Rounding and Estimating

When estimating a calculation the usual rule is to round each calculation to 1 significant figure.

1st what is a significant figure?
The word significant means: having meaning.

With the number 3 468 249, the 3 is the most significant digit, because it tells us that the number is 3 million and something. It follows that the 4 is the next most significant, and so on.

So if we want to round this number to 1 significant we round it to the nearest million as this is where the first significant figure is

Examples:
Round the following to 1 significant figure

4325 is 4000 (because the 4 is the 1st significant figure so we round it to the nearest thousand)

0.007821 is 0.008 the 7 is the 1st significant figure so we round it to 3 decimal places.

Now on to estimating.  To estimate we round to 1 significant figure then work out the calculation

Estimate the answer to 

First round each number to 1 significant figure so 19.4 becomes 20 and 0.0437
is 0.04

Now we need to make the denominator a whole number. We can do this by multiplying both 20 and 0.04 by 100.        

Divide everything by 4.

(Question taken from GCSE Bitesize http://www.bbc.co.uk/schools/gcsebitesize/maths/number/roundestimaterev4.shtml)

Ordering Fractions and Converting between fractions and decimals

When ordering fractions there are two main ways to do this

1) Convert each fraction to a decimal and then order

                                        or
2) Write each fraction over a common denominator.

Example of method 2)

 




















If we are to use method one we need to know how to convert fractions to decimals.  Some are easy such as 1/2 = 0.5 and 1/4 = 0.25.

If we come across a fraction that we do not know as a decimal and we cannot compare it to percentages to help us we can use division to help us.

 

Angles and Angle Theorems

Different sized angles have different names. These are illustrated below

 

There are many different angle theorems some are listed below:

 

Adjacent angles on a line add up to 180 degrees

 

 

 

 Angles at a point add up to 360 degrees

 

 

Angles in a triangle equal 180 degrees

Angles in parallel lines

Solving Equations

To solve equations we use inverses (do the opposite) to find out what the unknown is

One step equations

Examples:

1)        2x = 8              We divide both sides by 2 to get x on its own
             x = 4

2)      x + 5 = 9           We subtract 5 from both sides
               x = 4

3)     x - 2 = 12            We add 2 to both sides
             x = 14

Two step equations

Examples:

1)      2x + 1 = 5           First we subtract 1 to get 2x on its own
                      2x = 4          Then we divide both sides by 2 to get x
                 x = 2

2)       3x - 5 = 16         Add 5 to both sides
               3x =  21        Divide both sides by 3
                 x = 7

3)       7 - 4x = 3           Because we have a minus 4x we add this first 
                 7 = 3 + 4x   Now subtract 3
                 4 = 4x         Divide both sides by 4
                 x = 1

Unknowns on both sides

Examples
  
1)       2x + 3 = x + 5     Deal with the smallest x in this case x on the RHS
           x  + 3 = 5            Now continue as above
                 x = 2

2)      3x - 7 = 5x + 9     Be careful with negatives
              -7 = 2x + 9
             -16 = 2x    
              - 8 = x

Equations with Brackets

First multiply out the brackets then continue as above
Examples

1)      3(2x + 5) = 33
            6x + 15 =  33
                    6x = 18
                     x = 3

2)    2(x  + 7) = 3(2x - 2)
        2x  + 14 = 6x - 6
                 14 = 4x - 6
                 20 = 4x
                  5 = x
 

Probability

Probability all is all about the chance of something happening.

Probability Words
If a question asks you to describe the probability using words the words most often used are

Impossible - Absolutely no chance it will happen
Unlikely - there is a small chance it will happen
Even Chance - the probability is a half or 50 - 50
Likely - it is not definitely going to happen but there is a good chance
Certain - it will definitely happen.

These words can be displayed, with numbers along the probability scale.



The Probability Scale

Probability Calculations

Probability of something happening
If the question asks you to calculate the probability the answer will need to be given as a numerical answer usually a fraction but sometimes written as a decimal or percentage.

To calculate the probability of something happening we use the following formula




















Example:  The probability of getting a 5 when rolling an ordinary dice is 1/6 because there is 1 number 5 on the dice out of 6 possible different answers.

The probability if getting an Ace from a standard pack of cards is 4/52 because there are 4 aces and 52 cards this can be simplified to 1/13.  You could also say that it is 1/13 because there are 13 cards in a suit and 1 of them is an ace.

Probability it doesn't happen
The probability of something not happening is 1 - the probability it does.  This is because either that thing is going to happen or it will not happen and so it covers all possibilities and together they equal 1.

Example:  The probability it will rain tomorrow is 0.56 so the probability it won't rain is 1 - 0.56 = 0.44

Experimental Probability and Relative Frequency
Probability can be used to predict results or to see if something is fair.  For example if I flip a coin 10 times theoretically we should get 5 heads and 5 tails but in reality we may get 7 heads and 3 tails.  From 10 results this may not seem odd, but if we flipped a coin 1000 times and got 700 heads and 300 tails you may think the coin is biased.

Relative frequency is the probability that an experiment produces.  For example above where I flipped my coin 10 times the theoretical probability of a head is 0.5 but in the experiment I got a head 7 out of 10 times so the relative frequency or experimental probability is 0.7.

Transformations

Rotation:

When performing a rotation we need 3 pieces of information:

A centre of rotation (either given as a point or sometimes described as the origin which is the point (0,0))

A direction - either clockwise or anti clockwise (Unless the angle is 180 degrees where it doesn't matter)

An angle.

To perform a rotation use tracing paper and trace your shape, place your pencil on the centre of rotation and rotate the given angle in the given direction to see where the shape is rotated to.

Reflection:

When performing a reflection we need to know which line is the mirror line.  This will either be given on the question or expressed as an equation of a line.

Remember x = 2 is a vertical line which passes through the x axis at the point 2 and y = 3 would be a horizontal line which passes through the y axis at the point 3.

A reflection should look like a mirror image and a mirror can be used if you need extra help.

Translation:

A translation is where an object is moved (without reflecting, rotating or enlarging).  Usually the translation will be described by a vector, where the top number represents the x direction and the bottom number the y direction.

Enlargement:

There are two types of enlargement, one without a centre and one with.
A basic enlargement requires you to enlarge a shape by a given scale factor but it doesn't matter where the shape is drawn.  If a shape is enlarged by scale factor 3, each side of the original shape needs to be multiplied by 3 to give the enlargement.

An enlargement from a given centre requires you to draw an enlargement the centre also specifies where the shape should be drawn.

See the example below:

Sometimes an enlargement can make a shape smaller, but this is still called an enlargement, this happens when the scale factor is between 0 and 1.

Describing a transformation

When you describe a transformation you need to look at what has happened to the original shape.  Has it been turned (it's a rotation), flipped (a reflection), made bigger or smaller (an enlargement) or simply just moved (a translation).  You will also need to describe how so if it's a rotation remember the 3 bits of information needed 1) Centre 2) Direction 3) Angle.  If it is a reflection write down the mirror line as an equation.  If it is an enlargement write down the centre and the scale factor and if it's a translation write down the vector by which it has been translated.

Drawing Quadratic Graphs

A quadratic graph will always be a u or n shape depending on whether it is positive or negative.

Always draw your graphs using a pencil and they should be a smooth curve going through all your plotted points

To draw a quadratic graph you will be given an equation and you will need to use x values to create y values to enable you to draw your graphs.

See the example below

Ratio and Direct Proportion

Ratio is all about sharing and being fair.

We can simplify ratios in a similar way to fractions.  If we divide all parts by the same amount we are keeping the ratio equivalent.

Examples:

1) 10:20 divide both sides by 10 we get 1:2 this is simplified.
2) 27:36 divide both sides by 9 we get 3:4
3) 24:64:16 divide all parts by 8 we get 3:8:2

We can also divide in a ratio

Examples:

1) Share £200 between Anna and Bob in the ratio 1:3

There are 4 parts all together (1 + 3 = 4) so divide the original amount by the number of parts £200 divide 4 = £50 so Anna gets 1 part = £50 and Bob gets 3 x £50 = £150 their amounts add up to the starting amount.

2) Share £1000 between Carl, Dave and Emma in the ratio 2:3:5

2 + 3 + 5 = 10

£1000 divide 10 = £100

Carl gets 2 x £100 = £200

Dave gets 3 x £100 = £300

Emma gets 5 x £100 = £500.

Direct Proportion

Lots of questions involving direct proportion are either about ingredients or the cost of items.

Examples:
1) This is a list of ingredients needed to make 24 scones
600g flour
100g dried fruit
250g butter
Water to mix
a) How much dried fruit is needed for 6 scones?

1st see how the number of scones the ingredients are for relates to the amount in the question.

6 scones is a quarter of the original 24 so we divide the amount of dried fruit in the original list of ingredients.

100g divide 4 = 25g of dried fruit to make 6 scones

 

b) How much flour would you need for 36 scones?

 

36 scones is 1.5 times as many as the list of ingredients so multiply the amount of flour by 1.5 (or add half as much on again) 

 

600g x 1.5 = 900g of flour needed to make 36 scones

Long Multiplication and Division Methods

Here are 3 methods to perform Long Multiplication

Method 1 - Column Method

Add your answers together





Method 2 - Grid Multiplication

Method 3 - Chinese Multiplication or Lattice Method

You can use these methods when multiplying decimals.

For example 46 x 37 = 1702, so 0.46 x 3.7 = 1.702, how every many decimal places are in the question there are the same amount in the answer.

Long Division:

You can either do long multiplication the short division way or using the long division method.

Using the short division method:
I found this video on youtube which explains it well

Using the long division method
Again this video should help

When dividing decimals if the decimal is in the second number we must deal with this before attempting division.
5.39 divide 1.1.  To make the second number not a decimal we multiply by ten, but whatever we do to the second number we must also do to the first so our calculation becomes 53.9 divide 11 which is 4.9.  This is our answer.

Area of 2- D Shapes

To calculate the area of any shape, if you have squares inside the shape you can simply count the number of squares as area is the space inside a shape

Example:

Finding the area of a Rectangle (or Square)

If we don't have squares to count we have to find the area of different shapes using different formulas.

To find the area of square or rectangle we do length x width

Example:



























Area of Triangles:

To find the area of a triangle we must use a different formula

Area of a parallelogram

The area of a parallelogram is similar to finding the area of a rectangle, but again we must be careful with the height and make sure we use the height that is perpendicular to the base (Or the straight vertical line from the base)

Area and Circumference of circles

Remember the song ...

Circumference is Pi x diameter, Pi x diameter, Pi x diameter
Circumference is Pi x diameter
Area is Pi r squared!

Fractions decimals and percentages - Comparing & Converting

When comparing fractions decimals and percentages it is a good idea to convert them so they are all either a fraction, a decimal or a percentage.  If you convert them to fractions it easier to write them all with a common denominator making them easier to compare

Converting decimals to percentages and fractions

To convert 0.36 to a percentage is very easy.  If the decimal you are looking at is 0. something and has 2 decimal places then ignore your 0. and you have the percentage so 0.36 is 36%, if the decimal doesn't have 2 decimal places like 0.4 then add a zero to make 2 decimal places so 0.4 becomes 0.40 and so is 40%

Converting to fractions, you can either use your percentage because all percentages are out of 100 so 0.36 is 36% so is 36/100 and then cancel or use what you know about place value.

0.36, the last digit is in the hundredths column and so 0.36 = 36/100 which cancels to 9/25.

And so 0.4 the last digit is in the tenths column so 0.4 = 4/10 which cancels to 2/5.

Converting percentages to decimals and fractions

Percentages are out of 100 so every percentage can be written as a fraction out of 100 and cancelled appropriately.

If a percentage has 2 digits (in the tens and units column) then to write as a decimal put a 0. in front of the digit.

E.G 27% is 0.27, 56% is 0.56.

If the percentage is only 1 digit in the units column remember to add a zero first so 3% is 0.03 and 7% is 0.07.

If you look we are dividing each % by 100 to convert to a decimal.

Converting Fractions to percentages and decimals

This is the hardest to do sometimes.

 

If this can't be done simply remember a fraction is a division so 4/7 is 4 divided by 7 which gives you 0.571428 (to 6 decimal places) x by 100 to change to a  percentage which is 57% (to the nearest %)

Volume and surface area

To calculate the volume of a prism you have to find the cross sectional area

(This is the 2 identical areas usually at the front and back of the shape or can sometimes be the top and bottom)

and then multiply but the distance between these two areas usually the length or height of the prism


Volume = Cross sectional area x length

To calculate the surface area you need to calculate the area of each face (don't forget the ones you may not be able to see in the picture)

Example:




























Volume of a pyramid:

Number work

Adding & Subtracting:

When adding and subtracting numbers remember to line up the digits and when they involve decimals always line up the decimals.

Examples

Remember when adding to add the extras and carry on numbers and when subtracting that if the top number is smaller than the bottom you will need to borrow as in the examples above.

Rounding:
Rounding to the nearest 10.

458 rounded to the nearest 10 is 460. 
To round to the nearest 10 look at the units column and remember the rhyme.
5 and above give it a shove (round up), 4 and below down we go (round down)

Another example  19873 rounds to 19870.

Multiplying and dividing by 10, 100 and 1000

Transformations of trig graphs and Cast diagrams

Remember when transforming graphs

f(x) + a move the graph up (y direction) by a
f(x + a) move the graph across (x direction) by - a
af(x) multiply y values by a
f(ax) multiply x values by 1/a
-f(x) reflection in x axis
f(-x) reflection in y axis

Remember when transforming trig graphs
f(ax) means that in one usual complete cycle (0 - 360 degrees) there will be a complete curves

E.g:



















Example

CAST Diagrams

Fractions of shapes, Equivalent fractions and fractions of amounts

To find what fraction of a shape is shaded you need to work out how many equal pieces are in the shape and how many of them are shaded.



























Equivalent fractions are those that are equal.
Remember if you multiply the numerator (the top number) by some number as long as you do the same to the denominator (The bottom number) you will have an equivalent fraction.


















To find a fraction of an amount remember to divide the number by the denominator (the bottom number).  To find one quarter of 24 divide 24 by 4 to get the answer 6.

Linear Inequalities

When solving linear inequalities treat them like equations, solve them in the same way, using the same rules.

Make sure your answer is written with an inequality sign otherwise you could end up losing marks on an exam.

Examples:
1) Solve 2x + 3 < 14

Subtract 3 from both sides
                        2x    < 11

Now divide both sides by 2
                    x   < 5.5

This means that x can take any value below 5.5, but does not include the value 5.5.  If you were asked to state the largest integer (whole number) that belongs in this set you would give the answer 5 as this is the biggest integer that is less than 5.5


2)




























Sometimes you are asked to show inequalities on a number line




























When dealing with inequalities remember to try to avoid multiplying or dividing by a negative number.  Here's why ...

20 > 4, this inequality is true because 20 is greater than 4, now if I divide both sides by -2 the rules of equations would say my equation would be unaffected so lets try

-10 > -2, is this still true?

-10 is not greater than -2.  If you must divide or multiply by a negative number when using inequalities you must remember to turn the inequality sign.




 

 

Equations of straight line graphs

When calculating equations of straight lines
Remember:

y = mx + c for any straight line, where m represents the gradient and c represents the y intercept.

The gradient represents the steepness of the graph.

To calculate the gradient of a line remember it is the difference in y divided by the difference in x or in simple terms up divided by across.















The y intercept is where the graph crosses the y axis in the above example the graph crosses the y-axis at the point 2, so the y intercept is 2.

If we put these two pieces of information together we can write the equation of a straight line.

The equation of the line shown above in the form y = mx + c is

y = 0.5x + 2

Obtaining the gradient and y intercept from an equation

If the equation is written in the form y = mx + c we can read the gradient and intercept straight from it

E.G:
1) y = 3x + 5, gradient = 3, y intercept = 5

2) y = 4 - 5x, gradient = - 5, y intercept = 4

If the equation is not in the form y = mx + c we may have to rearrange

E.G

3) 2y = 8x - 4
if we divide everything by 2 we get
y = 4x - 2 and we can now obtain the gradient as 4 and y intercept as -2

4) 3y + 6x = 12

First move the 6x from the LHS by subtracting it from both sides

3y = 12 - 6x

then divide by 3 to leave you with

y = 4 - 2x, the gradient is -2 and the y intercept is 4.

 

 
 
 
 
 
 

Y9 - Christmas Test Review Homwork

Percentages of amounts:

To find 10% divide by 10

To find 1% divide by 100

To find 50% divide by 2

To find 25% divide by 4

 

Use these to help you calculate any percentage by building percentages

Example:

Find 32% of 80

 

10% of 80 = 80  ÷ 10 = 8

1% of 80 = 80  ÷  100 = 0.8

 

32% = 10% + 10% + 10% + 1% + 1%

        =  8     +  8    + 8     + 0.8  + 0.8

       = 25.6
 

 

Rounding:

When rounding remember the rhyme

5 and above give it a shove, 4 and below down we go …

Examples:

Round to 1 decimal place

a) 4.57

The number next to the first decimal place (the 7) is above 5 so we round the number in the first decimal place up to 7.

Answer: 4.6

 

Round to 2 decimal places

b) 0.83192

The number next to the second decimal place (the 1) is below 5 so we don’t change the number in the second decimal place,

Answer: 0.83

 

Significant Figures

A significant figure is any digit that is not a zero at the beginning or end of a number.

The number 560 has 2 significant figures the number 506 has 3 significant figures.

 

When rounding to a number of significant figures use the same rules as above but first decide what it is you are rounding to.

 

For example rounding 5673 to 1 significant figure, the 5 is in the first significant figure and is in the thousands column so we round to the nearest thousand the answer is 6000

 

0.3178  rounded to 1 significant figure is 0.3 as the first significant figure is 1 decimal place.

Multiplying and dividing by numbers between 0 and 1

Remember multiplying by 0.1 is the same as dividing by 10
dividing by 0.1 is the same as multiplying by 10.

Example:
68 x 0.1 = 68 ÷ 10 = 6.8

 

68 ÷ 0.1 = 68 x 10 = 680

 

ANGLES

 

Angles at a point = 360^o

Adjacent angles on a line = 180^o

Angles in a triangle = 180^o

Angles in a quadrilateral = 360^o

In Parallel lines

Corresponding angles are the ones on the shelf that you can slide up and down and are equal

Alternate angles are in the corners on opposite sides and are equal

 

DATA:

Averages & Range

Mean: Add up all the numbers and divide by how many are in the list

Median: The middle of an ordered list

Mode: The most popular number

Range: Highest subtract lowest

 

Pie Charts:

Find the total frequency

360 ÷ (Total frequency) = ?

Take each frequency and multiply by ? to work out the angle to draw.

 

If your angels don't add to 360 degrees you've done something wrong