Magic (actually Algebra) Tricks

….so what is going on with all of these? Can you use Algebra to show why they work?

Can you work out what is going on?

Think of a number…
Double
Halve
Take away the number you first thought of

Try with several different starting numbers.

Think of a small positive number

1. Square it.
3. Divide by your original number.
6. Divide by 6.

Try with several different starting numbers.

Multiply any two digit number by 11.
What do you notice?
Can you prove this result?

Solutions soon if you are puzzled.

For when you want a method to factorise quadratics….

By Colleen Young Posted in Algebra

Mathematics – Transition Time

Notes from Mr Barton

As we start a new academic year, it will be a time of change for many students. Perhaps you have completed GCSEs or equivalent qualifications (UK age 15-16) and are about to start on your A Levels or perhaps you have completed those and are about to start studying Mathematics at university.

To be in a position to begin your new courses well you should be thoroughly familiar with the essentials of the work you have studied to date. At whichever level you are studying your Algebra should be at a standard where you can manipulate expressions with ease.

Some resources to help you prepare and will be useful reference material for you during your course…

Make sure you have some useful apps on your phone if you don’t have them already. Mathscard app from Loughborough university is free and a handy reference guide of mathematical facts and formulae. Every student should have the Desmos app (free) and you could also get the  WolframAlpha app (low cost).

For students going on to A Level then these GCSE revision resources will be useful. The takeaways are really useful and Mohammed Ladak has picked out Transition Takeaways specifically chosen to help with A Level Maths preparation.

You could also look at Step Up to A Level Maths from The Centre of Innovation in Mathematics Teaching which helpfully lists skills you should be confident with and provides resources to support your study of these skills.

As you study your A level course you may find some of the material in the section below useful.

For many challenging questions to really get you thinking, try the brilliant Underground Mathematics site.

If you are preparing for university, then make sure your A Level knowledge is secure – perhaps check the Algebra Refresher from The Mathcentre which has many questions and the answers are at the end of the document. The The Mathcentre has an extensive collection of helpful resources for students of Mathematics.

Calculus workbook from Plymouth University

For a collection of forty mathematics activities bridging between A Level and University, try Carom Maths from Jonny Griffiths.

Check the List of Activities, how much do you know about Inequalities for example? For a complete PowerPoint with information and questions on Inequalities, choose Carom 1-2: Inequalities.

For older students AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’is very useful as is the excellent Math Centre site which includes extensive resources. The quick reference leaflets which are available on numerous topics are very clearly written and succinct, see these for example on the Product Rule and the Quotient Rule. There are also teach yourself booklets, revision booklets, videos and diagnostic tests. See also these very clear notes with exercises from Plymouth University. There are many free courses available from The Open University and MIT .

The HELM Project. If you have not come across the HELM Project before, the project was designed to support the mathematical education of engineering students and includes an extensive collection of notes which include clear worked examples. You can see on the list that a very small number of titles are ‘not ready yet’; for the sake of completeness the complete set is hosted by the Open University. To access the Open University resources you will need to create an account (easy and free), this will also give you access to the numerous free online courses.

See this Evernote shared notebook: Mathematics notes for many more useful links. Several universities have created very helpful Mathematics support which they have made available to all students. (You do not need an Evernote account to view the notebook).

If you are studying or about to study at university then have a look at Kevin Houston’s ‘How not to get a good mathematics degree‘ and ‘How to get a good mathematics degree‘. He also has provided a pdf file you can download: 10 Ways to Think Like a Mathematician. Kevin Houston works at the University of Leeds in the UK.

From Professor Stephen Chew of Samford university, this series of 5 videos looks at how to get the most out of studying (any subject, not just maths). Part 1 includes ‘beliefs that make you stupid’!
How to get the most out of studying (Part 1),  Part 2,  Part 3 Part 4Part 5

The Open University has several helpful publications for students of Mathematics. Many of these resources would be helpful for students still at school.

From John Kerl – see these excellent tips for mathematical handwriting.

For older students Peter Alfeld wrote this guide on Understanding Mathematics for his students at Utah University.

And finally – check the 11 Commandments of Mathematics!

Wishing Mathematics students everywhere – whatever stage you are at a very successful year.

Desmos and 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.

Simultaneous Equations

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.
We have:
h + 2q      =550 (1)
h + q   +t =600 (2)
—-2q  +t =500 (3)
Subtracting equation (3) from equation (2) gives
h−q=100 (4)

(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.

The gradient of a straight line

Use the excellent Desmos graphing calculator to explore the gradient of a straight line. Select the image then you can change the sliders for m and c to change the line and drag the points so you can verify the gradient of the line.

Select image for the Desmos page

To solve a quadratic inequality such as x2-6x+8>0 the best approach is to find the critical values of x which make the value of the function 0 and sketch the graph. Here we can factorise so we see that we require (x-2)(x-4)>0 We are looking for values of x to make x2-6x+8 positive. We see that y is positive when x>4 and when x<2.

Note the link to another Desmos page where you can look at further quadratic functions.

Select image to experiment

If you wish to see some worked examples and try some exercises you could check The Maths Teacher – See Algebra AS Level – Inequalities: Linear and Quadratic. Note the choice of video / transcript or go straight to the exercise with worked solutionsYou may find other useful resources here too.

You could of course use WolframAlpha to check the solution to any inequalities and generate as many examples as you want. Simply type in the inequality and plot as well as the he solutions will be returned.

Select image for query on WolframAlpha

Algebraic Fractions

A collection of resources for you to learn and practise algebraic fractions.

For a basic explanation see Mathisfun and note the questions to try at the end of that page.

The University of Hull Mathematics notes seem very clear, in the Algebra section we have clear notes and exercises with answers on Algebraic Fractons. The notes and exercises include adding, subtracting, multiplying and dividing algebraic fractions.

Algebraic manipulation from Just the Maths includes a section (1.5.4) on Algebraic fractions.

The mathcentre has a section on simplifying algebraic fractions. On the subject of algebra generally, the mathcentre has a very useful refresher booklet.

From the Centre for Innovation in Mathematics Teaching see section 10.13 of this chapter on equations for examples and exercises with algebraic fractions. You could check answers to the exercises on Wolfram Alpha. For example :

Select the image top check answer on WolframAlpha

By Colleen Young Posted in Algebra

Completing the square….

..a basic skill any student of Mathematics needs.

A few examples if you need a little reminder:

By Colleen Young Posted in Algebra

Solving Equations

CIMT have tutorials on equations: Linear Equations 1Linear Equations 2and Linear Equations with Brackets in their Interactive Resources

Try this Linear Equation Generator

Plymouth University Tutorial

Helm Project

For a tutorial on simple equations, these notes from Plymouth University are clear; see also Section 3 of the Helm Project on equations. Both of these resources include examples, exercises and answers.

Or try AJ Hobson’s Just the Maths (individual pdfs hosted by UEA) (or a complete pdf from the Math Centre:   AJ Hobson’s ‘Just the Maths’) where you will find clear notes, examples and exercises on many Mathematics topics including Algebra.

subtangent.com – Linear Equation Calculator

Duncan Keith’s Linear Equation Calculator

Choose the type of equation you require then the sequence of operations required to solve the equation.
Select Do it after each operation, for example -32 Do it were the keys selected to start the above problem.

The slideshow below shows how to use the calculator to solve equations where the unknown is on both sides.

The next sites work in a similar way to the subtangent resource.

Flashy Maths – Solving Equations (select the swf file to play online)

John-Paul Green’s Flashy Maths

There are 4 levels to choose from: the level 1 equations are of the type ax = b,
level 2, x±c = d, level 3, ex±f = g and level 4, hx±i = hx±j = k.
Choose the series of operations you require to solve the equation selecting apply at each stage to see the result of your chosen operation.
Choose numbers to add and subtract integers and letters to add and subtract variables.