Upload
others
View
2
Download
0
Embed Size (px)
Citation preview
S4 Formulae & Expressions.notebook
1
September 12, 2013
Aug 1215:21
Daily Practice 14.8.2013Please complete these in the back of your class work jotters...
Q1. Calculate the value of a house originally worth £120 000 that has appreciated by 19% over 2 years
Q2. Calculate the standard deviation of 2, 3, 8, 7, 5
Q3. 2/3 + 13/4
Q4. Write 809000 in scientific notation
Q5. Find the length of PQP
QS23.5cm
16cm1340
Aug 1413:52
Q1. Calculate the value of a house originally worth £120 000
that has appreciated by 19% over 2 years
Aug 1413:52
Q2. Calculate the standard deviation of 2, 3, 8, 7, 5
Aug 1413:52
Q3. 2/3 + 13/4
Q4. Write 809000 in scientific notation
Aug 1413:52
Q5. Find the length of PQ
P
QS23.5cm
16cm1340
Aug 1216:50
Learning Intention:
Today we are going to learn about Indices
S4 Formulae & Expressions.notebook
2
September 12, 2013
Aug 1216:51
An index, exponent or power is just the number of times a number is multiplying by itself.
For example: 35 = 3 x 3 x 3 x 3 x 3 35 is pronounced "3 to the power of 5"
k x k x k x k x k x k = k 6
Aug 1216:54
We already know that m x m2 = m3 and that m2 x m2 = m4.Indices
When you are multiplying powers of a number, you add the indices.
The general rule is: • ka x kb = ka+b
Examples:
(i) j2 x j5 (ii) 3g5 x 4g6 (iii) m 2 x m-1 x n 3
Indices 14.8.2013
Aug 1312:09
When you are dividing powers of a number, you subtract the indices.
The general rule is that
• ka ÷ kb = ka-b
Examples:
(i) a12 (ii) 3a5 (iii) 4x18
a7 a2 12x7
Aug 1312:06
Any number to the power of 0 is equal to 1 a0 = 1
The proof:
Any number divided by itself is 1
So a2 a2 = 1
But also a2 a2 = a0
Therefore a0 = 1
Aug 1312:10
A negative power such as 3-5 means 1
In general: k-m = 1
The proof by example:
a2 x a3 = a x a x a x a x a
35
km
a6
a6a2 x a3 a5
a6 a-1
= =
a x a x a x a x a x aBy dividing top and bottom by a
common factor: a5 you are left
with 1 a
Therefore a-1 = 1 a
We know that
But also,
Aug 1416:13
Daily Practice18.1.2013
Q1) 12/3 - 3/4
Q2) Calculate the value of an investment of £2500 that appreciates by 2.3% yearly over 3 years
Q3) State the median and quartiles of 16 5 17 8 3 14 5
Q4) If a box of washing detergent now weighs 1.5kg with 50% extra
free, what was its original weight?
Q5) Calculate the area of triangle ABC
15.8.2013
A
B C170
24cm
38cm
S4 Formulae & Expressions.notebook
3
September 12, 2013
Aug 1511:01
Q1) 12/3 - 3/4
Q2) Calculate the value of an investment of £2500 that appreciates by 2.3% yearly over 3 years
Q3) State the median and quartiles of 16 5 17 8 3 14 5
Aug 1511:01
Q4) If a box of washing detergent now weighs 1.5kg with 50% extra
free, what was its original weight?
Q5) Calculate the area of triangle
ABCA
B C170
24cm
38cm
Aug 1416:32
Learning Intention:
Today we are going to pracitise using the
laws of indices.
Aug 1311:40
Try these:
1) a2 x a7 x a4
2) k8 x k-7
3) b7 x b12 x 3b6
4) 2a3 x 5a4
5) 3v x 2v5 x 8v-7
6) v2 x v-8 x v3
7) 3m x 7m-5
8) 9z12 x 6z20 x a
9) k8 k7
10) b15 b2
11) 12a3 2a
12) 15v10 2v5
13) 8v2 v-8
14) 12m 4m-5
15)
16)
17)
18)
Aug 1311:51
Daily Practice 20.8.2013
Q1.�Simplify�the�surd�
Q2.�Write�as�a�single�fraction
Q3.�Simplify
Q4.�Find�the�median�and�quartiles�of�27 33 45 64 78 91
Q5.�Rationalise�the�denominator
+
x
Aug 1910:02
Today we are going to learn about taking "powers" of "powers" of a number.
Homework 1 (Indices)
online - due 27.8.2013
www.missdeely.weebly.com
S4 Formulae & Expressions.notebook
4
September 12, 2013
Aug 2015:07 Aug 1909:55
* (a2)3 = a2 x a2 x a2 = a x a x a x a x a x a = a6
So in general (ak)m = akm
Indices 20.8.2013
Examples:
Aug 2017:48
Daily Practice 21.8.2013
Q1.
Q2.
Q3.
Q4.
State the gradient and the y-intercept of the line -y = 3x + 2
Factorise 3x2 + x - 2
Simplify 9x2 - 1 3x2 + 4x + 1
Calculate the length of x
Aug 2113:44
State the gradient and the y-intercept of the line -y = 3x + 2
Factorise 3x2 + x - 2
Aug 2113:43
Simplify 9x2 - 1 3x2 + 4x + 1
Calculate the length of x
Aug 2015:14
S4 Formulae & Expressions.notebook
5
September 12, 2013
Aug 2010:06 Aug 1910:06
Aug 1909:58
power
root
Fractional Indices:
When the root is 2 (n = 2) it is just
Indices 21.8.2013
Aug 1909:56
Examples:
1) 2) 3)
Aug 2018:01 Aug 2018:01
S4 Formulae & Expressions.notebook
6
September 12, 2013
Aug 2117:07
Daily Practice18.1.2013 22.8.2013
Q1. Sketch the line y = x + 2
Q2. Calculate the volume of a cone with a diameter of 8.5cm and height 7cm
Q3. Calculate the value after 5 years of an investment of £4500 that grows by 8.3% per annum?
Q4. Multiply out and simplify
3k(6k 1) 2k2 + 5
7cm
8.5cm
Aug 2210:57
Sketch the line y = x + 2
6 5 4 3 2 1 0 1 2 3 4 5 6
6
5
4
3
2
1
1
2
3
4
5
6
x
y
Aug 2018:01
Write these with a fractional Index
Aug 2211:19
Aug 2109:09
Mixed Revision of Indices:
8) (z12 x 6z2 x a)2
9)
10)
11) 12a-3 2a
12) 36v10 2v-5
13) 8v2 12v-8
14) 20m-3 4m-5
15)
16)
17)
18)
1) 6a2 x 5a-7 x a4
2) 10k8 x k-7
3)
4) (2a3)2
5) 3v x 2v5 x 8v-7
6) v2 x v-8 x v3
7) (5q2)3 v3
x
12k8 4k7
Aug 2211:26
S4 Formulae & Expressions.notebook
7
September 12, 2013
Aug 2612:47
Fill in the blanks below:
(a) m2 x m7 x ___ = m‐1
(b) 24k20 ÷ 8k‐1 = ____
(c) ___(3m + 7) = 21m2 + 49m
(d) x ____ = 6
(e) x ___ = m
(f) (x + 2)( x ‐ __) = x2 ‐ x ‐ 6
(g) 45 = 90% = 1% ____ = 100%
Daily Practice 27.8.2013
Aug 2117:10
Today we are going to learn about
adding/subtracting algebraic fractions
Homework due today
Aug 2117:11
To +/ fractions, the denominators(bottoms) must be the same.
To make them the same, find the lowest common multiple
Example:
5 36 4
LCM = 12
x 2
x2
x 3
x 3
10 912 12 =
112
Aug 2117:14
To +/ algebraic fractions, the denominators must be the same.
To make them the same, find the lowest common multiple
Example:
1 3x 4x2+
Aug 2117:12
Examples: Simplify these expressions
1.
2.
3.+
+
4.
Aug 2717:06
Daily Practice21.9.2011 28.8.2013
Q1. What is the original price of a dress that was reduced by 25% in the sale and now costs £36.00?
Q2. Expand the brackets and collect like terms3m + 2m(1 - 5m) + 3m
Q3. What is the gradient of the line joining (5, 2) and (-3, 0)?
Q4. Factorise 3x2 + 7x - 20
S4 Formulae & Expressions.notebook
8
September 12, 2013
Aug 2813:49
Q1. What is the original price of a dress that was reduced by 25% in the sale and now costs £36.00?
Q2. Expand the brackets and collect like terms
3m + 2m(1 - 5m) + 3m
Aug 2813:49
Q3. What is the gradient of the line joining (5, 2) and (-3, 0)?
Q4. Factorise 3x2 + 7x - 20
Aug 2808:21
Today we will be practising adding and subtracting algebraic fractions
Homework: Algebraic fractions due 4.9.2013
Available on www.missdeely.weebly.com
Aug 2814:06
Aug 2612:47
Examples:
1. 3m - 2 m + 6 5 4
2. 8k - 3 2k + 76 4
+
4. 2 2k - 7 k k2
+
3. 3 xx x - 1
Aug 2717:08
4. 3.
S4 Formulae & Expressions.notebook
9
September 12, 2013
Aug 2814:18 Aug 2814:35
Aug 2809:28
Daily Practice 29.8.2013
Q1. Simplify m7 m3
Q2. State the gradient of the line joining (-5, 2) and (3, 7)
Q3. Simplify
Q4. Write as a single fraction
Q5. Simplify
Aug 2911:08
Q2. State the gradient of the line joining (-5, 2) and (3, 7)
Q3. Simplify
Aug 2911:08
Q4. Write as a single fraction
Q5. Simplify
Aug 2807:49
Today we will be learning how to multiply and divide algebraic fractions
S4 Formulae & Expressions.notebook
10
September 12, 2013
Aug 2809:31
To multiply fractions, multiply the top by the top and bottom by the bottom
To ÷ fractions, turn the 2nd fraction upside down and multiply
Before you multiply fractions, try to simplify by dividing the top and bottom by common factors (canceling out)
Aug 2715:33
Examples:
1. 4m 5 3 7m
2.
3.
x
5k2 3 v 7vk÷
3c2 30m2
5m 24cx
4.
Aug 2912:48
Daily Practice 3.9.2013Q1. Find the area of a sector with angle at the centre 480 and radius 12cm
Q2. Simplify 2k x 3k2 x 5k
Q3. Simplify 15m20 3m
Q4. Simplify the surd 32
Q5. Factorise 81x2 - y2
Aug 2911:28
Aug 2911:36
Today we will be learning how to complete the square.
Aug 2912:51
Completing the square
Completing the square means writing a quadratic expression
in the form
(x + p)2+ q or a(x + p)2 + q or p - (x + q)2
Square out the bracket and then equate the terms.
Examples: Write these expressions in the form (x + p)2+ q
1) x2 + 6x + 10 2) x2 - 2x + 3
S4 Formulae & Expressions.notebook
11
September 12, 2013
Sep 315:14 Aug 2913:20
Questions: Write these in the form (x + p)2 + q
1) 2x2 + 4x - 8 7) x2 - 10x - 5
2) 15 - 10x - x2 8) x2 + 18x - 81
3) 5x2 + 10x + 15 9) 3x2 - 12x + 2
4) x2 + 8x - 10 10) 2x2 - 3x - 6
5) x2 - 10x - 5
6) x2 - 40x + 1
Sep 214:11
Daily Practice 4.9.2013
Q1. Simplify h5 x h2 x h7
Q2. Write 56980000 in scientific notation
Q3. Factorise x2 + 7x 18
Q4. Calculate the area of an eighth of a circle with diameter 15cm
Q5. Simplify
Aug 2913:15
Examples: Write these expressions in the form a(x + p)2+ q
1) 3x2 + 6x + 2 2) 4x2 - 12x - 5
Aug 2913:19
Examples: Write these expressions in the form p - (x + q)2
1) 4 + 2x - x2 2) 4x - 6 - x2
Aug 2913:20
Questions: Write
1) 2x2 + 4x - 8 in the form
2) 15 - 10x - x2 in the form
3) 5x2 + 10x + 15 in the form
4) 3x2 - 12x + 2 in the form
5) 2x2 - 3x - 6 in the form
6) 13 + 4x - 2x2 in the form
a(x + p)2+ q
a(x + p)2+ q
a(x + p)2+ q
a(x + p)2+ q
p - (x + q)2
S4 Formulae & Expressions.notebook
12
September 12, 2013
Sep 214:13
Topics to revise:• Surds, Indices and scientific notation
• Algebraic expressions multiplying out, factorising, completing the square
• Adding, subtracting, multiplying, dividing and simplifying algebraic fractions
• Change the subject of a formula
• Find the gradient given two points.
• Calculate arc length, area of a sector and angle at the centre
• Volume of solids (cuboid, cone, pyramid, sphere, cylinder etc.)
.
Sep 414:42
Sep 414:31 Sep 212:35
Sep 310:46
Q1) Multiply out and simplify
(3x 1)(x + 2)
Sep 310:47
Q2) Multiply out and simplify
(3x 1)(x2 + 5x + 2)
S4 Formulae & Expressions.notebook
13
September 12, 2013
Sep 310:48
Q3) Factorise
81y2 100
Sep 310:48
Q4) Factorise fully
4y2 100x2
Sep 310:48
Q5) Simplify
4y2 x 3y2 x 7y4
Sep 310:49
Q6) Simplify
25y2 5y8
Sep 310:50
Q7) Factorise
4x2 15x + 9
Sep 310:50
Q8) Calculate the volume of a sphere with radius 12cm
S4 Formulae & Expressions.notebook
14
September 12, 2013
Sep 310:51
Q9) Calculate the volume of a cylinder with radius 1.5m and height 7m
Sep 310:51
Q10) Calculate the volume of a cone with radius of base 11cm and height 18cm
Sep 310:52
Q11) Calculate the area of a sector of a circle with radius 6.7cm and angle at the centre 560
Sep 310:52
Q12) Write 0.0003867 in scientific notation
Sep 310:53
Q13) Write x2 2x + 8 in the form (x p)2 + q
Sep 310:53
Q14) Calculate the gradient of the line joining A(0, 2) and B(3, 5)
S4 Formulae & Expressions.notebook
15
September 12, 2013
Sep 310:54
Q15) Simplify the surd
45
Sep 310:55
Q16) Simplify
45 + 2 5 + 245
Sep 310:56
Q17) Express x2 + 8x + 19 in the form y = (x + p)2 + q
Sep 413:15
Q18) Calculate the answer to 6.2 multiplied by 3.48 x 108 and write your answer in scientific notation
Sep 413:17
Q19) Calculate the gradient of the line joining (1, 7) and (1, 5)
Sep 413:17
Q20) Simplify m2 x 15m3
3m3
S4 Formulae & Expressions.notebook
16
September 12, 2013
Sep 413:18
Q21) Simplify 2m2 x 8m2m3
Sep 413:18
Q22) Calculate the volume of this cylinder in litres
48.3cm
37cm
Sep 413:20
Q23) Calculate the length of an arc with radius 9cm and angle at the centre 420
Sep 413:20
Q24) Multiply out and simplify (3m 1)(m2 + 2m + 5)
Sep 413:21
Q25) Factorise 3km2 + 15mk
Sep 413:21
Q26) 5k 3m 11 15kX
S4 Formulae & Expressions.notebook
17
September 12, 2013
Sep 413:23
Q27) (x + 2) (x + 5) 3 4+
Sep 413:23
Q28) 5km2 25m2h4h 3
Sep 413:26
Q29) Write 1288 million in scientific notation
Sep 416:19
Q30) p = 3.08 x 104
and q = 9.2 x 103
State the value of p + q
Sep 416:20
Q31) (3d3)2
Sep 416:21
Q32) 6r2s 2rs
S4 Formulae & Expressions.notebook
18
September 12, 2013
Sep 416:21
Q33) 6g2 x g2g3
Sep 416:22
Q34) Write this quadratic in the form (x + a)2 + b
x2 + 6x + 20
Sep 416:24
Q35) Write this quadratic in the form (x + a)2 + b
x2 + 4x + 5
Sep 416:25
Q36) Write this quadratic in the form (x + a)2 + b
x2 + 4x 4
Sep 416:25
Q37) 3 1 p (p 2)
Sep 416:28
Q38) x2 92x2 5x 3
Fully simplify
S4 Formulae & Expressions.notebook
19
September 12, 2013
Sep 416:29
Q39) 6x 18x2 5x + 6
Fully simplify
Sep 416:30
Q40) p2 4p2 5p + 6
Fully simplify
Sep 416:31
Q41) Express as a single fraction 5 3 x (x + 1)
Sep 416:33
Q42) Express as a single fraction x 2 x + 3 +
4 2
Sep 416:34
Q43) State the gradient which passes through
A(2, 4) and B(5, 6)
Sep 416:35
Q44) State the gradient which passes through
G(3, 7) and H(10, 7)
S4 Formulae & Expressions.notebook
20
September 12, 2013
Sep 416:35
Q45) A straight line with gradient zero passes through A(5, 4) and B(7, g)
What is the value of g?
Sep 416:36
Q46) Calculate the area of the sector
8004.5cm
Sep 416:38
Q47) Calculate the area of the spotty sector
1600
5.6cm
Sep 1211:02
Sep 1211:04 Sep 1211:07
S4 Formulae & Expressions.notebook
21
September 12, 2013
Sep 1211:08 Sep 1211:11
Sep 416:40
Q48) Calculate the shaded area
35050cm
75cm
Sep 416:42
Q49) Calculate the length of the minor arc ABA
B740
Sep 416:44
Q50) Calculate the perimeter of the sector AOB
12507cm
o
A
B
Sep 416:48
20cm
35cm
Q51) Calculate the volume of this square based pyramid
S4 Formulae & Expressions.notebook
22
September 12, 2013
Sep 416:48
Q52) A box in the shape of a cuboid is just big enough to hold 6 tennis balls. Each tennis ball has a diameter of 6.8cm. Calculate the volume of the cuboid
Sep 416:52
Q53) Rationalise the denominator
52
Sep 417:03
Q54) Calculate the total volume of this shape
Sep 417:03
Q55)
Sep 417:05
Q56)
Sep 417:07
Q57)
S4 Formulae & Expressions.notebook
23
September 12, 2013
Sep 417:08
Q58)
Sep 417:10
Q59) See Q58
Sep 417:10
Q60)
Sep 417:11
Q61)
Sep 417:12
Q62) Simplify
64 49