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The time taken for the moon to orbit the earth, the lunar sidereal period, is approximately 27.3 days. The time taken for the earth to orbit the sun, one solar year, is approximately 365.25 days. We can use this information to create the phases of the moon over one lunar month. (Note that one lunar month is the time taken between two identical phases, such as a full moon, and is actually greater than the lunar sidereal period due to the earth’s orbit of the sun).

For this investigation we will assume the earth’s orbit of the sun, and the moon’s orbit of the earth are circular (they are actually slightly elliptical).

The average distance of the earth to the sun is approximately 1.5 x 10¹¹ m. The average distance of the moon to the earth is approximately 3.84 x 10⁸ m.

The diagram below demonstrates how we see different phases of the moon:

The view when seen from earth is then:

To calculate the phases we need to use the following steps:

• Use the diagram below to find x : y in terms of theta.

• Use the diagram below to find expressions for alpha and beta after n days. You may assume that initially the sun, earth and moon are on the same horizontal line and it is a full moon. The orbits are anti-clockwise.

• Use Pythagoras to find an expression for the distance of the moon from the sun in terms of alpha and beta. Let this expression be equal to d.
• Use the cosine rule to find an expression for theta  in terms of d.

Putting all of this together we will have an expression for theta in terms of n. We can use Excel to help us calculate the values of x and y for various values of n. Note that the angles are measured in radians:

The values of x and y in this table are based on a moon radius, on paper, of 2.5 cm. Looking at the values we can see that this model gives a lunar month of between 29 and 30 days.

We can now use these values of x and y to create diagrams of phases of the moon. These can easily be created using Microsoft Word’s drawing tools. For example, when n = 6 we need a semi-circle (in Word use the Arc drawing tool) and an ellipse (use the Oval tool). Then just fill them with white and give them a black background:

The first eight phases of the moon will look like this:

In many multiple choice tests points are deducted for incorrect answers. For example, in a multiple choice test with five possible choices it is quite often the case that an incorrect answer would result in one quarter of a point being deducted from the final score. In a test with four possible choices every incorrect answer would result in one third of a point being deducted. Generally speaking, if there are n choices, each incorrect answer would result in 1/(n-1) points being deducted.

One argument for this policy is that in a test with, for example, 80 questions with five choices for each question, a person who knows absolutely nothing about the subject can be expected to get approximately 16 questions correct and 64 questions incorrect. This will give a final score of 16 – 0.25 x 64 = 0. Without this policy the person, who knows absolutely nothing about the subject, would still receive a score of 16 / 80 or 20%.

It is quite often the case that a person is able to eliminate some of the possible choices and thus stand a better chance of guessing correctly. We can set up a spreadsheet to investigate various scenarios:

The first column is the question number. The second column is the number of choices the test taker was able to eliminate (if the person knows the correct answer four choices can be eliminated etc.) The third column is the result of the test takers choice. He/she is either correct or incorrect, the probability of which is decided by how many choices can be eliminated.

Here are the formulae used in the spreadsheet:

The formula in column A simply adds one to the value in the cell above and displays the result. The formula in column B randomly chooses how many choices the test taker is able to eliminate. The formula in column C displays either “Correct” or “Incorrect” based on a probability which is calculated using the number of choices eliminated.

Note that we do not have to type every formula into every cell. Once we have typed the formula in cell A3 we can just copy and paste it into the rest of column A. Once we have typed the formulae into cells B2 and C2 we can copy and paste them into the rest of columns B and C.

We can now calculate our final score:

The score is calculated using the following formulae:

We can use a spreadsheet like this to investigate various “what if…” scenarios. For example, what if the test taker is able to answer half of the questions correctly but completely guesses the remaining questions? What if the test taker is able to eliminate only three choices from every question?

What is the best way of lacing a shoe? How can we minimise the amount of shoelace needed? We can use Pythagoras to investigate. Suppose the holes in each column are separated by a distance of 2 cm, and the two columns of holes are separated by a distance of 5 cm.

Diagrams and calculations created using Microsoft Word’s drawing tools and equation editor

It is often said that is is possible to connect any person in the world to any other person by an average of six degrees of separation. For example Peter knows Mark who knows Paul who knows Lisa who knows Louise who knows Andrew. We can use matrices to investigate this theory.

Five towns are connected by roads, as shown in the diagram:

The table shows how many direct routes there are between towns:

The adjacency matrix (matrix A) for the network is therefore:

The matrix A² will then represent the number of routes between two towns with one stopover. The matrix A+A² will represent the number of routes between two towns with at most one stopover. The matrix A³ will represent the number of routes between two towns with two stopovers. The matrix A+A²+A³ will represent the number of routes between two towns with at most two stopovers.

Now, suppose we replace our original network of towns with a network of people. If two people know each other they are connected by a line. Matrix A will then represent which people are separated by one degree of separation (i.e. who knows who directly). Matrix A² will then represent which people are separated by two degrees of separation (i.e. Bob is friends with Jill who is friends with Mark, so Bob and Mark are separated by two degrees of separation). Matrix A+A² will represent which people are separated by a maximum of two degrees of separation. Matrix A³ will represent which people are separated by three degrees of separation etc.

Note that with current matrix we are looking at, whether it is A² or A³ etc, the only thing we need to determine if Bob and Mark, for example, are separated by the current degree of separation is any number except zero.

This investigation works well using very large matrices. However, it is impossibly difficult to create an n x n matrix where n represents the population of the world, so you will have to scale down your model. 

Excel is very capable at handling matrix calculations involving large matrices. Click on the image for an example spreadsheet:

A stereoscopic image is designed so that when viewed a certain way (for example with a pair of special glasses) the viewer sees a slightly different image with each eye, producing a 3D effect. Here are two different methods of creating them. They both use Google Sketchup, which is free to download.

The first method requires the use of red and green lenses. These can easily be made if you have access to material suitable to be used as the coloured lenses. Here is an example image:

The green shapes will only be seen by the red lens (since the red lens makes the white background appear red). The red shapes will only be seen by the green lens (since the green lens makes the white background appear green). The blue shapes will be seen by both lenses.

Let’s explore the math behind this image…

When we focus on something, we see double of whatever is closer or further away than what we are focusing on. Try it and see. Put your finger between your eyes and the computer screen. Focus on your finger. You will see two screens behind your finger. Now, focus on the screen, you will see two fingers between the screen and your eyes.

Suppose we wish to create a 3D effect with a circle shape and a triangle shape. We would like the triangle to appear further away than the circle.  Suppose the viewer is 60 cm from the paper. You wish to create an effect where the triangle is 40 cm behind the circle. Let’s assume the distance between the viewer’s eyes is 6 cm:

Viewed from above

According to this diagram, the triangle will appear to the left of where the left eye is looking and to the right of where the right eye is looking. We will therefore see two triangles. Using similar triangles we can calculate the value of x, the distance between the two triangles:

$latex \frac{6}{60}=\frac{x}{40}$

So the value of x will be 4 cm.

Next, we need to consider the size of the circle and the triangle. Suppose we wish each to have a height of 10 cm. We can just draw a circle of diamter 10 cm on the paper, however, we want the triangle to appear further away so it should be smaller:

Viewed from the side

Again by similar triangles we have:

$latex \frac{h}{60}=\frac{10}{100}$

So the value of h, the height of the image on the paper, will be 6 cm.

By a similar method it can be shown that the spacing of the two triangles on the paper will not be 4 cm, as calculated previously, it will actually by 2.4 cm, since distances further away appear smaller. Our result will be (scaled down to make it fit the computer screen):

The effect works better if you have many shapes on screen, as in the stars example.

Why did I use Google Sketchup to create these images? When drawing two overlapping shapes Google Sketchup automatically creates a shape from the overlapping region, making it easy to fill with a different colour. Google Sketchup also has a ruler which you can use to make your diagrams accurate.

The second method is simpler to explain. Create a 3D image in Google Sketchup and export the image as a 2D graphic. Then, move the camera slightly to the left and export the image again as a 2D graphic. Put the two images side-by-side, cross your eyes so the two images overlap and relax until the image focuses:

A fractal tree is “grown” by repeatedly replacing each branch with smaller branches:

The total number of branches and the total length of all of the branches follow geometric series.

We can use Google Sketchup, a free 3D modelling tool, to create fractal trees. We start with the basic design of a branch on the left. We make two copies and resize to 70% (for example) of the original size. We then rotate them both and position them, as in the diagram on the right. We then group these three pieces together. This will be our building block (the diagram on the right that is):

Note that the circle and the line make it easier to position the branches in Google Sketchup. Just drag the mid-point of the base of each of the small branches to the end of the line inside the circle on the large branch. Also, when rotating the branches you can use the circle and line as reference points.

Make two copies of the design on the right which are both 70% smaller. Replace the two branches of the large design with the two smaller copies. Your tree should now have grown by four extra branches.

Keep repeating the previous step again and again, each time making only two copies and replacing the branches on the larger design with the two copies. This is the easiest way to grow the tree. See the attached Google Sketchup file for an example.

Experiment creating different trees by changing the scale of the new copies, the angle of each branch and the initial number of branches (this example has two branches initially).

Calculate the total number of branches and their total length after n iterations…

Rogue waves are described as unusually large waves for a given sea state. They can often appear out of nowhere. One theory is that they are caused by the chance superposition of many smaller waves.

The principle of superposition states that if two or more waves occupy the same medium, then the total displacement of that medium is the sum of the displacement of the individual waves.

In the diagram above, the black wave is the sum of the three orange waves

We can set up a spreadsheet to simulate the formation of a rogue wave.

Before we begin our spreadsheet we should consider one more fact: the only thing that affects the speed of a wave is the physical properties of the medium through which it is travelling. This means that all of our waves should be travelling at the same speed. However, this does not mean they should be travelling in the same direction.

The easiest way to create a rogue wave is to use cosine functions of the form

$latex y=A cos (\frac{x-Bt}{C})$

to describe our smaller waves. The rogue wave will then occur when t = 0 centred on the line x = 0. The value of A will equal the amplitude of each wave, the value of B (-1 or 1) will affect the direction of each wave, and the value of C will affect (but not equal) the wavelength of each wave. Changing the value of t will move the wave left or right.

In this example we will just use three waves. Large waves will be common in this simulation. To get a more interesting simulation you should use thirty or forty waves.The more waves you have, the less the chance of a rogue wave occuring, however, a rogue wave will always occur at t = 0.

Here is a screenshot of the spreadsheet. The values of A, B and C for each wave have been randomly chosen.

With careful thought it is possible to calculate every value using only three formulae. Lets take a closer look.

In cell B5 (which is merged with C5 and D5) enter the following formula:

=B$4*COS(($A5-C$4*$C$1)/D$4)

Study this formula carefully. Note the use of dollar signs. Remember, a dollar sign in front of a letter results in the column reference staying the same when copied and pasted to another cell. A dollar sign in front of a number results in the row reference staying the same when copied and pasted to another cell. This formula can be copied and pasted to every cell for all three waves and it will work. Study it and understand why.

In cell A5 type -300. In cell A6 type

=A5+0.2

The rest of the x coordinate cells can now be filled in by copy and pasting cell A6 into them.

Copy and paste cell A6 into the cells below until the value of the x coordinate is equal to 300 (that’s approximately the next 3000 rows).

Copy and paste cell B5 into every other cell for waves 1, 2 and 3 (that’s approximate 3000 rows for each wave).

The only thing left to do is calculate the resulting wave. This can easily be done using the SUM function.

In cell K5 type

=sum(B5:J5)

Note that instead of typing “B5:J5″ you can just highlight the cells instead.

Copy and paste this formula into the cells underneath (approximately the next 3000 of them).

All that is left to do is to plot the graphs.

Highlight cells A5 to K3005 and select Insert > Scatter> Scatter with smooth lines. Fix the axes so the scales do not change by right clicking them and inputting appropriate values. Tidy the chart up and you should have something similar to:

Note that it is clearer to see the resulting wave if you choose your colours carefully.

Now, increasing or decreasing the value of t will move your waves left or right, depending on the value of B.

It is possible to make an animation out of this by copying and pasting each frame into Microsoft Paint, saving as a bitmap image, then importing them all into Windows Movie Maker. You may need a few hundred frames, but this can easily be achieved with a couple of hours of focus. All of these programmes should be already installed on any windows machine. Windows 7 users may have to use Windows Live Movie Maker.

This tutorial is a simple example with only three waves. It works much better if you have thirty, forty or even fifty waves, as you can see in this video I made for my physics class. Fast forward to 30 seconds to see the waves in action.

A cartogram is a map where the land area is proportional to a certain variable, such as population. We can use Microsoft Word’s drawing tools to easily create a population cartogram.

Turn on Microsoft Word’s grid (Page Layout > Align > Grid Settings).

Choose a suitable size for the grid spacing. The horizontal and vertical spacing should be equal. It is better for the spacing to be too small than too big. If we are making a cartogram for Europe, for example, a spacing of 0.2 cm would be suitable. We can always enlarge our diagram at the end if it is too small. Choose a scale for your diagram. In this tutorial we will use one square = one million people.

Search the internet for an outline image of the country you wish to create. Research its population. Copy and paste the image into your Word document. Resize the country so that it approximately covers the correct number of squares based on its population. Click on the image to enlarge.

Note that to move the image wherever we want just click on it, then select Format > Text Wrapping > In Front of Text.

Now, we need to approximate the shape of the country using small squares. The size of each square should equal the size of a square on the grid.

Draw a square equal to the size of a square on the grid by selecting Insert > Shapes > Rectangle. This square will represent one million people. Copy and paste the square as many times as you need to until the number of squares accurately represents the population. Position these squares over the top of your outline map so that they approximate its shape. Click on the image to enlarge.

Now, you may wish to erase all of the lines around the individual squares and also give the country a border.

Select Insert > Shapes > Freeform and draw  around the outline of the shape. Then, erase the outline image. Click on the image to enlarge.

Now, add more countries to your map!

Click here for a student sample of a cartogram.

A hypotrochoid is the curve traced out by a fixed point on a circle rotating around the inside edge of a larger circle. There are a variety of ways we can create them in class.

One way is by using Scratch, a free computer programming language for young learners.

 Another way is by using a graphical calculator with the parametric equations

$latex x=(R-r)cos(\theta)+d cos(\frac{(R-r)\theta}{r})$

$latex y=(R-r)sin(\theta)-d sin(\frac{(R-r)\theta}{r})$

By far the most beautiful and accesible way, however, is by using Excel. We can create a spreadsheet based on the above parametric equations.

The equations can be derived by answering the following questions based on the diagrams:

1. Write down an expression for the circumference of the outer ring.
2. Write down an expression for the circumference of the innter cog.
3. Write down an expression for the distance that the inner cog has travelled around the outer ring (arc AB).
4. Write down an expression for what fraction of a revolution the inner cog makes during this time.
5. Write down an expression for angle alpha (be careful, the inner cog is travelling on a curved surface).
6. Write down an expression for the coordinates of the centre of the cog.
7. Write down an expression for the coordinates of the pencil.

Click on the image to download the spreadsheet:

Research has shown that traffic jams can be caused by the bad driving of just one driver. Slowing down, even just a little, can cause the driver behind to overreact and slow down a little more, the next driver overreacts and slows down even more… Eventually a traffic jam is created.

We can set up a mathematical model to simulate this.

For this investigation we will use Microsoft Excel. Our model will use the following rules:

• Each car must travel at a safe distance behind the car in front. This safe distance will be measured in time and based on the car’s speed. For example, if the safe distance is 2 seconds, and a car is travelling at 8 m/s, then the car must be 16 metres behind the car in front
• If the distance to the next car is less than the safe distance then the car will decellerate. The larger this difference, the larger the magnitude of the decelleration.
• If the distance to the next car is greater than the safe distance then the car will accelerate. The larger this difference, the larger the magnitude of the acceleration.
• It takes each driver a certain amount of time to react to the car in front. For example, if a driver’s reaction time is 0.5 seconds, at t=2.5 seconds the driver will react to what happened at t=2 seconds.

In Excel, create boxes for the following variables: Time Interval, Initial Spacing, Reaction Time, and Safe Distance. Give each of these variables values. These can be changed later. Click on the image to enlarge:

Since the acceleration of each car will be constantly changing, we will approximate their motion as small time intervals of constant acceleration.

The initial spacing will determine the distance between each car at t = 0 sec.

Reaction time and safe distance are described above.

Create columns for the time, and the position, velocity, acceleration and safe distance for each car. In this example we will just use three cars. In cell A5 enter zero. In cell A6 enter =A5+B$1.

Note: We need the dollar sign in B$1 so that when we copy and paste the formula to a different row, the reference to cell B1 will not change. You can read more about relative and absolute references here.

Now, copy and paste the formula from cell A6 into cells A7 to A20. We will extend the number of rows later. Click on the image to enlarge.

Next, we need to set the initial values of the position, velocity, acceleration and safe distance of each car.

Car 3 will be our lead car. This means it is free to accelerate. Let’s give it an acceleration of 1 m/s/s.  The initial acceleration of every other car will be zero. The initial velocity of every car will be zero. The initial position of each car will be based on the value of initial spacing, with car 1 having an initial position of zero. The safe distance of each car is given by the equation :

safe distance in metres = (safe distance in seconds) * velocity + car length

Enter the following formulae into the following cells. Copy and paste the equations into the given cells. Note the use of dollar signs. A dollar sign in front of the letter indicates that the column reference will not change when copied and pasted to another cell. A dollar sign in front of the number indicates that the row reference will not change.

In the other cells on row 5, enter the values from the previous paragraph. Your sheet should now look identical to the image attached. Note that cells E5, F5, I5, M5 and J5 should all contain formulae. Click on the image to enlarge.

Now, we need to complete the second row of our data. We need to start thinking about our reaction time. For this we will use Excel’s IF function. An IF function takes the format:

 =IF(condition,value if true,value if false)You can read more about the IF function here.

We also need to decide how to calculate our acceleration. If a car is too close to the one in front it needs to decellerate. If it is too far behind the one in front it needs to accelerate. The simplest way of modelling this is:

acceleration = distance to car in front – safe distance

Since we are only on our second row of data, a driver can only react to the car in front if the reaction time is equal to the time interval we are using to approximate the motion.

In the acceleration cell for car 1 (cell D6), enter the following:

=IF($F$1=1,F5-B5-E5,0)

Translated into English this reads “if the reaction time is equal to one time interval, set the value of acceleration as (distance to car in front – safe distance) using the values in the previous row, otherwise set it to zero”.

Copy and paste this equation into the acceleration cell for car 2 (cell H6). Let car 3′s acceleration again be 1 m/s/s.

To calculate the new position and new velocity of each car we need to look to physics:

$latex s=ut+\frac{1}{2}at^2$

$latex v=u+at$

To calculate the new position of car 1, in cell B6 enter:

=B5+C5*$B$1+0.5*D5*$B$1^2

To calculate the new velocity of car 1, in cell B7 enter:

=C5+D5*$B$1

These are based on the motion equations above.

Copy and paste these position and velocity equations into the position and velocity cells of cars 2 and 3. To calculate the new safe distances we can just copy and paste the safe distance cells from the first row of data. Click on the image to enlarge.

The formulae we have entered for the position, velocity and safe distance for each car will not change from row to row. Copy and paste them into the relevant cells in the third row of our data (which is actually row 7 in our spreadsheet).

Now, we need to consider the case where our reaction time is 1 OR 2 time intervals. If it is one interval we can use the previous row of data to calculate the acceleration of each car. If it is two intervals we need to use the first row of data to calculate the acceleration.

Copy and paste the acceleration cell from the second row of car 1 (cell D6) to the third row of car 1 (cell D7).

The formula in cell D7 should now read:

=IF($F$1=1,F6-B6-C6*$H$1,0)

We need to change this formula to take into account the case where the reaction time is equal to two time intervals.

Change the formula in cell D7 to the following:

=IF($F$1=1,F6-B6-C6*$H$1,IF($F$1=2,F5-B5-C5*$H$1,0))

 Translated into English this reads “if the reaction time is equal to one time interval, set the value of acceleration as (distance to car in front – safe distance) using the values in the previous row, otherwise if the reaction time is equal to two time intervals, set the value of acceleration as (distance to car in front – safe distance) using the values in the previous previous row, otherwise set the value to zero”.

Copy and paste this equation into the acceleration cell for car 2 (cell H7). Let car 3’s acceleration again be 1 m/s/s. Click on the image to enlarge.

For the next row of data, the position, velocity and safe distance formulae will not change.

Copy and paste them from the relevant cells on the third row of our data to the relevant cells on the fourth row.

We will need to update our acceleration formula to take into account the case when the reaction time is equal to three time intervals.

Copy and paste the acceleration cell from the third row of car 1 into the fourth row of car 1. Then change it to the following:

=IF($F$1=1,F7-B7-C7*$H$1,IF($F$1=2,F6-B6-C6*$H$1,IF($F$1=3,F5-B5-C5*$H$1,0)))

Copy and paste this cell into the acceleration for car 2. Again, let the acceleration of car 3 equal 1 m/s/s.

For this example we will assume that the reaction time can only equal one, two or three time intervals. To take into account longer reaction times, just follow the same steps, updating the acceleration formula each time.

We can now copy our entire fourth row of data into all of the remaining rows. Click on the image to enlarge.

This is all we need to know to create our data. Obviously we will need to extend the simulation past three seconds. It will also be more interesting if we include more than three cars. The final Excel file can be downloaded here.

Now we need to think how we can display this data in a way that is easy to understand… This will come in part 2.