Spirograph by Nathan Friend

Inspirograph by Nathan Friend

As a child my Spirograph was definitely a favourite toy so I was delighted to find this digital version, Inspirograph by Nathan Friend. Try altering the gears so that the fixed and rotating gear are the same size, or make one size a factor of the other, make the two sizes have a common factor, or not! Investigate. You can change the colours too and create a work of Art!

The Nrich problem ‘Making Maths: Planet Paths‘ challenges students to draw some planet paths using a Spirograph. In case there is no Spirograph to hand they give instructions for making a simple one.


Spirograph – Desmos

Alternatively, try an online version. Try Spirograph on the Desmos Graphing calculator.

For GeoGebra fans there are various applets available, including this which allows colour changes.

Trigonometry Resources

Graphing sine function

I wrote an earlier post on some Trigonometry resources. To learn more about Trigonometry including graphing the trigonometric functions and how they are defined for angles greater than 90° try these applets from John Page’s Math Open reference. There are graphing applets for sine, cosine and tangentNote that you can drag point A to alter the angle and see how this relates to the graph; try Progressive mode to see the graph develop. Choose the Full screen option for a very clear display.

You will find many other excellent applets on this site, see the full index here.

Trig equation

This Evernote shared notebook (after selecting the link just choose to view the notebook) shows the three trigonometric functions plotted with the Desmos graphing calculator, each note has a link to a graph page. You could use these to solve simple equations like sinx = 0.2 for example. Simply adjust the slider then click on the points of intersection.

Polar Coordinates

Polar coordinates are simply a way of defining the position of a point in 2 dimensions. The distance from the origin (r) and the angle made with the x axis (measure in an anti-clockwise direction) define the position of the point. The point in the above diagram is a distance 3 from the origin and the angle made with the x axis is 30°.

Desmos - polar coordinates
Try this page on Desmos to experiment with plotting points. 

For some excellent resources on polar curves see these from the mathcentre. (Notes & examples)

You can very easily experiment with families of polar curves using the excellent Desmos graphing calculator. Click on the image below and experiment with the sliders.

Desmos domain and sliders

Desmos polar curve, click on the image to experiment

It is possible to see how polar curves are traced out by using a slider in the domain on the Desmos graphing calculator. Try this example showing r=acoskθ.
Further examples: r=acos2θ,   r=a(1-cosθ)  r=ae-kθ 

Alternatively, try this Polar Grapher; use the slider to change the angle and you will see how the curve is traced out. Note the value of R is displayed so you can easily see if it is positive or negative.
Polar Grapher

See also The Polar Gallery from mathsdemos.org. If you scroll down the page you will see that you can download a set of 14 Excel files which will allow you to experiment with several families of polar curves.

A variety of online resources…

As you will see from the many links on this blog there are many high quality Mathematics resources available on the Internet. The collection here was inspired by a recent test taken by my Year 10 (ages 14-15) students. You can use these to explore many topics you study.

There are many excellent reference materials online, see the reference page on this blog.
You could look up expression, equation and formula in Jenny Eather’s dictionary  for example to help you understand the difference between these terms. On the subject of expressions, equations and formulae, a related term you should be familiar with is identity. You could use this glossary for teachers to look it up.

You can obviously use a calculator but if you don’t have one to hand you could use WolframAlpha or one of the many online calculators available such as this one from Mathisfun.

To solve a quadratic equation using the formula try this calculator from Math Warehouse.
You could also plot the graph, note the roots where the curve intersects the x axis, ie where y=0.

WolframAlpha can always be used to check algebra, for example you may wish to check the solution to a pair of simultaneous equations.

Trial and Improvement is a method for solving equations that you cannot solve exactly.
The spreadsheet illustrations below show the solution of x3−x = 50.

The spreadsheet used here is of the extensive collection on  Mike Hadden’s MathsFiles site (Trial and Error1dp).
On the subject of Excel, you could use it to plot points and draw a graph; you could then fit a trend line as shown in the example below. Examining the numbers here you should recognise the powers of 2, we have  y = 1000×2x.


Mathisfun has many useful Geometry resources for example you could explore quadrilaterals with this interactive and this area calculation tool allows you to check the area of several common shapes.

Trigonometry – you can use this right-angled triangle calculator by Joe Barta to check any trigonometric calculations. Simply enter the known values and state the accuracy required.

To calculate the area of a triangle if you know two sides and the included angle you could use this WolframAlpha widget.

Try this widget for the cosine rule.

Communicating Mathematics online can be tricky! An online whiteboard can be the answer, I have used Scriblink here:

(For other online whiteboard resources see this post)

Trigonometry Resources


Interactive Mathematics has clear explanations of some problems involving right-angled triangles.

For more notes and examples see:

  • CIMT GCSE section chapter 4, section 4.4 onwards is on trigonometry. The notes and exercises do not require a password to access them. Schools can obtain a password for the answers.
  • University of Plymouth – Trigonometry package (one of a series of packages which may be either worked through on line or downloaded and used on any computer.
  • University of Leeds Maths Solutions – Trigonometry

David Smith’s The Maths Teacher site has a great collection of videos for both GCSE (age 14-16) and A Level (16-18). Transcripts are available for each lesson, also exercises with worked solutions. Many of the GCSE videos would also be useful for younger students. These resources include trigonometry at GCSE (see Geometry & Measures) and A Level.

For some advanced resources including graphs and the unit circle definitions, see this post on Mathematics for Students and for some calculators this page.

There are many further sources of notes available – see the resources on the Notes page.

For checking vocabulary – see the resources on the Reference page.