S4 Formulae & Expressions.notebook

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


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


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


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


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


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


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


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


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%


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


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.


9

September 12, 2013

Aug 2814:18 Aug 2814:35

Aug 2809:28


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


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


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


Q1. Simplify h5 x h2 x h7


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


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)


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


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)


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


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


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


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


x2 + 4x + 5

Sep 416:25


x2 + 4x 4

Sep 416:25

Q37) 3 1 p (p 2)

Sep 416:28

Q38) x2 92x2 5x 3

Fully simplify


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)


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


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


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)


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

S4 Formulae & Expressions.notebook

Documents

Transcript of S4 Formulae & Expressions.notebook