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

 

 

 

 

 

 

 

 

Percentages of amounts

Finding 20% of 50

Let's start by working out 10% of 50

10% of 50 = 50 ÷ 10 = 5

Now working out 20% is easy:

20% = 10% + 10% = 5 + 5 = 10

Let’s look at a harder example

Find 35% of 80

Let’s start again by working out 10%

10% of 80 = 80 ÷ 10 = 8

Now work out 5%:  5% is half 10% so we can divide our answer for 10% by 2

5% = 8 ÷ 2 = 4

Now working out 35% is easy:

35% = 10% + 10% + 10% + 5% = 8 + 8 + 8 + 4 = 28