Surds are numbers left in square root or cube root format. We leave them as surds because in decimal form they go on forever, so it uses up lots of ink to write them and accuracy is quickly lost. There are lots of tricks to simplify surds and these two videos from maths520 show them clearly. This topic is important for Higher GCSE students.
Have you got it? Try these questions on BBC Bitesize. then continue to these. Also try the jigsaw.
This video is quite long so you might want to watch it in two sittings, but it does explain clearly what higher GCSE students need to know about transformation of graphs. Thank you Ukmathsteacher!
Maths is fun has a good explanation of this with some nice interactive activities and questions. Bitesize activities are here.
Here’s a great music video from the Proclaimers.
Listen carefully to the lyrics and add up how many miles are mentioned altogether!
Scroll down to see the answer.
(8×500)+(4×1000) = 8000 miles
Thanks to Ron Barrow for this helpful example of how to use probability tree diagrams. 158,411 views is impressive! You need to know this if you are taking the GCSE Higher paper.
This video by Luke Redding is also very clear and takes the topic a bit further because it includes experiments where the item is not replaced.
Maths is fun also explains this well and includes some interactive questions. GCSE Bitesize is another good site to test yourself on this.
Some students find it incredibly difficult to visualise nets being folded up into 3 dimensional shapes. The best way to gain confidence with this is having fun making lots of different shapes and I have already blogged about an excellent site for this where you can print off all sorts of nets and make some amazing shapes. With exams rapidly approaching you may not have time for that so here is a page from Nrich where you can watch 24 different nets being folded up to make 3d shapes. Before you press play each time try to work out what the shape will look like when it is folded, then see if you were right.
Here is a great phone app that will help you with your arithmetic so you don’t need to be afraid when you are faced with that non-calculator exam. It’s called Maths Tricks and shows you lots of short cuts to performing calculations and gives you endless practice to improve your speed and accuracy. Best of all it’s free!
https://play.google.com/store/apps/details?id=com.dexterltd.maths.tricks_lite
You can find it on Google Play. I am recommending this because it is good, not because I have any connection to the app.
If you are studying GCSE you need to be able to draw and recognise graphs of simple functions, both straight lines and curves. If you are at college your computer might have software on it such as Omnigraph. If you are at home you can download free open source software that does the same job, called “Graph”. You can read more about Graph and download it here. When you have downloaded it experiment with different functions and see what happens to the graph. To enter x² you have to use x^2.
Here are some ideas of functions to try.
y=x, y=2x, y=3x, y=4x etc
y=x+1, y=x+2, y=x+3 etc
y=-x, y=-2x, y=-3x etc
y=x-1, y=x-2, y=x-3 etc
y=x^2, y=x^2+1, y=x^2+2 etc
y=x^2+x, y=x^2 +2x, y=x^2 +3x etc
y=x^3, y=x^3 +1, y=x^3 +2 etc
Here is a crossword to help you with some of the important vocabulary you need for statistics at GCSE level. You can do the interactive version or print off a paper copy and check you answers later on the interactive version. There are a few non-mathematical clues!
Do you understand the difference between a formula, expression, identity and equation?
A formula is a rule written using symbols that describe a relationship between different quantities. Typical maths formulae include
A = πr² (area of a circle)
C=πd (circumference of a circle)
An expression is a group of mathematical symbols representing a number or quantity. Expressions never have equality or inequality signs like =, >, <, ≠ ,≥ ,≤. Some examples
3a
3xy + 4x
t² + t³
An identity is an equation that is always true, no matter what values are chosen.
Examples
3a + 2a = 5a
x²+x² = 2x²
5 x 10 = 10 x 5
An equation is a mathematical statement that shows that two expressions are equal. It always includes an equals sign.
Examples
x² =100
3x(x+5)= 42
(x+3)(x-2)=0
Use this exercise to make sure you understand the difference.
GCSE students need to be able to work out the equation of a graph from what it looks like.
If it’s a straight line graph you just need to look for two things.
1. The Intercept. This is where the line crosses the y axis.
2. The gradient. This is the steepness of the line. If the line goes up from left to right it will be positive. If the line goes down from left to right it will be negative. The larger the number the steeper the line.
This example shows the line y=2x-4. The line goes up two units for each unit it goes across. The gradient is 2÷1=2. It crosses the y axis at -4, so the intercept is -4.
Mathematicians use y=mx+c as the general formula for any straight line. The gradient is m and the intercept is c.
Try this exercise to see if you can match the graphs with their equations.
Try this exercise to see if you can match the equations with the correct gradient and intercept.
Do you know all the important words for shape? This is a simple matching game to help you. Each time you play you will get a different selection of words.
Here is a similar exercise with pictures
Each number in a sequence is called a “term”. In the sequence 3, 6, 9, 12, 15 the first term is 3 and the 5th term is 15.
You could call this sequence “the three times table”. In algebra we describe it as 3n.In other words the first term is 3×1, the second term is 3×2 etc.
3n+ 4 describes the sequence 7, 10, 13, 16, 19… because the first term is 3×1+4=7, the second term is 3×2+4=10 and the third term is 3×3+4=13. Notice that because n is multiplied by 3 the sequence goes up in 3’s.
Have a go at matching these nth terms with the right sequence.
Great video from Numberphile that explains the controversy!
In the last exercise you learnt how to factorise quadratic expressions. We will now use this in order to solve simple quadratic equations.
Suppose x²+9x +20 = 0
If we factorise we get (x+4) (x+5) = 0
In other words, two numbers multiply together to make 0. This means one of those numbers must be 0!
So we know EITHER x+4 = 0 OR x+5 = 0
If x +4 = 0 x = -4
If x+5 =0 then x=-5
So the solution is x = -4 or -5
Remember quadratic equations will nearly always have 2 solutions.
Try this- you will probably need pencil and paper to factorise the equations first.
To solve simple quadratic equations you need to be able to factorise quadratic expressions, like x²+9x +20
To do this look for a pair of numbers that add up to 9 and muliply together to make 20.
If you can’t find the right pair, write down all the pairs of factors of 20.
1 x 20
2 x 10
4 x 5
Now we can see the correct pair is 4 and 5.
So x²+9x +20=(x+4)(x+5)
Check this by multiplying out the brackets.
Lets try one involving negative numbers.
x² -x -12
The pairs of factors of -12 are
-12 x 1
-6 x 2
-4 x 3
-3 x 4
-2 x 6
-1 x 12
The pair that add up to -1 (because there is -x in the expression) are -4 and 3
So x² -x -12=(x-4)(x+3)
Substitution in maths means swopping the letters for the right numbers so you can work out the value of an expression. Don’t forget that ab means a multiplied by b and c/d means c divided by d. Have a go at this interactive worksheet to get the idea.
http://www.mathswithgraham.org.uk/potatoes/numeracy/substitution.htm
Try this interactive worksheet to teach yourself factorising.
If you would rather use pen and paper there is a paper version here. Check your answers using the interactive version.
Simple equations
Can you solve the simple equations Test your knowledge on this science quiz to see how you do and compare your score to others. Quiz by mathswithgraham
