numbers.

4. Two towns Page 27

Landscape Distance

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.

RabbitMath Grade 11 August 2019

5. Parabola and rotating line Page 33

We have a parabola and a family of lines passing

through the origin. How many times does each line

intersect the parabola? Which lines are tangent to the

parabola?

Plot the diagram and use Python to animate it.

6. Parabola and circle Page 37

A circle of radius 1 with centre (0, –1) sits below the

parabola 𝑦 = 𝑥2 and intersects it at the origin.

If we let the circle move up the y-axis, it will intersect

the parabola the number of intersections will change

and as it moves it will take the values 0, 1, 2, 3 and 4.

Your job is to report the number of intersections in

terms of the position of the centre (0, c) of the circle. .

7. Roofing Page 41 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.

RabbitMath Grade 11 August 2019

8. Money machine Page 45

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.

9. Tire Page 52

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 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. This will allow us to conclude that the graph of P against time t is exponential.

10. Exponential growth&decay Page 57

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.

RabbitMath Grade 11 August 2019

a1

d d d d d

a2

a3

a4

a10

11. Arithmetic and Geometric growth Page 62

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.

12. The scholarship problem Page 67

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.

13. Circle and spring Page 74

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 we include, 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 take measurements of period and amplitude..

RabbitMath Grade 11 August 2019

14. Tides Page 79

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. As a treat, we investigate 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.

15. Music—the magic of 12 Page 86

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.

C C# D Eb E F F# G G# A Bb B 0 16.35 17.32 18.35 19.45 20.60 21.83 23.12 24.50 25.96 27.50 29.14 30.87 1 32.70 34.65 36.71 38.89 41.20 43.65 46.25 49.00 51.91 55.00 58.27 61.74 2 65.41 69.30 73.42 77.78 82.41 87.31 92.50 98.00 103.8 110.0 116.5 123.5 3 130.8 138.6 146.8 155.6 164.8 174.6 185.0 196.0 207.7 220.0 233.1 246.9 4 261.6 277.2 293.7 311.1 329.6 349.2 370.0 392.0 415.3 440.0 466.2 493.9 5 523.3 554.4 587.3 622.3 659.3 698.5 740.0 784.0 830.6 880.0 932.3 987.8 6 1047 1109 1175 1245 1319 1397 1480 1568 1661 1760 1865 1976 7 2093 2217 2349 2489 2637 2794 2960 3136 3322 3520 3729 3951 8 4186 4435 4699 4978 5274 5588 5920 6272 6645 7040 7459 7902