Desmos – function notation
Desmos can be used very simply to illustrate function notation and note the use of Desmos as a calculator to evaluate the value of a function for a given input. (Select image to go to the Desmos graph page).
The University of Plymouth have a series of very clear notes with examples, exercises and solutions; see this booklet on Functions.
University of Plymouth Notes
Helm Notes – Functions
The Helm Project – Basic Functions and One-to-one and inverse functions
You can use the Desmos graphing calculator to support your study and understanding of functions. Using function notation you can enter composite functions as shown in the next two illustrations. Selecting each image will take you to the relevant Desmos graph page.
Composite Functions with Desmos – select image
Composite functions with Desmos – select image
For further exploration of composite functions check the pages in this slideshow.
You can also easily explore transformations of graphs with Desmos.
Note the new page GCSE Revision – some useful resources for UK Year 11 GCSE students.
Lots of questions by topic.
MathisFun Easter Puzzle
An Easter puzzle from MathisFun – as an excuse for solving simultaneous equations. We could of course use algebra. Using the notation, h, q and t for the egg Horace wants, the egg with the small square pattern and the egg with the stripey pattern respectively.
h + 2q =550 (1)
h + q +t =600 (2)
—-2q +t =500 (3)
Subtracting equation (3) from equation (2) gives
(1)−(4) gives 3q=450 so q=150 and h must be 250 ($2.50).
We could check our solution on WolframAlpha of course:
Select image for WolframAlpha query
Did you know that you can easily invert matrices and solve simultaneous equations using Excel?
Select image for Excel file
Select the image or this link for the Excel file. Excel simultaneous equations
To enter the MINVERSE function in the example above, select cells C7:E9, enter the MINVERSE function as shown then press CTRL + SHIFT + ENTER; similarly to enter the MMULT function in this example, select cells H7:H9, enter the MMULT function as shown and then press CTRL + SHIFT + ENTER. (Using formulae like these demonstrates a neat Excel technique, to learn more see MrExcel on Array Formulae).
For notes / examples / tutorials on Simultaneous Equations try the mathcentre resources or this workbook from Plymouth University. For more on solving three equations in three unknowns for older students see Simultaneous Linear Equations from AJ Hobson’s ‘Just the Maths’
Since it’s Easter an updated version of the best Easter Eggs of all – from WolframAlpha.
If you want some more puzzles, MathisFun has plenty more, or try some of the puzzles here.
Box&Whisker – WolframAlpha
Box Plots show summary information for data – the box shows the median and quartiles and the whiskers are extended to all points which are not outliers.
A very clear description for younger students can be found on BBC Bitesize. For A Level Students the Centre for Innovation in Mathematics Teaching includes Box Plots in Chapter 3 of their Statistics text (page 26 of the pdf file, which is page 76 of the text).
Box Plots are particularly useful when comparing one or more distributions;
CIMT Statistics Text
To help older students think further about about outliers and skew, have a look at this excellent illustration of Nobel Prize data from Plotly. Who is that outlier for the Nobel Peace Prize?!
Plotly – Nobel Prize winners by field
Note that it is possible to display points with Box Plots in Plotly, the following data and charts demonstrates skew well:
You can learn more about Plotly for graphing and sharing your data here: Learn and explore Plotly
Click on the image and then on the slider!
Why not send your mathematical friends (or anybody else!) a Desmos Valentine? The wonderful team at Desmos have made their brilliant math-o-grams available earlier this year….and watch out for new designs coming soon.
The math-o-grams are really easy to create; why not give it a try? Just select your design, add your message and share!
If you want to check your calculations for a regression line and see the line then Excel and WolframAlpha are both very easy to use. In Excel – just add the data, insert a chart – choose Scatter, then add a trend line.
…or let WolframAlpha do all the work for you (including a plot)!
You could also use Desmos. Their video gives clear instructions:
For another example with an A Level question (using the same data as for Excel and WolframAlpha above) check this slideshow.
Try this least squares demonstration on GeoGebra.