Transcript Formulas Powerpoint (level 6)

Formulas
Simple Formulas
• You might be given a formula and asked to
substitute numbers, e.g.
• E = mc2
• Find E when m= 90 and c = 3,000,000
• E = 90 X (3,000,000 X 3,000,000)
• E = 90 x 9,000,000,000,000
• E= 810,000,000,000,000
Making a formula
• Charlene has joined a swimming club. She had
to pay £25 to join the club. She also pays
£1.50 every time she goes swimming. What is
the formula (t = total cost and n = number of
swims)
• T= £25 + £1.50 x n
• Or t= 25 + 1.5n
• How much is it for 30 swims?
• T= 25 + (1.5 x 30)
• T= 25 + 45
• T= £70
Lets try these;
1. A goods train has an engine 6m long. Each
wagon is 8m long. Write down a formula for
the total length of the goods train (T= total
length, n = number of wagons). Use this
formula to find the total length of a train with 20
wagons
2. Year 9 are having a party. It costs £90 to hire a
disco and £3 per pupil for refreshments. Find a
formula for the total cost of the party (T = total
cost, n = number of pupils). How much would it
cost if there are 120 pupils in year 9?
Answers
• T = 8n + 6
• T = 166m
• T = 3n + 90
• T = £450
Formulas with n2
• When you are given a sequence of
numbers sometimes you can identify
patterns e.g.
• 1,4,9,16,25……
No.
1
2
3
4
5
Sequence
1
4
9
16
25
How do you get from 1 to 1, 2 to 4, 3 to 9, 4 to 16 and
5 to 25
You square these numbers 3 x 3 =9
Formulas with n2
We must find if there is a pattern in this sequence,
we do this by taking away
Sequence
1
4
3
9
5
16
7
25
9
2
2
2
Because we had to find the difference twice this means that the number
needs to be squared.
We half the number we end up with, in this case 2 to find out if we multiply
this squared number
What is our formula?
Formulas with n2
N
1
2
3
4
5
Sequence
1
4
9
16
25
n2
1
4
9
16
25
Our formula must be 1n2
Formulas with 2n etc
• Is there a pattern in this sequence:
• 3,5,7,9,11…..
Sequence
3
5
7
9
Difference
2
2
2
11
2
Because the difference is two you must multiply the number by two,
Number
1
2
2n
2
4
3
6
4
5
8
These numbers are all one short of our sequence, so our formula
must be 2n + 1
10
Finding the nth term
• You are given this pattern
• 6,15,28,45,66….
• First you want to find the differences in these
numbers
• 6
,15,
28,
45,
66
•
9
13
17
21
•
4
4
4
• This tells us that we have 2n2 in our formula
Finding the nth term
• We then check if that gives us the sequence
number
N
1
2
3
4
5
Sequence 6
15
28
45
66
2n2
2
8
18
32
50
Rest
4
7
10
13
16
We need more in our formula, so the next step is to find how much more
Finding the nth term
N
1
2
3
4
5
Sequence
6
15
28
45
66
2n2
2
8
18
32
50
Rest
4
7
10
13
16
3
3
3
This means we must also multiply each number (n) by 3
3
When 3n is added to the 2n2 number we are 1 short e.g. 3x1is 3 +
2= 5, but the sequence number is 6
What is our final formula?
2n2 + 3n +1
Lets try these;
• Find the formula for the nth term and the
6th term in the sequence:
• 3,7,13,21,31,..
• 6,11,18,27,38,…
• 3,10,21,36,55,…
• 6,15,28,45,66,…
• 9,20,37,60,89,…
• 2,4,7,11,16,….
Answers
•
•
•
•
•
•
•
•
•
•
•
•
N2 + n + 1
43
N2 + 2n + 3
51
2n2 + n
78
2n2 + 3n + 1
91
3n2 + 2n + 4
124
N2/2 + n/2 + 1
22
Trial and Improvement
• This is when you try a number to see how close you are to getting
the answer
• E.G.
• Solve x2 + 3x = 82 (to 1 d.p.)
• Lets try x = 7
• 49 + 21 = 70 (this is too small)
• Lets try x = 8
• 64 + 24 = 88 (this is too big, but is closer to our answer)
• Lets try 7.6
• 57.8 + 22.8 = 80.6 (this is 1.4 too small)
• Lets try x = 7.7
• 59.3 + 23.1 = 82.4 (this is 0.4 too big, but our closest answer to
1d.p.)
• Our answer is x = 7.7
Lets try these:
•
1.
2.
3.
4.
5.
6.
Solve x to 1 d.p.
X2 + x = 79
X2 + 2x = 19
X2 + 4x = 93
39 = x2 + 3x
X(7 + x) = 11
X (24 + x) = 110
Answers
1.
2.
3.
4.
5.
6.
8.4
3.5
7.9
4.9
1.3
3.9