School Students Algebra Quadratics
Difference of two squares Common Factors
Adding and subtracting negative numbers
I will just give simple examples using a vertical number line.
Moving upwards is adding
Moving downwards is subtracting
Example One: 2 - 3
So start at 2 and move down 3
5
4
3
2 Start
1 One place
0 Two places
-1 Three places End So the answer is -1
-2
-3
-4
-5
Example Two: - 2 - 3
So start at - 2 and move down 3
5
4
3
2
1
0
-1
-2 Start
-3 One place
-4 Two places
-5 Three places End So the answer is -5
Example Three: - 4 + 3
So start at - 4 and move up 3
5
4
3
2
1
0
-1 Three places End So the answer is -1
-2 Two places
-3 One place
-4 Start
-5
Back to quadratics Example Three
Notice when multiplying with negative numbers
Negative times Negative makes Positive - 5 x - 5 = + 25
Positive times Positive makes Positive + 5 x + 5 = + 25
Positive times Negative makes Negative - 5 x + 5 = - 25
Back to quadratics Example One
Back to quadratics Example Three
A factor is a number which divides equally into another number. This then leads you to two numbers that multiply together to give you the original number.
How do you find the factors of a number? There are a few ways so we will work through an example:
Finding the factors of 48.
First divide by 2 until you do not get a whole number as answer,
Then divide by 3 until you do not get a whole number as answer,
Then divide by 5 until you do not get a whole number as answer,
Then divide by 7 until you do not get a whole number as answer,
Then divide by the next prime number etc until you get the answer one.
2 | 48
2 |24
2 |12
2 |6
3|3
1|1
So now we know that 2 x 2 x 2 x 2 x 3 x 1 = 48, now we just take the numbers in different groups
for example: 4 x (2 x 2 x 3 x 1 ) = 2 x 2 x 2 x 2 x 3 x 1
1 x (2 x 2 x 2 x 2 x 3) 2 x (2 x 2 x 2 x 3 x 1)
4 x (2 x 2 x 3 x 1 ) 8 x (2 x 3 x 1) 16 x (3 x 1 )
So we get the list of numbers that multiply to get 48 are:
1 x 48, 2 x 24, 4 x 12, 8 x 6, 16 x 3,
Another way is starting with 1 then 2 then 3 and so on finding whole numbers that multiply to give the original number noting the whole number answers as a pair. Stop when one of the numbers is already in a pair.
So using 48 again
1 times 48 is 48 so the first pair is (1 x 48),
2 times 24 is 48 so the second pair is (2 x 24),
3 times 16 is 48 so the third pair is (3 x 16),
4 times 12 is 48 so the fourth pair is (4 x 12),
5 times there is no whole number, therefore no pair
6 times 8 is 48 so the fifth pair is (6 x 8)
7 times there is no whole number, therefore no pair
8 times 6 is 48 we have 6 again which was in our fifth pair so we can stop.
1 x 48, 2 x 24, 4 x 12, 8 x 6, 16 x 3,
