**HIGH
SCHOOL MATH ACTIVITIES **

I've gathered together a few of what I think are interesting math activities that I created and used over the years. Feel free to use, as is or modified to fit your needs, any of the worksheets. If you have any comments, questions or suggestions, you can contact me at bkiggins@kw.igs.net .

- Finite differences
- Powers of Three
- Graphing Project
- Pop Can Assignment
- Measuring the Acceleration due to Gravity
- Using Trig to Calculate Pi
- Tangent Experiment
- Sloping Letters
- Lotteries and Other Probability Stuff
- Voronoi Diagrams
- Pi in a Can
- Benford's Law
- The Mandelbrot Set and Complex Numbers
- Stylometry
- Drill Sheets

In the grade 9 course we spend a little time examining finite differences as a way of deciding whether a particular equation is that of a linear relation or a quadratic. I've extended that idea to include 1) how the finite difference is related to the coefficients of the relation 2) extending finite differences to cubic relations and beyond and 3) proving what is observed algebraically. The worksheets only go as far as proving the quadratic case but, if you are familiar with Excel, you can modify the spreadsheet and examine cubic and higher order polynomials ( the algebra begins to get quite messy ).

I've created an activity with which a student, using Excel, can explore finite differences.

I first ran across the fraction 1/243 in one of the Rama series of books by Arthur C. Clarke. Now 243 is the fifth power of 3. The decimal equivalent forms a very curious pattern as I describe in the notes. I then created an Excel spreadsheet to calculate decimal equivalents of the reciprocals of different powers of three to help analyze the patterns ( the notes explain how to create this spreadsheet ). I then could extend the work to explore powers of other repeating decimals. The only limitation is the size of your spreadsheet and your imagination.

As part of an analytic geometry I would often assign as a project the creation of a picture, a picture constructed from the equations in the geometry section. In some courses this mean only linear relations, or linear and quadratic functions or, in senior courses, trig and conic sections functions. A typical assignment is included. I would create an example ( to save space I've omitted the picture for this example,- you're welcome to try and produce it from the equations ). I would hand out the instructions, with the example, a blank table and a piece of graph paper, and, of course, a due date . Any picture in good taste was acceptable - geometric designs, copies of logos, whatever. The majority of the marks were for the math but I always included a small part for creativity and neatness.

Much along the same lines, I once had senior students, once we finished the section on ellipses, try to construct the Toyota symbol as a combination of three ellipses ( best done on a computer ).

This was an assignment I used at various levels in a measurement unit. I would hand it out at the start of the unit and expect it to be handed in at the end of the unit. The numbers I used for the thickness of the aluminium were obtained from an article in Scientific American on the construction of beer cans, but it was quite awhile ago and I don't have a reference for the article. If you have a metal shop you could have a can cut open and the dimensions measured with a micrometer (there might even be a student in the math class taking a metal shop course).

5.
**Acceleration due to Gravity**

This is an experiment not unlike many you can find around. It combines several skills : rearranging equations, properties of a quadratic, graphing, use of a graphing calculator ( or spreadsheet if you wish ). Divide the class into groups of two and then average all the results - don't be surprised if you get a reasonably accurate value for g. You can discuss errors and how to improve the experiment. If this were done in a senior grade, you further change the formula using logarithms so that you can get a straight plot ( this I have done examining Hooke's Law , but it requires more equipment which I had to borrow from the science department).

This is an exercise that shows a use for basic trigonometry. Like the previous experiment, it also involves formula manipulation, a skill with which I've always found students have difficulty. There is no experimental error here so the result should be very good ( ~3.14159265 ).

This is an introductory look at tangent s as a measure of slope. The activity may have to be modified depending on the building in which you are teaching. It does get students out of the classroom and working in groups. In the tech department you may be able to find the building code specifications for staircases in your locale - then compare the class results with the code.

This is the start of a simple grade 9 introduction to the difference between positive and negative slopes ( also the slopes of horizontal and vertical lines ). It can easily be extended by having the students make up their own phrase ( always something in good taste ) and the corresponding description using slopes.

I used lotteries at the junior and senior levels. At the junior level it is fairly easy to explain the basis of probability. Then I can use the multiplication principle as an example of fraction multiplication. I've attached a PowerPoint slide show I created to illustrate the chance of winning a lottery - the last page is a worksheet the students would have in order to follow along with the slide show. At the senior level I've created a variety of assignments ( I changed the lotteries I used from year to year ) primarily based on calculating the expected value of a set of lotteries.

I first came across Voronoi diagrams in an article in a TI newsletter. At the simplest level, these are diagrams that divide a set of nodes into areas of proximity. They have a number of applications ( http://www.ics.uci.edu/~eppstein/gina/scot.drysdale.html ). For example, consider a small town with 5 schools. Where would draw the boundary lines if your only consideration was each student should go to the school closest to him or her. ? The boundary lines are the perpendicular bisectors of the line joining each pair of schools. This makes a great exercise in analytic geometry. If you have access to Sketchpad, the students could try to create a simple Voronoi diagram ( it quickly becomes quite complex as more nodes are added ).

This is a simple grade 9 or 10 exercise that combines measurement skills , graphing skills and the concept of slope. You need a variety of different diameter cans. You might assign this as a take-home project in which each student had to find the different cans at home or you might have different cans brought into the class ( once the activity is done, the cans could be donated to the local food bank ). The students, working in groups of two, need to determine as accurately as possible the diameter and circumference of each type of can. If this is to done in class you would need rulers and string ( wrap the string around the can to get the circumference, then lay it out against a ruler ). Each group of students creates a table of circumference versus diameter, then plots a graph, with diameter along the horizontal axis. The points should fit a linear relation fairly closely. For analysis, depending on the students and the course, you could do one or more of the following :

- leave it there, simply as an exercise in making a table and graphing
- calculate the slope, showing that it equals pi
- do curve fitting with the TI-83 or in Excel
- discuss direct variation as the line should go through (0,0)
- do error analysis, suggesting ways to improve the results ( i.e. averaging group results )

Benford's Law describes the distribution of first digits in a random numerical
list. For example, if you have a list of the lengths of the major rivers of the
world, the first digit *n* will occur with a frequency of the logarithm of
(*n *+ 1) divided by *n* ( see
http://en.wikipedia.org/wiki/Benford's_law ) . The students might be
interested to know one application of this is for forensic accounting ( a real
set of accounts should follow Benford's Law, while a fake set likely would not
).

13.
**The Mandelbrot Set and Complex Numbers **

The Mandelbrot set is an example of a fractal ( See http://en.wikipedia.org/wiki/Mandelbrot_set ). You have to be careful here as a discussion of fractals could branch out in so many ways that it could generate an entire course on its own. So this exercise is restricted to how the Mandelbrot picture is constructed on the complex plane. It gives students a chance to practice elementary complex arithmetic. If you have access to the internet there are many sites the students can visit to see fractal pictures. There is an excellent freeware program, Fractint, which they can download at home ( not likely allowed on a school network ). If you want to go further, I based my exercise on the work of Robert Devaney ( http://math.bu.edu/DYSYS/ ) - there is lots and lots that can be created from the information given there.

Stylometry is the statistical study of written material. It turns out that the style of a particular author shows up in the distribution of word length, sentence length, in the repetition of certain words, etc. . There have been many books written on the subject ( a good starting point would be http://en.wikipedia.org/wiki/Stylometry ) . Stylometry is sometimes used to settle disputes of authorship ( from the bible to Shakespeare to the courtroom ). As an exercise in graphing and descriptive statistics I would have students read passages from different texts ( or passages from the same text but in different chapters or sections ) and do some or all of the following ( depending on time ) :

- graph the distribution of word length
- graph the distribution of sentence length
- find average word length and sentence length
- compare graphs to help identify which passages are from the same author
- for an author who writes under his or her own name but also under a pseudonym, determine if there is a change in style
- look at the different methods used to calculate a readability level ( see http://en.wikipedia.org/wiki/Readability )

A favourite of mine was to use the short story The Feeling of Power by Isaac Asimov ( http://en.wikipedia.org/wiki/The_Feeling_of_Power ). It is a story about a time in the future when all calculations are done by machine and a technician discovers ( or rediscovers ) how to do basic arithmetic calculations by hand.

These are Excel worksheets I made up to help students who needed review of some basic arithmetic operations. It is easy to change the numbers and operations so that an infinite number of different questions can be created. You can use these as timed drills at the start of class, no recording of results, repeating the same sheet ( with different numbers and/or operations ) students can try to better their time.