get sharp. be sharp. · changing interest rates 36 trigonometry 37 special angles 38 analytical...

GET SHARP. BE SHARP.

G R A D E 8 – 1 2

www.besharp.co.za

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© 2019 Student Edge

The Sharp Logo, Sharp RoBoHoN and affiliated trademarks are the property of Sharp Corporation. LEARNER MANUALEL-W535SA

Written by Catherine Reynders


SHARP EL-W535SA


© 2019 Student Edge 3

This manual is designed to ensure you make use of all the features and functions available on the Sharp EL-W535SA.

Whether you are in grade 8, 9,10,11 or 12 we have material available for you. As you move through the grade 8 to 12 curriculum, continuously refer back to your manual, this will ensure that you are

sharpening your calculator skills in conjunction with each topic.

RoBoHoN is a Mathematics genius. He will be beside you every step of the way!

Use this manual to make sure you are prepared when the opportunity of a life time presents itself.

Good Luck for the year ahead!

Welcome to the Get Sharp, BE SHARP Manual!


SHARP BASICS

Table of Contents

SECTION 3

SECTION 2

SECTION 1

2nd F 8

ALPHA 8

ON / C 8

MODE 8

1. Activate normal mode

2. Activate stat mode

3. Activate table mode

4. Activate drill mode

ARROWS 9

Left and Right 9

Up and Down 9

BS 9

CHANGE 10

ANS 10

DRILL MODE

LET’S FIRST LOOK AT MATH 11

LET’S NOW LOOK AT OUR TABLE FUNCTION 12

GRADE 8

MULTIPLES 13

FACTORS 13

DIVISORS 14

INTEGERS 14

EXPONENTS 14

ROOTS 15


COMMON FRACTIONS 16

PERCENTAGES (5 Points) 18

DECIMAL FRACTIONS 19

FINANCE 19

PERIMETER 21

AREA 21

THEOREM OF PYTHAGORAS 21

DATA HANDLING 22

GRADE 9

FINANCE 24

FUNCTIONS 25

GRADE 10

F(X)–NOTATION 28

FUNCTIONS 28

Quadratic / Parabola Function 28

Hyperbola Function 30

Exponential Function 31

FINANCE 34

Compounding Periods 34

Timelines – Deposits and Withdrawals 36

Changing Interest Rates 36

TRIGONOMETRY 37

Special Angles 38

ANALYTICAL GEOMETRY 39

Distance 39

Gradient 39

Midpoint 40

DATA HANDLING 40

The 5 Number Summary 40

SECTION 5

SECTION 4

56


GRADE 11

QUADRATIC FORMULAE 42

SURDS 42

Simplifying Surds 42

Adding and Subtracting like Surds 43

Multiplying Surds 44

Dividing Surds 44

Rationalising Denominators 44

FINANCE 44

Nominal and Effective Interest Rates 44

TRIGONOMETRY 45

Sine Rule 45

Cosine Rule 46

Area Rule 47

DATA HANDLING 48

Standard Deviation 42

SECTION 6


SECTION 7 GRADE 12

LOGARITHMIC FUNCTIONS 50

FINANCE 51

Solving for n 51

Present Value Annuity 52

Future Value Annuity 53

DATA HANDLING 54

Bivariate Data 54

Correlation Co-Efficient 55

PROBABILITY 55

Factorials 55

Combinations 56

Permutations 56


2nd F 2nd F gives you access to the orange functions associated with each key.

ALPHA ALPHA gives you access to the green functions associated with each key.

ON/CLet’s begin by turning our Sharp on,

Key Sequence:

and now turning our Sharp off.

Key Sequence:

This key is also used to clear our current calculation and begin a new one. It clears all internal values, including the last ANS and statistics.

Key Sequence:

MODEThere are four different modes available to you on your Sharp.

1. To activate normal mode

Key Sequence:

2. To activate stat mode

Key Sequence:

3. To activate table mode

Key Sequence:

4. To activate drill mode

Key Sequence:

Normal mode is the mode used mostly. Once in modes 1–3 we can return tonormal mode using our HOME key.

SHARP BASICS

NORMAL MODE: Basic Arithmetic

STAT MODE:Statistical Calculations

TABLE MODE:Sketching Functions

DRILL MODE:Drill And Practice

2nd F

ON/C

OFF

5 + 2 = ON/C

MODE 0

MODE 1

MODE 2

MODE 3

SECTION 1


Let’s Try!Activate stat mode

Key Sequence:

Now return to normal mode

Key Sequence:

ARROWSLeft and Right

These keys are used to manipulate your cursor through your calculation shown on the screen.

Let’s Try!

Key Sequence:

Up and Down

These keys are used to access previous calculations. They allow you to scroll through your calculation steps one-by-one.

Key Sequence:

Now that we have entered different sums, we can look back on our calculations.

Key Sequence:

BS

We can correct our mistakes using the backspace key. Simply manipulate your cursor to the mistake and then backspace to correct it. The BS key deletes the symbol to the left of the cursor.

Key Sequence:

The DEL key deletes the symbol at the cursor.

Key Sequence:

• • • 7

MODE 1

HOME

6

6 9

96

3 + 1 = • • • 4

5

5

5

+

+

+

8

8

8

-

-

-

=

= =

DEL =

BS

2nd F=

5 - 2 = • • • 3

1 + 1 = • • • 2

• • • 4

• • • 4 • • • 230 461,68


Q: Why sh

ould you

wear g

lasses

durin

g math cl

ass?

A: It im

proves

di-v

ision

.

CHANGE

This key allows us to change the form of our answer between mixed number, fraction and decimal.

Key Sequence:

ANSThis key allows us to access the answer to our previous calculation.

Let’s Try!

Key Sequence:

Now enter the following:

Key Sequence:

=

=

=ANS

1

1

5

.

+

-

2

2

ALPHA

5 CHANGE• • •

• • • 3

• • • 2

• • • • • •114— 1,2554—CHANGE

This subtracts the previous

answer we calculated.


Drill mode is available on your calculator and will help you to SHARPEN your basic arithmetic skills.

Activate drill mode on your Sharp.

Key Sequence:

Drill mode now gives us two options:

Math: Addition, subtraction, multiplication and division

Table: Multiplication tables

LET’S FIRST LOOK AT MATH

Key Sequence:

Use the up and down keys to select the amount of questions you would like to be asked. You have the options of 25, 50 and 100 questions. Let’s choose 25 questions.

Use the left and right keys to pick the operation you would like to be tested on. You have the options of addition, subtraction, multiplication, division and mixed questions. Let’s pick the final option of mixed operations.

Key Sequence:

We now need to answer the question displayed on the screen. Submit your answer by pressing the = key. Continue answering the questions that follow.

Once all 25 have been answered it will display the percentage of questions you answered correctly. By pressing the = key it will now return to the math function where you can reselect the number of questions to be asked, as well as the operation to be questioned.

DRILL MODE

MODE 3

0

=

SECTION 2


LET’S NOW LOOK AT OUR TABLE FUNCTION

Key Sequence:

Use the up and down keys to select the multiplication table you would like to be tested on.

The left and right keys allow us to decide whether we would like the multiplication questions to be asked serially (in order) or randomly.

Please select the 5 times table, to be asked randomly and press your = key.

Key Sequence:

Answer the question displayed on the screen. Submit your answer by pressing the = key. Continue answering the questions that follow.

Once all the questions have been answered it will display the percentage of questions you

answered correctly. By pressing the = key it will now return to the table function where you can select another multiplication table and once again decide whether you would like to be asked serially or randomly.

REMEMBER We press ‘home’

to return to normal mode.

MODE 3 1

=


MULTIPLES

Finding multiples of any number is made easy using Sharp.

Find the multiples of 5.

Key Sequence:

We can also easily find the lowest common multiple between two numbers.

Find the lowest common multiple between 6 and 18.

Key Sequence:

FACTORS

Breaking a number up into its prime factors has never been easier. We do not need to use the ladder method. We can simply use the P.FACT key.

Example: Find the prime factors of 63.

Key Sequence:

GRADE 8

= = = = =+5 5• • • 5 • • • 10

6 3 = • • • 32 x 72nd F P.FACT

6 1 8 = • • • 182nd F LCM

68 7

9 3

3 3

1

LADDER METHOD

= 7 x 3 x 3

= 7 x 32

Q: Why did the learner eat her math book?A: She heard it was a piece of cake.

SECTION 3


DIVISORS / HIGHEST COMMON FACTOR

Find the greatest common divisor of 12 and 18

Key Sequence:

INTEGERS

Now that you are in grade 8 – you begin to work with both positive and negative whole numbers. When entering a negative number into our Sharp we use the (-) (negative key) and NOT the – (subtraction key).

Let’s Try!

Calculate the following: -5+8

Key Sequence:

Calculate the following: 8-5

Key Sequence:

EXPONENTS

We can easily calculate the square or cube of a number using our Sharp.

Calculate 42

Key Sequence:

Calculate 33

Key Sequence:

Numbers can be multiplied by themselvesany number of times. We can use the yx key to help us with these calculations. This key calculates exponential values.

Divisor: A number that divides into

another without a remainder.

Cubing: multiplying

an integer with itself

three times.

NOTE

THE

DIFFERENCE!

Squaring: the product of

an integer with itself.

1

(-)

8

4

3

2 1

5

-

x2

x3

8

+

5

8 =

=

=

=

• • • 6

• • • 3

• • • 3

• • • 16

• • • 27

2nd F

2nd F

GCD


Calculate 4 8

Key Sequence:

You would have learnt your exponent laws by now. Do not hesitate to use brackets on your Sharp.

Calculate –2 2

Key Sequence:

Calculate (–2)2

Key Sequence:

Calculate ( 2 2) 4

Key Sequence:

ROOTS

Calculate 2 25

Key Sequence:

Calculate 3 512

Key Sequence:

Q: Have you heard

of a mathematical

plant?

A: You can identify it

by its square roots.

4

(-)

(

(

2

(-)

2

x2

2

x2

8

)

)

x2

yx

=

=

=

4 =

• • • 65 536

• • • -4

• • • 4

• • • 256

yx

2 5

5 1

=

2 =

• • • 5

• • • 82nd F 3

46


c

48—

154—

14—

6 5 68 1 = • • • 32nd F

a/b 84 = • • •12—

a/b 415 = • • •34—3

• • •134—

13 4 =2nd F • • •14—3ab/c

CHANGE

x

x

Sometimes we deal with more than just a square or cube root. allows us to calculate the xth root of a number.

Let’s Try!

Calculate 8 6561

Key Sequence:

COMMON FRACTIONS

Don’t be afraid to use your Sharp when working with fractions. a/b inputs proper or improper fractions which consist of a numerator and denominator.

If you enter a fraction into your Sharp, it will simplify it for you.

Let’s Try!

Simplify

Key Sequence:

We can enter improper fractions and our Sharp will convert them to mixed numbers and vice versa.

Write as a mixed number.

Key Sequence:

Entering a mixed number requires the use of the a b key. Use the change key to change your answer from a mixed number to an improper fraction. If pressed for a second time it will give you your answer in decimal form.

Write 3 as an improper fraction.

Key Sequence:


Calculate +

Key Sequence:

Calculate 5 – 2

Key Sequence:

Multiplication and division of fractions follow the same key sequence structure.

Calculate x 3

Key Sequence:

Calculate ÷

Key Sequence:

Let’s link what we’ve learnt about exponents to our new knowledge of fractions.

Simplify

Key Sequence:

Simplify ( )

Key Sequence:

25—

92

108——

48—

4

38—

12—

68—

13—

12—

14—

23—

a/b1 a/b6+2 8 = • • •14—1

1 15 - =2 42 • • •14—3

a/b3 23x8 3 = • • •38—1

a/b2 a/b1÷5 3 = • • •15—1

x29 01a/b 8 = • • •34—

4( 4yx)a/b 8 = • • •116—

• • •54—CHANGE




2nd F

2nd F 2nd F

ab/c

ab/c ab/c

NOTE:

Your Sharp will

always give answers

as fractions in their

simplest form.

44


PERCENTAGESPercentage of a Value:

Calculate 70% of 200.

Key Sequence:

A value expressed as a percentage of another:

What percentage of 330 is 231?

Key Sequence:

Percentage Increase:

Tracy currently earns R60.00 per hour. If she receives an increase of 15%, what will she now be earning per hour?

Key Sequence:

Percentage Decrease:

Brett receives 20% discount on a laptop costing R4500.00. How much does he pay for his laptop?

Key Sequence:

Original Pricing:

Catherine received a 5% discount which amounted to R125.00. What was the original price of the item?

Key Sequence:

6 0 + 1 5 % • • • 69

2 0 0 x 7 0 % • • • 1402nd F

2 3 1 ÷ 3 3 0 % • • • 702nd F

4 5 0 0 - 2 0 % • • • 36002nd F

1 2 5 ÷ 5 % • • • 25002nd F

“OF” means“MULTIPLY”

Always enter your

percentage second.

© 2019 Student Edge42

DECIMAL FRACTIONS

Use your Sharp to help you with converting fractions to decimals and vice versa.

Write as a decimal

Key Sequence:

The decimal point key enters a decimal point.

Write 0.25 as a fraction

Key Sequence:

Basic arithmetic is also possible.

Calculate 9.9 – 8.03

Key Sequence:

FINANCE

Calculating simple interest is really…SIMPLE.

If I invest R5000.00 at a simple interest rate of 12% per annum, how much money will I receive after a 6-year period?

Before we place any interest rate into a financial formula, we need to convert the interest rate to a decimal.

Key Sequence:

19

14—

.

.

a/b

2 0

9

0

1

1

.8-

5

=

1 =

9

2

4

÷ 0

0 3 =

=

• • •87100—1

• • •14—

• • •187100—CHANGE

• • • 0,25

• • • 0,12

CHANGE

CHANGE

• • • 1,87CHANGE

NB!


We can now enter all our information into our simple interest formulae.

Key Sequence:

I received R84.00 after investing an initial amount at a 10% simple interest rate 2 years ago – what was my initial investment?

Key Sequence:

CALCULATING P

8 .1 )a/b x( =4 1+ )( 20 • • • 70

5 0( x0 1+ )0 .1 60 2( ) = • • • 8600

Be careful when

manipulating your

formulae!

CALCULATING A

P= A1+in

To avoid converting the

percentage to a decimal

Now to represent the percentage

key in your formulae press

1 1a/bA

0 0 STO

ALPHA A


PERIMETER

π is used for the first time in grade 8. It is important to realise that π is just a number. Pressing π automatically enters the value 3.141592654.

Calculate the perimeter of a circle with a radius of 2 cm.

Key Sequence:

AREA

Calculate the area of a circle with a radius of 2 cm.

Key Sequence:

THEOREM OF PYTHAGORAS

Calculate the length of x

Key Sequence:

21

• • • 12,57CHANGE

2 2π =

=

x x • • • 4π

)

• • • 12,57CHANGE

π ( x2x 2 • • • 4π

• • • 6,40CHANGE

√ (x2 =5 )+( 4) x2• • • 41

Be careful when

manipulating your

formulae!

5

4

x

40


Key Sequence:

DATA HANDLING

We can use our Sharp to calculate mean and median.

Consider the following data set: 50; 75; 75; 75; 80; 80; 95 and calculate the mean and median.

When calculating the mean and the median, the data values as well as the frequencies of the data values are important.

Let’s Try!1. Activate your stat mode

Key Sequence:

22

• • • 24,68CHANGE

√ -) =2 4x2 x2( (5 ) • • • 609

MODE 01

4

x

25

Q: Why was the

obtuse angle so

upset?

A: He knew he would

never be right.

Be careful when

manipulating your

formulae!


2. Enter your data values.

Key Sequence:

3. Our data has now been entered and we can search for the mean and median.

Key Sequence:

Scroll up and down to find the answers.

To get back to your data table (to edit any values)

Key Sequence:

To activate normal mode

Key Sequence:

The data remains stored in your stat mode. To delete the data, return to your data table and then clear.

Key Sequence:

ON/C DATA

HOME

MODE 01 2nd F CA

0ALPHA STAT

05 =

57 =3

08 =2

59 =

DATA

(x,y)’

(x,y)’

The default frequency is 1 – the key allows us to enter the corresponding frequencies for the data

values which are NOT 1.

(x,y)’

Symbol meanings = number of data values entered = mean / average value of data set

Med = median

-xn n 7

x 75.7142857

Med 75

-

38

© 2019 Student Edge 24 37

Most of your basics have been covered in the grade 8 section. Please refer back to these notes as you work through the various sections.

FINANCE

In grade 9 we begin to look at compound interest. Let’s get started.

A = P (1 + i)n

If you invest R5200.00 at a compound interest rate of 18% per annum, how much will your investment be worth after 3 years?

Before we place any interest rate into a financial formula, we need to convert the interest rate to a decimal.

Key Sequence:

Now we can continue substituting into our formulae.

A = 5200(1 + 0.18)3

Key Sequence:

A teacher’s salary is R4000.00. What was his salary 2 years ago if his salary increment was 7% per annum?

4000 = P (1 + 0.07)2

4 000(1 + 0.07)2

Key Sequence:

CALCULATING P

CALCULATING A

P =

If you are dealing with

different compounding periods please refer to the

Grade 10 finance section.

• • • 0,18CHANGE1 =0÷ 0

• • • 8543,77

• • • 3493,75

2nd F =x3

8 1

4 1a/b 70 0( 00 +0 . ) =x2

5 +( )0 .1 82 00 1

GRADE 9

SECTION 4


Tokollo invests R73 500.00 and after 9 years her investment was worth R101 000.00. What was the interest rate she received?

101 000 = 73 500(1 + i)9

101 00073 500

Key Sequence:

Remember that we always convert our interest rate to a decimal before substituting into our financial formulas – and so now we need to convert this decimal back to a percentage.

Key Sequence:

FUNCTIONS

Let’s use our Sharp to help us sketch straight line functions. Our Sharp generates co-ordinate points which we can then plot.

Sketch f(x) = 2x+1

1. Activate your table mode

Key Sequence:

2. It now requests that you enter your function

Key Sequence:

i = 9 – 1

• • • 0,035942nd F 3√ –x a/b 0

1

0

0

0

=

0 071

x

51

0

0 - =19

MODE 2

x + 1 =2 ALPHA

PLEASE NOTE: We ALWAYS use ALPHA X as the

X to represent our independent

variable.

CALCULATING i

• • • 3,59

36


3. It now allows us to enter a second function. We do not have a second straight-line function which we would like to plot and so we can simply skip this step. If given a second straight-line function you would now enter the second function following the key sequence in step 2.

Let’s skip this step.

Key Sequence:

4. We are now able to enter our x-value that we would like our function to start at. In order to get an accurate sketch, it is best to begin on the negative x-axis.

Key Sequence:

5. The x-step is the scale of your x-axis. A step of 1 is normally sufficient when sketching a straight-line function.

Key Sequence:

6. The table that can be seen shows the x co-ordinates with the corresponding y co-ordinates of your function, which can now be plotted. Use your scrolling keys to move through your table.

7. Once we have our co-ordinates to plot, we are able to return to step 1 in order to edit any input information which might have been incorrectly entered. Edit any information needed and press = to move through steps 2 – 6.

Key Sequence: ON/C

=

(-)

1

5

=

=

-5x

We want points to plot from -5 upwards

y

35


8. Your data remains stored in table mode. Please clear the data to allow for a new input function.

Key Sequence: REMEMBER

We press ‘home’ to return to

normal mode.

2nd F CA

Q: How can you

make seven even?

A: Simply remove

the “S”.

(0,1)

(-4,-7)

-1 10-5 6-3 8-7 4-9 2-2 9 -6 5-4 7-8 3-10 1

-2

-1

-3

-4

-5

-6

-7

6

7

5

4

3

2

1

34


F(X)-NOTATION

Find f (2) where f(x) = – x2 + 2x + 5

Key Sequence:

FUNCTIONS

We can use our Sharp to help us sketch functions.

Quadratic / Parabola Function

Sketch f (x) = x2 + 2

1. Activate table mode on your Sharp.

Key Sequence:

2. We now need to enter our function.

Key Sequence:

3. We do not have a second function we would like to enter – and so, just as with straight-line functions (Grade 9), we can skip this step.

Key Sequence:

4. We are now able to enter our x-value that we would like our function to start at. In order to get an accurate sketch, it is best to begin on the negative x axis.

Key Sequence:

• • • 5(-) 2x2 5 =2 2+ +( () )

MODE 2

x x2 + 2 =

=

(-) 5

ALPHA

Make use of bracketsduring substitution.

I’ll do algebra, I’ll do trig, and I’ll even

do statistics, but graphing is where I

draw the line!

GRADE 10

SECTION 4

PLEASE NOTE: We ALWAYS use ALPHA X as

the X to represent our independent variable.

=

© 2019 Student Edge32 29

5. The x-step is the scale of your x-axis. A step of 1 is normally sufficient when sketching as it gives us a range of co-ordinates to plot.

Key Sequence:

6. The table that can be seen shows the x co-ordinates with the corresponding y co-ordinates of your function, which can now be plotted. Use your scrolling keys to move through your table.

Final Sketch

7. Once we have our co-ordinates to plot, we are able to return to step 1 in order to edit any input information which might have been incorrectly entered. Edit any information needed and press = to move through steps 2 – 6.

Key Sequence:

1 =

-1-3 42-2-4 31

-2

-1

-3

-4

-5

5

4

3

2

1

Who said mathletes can’t

dance?!

ON/C

x ANS

-1 2

0 3

1 3


8. Your data remains stored in table mode. Please clear the data to allow for a new input function.

Key Sequence:

Hyperbola Function

Sketch f (x) = + 2

Key Sequence:

Look at the table which is now displayed. Using your up and down arrows move through the table.

Final Sketch

This indicates that x = 1 is an ASYMPTOTE

x ANS

-1 1

0 0

1 ..........

2x-1——

2nd F CA

2 5- + 2 = = 1(-)x =1 =ALPHA

-1-5 6-3 8-7 42-2-6 5-4 7-8 31

-2

-1

-3

-4

4(3,3)

(-1,1)

3

2

1

a/b

31


Exponential Function

Sketch f(x) = 3x

Key Sequence:


Final Sketch

Trigonometric Functions

X ANS

-1 0.333333

0 1

1 3

yx = (-) 5 = 1 =x =3 ALPHA

(1,3)

(0,1)

-360 -180 180

10

5

-5

-10

360˚ Degrees

0 90˚

1

cos x

tan x

x

-1

180˚ 270˚ 360˚

0 90˚

1

sin x

x

-1

180˚ 270˚ 360˚


We can use our Sharp to help us sketch trigonometric functions.The same process is followed however, it is important to note that our x-axis scale should be in steps of 45 or 90. This is due to the fact that our x-axis units are now degrees.

Sketch f(x) = sin x

Activate your table mode.

Key Sequence:

Enter the above function.

Key Sequence:


Final Sketch

X ANS

-90 -1

0 0

90 1

Let’s Try!

= (-) 3 46 50 ==x =SIN ALPHA

MODE 2

-360 -180 180

2

1

-1

-2

360


This indicates that x = -270 is an ASYMPTOTE

Clear your table.

Key Sequence:

Sketch y = 2 tan x

Enter the above function.

Key Sequence:


Final Sketch

X ANS

-360 0

-315 2

-270 ..............

= 3= (-) 46 50 ==xtan ALPHA2

(45,2)

(-45,-2)

2nd F CA

-360 -180 180 360

2

-2

-4

4


FINANCE

Compounding Periods

We are now introduced to questions where the interest is compounded.• Annually• Semi-Annually• Quarterly• Monthly

R200.00 is invested at 12% per annum compounded semi-annually. What will the investment be worth after 6 years?

Substitute your values into your financial formulae.

Key Sequence:

How much money must be invested at 12% per annum compounded quarterly to receive R239.10 after 4 years?

Substitute your values into your financial formulae and manipulate it to make P the subject.

Key Sequence:

0.122

2 x6A = 200 (1 + )

0.124

4x4(1 + )239.10 P =

• • • 402,44a/b0 0 1 0 . 1 )2( 2+ yx 2 6 =x2

• • • 149,0003 9 1 a/b ( 1 )+. . 1 2 4a/b0 yx 4 4 =x2

CALCULATING P

CALCULATING A


Calculate the compound interest rate if an investment of R149.00 yields R239.10 after 4 years with interest added quarterly.

Substitute your values into your financial formulae and manipulate it to make i the subject.

Key Sequence:

Remember that we always convert our interest rate to a decimal before substituting into our financial formulas – and so now we need to convert this decimal back to a percentage.

Key Sequence:

Catherine’s parents opened a savings account with R5000.00 at 9% compound interest when she was born. How old is she now if her saving

has grown to R23 586.00?

In Grade 10 we find the value of n by trial and improvement.

Substitute your values into your financial formulae and manipulate it to look like the following.

We now substitute different values for n until (1 + 0.09)n is the closest value to 4.7172

Key Sequence:

16i = 4 x ( -1)149239.10

= (1 + 0.09)n

500023586

4.7172 = (1 + 0.09)n

CALCULATING i

• • • 0,11990x ( 6 3 9 . -11 a/b 1 4 92 1 ) =4 2nd F √ –x

• • • 121 0 =x 0

Let’s Try!

• • • 1,4115

• • • 3,9703

• • • 4,3276

• • • 4,7171

• • • 5,1416

=1 +

BS

BS

BS

BS

.

6

1

1

1

9

=

=

=

) yx 40

1

BS

BS

BS

0

=

7

8

9

(

CALCULATING n

She is 18 years old


Timelines – Deposits and Withdrawals

Anusher invests R8000.00 for 8 years at an interest rate of 11% per annum compounded monthly. 5 years later a second amount of R5000.00 is deposited.

Calculate the final amount at the end of the investment period.

Timeline:


Key Sequence:

If we treat the R5000.00 as a withdrawal – the only change is that we subtract instead of add.

Key Sequence:

Changing Interest Rates

R6000.00 is invested for 9 years. During the first 6 years the interest rate is 6% per annum compounded quarterly. There after the rate is 11% per annum compounded

monthly. Calculate the final amount at the end of the investment period.

Timeline:

0.11 12

8x12A = 8000 (1 + ) + 5000 (1 + )0.11

12

3x12

0.11 12

8x12A = 8000 (1 + ) – 5000 (1 + )0.11

12

3x12

0.064

0.1112

• • • 26 154,43

• • • 12 265,64

1 20 0 ( + 0 . ) +1 10 a/b1 yx 58 01 02 (x 08

)+ 0 1 a/b 1 2 3. yx1 x 1 =21

1 20 0 ( + 0 . ) -1 10 a/b1 yx 58 01 02 (x 08

)+ 0 1 a/b 1 2 3. yx1 x 1 =21

T7T3 T5T1 T6T2 T4T0 T8

8000 5000

T7 T8T3 T5T1 T6T2 T4T0 T9

6000



Key Sequence:

TRIGONOMETRY

Let’s start off with the basics. We find ourselves in two situations.• Situation 1: We are given an angle and asked to calculate the ratio of sides

Evaluate sin 30˚

Key Sequence:

Evaluate cos 45˚

Key Sequence:

Evaluate tan 60˚

Key Sequence:

• Situation 2: We are given the ratio of sides and asked to calculate the angle

Given sin Ø = find the value of Ø

We use the sin–1 key to isolate Ø

Key Sequence:

Given tan Ø = –1 find the value of Ø

Key Sequence:

0.06 4

6x4A = (6000 (1 + ) ) 0.11

12

3x12x (1 + )

Ø = sin–1 ( ) Ø = 30˚

1 2

0 46 0 0 1 + 0 ) ). a/b0 6( yx x6 4x(

• • • 11 912,44)1 + . 1 a/b 1 2 30 yx1 x 1 =2(

6 0 ) =( • • •tan 3

4 5 ) =( • • •cos 22

3 0 ) =( • • •sin 12

12—

• • • 301 )( a/b 2 =2nd F sin-1

• • • -45(-)( 1 ) =2nd F tan-1


Special Angles

Diagrams are to be used when calculating special angles.

We can use our Sharp to double check our answers. Enter the trigonometric ratio and the corresponding angle into your Sharp.

45˚ is a special angle. If we would like to evaluate the ratio of sides, we can use our diagrams or alternatively our Sharp.

Key Sequence:

45º60º

30˚

1

1

1

3

22

UsingSharp

cos(45˚) = 22

Usingdiagram

cos(45˚) = 12

NOTE: Your Sharp rationalises your denominators.

22

12

x ( 22

=

(

4 5 ) =( • • •cos 22


Q: Why was the

obtuse angle so

upset?

A: He knew he would

never be right.

39

ANALYTICAL GEOMETRY

Distance

The points A(-1; -4) and B(2; 3) lie on a line. What is the distance between points A and B?

Key Sequence:

Gradient

The points A (-1; -4) and B (2; 3) lie on a line. What is the gradientbetween points A and B?

Key Sequence:

The points A (-1; 4) and B (-1; -3) lie on a line. What is the gradientbetween points A and B?

Key Sequence:

This means we have a vertical line. The gradient is UNDEFINED!

73

13

Do not forget about this fact!

dAB = (– 1 – 2)2 + (– 4 – 3)2

• • • 58

• • • 2

• • • ERROR 02

• • • 7,62CHANGE

• • •CHANGE

4 )(-)

(-)

(-)

(-)

(-)

(-)

=

(-) =

1

4

4

2

3

( )

x2

1

1

+

-

- 1

(

2

( )

x23-

-

- 3

-) =(

a/b

a/b

√ –

MAB = – 4 – 3– 1 – 2

MAB = 4 – ( – 3 )– 1 – ( – 1 )

MAB = 4 – ( – 3 )0

Don’t be afraidto use brackets


Midpoint

Find the midpoint between points A (-5; 6) and B (10; -2)

Key Sequence for x co-ordinate:

Key Sequence for x co-ordinate:

DATA HANDLINGThe 5 Number Summary

Calculate the 5 number summary for the following set of data:52; 61; 62; 65; 65; 82; 85; 91; 91

You learnt how to enter a data set in grade 8. Let’s enter the following data set.

Key Sequence:

Our data has now been entered and we can search for the 5 number summary.

Key Sequence:

MidAB = ( ; )6 + ( – 2 ) 2

– 5 + 10 2

( 2 ) 2 =+6 • • • 2(-)

+ 1 0 2 =5 • • •CHANGE(-) 52

2 =5

1 =6

5 =8

2 =6

2 =8

MODE 1 0

0STATALPHA

5 2 =6 (x,y)’

2 =19 DATA(x,y)

a/b

a/b

To clear data previously entered

Key Sequence: CA DATA2nd F


Use the up and down arrows to move through your table.

We can now use the above answers for further calculations.Ensure you are in normal mode.

Calculate the range of the data set.Range = xmax – xmin

Key Sequence:

Calculate the inter-quartile range of the data set.

Inter-quartile Range = Q3 – Q1

Key Sequence:

Are there any outliers in the above data set?

Q1 – 1.5(IQR)

Key Sequence:

Q3 + 1.5(IQR)

Key Sequence:

There are clearly no outliers.

5 = • • • 21,75CHANGE CHANGE6. 2 5 ( . 5 x- . )116

• • • 26,5

• • • 39

CHANGE CHANGE-

-

6

5

.

=

=1

2

58

1

8

9

= • • • 127,75CHANGE CHANGE5+ . ( . x 2 61 )588

xmin 52

Q1 61.5

Med 65

Q3 88

xmax 91

Q: Do you know

a statistics joke?

A: Yes, but it’s

mean.


QUADRATIC FORMULAE

The quadratic formula helps us solve any quadratic equation.

Solve the following 3x2 – 4x – 7 = 0 correct to two decimal places.

Key Sequence:

We now have our first solution. To obtain the second solution use your scrolling keys to change the + (plus) to a – (minus) in your formula.

Key Sequence:

Note: when we get an error on our Sharp it means that there are NO REAL roots for this equation. We therefore have no solution.

SURDSSimplifying SurdsYour Sharp automatically writes surds in their simplest form.

Write 32 in its simplest from

Key Sequence:

– b ± b2 – 4ac 2a

x =

4 -(-) ( 4 + ( ( 7(-) X2 (-) )) 4 )x x (-)3a/b √ –

• • • 2,33CHANGE

x 3 =2 • • • 2 13

• • •CHANGE 73

-BS = • • • –1

2 =3√ – • • • 4 2

a = 3 b = -4 c = -7

allows for two solutions for x

Method:32 16 x 2

4 2==

GRADE 11

SECTION 5


Always enter your

percentage second.

43

Adding and Subtracting like Surds

Calculate 2 + 5 2

Key Sequence:

Calculate 3 3 - 3

Key Sequence:

Note: Calculators are often not allowed when working with surds. Be smart and show your steps! DO NOT just enter the entire question on your calculator. Enter it surd by surd showing simplification.

Calculate 8 + 2First simplify your 8

Key Sequence:

Now enter your like terms

Key Sequence:

√ – 2 25 =√ – • • • 6 2

=√ – 8 • • • 2 2

- 3 =√ –

√ –

√ –3

2

3

2

• • • 2 3

For Example

How the steps in your

book SHOULD look

How the steps in your

book SHOULD NOT look

8 + 2

= 2 2 + 2

= 3 2

8 + 2

= 3 2

+

+ 2 =√ – • • • 3 2


Multiplying Surds

Calculate 2 3 x 5 5

Key Sequence:

Dividing Surds

Key Sequence:

Rationalising DenominatorsYour Sharp automatically rationalises denominators in all calculations. You can therefore enter any fraction with an irrational denominator and it will rationalise it.

Key Sequence:

FINANCE

Nominal and Effective Interest Rates

In grade 11 we are introduced to nominal and effective interest rates.

Note: always change your percentage to a decimal before substituting.

Convert an effective interest rate of 15.3% p.a. to an annual nominal interest rate compounded quarterly.

Substitute what you have been given into your formula, and manipulate it so that inom is the subject.

44

x =√ – √ –3 5 52 • • • 10 15

Calculate 105

Example: 23

5 =√ – √ –1 0 • • • 2a/b

2 =√ – 3 • • • a/b 2 33

inom = 4 ( 4 1.153 – 1)

ief f = (1 + ) – 1inomm

m

0.153 = (1 + ) – 1inomm

m

ieff = effective interest rateinom = nominal interest ratem = compounding period

Be rational Get real


Key Sequence:

Now we need to change our answer to a percentage!!!

Key Sequence:

Convert a nominal rate of 18% p.a. compounded monthly to an effective rate.

Key Sequence:

Now, we need to once again change our answer to a percentage.

Key Sequence:

TRIGONOMETRY

Sine Rule

Solve ∆ABC

We can easily find BB = 180˚ – 48˚ – 67˚ (Interior angles of a triangle)B = 65˚ To find sides a and c we apply the sine rule.

ief f = (1 + ) – 10.18

12

12

=b

sin B=

asin A

csin C

14sin 65˚

csin 67˚

= =a

sin 48˚

• • • 0,1444 4 . 1 5 -3( 1 1 ) =2nd F √ –x

• • • 0,1956( + 1. 1 8 ) yx1 21 -0 1 2 =

• • • 14,49x 01 0 =

• • • 19,56x 01 0 =

a/b

ˆˆˆ A

C

48˚

67˚B

c 14

a

A

CB

c b

a


Let’s begin with a

Key Sequence:

The same key strokes are followed to find the length of c.

Key Sequence:

Cosine Rule

Find the length of q.

Key Sequence:

ORKey Sequence:

a2 = b2 + c2 – 2bc cos A

14sin 65˚sin 48˚x 14

sin 65˚

=a

sin 48˚

a =

14sin 65˚sin 67˚x 14

sin 65˚

=c

sin 67˚

c =

• • • 11,48

• • • 14,22

a/b

a/b

4

6

1

1

8

7

4

4

)

)

6

6

(

(

(

(

x

x

5

5

)

)

=

=

sin

sin

sin

sin

P

RQ

10 q

15

A

CB

c b

a

q2 = (10)2 + (15)2 – 2(10)(15) cos 30˚

q = (10)2 + (15)2 – 2(10)(15) cos 30˚

√ – ( ) 1 0 2( 1 5 1x () (+ - 0 0x 35 )x1 ) = • • • 8,07x2 x2 cos

• • • 65,19( ) 1 0 2( 1 5 1x () (+ - 0 0x 35 )x1 ) =x2 x2 cos

• • • 8,07√ – =ANSALPHA

30˚


Area Rule

Find the area of ∆PQR

Key Sequence:

P

RQ

10 q

15

A

CB

c b

øa

PLAN(P+L)(A+N)PA+PN+LA+LN

PLAN(P+L)(A+N)PA+PN+LA+LN

your plan has

been foiled.

1 2

Area ∆ABC = ab sinC

1 2

Area ∆PQR = pr sinQ

1 2

Area ∆PQR = (10)(15) sin30º

30˚

• • • 37,5CHANGE

( )x x 1 0 1 5 01 3x =2a/b sin • • • 37 12

CHANGE • • •752

Squaring: the product of

an integer with itself.


DATA HANDLING

Standard Deviation

In grade 11 we deal with the concepts of variance and standard deviation. Once again this is made effortless using your Sharp.

Find the variance and standard deviation of the following data set,

Activate stat mode.

Key Sequence:

Note: If there is data present from a previous entry press the following to clear.

Key Sequence:

We now enter our data as discussed in grade 8.

Key Sequence:

x Frequency

4 9

5 11

6 15

7 19

8 12

STATALPHA

DATA

MODE

CA DATA2nd F

(x,y) 4

0

9 =

(x,y) 5 1 1 =

(x,y) 6 1 5 =

(x,y) 7 1 9 =

(x,y) 8 1 2 =

1 0


Scroll through the list to find the information needed.

Note: Take note of the differing symbols distinguishing between

Population Sample

Variance σ 2x s2x Standard Deviation σ x sx

n Number of input data 66

x Mean / Average 6.21212121

sx Standard deviation of a sample 1.30696538

s2x Variance of a sample 1.70815851

σ x Standard deviation of a population 1.29702634

σ 2x Variance of a population 1.68227732

∑x Sum of the data 410

∑x2 Sum of the data raised to the second power 2658

xmin Minimum data value 4

Q1 Quartile 1/25th Percentile 5

Med Median 6

Q3 Quartile 3/75th Percentile 7

xmax Maximum data value 8

12


LOGARITHMIC FUNCTIONS

Note: we can convert a logarithmic function to an exponential function.

When dealing with logarithmic functions we have to take into consideration certain restrictions.• x > 0, x must be positiveLet’s see what happens when we have a negative value for x

Calculate log(–3)

Key Sequence:

• a > 0 and a ≠ 0If we do not adhere to the above restrictions we will once again get an error.

Calculate log–24

Key Sequence:

It is important to understand the difference between the key and the key.

Find the value of log1010

Key Sequence: =

y = logax

ay = x

logaxlog

log ( (-) 3 ) = • • • ERROR 02

2nd F logax (-) 2 4 = • • • ERROR 02

2nd F logax 1 0 1 0 • • • 1

GRADE 12

SECTION 6

Q: Why was the math book upset?

A: It had so many problems it needed

to solve.


Find the value of log1010

Key Sequence:

When a log has a base of the value 10, we can simply use the key. This key automatically inserts a base value of 10.

Calculate log 2

Key Sequence:

Logs are extremely useful when solving exponential equations where the bases are NOT THE SAME.

Solve for n: 2n = 3

We begin by ‘logging both sides’ and then manipulating our formula following the log laws.

Key Sequence:

FINANCE

Solving for n

Now that we have covered logarithms, we can accurately solve for n when dealing with our financial formulas.

log ( 1 0 ) = • • • 1

log 2n = log 3

log 3

log 2n =

2nd F logax 1 4 2 = • • • – 12

14

log

a/b

3 2 = • • • 1,58a/b log log

I ate some pie…and it was delicious!

Our base is not a value of 10, therefore

we cannot simply use the log key.

-1 23 ∑ π


You inherit R55 000.00 and deposit the money into an investment account at an interest rate of 11.25% p.a. compounded monthly. After how many years and

months will the investment have grown to R180 000.00?Substitute the given information into your compound interest formula and manipulate it as follows:

Key Sequence:

The above answer is now in years. To calculate the number of months – subtract the whole number from your answer and multiply by 12.

Key Sequence:

Present Value AnnuityWe use our present value annuity formulae when regular fixed payments are made to repay a loan.

A monthly payment of R2000.00 is made for 20 years to repay a loan. If the interest rate is 8.5% p.a. compounded monthly, calculate the size of the loan.

Substitute all given information into your formula.

Key Sequence:

180 000 = 55 000 (1 + ) 0.112512

12n

= (1 + ) 0.112512

180 00055 000

12n

= 12n log (1 + ) 0.1125

12

log ( ) 180 00055 000

( 0 1 8 0 5 5 00 0 ) 1( 2+ 0 51 1.

• • • 127,058

• • • 10,58820118

1 2 )

÷

=

=21

0 0a/ba/b

a/blog log

• • • 0,588201184- 0 =1

• • • 7,058414217x 2 =1

x 1 – (1 + 1)-n

iP =

+ 02 0 0 1 - ( 5 )1 .0 0( a/b yx18 2a/b

0 5.0 a/b 18 2 • • • 230 461,68)( 2 x 2 )0 1 =(-)

P = 0.085

12

1 – (1 + )0.08512

–(20 x 12)2000

10 years and 7 months


What monthly payment is needed to pay off a 6-year loan of R180 000 at 11% p.a. compounded quarterly?

Substitute all given information into your formula, and manipulate it in order to make x the subject.

Key Sequence:

Future Value Annuity

We use the future value annuity formula when cash is set aside for a particular purpose so that the correct amount will be available when needed.

Determine the final amount saved if R500.00 is invested at the end of each month for 8 years at an interest rate of 5% p.a. compounded monthly.

Substitute all given information into your formula.

Key Sequence:

• • • 10 344,35( 6yX x ) )(-) 4 =

• • • 58870,260 50 2. 1 =

0 1 8 0 0 .0 1 4 1( -0 1a/ba/b

+ )4.( 101 a/b1

x

0 5 0 1 0 .( 5 1 yX2 ) ) ( 8 1 2x( 0a/ba/b

a/b - )1+

x (1 + i) n – 1

iF =

180 000 = 0.11

4

x 1 – (1 + )0.114

–(6 x 4)

F = 500 (1 + ) – 10.05

12

0.0512

(8 x 12)

x = 180 000 x 0.11

4

1 – (1 + ) 0.114

– (6 x 4)

Ensure you always use a PAIR of brackets. If you open a bracket and forget

to close it you will get an error.

Q: What do

you call friends

who love math?

A: AlgeBROS!!!!


What monthly installment is needed if an amount of R160 000.00 is wanted after 5 years, earning an interest rate of 8% p.a. compounded quarterly?

Substitute all given information into your formulae, and manipulate it in order to make x the subject.

Key Sequence:

DATA HANDLINGBivariate Data

We can use our Sharp to calculate the formula of the line of regression, otherwise known as the line of best fit.

Find the formula for the line of regression using the data stated below.

Activate state mode

Key Sequence:

Select the regression line option

Key Sequence:

Enter your bivariate data

Key Sequence:

• • • 6 585,07x –4 =1 )

5yX0 1 6 0 0 .0 8 4 (( 10 0a/ba/b . )48+ 00 a/bx

160 000 = 0.08

4

x (1 + ) – 10.084

5 x 4

x = 160 000 x 0.08

4

(1 + ) – 1 0.084

5 x 4

y = a + bx

x 2 4 9 10

y 5 7 11 15

MODE

DATA

(x,y) 2

1

1

5 =

(x,y) 4 7 =

(x,y) 9 1 1 =

(x,y) 10 1 5 =

To clear data previously entered

Key Sequence: CA DATA2nd F


We can now search for the answers from the data entered.

Key Sequence:

Scroll through the table to find the value of a and b.

Finally, y = 2.587 + 1.106x

Correlation Co-Efficient

Using the data values previously entered we can find the correlation co-efficient. This is the value represented by r.

Follow the exact steps from the previous example.

r = 0.96

PROBABILITY

Factorials

The factorial of n can be written as:

Calculate 4!

Key Sequence:

^

STATALPHA 1

a 2.586592179

b 1.106145251

r 0.963346831

n! = n x (n – 1) x (n – 2)…

4! = 4 x 3 x 2 x 1

2nd F • • • 244 =n!

–1.0 – 0.5 0.0 + 0.5 + 1.0

WeakWeak

Correlation Co-EfficientShows Strength & Direction of Correlation

Negative Correlation Positive Correlation

Find y when x = 2>

2nd F2 y'ON/C

• • • 4,80


nCr =n!

r! (n – r)!

=15!

6! (15 – 6)!

nPr =n!

(n – r)!

2nd F • • • 40 3208 =n!

2nd F nCr • • • 5 005=1 5 6

2nd F 2nd F 2nd Fn! n! n! • • • 5 005=1 5 6 1( 5a/b )– 6

In how many different ways can 8 people stand in a queue?

Key Sequence:

Combinations

The number of ways of selecting r objects from a set of n objects where order is NOT IMPORTANT!

In how many different ways can we select 6 learners from a class of 15?

We can substitute into our formula.

Key Sequence:

OR

We can simply use the combination key to do the exact same calculation.

Key Sequence:

Permutations

The number of ways of selecting r objects from a set of n objects where order IS IMPORTANT!

• Scenario 1: All objects in the set are different.

How many 4-digit pin codes can be formed with digits 0-9 where no digit is repeated?


= 10!(10 – 4)!

= n!r1! x r2! x…

= 6!3! x 2!

2nd F nPr • • • 5 040=1 0 4

2nd F 2nd Fn! n! • • • 5040=1 0 1( 0a/b )– 4

2nd F 2nd F2nd Fn! n!n! • • • 60=x6 23a/b

We can substitute into our formula.

Key Sequence:

We can simply use the permutation key to do the exact same calculation.

Key Sequence:

• Scenario 2: Some objects in the set are identical.

Consider n objects of which r1 are identical objects of 1 type, r2 are identical objects of a 2nd type, r3 of a 3rd type, and so on.

Then,

Find the number of permutations of the letters in the word BANANA.

Key Sequence:


I SURVIVEDMATHEMATICS…Thanks to MY

SHARP!


The Sharp Logo, Sharp RoBoHoN and affiliated trademarks are the property of Sharp Corporation.


get sharp. be sharp. · changing interest rates 36 trigonometry 37 special angles 38 analytical...

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