vms x Year 8 Mathematics Equations http://www.mathsisfun.com/fractions.html Equations

Learning Intentions Pupils should be able to: Simplify algebraic expressions by collecting like terms and removing brackets Interpret and apply simple rules expressed in words to generate a numerical solution Generate and use a simple formula to find connections between two or more variables Formulate and solve linear equations originally expressed in words Solve equations involving brackets, terms on both sides, re-ordering and collecting like terms

Collecting Like Terms A Mathematical expression uses numbers and letters to represent numbers Expressions are used to solve problems Example: We represent the following as 4a + 3c. (4 apples and 3 cars

More Terms Sometime we need to combine all the terms together Simplify the following: 3c + 2a + 4c + 3a

Simplify 4a + 5a – 2a 10x – 2x + 3x 5a + 4x – 2a + 5x 7a – 3x + 2x – 8a 9x2 – 3x + 5x + 2x2 -3x2 + 5x – 5x2 + 10x2

Brackets! Sometimes we use brackets Simplify the following: 2(3a + 2p) =

Simplify 4(3x + 4b) 5(x + 4) 3(2x – 5) 2x + 3(x + 2) 5(2x + 4) – 8x

Solving Equations An equation means that two things are equal. A pair of scales is sometimes used to show equations because when both sides balance they are equal. If you put something else on one side, you must put the same amount on the other side or the scales are no longer balanced. If you take something off one side, you must take the same amount off the other side or the scales are no longer balanced.

x x Solving Equations Look at the scales We can write: If you take two marbles off each side you get 2x = 4 If you now take each side and divide by two you get x = 2 x x

Show your working out 2x + 2 = 6 ( – 2) 2x = 4 ( ÷ 2) x = 2 Lets look at that equation again 2x + 2 = 6 ( – 2) 2x = 4 ( ÷ 2) x = 2

Another example Solve: 2 = 3x – 16

What about negatives! Solve: 3x + 15 = 3

Who has the most? Solve: 6x + 26 = 4x

Try that again Solve: 4 = 16 – 3x

What about fractions? Solve: 7x + 20 = 26

And brackets! Solve: 3(2x + 1) = 15

Some more! Solve 3(4 – 3x) = 3

More terms Solve: 4x + 3 = 2x + 11

More brackets Solve: 8(x + 1) = 2 (x – 16)

Help! Remember when solving equations DO THE SAME TO BOTH SIDES Expand brackets and Simplify Deal with LETTERS first (keep on side where are most) Deal with NUMBERS after (remove numbers on same side as letters) get ‘single letter = ’ (leave as fraction if you need to)

Problems Sometimes we are given practical examples. We need to create the equation and then solve it. Example: A bus costs £200 to hire for the day. A hockey club charges £10 for each non member (n) and £6 for each member (m) to go on an outing. Write down an equation for the cost of hiring the bus If 20 members go on the outing, how many non- members need to go if the club is not to lose money?

Where did the equals go? Sometimes the answer to a problem has a number of solutions. If the cost of a cinema ticket is £6, then you must have £6 or more to go (everyone wants popcorn and coke!) We can write this as: Money ≥ 6

Using the number line We often illustrate this answer using a number line

Too big Sometimes vehicles must be less than a certain height to let them into a car park. The height of the vehicle may need to be less than 2 m. We can write this as height < 2 m And show it on the number line: Remember: more ink – solid, less ink - hollow

Both Sides Sometimes two inequalities are combined into one. For example: x > 2 x ≤ 8 Are often combined to: 2 < x ≤ 8 We represent this on the number line as:

Solving Inequalities Sometimes we need to simplify inequalities. We do this the same way as equations – it’s just the sign is different! Example: x – 5 < 3 x – 5 < 3 (+5) x < 8

Sequences A mathematical sequence is a list of numbers that are connected together in some way. For example: the numbers 2, 4, 8, 16, 32, 64, .. form a sequence Each number is double the value of the previous number

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Linear Sequences In a linear sequence the next number is found by adding a fixed value (common difference) to the previous number For example: The numbers 2, 6, 10, 14, 18, 22 form a linear sequence The next number is found by adding 4 each time

Linear Sequences Sequence values can be written in a table indicating the position in the sequence (n) and the value For example, the previous sequence can be put into a table as shown below n 1 2 3 4 5 6 Value 10 14 18 22

Finding the Formula To find the formula for a linear sequence we need to: Find the common difference Write the expression Modify the rule to find the first term Write down the rule Check your answer!

Matchsticks Here is a pattern of triangles that is built using matchsticks.

Matchsticks Here is a pattern of triangles that is built using matchsticks.

Matchsticks Here is a pattern of triangles that is built using matchsticks. Sequence 1 2 3 4 5 6 7 Matchsticks

Differences We can calculate the difference between two patterns Lets look at the first pattern again Sequence 1 2 3 4 5 6 7 Matchsticks 9 11 13 15 +2 +2 +2 +2 +2 +2

Common Difference We can see that the difference between successive values is 2. This can help to create a formula. We will multiply the sequence number by 2. This is nearly right. Can you explain how to get the correct sequence? Sequence 1 2 3 4 5 6 7 Matchsticks 9 11 13 15 Add 2 8 10 12 14

Simplifying our Wording The rule is: The number of matches is found by multiplying the sequence number by 2 and adding 1 If we use n for the sequence number, the formula becomes: Number of Matchsticks = 2n + 1

The Final Sequence This is still too long We will use n to represent the position in the sequence Number of Matchsticks = 2n + 1 So for the third sequence the number of matches is Number of Matchsticks = 2 x 3 + 1 = 7 Is it correct?

Second Pattern Can you find the formula for this sequence?

Matchsticks 2

Matchsticks 2

Matchsticks 2

Matchsticks 2

Matchsticks 2

Matchsticks 2

Matchsticks 2 Sequence 1 2 3 4 5 6 7 Matches

Matchsticks 2 Number of Matchsticks = 8n + 4 Sequence 1 2 3 4 5 6 7 Matches 12 20 28 36 44 52 60

Finding the Formula Find the formula for the sequence 1, 4, 7, 10, … Common difference = 3 Expression = 3n Modify the rule to find the first term when n = 1, 3n = 3 we need to subtract 2 Write down the rule 3n – 2 When n = 3, 3n – 2 = 3 x 3 – 2 = 7 