Constructions

bisect-angle-math-open-ref
Another post to clarify the requirements for the UK GCSE Mathsematics specification. The constructions required are as follows:

Use the standard ruler and compass constructions

Math Open Ref Instructions
Note that all these Math Open Reference demonstrations are available as a printable atep by step instruction sheet, see for example the guide to bisecting an angle.

The AQA specification also includes a note that Constructing a 60° angle is required. This is shown here by constructing an equlateral triabgle.
​Students will be asked to use these constructions  to construct given figures and solve loci problems. Students should also know that the perpendicular distance from a point to a line is the shortest distance to the line.
The constructions shown here are all from the excellent Math Open Reference by John Page. (There are many more constructions given on the site which are not a GCSE requirement). Another very useful source for demonstrations comes from BBC Bitesize on Loci and Constructions which gives step by step diagrams and instructions.BBC Bitesize Constructions
The BBC site also has clear examples of solving problems using constructions.

 

 

Transformations of Graphs

Translations Graphs

In this post we will look at reflections and translations (the requirement for the UK GCSE syllabus). For GCSE students are required to sketch translations and reflections of a given function.

Experiment with the following Desmos pages to examine these transformations, f(x) + a, f(x + b), -f(x) and f(-x) where a and b are integers.

Translations

Reflections - Graphs

Reflections

Equations of Transformed Graphs

Note that you can work out the equation of the transformed graphs.
Suppose we wish to work out the equation of  f(x) = x2−5x+4 after reflection in the x axis.
To reflect in the x axis the transformation is –f(x)
So f(x) becomes  –f(x) and we have  – (x2−5x+4) = – x2 +5x-4

To work out the equation of  f(x) = x2−5x+4 after reflection in the y axis.
To reflect in the y axis the transformation is f(–x)
So f(x) becomes  f(–x) so we replace x by –x
and we have (–x)2−5(–x)+4 =  x2 +5x+4

Suppose we wish to work out the equation of  f(x) = x2−3x+2 after a translation of 3 units to the left parallel to the x axis as shown in the first image in this post.
f(x) = x2−3x+2, we need f(x+3) for the equation of the transformed graph
f(x+3) = (x+3)2−3(x+3)+2 = x2+6x+9−3x−9+2 = x2+3x+2

Diagnostic Questions

GCSE Diagnostic Qns
To use the links here you will need to be logged in to the brilliant Diagnostic Questions site. Students, just in case your school does not use Diagnostic Questions you can create your own free account here. On Craig Barton’s Diagnostic Questions site you can find collections of GCSE 2017 examination questions from AQA, OCR and Pearson Edexcel (scroll down each of the pages linked to for numerous quizzes on different topics on the GCSE syllabus).

Craig Barton who is the creator of the excellent Diagnostic Questions site has compiled a collection of some of the worst answered new specification GCSE questions on Diagnostic Questions. See also his Question of the Week selection.

Each set consists of 10 questions chosen from all 3 awarding bodies. As well as the online version of each quiz, the design team have also created a booklet in case you wish to print it out.

Badly Answered GCSE Questions – Foundation 1
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 1
— Try the questions online or download the booklet

Foundation Badly Answered Questions – Set 2
Online       Paper

Higher Badly Answered Questions – Set 2
Online      Paper 

Badly Answered GCSE Questions – Foundation 3
— Try the questions online or download the booklet
Badly Answered GCSE Questions – Higher 3
— Try the questions online or download the booklet

You can also learn on the move with the free Diagnsotic Questions app.
GCSE 2017

Note that Advanced Level Questions are also available.

University Mathematics Preparation

STEP (Sixth Term Examination Paper) Mathematics is a well-established mathematics examination designed to test candidates on questions that are similar in style to undergraduate mathematics. For students studying for STEP papers try this excellent portal from stepmaths.co.uk which has (all free) access to STEP questions and solutions. Create an account, login and you have access to a complete library of resources.

The resources are very clearly presented. For each question you have access to a pdf with the question, Examiners’ Report and both an Official and thanks to Peter Mitchell a fully worked handwritten solution.

Note that you can also download a copy of  Stephen Siklos’ Advanced Problems in Mathematics and Core MathematicsAdvanced Problems in Mathematics is excellent preparation for ANY undergraduate Mathematics course.

Following each question, you will find a discussion and a full solution. The clear Contents page lists all 43 problems. Each problem has been given a title and a rough indication of the mathematical content which means you can pick out questions by topic.

Stephen Siklos Advanced Problems in Mathematics and Core Mathematics

See also, from Cambridge University, their STEP Support Programme.
STEP CU

And from Nrich, Prepare for University.

Projectiles

demos-pagesA project – note the new tab Demos – a place for some favourite demonstrations / simulations. Currently just two pages but note the numerous PhET simulations for a variety of subjects to explore.

The most recent addition is the PhET Projectiles Simulation.

 

projectiles

Use this excellent PhET simulation to explore the path of a projectile. Try changing the angle to investigate the relationship between the angle of projection and the horizontal distance (range) travelled.

Have a look at these problems on Underground Maths to extend your thinking:
Where did it land? and Maximum Angle Throw. Note the questions given on these problems and things you might have noticed.
projectilesFor amusement try projectiles of different types!

You will find some very useful notes on Mechanics on Mr Barton’s website.

There are numerous PhET simulations covering Physics, Biology, Chemistry, Earth Science and Mathematics. Note the growing collection of HTML5 versions which will work across all platforms and devices. The Projectiles simulation here is currently a Flash resource.

You can download an app for iOS also for Android.
PhET Balancing Act working nicely on my phone!
balancing-act

Mathematics Challenges

UK Maths Challenges
You can practice for the UK Maths Challenges with these past papers. Questions and full solutions are provided. You can find British Mathematical Olympiad papers here.

You could generate a random quiz, using Mathster’s UKMT Mathematics Challenge Online Quiz. Choose Junior, Intermediate or Senior and one of three difficulty levels; you can also choose the number of questions, a time limit and the order the questions are presented in – random or in order of difficulty.

Note Geoff Smith’s advice for young mathematicans and as he says you will find thousands of questions are available at all levels on the Art of Problem Solving site.

Nrich have a series of short problems based on the UK Junior and Intermediate Challenges.Nrich Short Problems


Nrich publish new problems every month. Why not try and get a solution published on their website? There is a menu specifically for studentsYou can sign up for an Nrich student newsletter if you want to be notified of new developments on the site.

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish.

If you have not tried Nrich problems before you may find their recommended starter problems good to try. You can search for problems by topic if you wish

Brilliant
Brilliant problem

Signing up to Brilliant (including an easy option for sign in for Facebook users) will allow you to join an international community and try numerous.questions at various levels.

Circles & Teddy Bears

teddy-bear-desmos

Have a look at the teddy bear – can you identify all the equations from this list?
This problem comes from the excellent Underground Maths site.
You can view the Teddy Bear on Desmos by selecting the image above.

desmos-toggleNote that on Desmos you can choose to display or hide a graph.

As with all problems on the site you can see the question and a very full solution with all reasoning explained.

For further notes on the equation of a circle, check these Helm Project notes and exercises on the circle.

Top Tools for Learning 2016

The Top Tools for Learning 2016, inlcuding top tools for Education. This is what the educators think. Students what do you think? What are your favourite tools for learning?

Mathematics, Learning and Technology

Jane Hart is the Founder of the Centre for Learning & Performance Technologies and 2016 marks the 10th year of her annual Top 100 Tools for Learning list. Jane has put together all the presentation slidesets as well as an alphabetical list of ALL the tools which have appeared on any of the lists.

The 2016 slideset is shown here.

Note from Jane’s overview she has done a finer analysis for 2016 including the Top 100 Tools For Education (for use in primary and secondary (K12) schools, colleges, universities and adult education.)

Back in April, I wroye about my own choices for 2016 and I am always interested to see where my own choices are in Jane’s list.

CY 2016 votes Education Personal Learning & Productivity Place in Top 200 2016  Place in Top 100 Tools for Education 2016
Evernote x x  17  27
WordPress x x  9

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