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An Interior Angle is an angle inside a shape. Example: ...

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Triangles are congruent when they have
exactly the same three sides and exactly the same three angles.

What is "Congruent" ... ?

It means that one shape can become another using Turns, Flips and/or Slides:

(see Congruent for more info)

Congruent Triangles

When two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

The equal sides and angles may not be in the same position (if there is a turn or a flip), but they are there.

Same Sides

When the sides are the same then the triangles are congruent.

For example:

is congruent to:

and also to:

because they all have exactly the same sides.

But:

is NOT congruent to:

because the two triangles do not have exactly the same sides.

Same Angles

Does this also work with angles? Not always!

Two triangles with the same angles might be congruent:

is congruent to:

only because they are the same size

But they might NOT be congruent because of different sizes:

is NOT congruent to:

because, even though all angles match, one is larger than the other.

So just having the same angles is no guarantee they are congruent.

Other Combinations

There are other combinations of sides and angles that can work ...
... read more at How To Find if Triangles are Congruent

Marking

When two triangles are congruent we often mark corresponding sides and angles like this:

is congruent to:

The sides marked with one line are equal in length. Similarly for the sides marked with two lines. Also for the sides marked with three lines.

The angles marked with one arc are equal in size. Similarly for the angles marked with two arcs. Also for the angles marked with three arcs.

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Triangles are congruent when they have exactly the same three sides and exactly the same three angles. It means that one shape can become another using Turns, Flips and/or Slides: (see Congruent for more info) When two triangles are congruent they will have exactly the same three sides and exactly the same three angles.

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Histograms - Representing data - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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Learn about and revise how to display data on various charts and diagrams with this BBC Bitesize GCSE Maths Edexcel study guide.

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The number of umbrellas sold and the amount of rainfall on 9 days is shown on the scatter graph and in the table.

The graph shows that there is a positive correlation between the number of umbrellas sold and the amount of rainfall. On days with higher rainfall, there were a larger number of umbrellas sold.

However, it is important to remember that correlation does not imply causation . If data plotted on a scatter graph shows correlation, we cannot assume that the increase in one of the sets of data caused the increase or decrease in the other set of data - it might be coincidence or there may be some other cause that the two sets of data are related to.

Types of correlation

Graphs can either have positive correlation, negative correlation or no correlation.

Positive correlation means as one variable increases, so does the other variable. They have a positive connection.

Negative correlation means as one variable increases, the other variable decreases. They have a negative connection.

No correlation means there is no connection between the two variables.

Lines of best fit

A line of best fit is a sensible straight line that goes as centrally as possible through the coordinates plotted. It should also follow the same steepness of the crosses.

The line of best fit for the scatter graph would look like this:

Interpolation and extrapolation

From the diagram above, we can estimate how many umbrellas would be sold for different amounts of rainfall. For example, how many umbrellas would be sold if there was 3mm of rainfall? What if there was 10mm of rainfall?

To estimate the number sold for 3mm of rainfall, we use a process called interpolation . The value of 3mm is within the range of data values that were used to draw the scatter graph.

Find where 3 mm of rainfall is on the graph. Draw a line by going across from 3 mm and then down.

An estimated 19 umbrellas would be sold if there was 3 mm of rainfall.

If there was 10mm of rainfall, we could extend the graph and the line of best fit to read off the number of umbrellas sold. This gives a value of approximately 64 umbrellas sold.

This process is called extrapolation , because the value we are using is outside the range of data used to draw the scatter graph. Since 10mm is much higher than the highest rainfall recorded, we cannot assume that the line of best fit would still follow the pattern when the rainfall is 10mm, so the value of 64 umbrellas is not a reliable estimate.

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Scatter graphs - Representing data - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

# data[].attributes.attachment_link.description
Learn about and revise how to display data on various charts and diagrams with this BBC Bitesize GCSE Maths Edexcel study guide.

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Practise your approximation and rounding skills with this online, self-marking exercise.

This is level 6; Rounding numbers to three significant figures. You can earn a trophy if you get at least 17 correct.

Description of Levels

Close

Level 1 - Rounding numbers to the nearest whole number

Level 2 - Rounding numbers to one decimal place

Level 3 - Rounding numbers to two decimal places

Level 4 - Rounding numbers to one significant figure

Level 5 - Rounding numbers to two significant figures

Level 6 - Rounding numbers to three significant figures

Level 7 - Rounding numbers to the nearest ten, hundred etc

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Log in Sign up

Curriculum Reference

Example

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

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Close

# data[].attributes.attachment_link.description
Practise your approximation and rounding skills with this online, self-marking exercise. This is level 6; Rounding numbers to three significant figures. You can earn a trophy if you get at least 17 correct.

# data[].attributes.attachment_link.readable_text
Practise your approximation and rounding skills with this online, self-marking exercise.

This is level 6; Rounding numbers to three significant figures. You can earn a trophy if you get at least 17 correct.

Description of Levels

Close

Level 1 - Rounding numbers to the nearest whole number

Level 2 - Rounding numbers to one decimal place

Level 3 - Rounding numbers to two decimal places

Level 4 - Rounding numbers to one significant figure

Level 5 - Rounding numbers to two significant figures

Level 6 - Rounding numbers to three significant figures

Level 7 - Rounding numbers to the nearest ten, hundred etc

More on this topic including lesson Starters, visual aids, investigations and self-marking exercises.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Log in Sign up

Curriculum Reference

Example

Don't wait until you have finished the exercise before you click on the 'Check' button. Click it often as you work through the questions to see if you are answering them correctly. You can double-click the 'Check' button to make it float at the bottom of your screen.

Answers to this exercise are available lower down this page when you are logged in to your Transum account. If you don't yet have a Transum subscription one can be very quickly set up if you are a teacher, tutor or parent.

Log in Sign up

Close

# data[].attributes.attachment_link.description
Practise your approximation and rounding skills with this online, self-marking exercise. This is level 6; Rounding numbers to three significant figures. You can earn a trophy if you get at least 17 correct.

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A long video - but explains it well

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How to write numbers in standard form.wmv

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Practise solving equations here

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IXL | Solve linear equations with variables on both sides | Algebra 1 math

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Improve your math knowledge with free questions in "Solve linear equations with variables on both sides" and thousands of other math skills.

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Plotting inequalities (practice) | Khan Academy

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Sometimes we need to solve Inequalities like these:

Solving

Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign:

We call that "solved".

Example: x + 2 > 12

Subtract 2 from both sides:

x + 2 − 2 > 12 − 2

Simplify:

x > 10
Solved!
How to Solve

Solving inequalities is very like solving equations ... we do most of the same things ...

... but we must also pay attention to the direction of the inequality.

Direction: Which way the arrow "points"

Some things can change the direction!

Safe Things To Do

These things do not affect the direction of the inequality:

Add (or subtract) a number from both sides

Multiply (or divide) both sides by a positive number

Simplify a side

Example: 3x < 7+3

We can simplify 7+3 without affecting the inequality:

3x < 10

But these things do change the direction of the inequality ("<" becomes ">" for example):

Here are the details:

Adding or Subtracting a Value

We can often solve inequalities by adding (or subtracting) a number from both sides (just as in Introduction to Algebra), like this:

Example: x + 3 < 7

If we subtract 3 from both sides, we get:

x < 4

And that is our solution: x < 4

In other words, x can be any value less than 4.

What did we do?

We went from this:

To this:

x+3 < 7

x < 4

And that works well for adding and subtracting, because if we add (or subtract) the same amount from both sides, it does not affect the inequality

Example: Alex has more coins than Billy. If both Alex and Billy get three more coins each, Alex will still have more coins than Billy.

What If I Solve It, But "x" Is On The Right?

No matter, just swap sides, but reverse the sign so it still "points at" the correct value!

Example: 12 < x + 5

If we subtract 5 from both sides, we get:

7 < x

That is a solution!

But it is normal to put "x" on the left hand side ...

... so let us flip sides (and the inequality sign!):

x > 7

Do you see how the inequality sign still "points at" the smaller value (7) ?

And that is our solution: x > 7

Note: "x" can be on the right, but people usually like to see it on the left hand side.

Multiplying or Dividing by a Value

Another thing we do is multiply or divide both sides by a value (just as in Algebra - Multiplying).

But we need to be a bit more careful (as you will see).

Positive Values

Everything is fine if we want to multiply or divide by a positive number:

Example: 3y < 15

If we divide both sides by 3 we get:

y < 5

And that is our solution: y < 5

Negative Values

When we multiply or divide by a negative number
we must reverse the inequality.

Why?

Well, just look at the number line!

For example, from 3 to 7 is an increase,
but from −3 to −7 is a decrease.

−7 < −3

7 > 3

See how the inequality sign reverses (from < to >) ?

Let us try an example:

Example: −2y < −8

Let us divide both sides by −2 ... and reverse the inequality!

−2y< −8

y > 4

And that is the correct solution: y > 4

(Note that I reversed the inequality on the same line I divided by the negative number.)

So, just remember:

When multiplying or dividing by a negative number, reverse the inequality

Multiplying or Dividing by Variables

Here is another (tricky!) example:

Example: bx < 3b

It seems easy just to divide both sides by b, which gives us:
x < 3
... but wait ... if b is negative we need to reverse the inequality like this:
x > 3
But we don't know if b is positive or negative, so we can't answer this one!

To help you understand, imagine replacing b with 1 or −1 in the example of bx < 3b:

The answer could be x < 3 or x > 3 and we can't choose because we don't know b.

So:

Do not try dividing by a variable to solve an inequality (unless you know the variable is always positive, or always negative).

A Bigger Example

First, let us clear out the "/2" by multiplying both sides by 2.

Because we are multiplying by a positive number, the inequalities will not change.

x−3 < −10

Now add 3 to both sides:

x < −7

And that is our solution: x < −7

Two Inequalities At Once!

How do we solve something with two inequalities at once?

Example:

First, let us clear out the "/3" by multiplying each part by 3.

Because we are multiplying by a positive number, the inequalities don't change:

−6 < 6−2x < 12

Now subtract 6 from each part:

−12 < −2x < 6

Now divide each part by 2 (a positive number, so again the inequalities don't change):

−6 < −x < 3

Now multiply each part by −1. Because we are multiplying by a negative number, the inequalities change direction.

And that is the solution!

But to be neat it is better to have the smaller number on the left, larger on the right. So let us swap them over (and make sure the inequalities point correctly):

Summary

Many simple inequalities can be solved by adding, subtracting, multiplying or dividing both sides until you are left with the variable on its own.

But these things will change direction of the inequality:

Multiplying or dividing both sides by a negative number

Swapping left and right hand sides

Don't multiply or divide by a variable (unless you know it is always positive or always negative)

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Sometimes we need to solve Inequalities like these: Our aim is to have x (or whatever the variable is) on its own on the left of the inequality sign: We call that "solved". Subtract 2 from both sides: x + 2 − 2 > 12 − 2 Simplify: x > 10 Solved!

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The graph of \(y = 2x + 1\) crosses the \(y\) -axis at (0, 1). The graph of \(y = 2x - 2\) crosses the \(y\) -axis at (0, -2). The constant term in the equation (the + 1 or ‒ 2) shows the point where the graph crosses the \(y\) -axis.

Gradient is a measure of steepness.

As you move along a line from left to right, you might go up, you might go down or you might not change at all.

Gradient \(= \frac{\text{change up}}{\text{change right}}\) or \(\frac{\text{change in y}}{\text{change in x}}\)

Gradients can be:

Fairly flat lines will have low gradients like \(\frac{1}{2}\) or \(- \frac{3}{4}\) and equations like \(y = \frac{1}{2}x + 1\) or \(y = − \frac{3}{4} x + 2\) .

To work out a gradient, use the scales of the axes and find how many units you go up or down for each unit you move right.

Work out the equation of this graph.

Gradient \(= \frac{\text{change up}}{\text{change right}}\)

The gradient is the same all along the line, so it doesn't really matter where you start or finish, but it is generally a good idea to use two points on the line that are far apart.

Work out the equation of this graph.

Gradient \(= \frac{\text{change up}}{\text{change right}}\)

Using (0, 1) and (4, −7), as we move along the line from left to right, we move 8 units down (from 1 to −7). We also move 4 units right (from 0 to 4).

Work out the equation of this graph.

Gradient \(= \frac{\text{change up}}{\text{change right}}\)

For example, using (0, −2) and (6, 0), as we move along the line from left to right, we move 2 units up (from −2 to 0) and 6 units to the right (from 0 to 6).

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y = mx + c - Straight line graphs - Edexcel - GCSE Maths Revision - Edexcel - BBC Bitesize

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The graph of \(y = 2x + 1\) crosses the \(y\) -axis at (0, 1). The graph of \(y = 2x - 2\) crosses the \(y\) -axis at (0, -2). The constant term in the equation (the + 1 or ‒ 2) shows the point where the graph crosses the \(y\) -axis.