Hopefully you already know how to change a fraction to a decimal.
For instance ²⁄³ =2 ÷ 3= 0.6666… This is a recurring decimal. But changing a recurring decimal back to a fraction is a little more complicated.
UK maths teacher have produced a great video on this.
If you prefer a written explanation Study maths have produced a good introduction of how to convert a recurring decimal to a fraction.
Here is a useful quiz from amrdolan to test your knowledge of gradients.
This is one in a large series of short videos from NCETM showing how people use maths at work. See the others here.
This is one in a large series of short videos showing how people use maths at work. See the others here.
A jigsaw to revise linear graphs and their equations. Do you remember y=mx+c? m is the gradient, c is the intercept on the y axis. If you have forgotten this look here first.
Some of these equations need re-arranging so you can find the gradient and intercept, but others are already in the y=mx+c format.
I think this is one of the best jokes yet! Do you get it?
This jigsaw will help you revise simplifying expressions, inequalities, expanding brackets and factorisation. Is this the worst joke yet?
A jigsaw to help revise simple equations. Hope you like the joke!
If you can factorise a quadratic that is the easiest way to solve it. Some quadratics don’t factorise, so then we can use the formula. This video from jayates shows how to do it. If you are studying GCSE Higher you are currently given the formula in the exam so you don’t have to learn it off by heart, but don’t forget to refer to the formula sheet at the front of your exam paper when you need it.
Maths is fun explains this well but don’t worry too much about the “imaginary numbers” at the end. If you are doing Higher you don’t need to know that yet.
Here is a worksheet that you can print off and practice solving quadratics with the formula.
This video shows you how to solve simultaneous equations using algebra.
This video shows how to solve simultaneous equations using a graph.
Now you try!
Study Maths (more examples and interactive worksheets)
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.
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
3xy + 4x
t² + t³
An identity is an equation that is always true, no matter what values are chosen.
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.
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.
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.
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.
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.