Navigating - PowerPoint Presentation

Slide1

Navigating Your Silver Casio Calculator

Dr J Frost (jfrost@tiffin.kingston.sch.uk)www.drfrostmaths.com

Last modified:

30

th

August 2015Slide2

For details on statistical calculations, matrices, solving polynomials/simultaneous equations and complex numbers, press the ‘Mode’ button.

Click a button.Slide3

You didn’t press a button for which information is provided. Click the button below to go back.

< ReturnSlide4

Mode Menu

< Return

1

COMP

Puts the calculator in normal ‘computation’ mode.

You would need to do this if you were previously using stats/table mode and want to revert back to regular calculations.

3 STATS

> Go

7 TABLE

> Go

Allows you to calculate various statistics based on a table of data, e.g. mean, variance, standard deviation, the equation of the line of best fit, strength of correlation, etc.

Allows you to generate a table of values for a given function (useful for sketching).

2 CMPLX

> Go

Allows you to add, multiply, times and divide complex numbers, as well as convert between Cartesian and modulus-argument (i.e. polar) form.

4 BASE-N

> Go

Allows you to convert a number from decimal (e.g. 35) to binary (100011) and other bases.

5 EQN

> Go

Solve quadratics, cubics, and up to 3 simultaneous linear equations.6 MATRIX> GoAdds, subtracts and multiplies matrices, as well as find the determinant and inverse.8 VECTORAdds, subtracts and finds the dot product of vectors. Calculator fairly pointless here!Slide5

Special Buttons

SHIFT

If you press a button after pressing SHIFT, it will use the operation indicated by the

gold text

above that button.

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ALPHA

If you press a button after pressing ALPHA, it will use the operation or letter indicated by the

red text

above that button.

The letter

X

is particularly useful for entering a function. Click the ‘MODE’ button then ‘TABLE’ for more information.Slide6

Arrow Buttons

You can use the up and down arrow buttons to retrieve previous calculations (a bit like your internet browser’s ‘Back’ and ‘Forward buttons!)

You’ll need the left and right button for example when entering a fraction, and want to switch between numerator and denominator.

The arrow buttons are also used when navigating a table (e.g. in Statistics mode)

< ReturnSlide7

On

On

Engineers are yet to discover the true nature of this button, which has eluded mankind for centuries.

B

ut some mathematicians have theorised that pressing this button turns the calculator on.

< ReturnSlide8

Multi-Statements

:

The semi-colon allows you to write multiple different expressions, and evaluate them one at a time.

[2] [+] [3] [ALPHA] [:] [4] [x] [7]

[=]

 5

[=]  28

< ReturnSlide9

The Absolute Function

Abs

The absolute/modulus function makes a negative number positive, and a positive number remains positive.

On its own it has limited use, but is useful if you want to plot a table of values, e.g. for

It’s particularly useful for C3/C4 at A Level, if you want to check your sketch for a function (involving the modulus function) is correct by generating a table of values.

You can also use it to solve equations involving

:

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“Solve

”

[Abs] [ALPHA] [x] [ALPHA] [CALC] [(] [ALPHA] [x] [-] [2] [)] [x

2

] [SOLVE]

 Slide10

The Reciprocal Function

x

-1

From Laws of Indices, you may have learnt that

This is known as the ‘reciprocal’ of x.

[6] [x

-1

] = 1/6

[1/7] [x

-1

] = 7

< Return

It can also be used to find the inverse of matrices

(see [MODE] -> [MATRIX])Slide11

The Factorial Function

In general,

is the product of 1 to

.

gives the number of ways of arranging

objects in a line. The factorial function tends to also crop up in Calculus and Number Theory.

< Return

x

-1

x

!Slide12

The Logarithm Function

log





Just as the ‘square root’ function is the opposite of ‘squaring’, log

2

for example is the opposite of finding 2 to the power of something.

log

2

32 = 5, because 2

5

= 32

l

og

3

81 = 4, because 3

4 = 81Use the arrow keys to move between the boxes after pressing the button.When you use the second log button with no ‘base’, it uses base 10.

< Return

logSlide13

Fractions





When you have more complicated calculations to do on a calculator that involve a division, it’s ‘safer’ to use a fraction because you don’t have to worry about BIDMAS.

For example, to evaluate:

You can enter this exactly as it appears using the fraction button, using the arrow buttons to move up and down. This avoids the problem of 4.7/0.3 being evaluated first.

Using SHIFT on this button allow you to have

mixed numbers

.

< ReturnSlide14

Root Functions

3

√



Use these buttons to get various roots of a number.

e.g.

< Return



√



√

Slide15

Recurring Decimals

This button allows you to enter recurring decimals. Your calculator will convert them to fractions.

Recall that

Your calculator will convert this to

.

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Powers

Examples:

< Return

You might wonder why we need the

,

and

buttons given that we can use the generic

one?

This is because in MATRIX and COMPLEX modes, we can’t use any arbitrary power: we can only use the individual

,

and

buttons, e.g. the first for inverting matrices.

 Slide17

Natural Logarithm

ln

This finds log

e

of a number, where e is Euler’s Constant (2.71...)

See the log button for more information.

This is hugely useful in Integration and Differentiation, which you learn about at A Level.

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Euler’s Constant

e



Euler’s Constant e is equal to 2.71828...

This first button allows you to do e to some power. Using e

1

allows you to see the value of e.

e can also be found above the [

] button by using [ALPHA].

e arises in many different places in maths, notably calculus, where

If the probability of winning the lottery is 1 in 14 million, and you buy 14 million random tickets, the probability that you don’t win the lottery at all is roughly 1 in e.

< Return

eSlide19

Degrees, Minutes, Seconds

When you have some

angle

or

time

as a decimal, press this key to convert it to degrees, minutes (a 60

th

of a degree) and seconds (a 60

th

of a minute).

< Return

or…

This makes sense as 4.75 hours is 4 hours and 45 minutes.

Fun fact:

Whereas the ‘decimal’ system is base 10 (i.e. each digit can have one of 10 values: 0 to 9), the ‘

sexagesimal

’ system is base 60. Subdivisions of hours and degrees are in

sexagesimal. Slide20

Factorise

FACT

This finds the prime factorisation of a number.

You need to enter the number first, then press =. THEN use the FACT button.

< Return

[120] [=] [FACT]

 2

3

x 3 x 5Slide21

After pressing [hyp], use either the sin, cos

or tan button (or inverse sin/cos/tan) to get their ‘hyperbolic’ equivalents: sinh, cosh,

tanh.

is the parametric form of a hyperbola with Cartesian equation

, just as

is the parametric form of a circle with equation

.

These are defined as:

These are useful as solutions to certain differential equations. For example, if you hang a rope between two points so that it forms a ‘u’ shape (known as a

caternary

), its shape can be given by

.

Hyperbolic Functions

hyp

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Trigonometric Functions

sin

Trigonometry allows you to find missing sides and angles on triangle. For right-angled triangles, sin,

cos

and tan give the ratio of different pairs of sides.

For example, to solve the following problems...

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cos

tan

s

in

-1

x

3

x = 3sin60

60

°

3

4

y

= cos

-1 (3/4) ySlide23

Brackets

(

Brackets are hugely handy in ensuring operations in your expression are evaluated in a certain order. Recall that in ‘BIDMAS’, ‘Brackets’ comes first.

< Return

)

 3

(because the x is done first)

 4

(using the brackets ensures + is done first)

 Slide24

Storing values in variables

STO

In algebra we use variables to represent values. We can use the letters

A

,

B

,

C

,

D

,

E

,

F

,

X, Y on the calculator for this purpose.

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Store store 3 + 5 in memory as ‘A’:(Note, don’t press the ALPHA

button after pressing STO)[3] [+] [5] [STO] [A]To evaluate 10A:[10] [x] [A] [=]

You may also wish to investigate the ‘CALC’ button.Slide25

Engineering Notation

ENG

Engineering notation is similar to standard form, except the power of 10 can only be a multiple of 3.

< Return

Slide26

Percentages

%

The % button is of fairly limited usefulness. It converts a percentage into its equivalent decimal (by dividing by 100).

[90] [x] [40] [%] = 36

(this found 40% of 90)

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Comma

,

The comma is used for used in generating random integers, and converting between rectangular and polar coordinates.

Click the RANDINT, REC or POL buttons for more information.

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Converting between decimal/surd/fraction

This very useful button converts your number between different forms. S stands for ‘Surd’ and D for ‘Decimal’.

The button also converts expressions involving fractions and constants (e.g.

) into decimal form, and back again.

[

√

] [8] [



]



[

]

 8.88576...

[4] [

] [9] 

[

] 0.4444...

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Improper Fractions and Mixed Numbers

This allows you to convert between improper fractions and mixed numbers.

[24] [

] [16]



[

] 

< ReturnSlide30

Independent Memory

M+

The independent memory is useful if you’re trying to keep a running total of calculations.

Once entering an expression, press

[M+]

instead of

[=]

to add your result from the running total.

To subtract the result, use

[M-]

To display the currently stored total, use

[RCL] [M]

< Return

M-

M

(Your value will be preserved when the calculator is turned off.

See the [CLR] button to see how to wipe the value.)Slide31

Clear Memory

CLR

This allows you to delete the values you’ve stored for variables and in independent memory.

< ReturnSlide32

Permutation Function

nPr

This function used in ‘Combinatorics’ (the study of arrangements of items and structures), allows us to find the number of ways of picking r objects from n, and putting them in a line.

Example:

We have 5 cards with the letters A, B, C, D, E.

We want to put 3 in a line. This gives words such as ABC, AEC, DEA, etc. How many possibilities are there?

[5] [

nPr

] [3]

 60

This function tends not to be used very often – the ‘choose’ function (

nCr

) is much more common.

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Choose Function

nCr

This function used in ‘Combinatorics’ (the study of arrangements of items and structures), allows us to find the number of ways of choosing r objects from n, such that the order of the items doesn’t matter.

Examples:

“How many different possible lottery tickets are there?”

You choose 6 numbers from 49. So:

[49] [

nCr

] [6] [=]



13983816

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Polar and Rectangular (

Catersian

)

Coords

Pol

Cartesian coordinates are represented by x and y values (and any further dimensions).

Polar coordinates however are represented by the distance of the origin, and the angle anticlockwise from the x-axis.

< Return

Rec

(√3,1)

2

30



x

y

In Cartesian coordinates:

(√3,1)

In Polar coordinates:

(

2

,

30)To convert Rectangular to Polar:[POL] [√][3] [,] [1] [=]To convert Polar to Rectangular:[REC] [2] [,] [30] [=]Slide35

Statistic

STAT

Allows you to calculate a statistic (such as mean, variance, correlation strength) based on a data set you’ve entered. Click on the MODE button from the calculator display and then ‘Stats’ for more information.

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Rounding

Rnd

Rounds a number according to the current accuracy set on he calculator.

< ReturnSlide37

Random Numbers

RAN#

This will give you a three-digit random number between 0 and 1.

To find a random number between 0 and 5:

[RAND] [

] [5] [=]

 3.78

Gives you a random integer (whole number) between a and b. Since this is in red, you need to use the ALPHA button to access it.

Random integer between 1 and 6:

[ALPHA] [

RanInt

] [1] [,] [6] [=]  4

< Return

RanInt

To get a list of random integers, just put your calculator in TABLE mode, then use the function

 Slide38

Pi



Pi is typically used in calculations to do with circles.

It is a constant with the value 3.1415...

< Return

3

Using

:

[2] [x] [3] [x] [



] [=]

Using

[



] [x] [3] [x

2

] [=]

Circumference

AreaSlide39

Standard Form

x10

x

Standard Form allows us to represent large or small numbers without having to use lots of digits.

Your calculator will automatically put your number in standard form if it can’t fit your number on the screen.

[3.2] [

] [5] [=]

 320000

< ReturnSlide40

The Answer Button

ANS

This incredibly handy button allows you to use your previous answer in a subsequent calculation.

[3] [x] [2] [=]

 6

[ANS] [+] [1]  7

At A Level, it is incredibly useful for iterative formulas:

Suppose

, and you start with

.

[3] [=]  3

[2] [+] [1] [/] [ANS]  2.333...

[=]  2.428...

[=]  2.411...

As you can see, we can keep hitting the = key to perform further iterations.

< ReturnSlide41

Stats Mode

< Return

1 - VAR

> Click to see how to enter your data.

This mode allows you to calculate various statistics based on a table of data, e.g. mean, variance, standard deviation, the equation of the line of best fit, strength of correlation, etc. You’ll be presented with various options:

For your single variable, calculates things like mean, standard deviation, variance, etc.

Single Variable (X)

Use when you have just one variable, e.g. height, weight, shoe size.

Two Variables (X, Y)

Use when you have a scatter diagram, e.g. hours revised against test score.

A + BX

Assumes your data points roughly follow a straight line, i.e. have a

linear

relationship. e.g. will find a

straight line of best fit

for you. Use if you’re trying to find the

Product Moment Correlation Coefficient

(which assumes a linear relationship).

_ + CX

2

Assumes y has a

quadratic relationship to x, i.e. Your points roughly fit onto a parabola.ln XAssumes your data follows the model y = a ln X + by = a + bxy = a + bx + cx2Slide42

Stats Mode – Entering Data

< Back

A table should appear.

Enter each X value in your data, pressing [=] after each one. If you have two variables, your Y value will temporarily be set to 0.

If you have a second variable, use the arrow keys to move to the top of the Y column. Now enter your Y values using [=] again.

Once you’ve finished entering your data, press the [AC] button to

go back to calculation entry mode

,

so that you can now calculate statistics based on

your table.

You can modify your table again using SHIFT -> [1] and selecting ‘Data’.

> Click to see how you now calculate statistics based on your table.

AC

Important Note: If you want a frequency column (which is NOT the same as having a second variable

!), first press SHIFT -> SETUP, press down, select STAT and turn frequency on. You will not need to do this setup again even after your calculator is switched off.

 Slide43

Stats Mode – Calculating Statistics

< Back

Presuming you have just pressed the [AC] button while in Stats mode:

< Home

1

|STAT|

Use the |STAT| button (SHIFT and 1). This will present a number of options...

Sum

Finds the sum of the values of your variables. e.g.



x

,



x

2

(useful when calculating variance),



y

,



xy, etc.VarAllows you to calculate the mean of x or y, the number of items n, and the population or sample standard deviations.RegWill find the a, b (and c) in your line or best fit, whether a + bx (if a straight line) or otherwise.Will also find your correlation coefficient r (known as the PMCC for the linear case).MinMaxUnsurprisingly, will find the maximum or minimum X or Y value.Once you’ve chosen a statistic to use, it’ll appear in your calculation area. You can always combine multiple together. Once done, press [=]< Home> PracticeDistrAllows z-table calculations.> GoSlide44

Stats Mode – Exercise

< Back

< Home

Use your calculator to directly calculate the following statistics.

Age

of dwarf (x)

Orcs

killed in battle (y)

46

1423

57

1203

26

697

105

1948

A formula for estimating the number of

orcs

killed (y) using the age of the dwarf (x).

(Use

Reg  a to find the y-intercept and Reg

 b to find the gradient of your line of best fit)The Product Moment Correlation Coefficient.(Use Reg  r. -1 means perfect negative correlation, 0 means no correlation, and 1 mean perfect positive correlation)The average number of orcs killed in battle.(Use Var  .)    Click to RevealClick to RevealClick to RevealSlide45

Stats Mode –

z-table calculations

< Back

< Home

You should have just selected ‘

Distr

’ in the STAT menu.

To calculate

, select ‘P(‘ in this menu.

Enter your z-value (e.g. 1.2) then close the bracket and press [=].

Voila!

Example

: Given that IQ is normally distributed with mean 100 and standard deviation 15, determine the probability that a randomly chosen person’s IQ is less than 137.5.

SHIFT -> [1] (stat)

Distr

-> P(

Enter: “P(2.5)” (as 137.5 is two and a half lots of 15 above 100)

Press [=]. Displayed result is 0.88493.Slide46

Table Mode

< Back

In some exam questions you’re asked to calculate a table of values for a given function:

x

-1

-0.5

0

0.5

f(x)

1.5

0.75

0.5

0.75

f(x) = x

2

+ 1/2

Your calculator can do this for you. Once in table mode, your calculator display should look like this:

Now input some expression in terms of X. You can use [ALPHA]



[X] to use X in your expression.

> NextSlide47

Table Mode

< Back

Now press [=]. You will be asked for the ‘Start’ number.

In our table, the first value of x is -1. Type in -1 and press [=]

You will now be asked for the ‘End’ number. In our table above, the last value of x is 1.

Type 1 then press [=].

Finally you’re asked for the

step size

. This is how much x is increasing by each time. In our table, it’s 0.5.

Once you press equals, you’ll be presented with a nice looking table.

You can use the arrow keys to scroll through it.

x

-1

-0.5

0

0.5

1

f(x)

1.50.750.50.75

1.5

< ReturnSlide48

Secret Menu!

7

Hold [SHIFT] and [7] and then press [ON].

Now press [9], then [SHIFT] 5 times.

After waiting for the messages to display, press [AC]. You can change the screen contrast, and pressing [AC] again activates a button test – pressing each button (in the correct order!) displays a different integer.

< ReturnSlide49

Solving Equations

SOLVE

Your calculator can solve any equation.

< Return

However

note its limitations:

It effectively uses ‘trial and error’ to get an answer, so will not give you an ‘exact’ answer (e.g. if you solved

, it wouldn’t be able to give the ‘exact’ answer

). But it is still useful to verify exact answers you have found yourself.

It will only find one solution

. If you’re finding roots to quadratic or cubic equations, use the MODE -> EQN mode instead, which will give ALL solutions.

Sometimes it fails to find a solution despite one existing

.

Example: Solve

when in degrees.

Use the ALPHA button to get

and

into your equation. Note that the equals symbol is above the ‘CALC’ button.

[ALPHA] [X] [x

2] [ALPHA] [=] [SIN] [ALPHA] [X]Then press the normal [=]Give the calculator a starting value to try, then press =.The calculator should give an answer of 0.017453…The indicates the difference between the LHS and RHS of your equation. If this is 0, then the solution is very accurate.  Slide50

Easy substitution

CALC

This allows you to more easily substitute values into an algebraic expression.

< Return

Suppose you wanted to evaluate

when

and

Method 1

: Store values into variables first.

[-1] [STO] [A]

(stores to A)

[2] [STO] [B]

[3] [ALPHA] [A] + [2] [ALPHA] [B] [=]

Result of 1 given.

Method 2

: Using CALC button.

[3] [ALPHA] [A] + [2] [ALPHA] [B

] [CALC]

Calculator will ask for value of each variable. Press = after entering each.Slide51

Complex Numbers

< Return

You can multiply, add, subtract and divide complex numbers, and raise to any power up to 3. You can also convert between Cartesian and modulus-argument form.

The value

can be found above the ENG key: use the ALPHA button to get it. You will only be able to use

while in the complex mode.

Calculate

[(] [1] + [4] [

i

] [)] [1] [-] [

i

] [)]

Calculate

[

[1] [+] [

i

] [=]

Put

into modulus-argument (polar) form.

You can use [SHIFT] [2] to get various functions involving complex numbers.

[1] [+] [i] [=] [SHIFT 2] [] [=]Convert to Cartesian form (i.e. ) will give you  Find [SHIFT 2] [arg] which gives 53.13Alternatively, [3] [+] [4] [i] [=][SHIFT 2] [] [=]which gives  Slide52

Changing Base

< Return

Our normal number system is ‘base 10’ (i.e. the ‘decimal number system’) because each digit has two possible values (0 to 9). However there are other common bases, base 2 (binary), base 16 (hexadecimal, used for example in colour codes in web design), and base 8 (octal).

Your calculator can also do binary operators ‘and’ (

) ‘or’ (

) useful for Computing.

In general:

Press one of the [DEC] [HEX] [BIN] or [OCT] buttons for the number system you wish to convert from.

Enter number and press [=]

Press

Press

one of the [DEC] [HEX] [BIN] or [OCT] buttons

to convert.

“Convert 15 (in decimal) to binary.”

Ensure calculator display says ‘Dec’ (if not, press [x

2

] button which has ‘Dec’ above it).

[15] [=][BIN] (located above log)“Convert 74AC to decimal.”[HEX] [7] [4] [A] [C] [=] [DEC]

(note that you don’t need to press alpha to get either A or C)“Calculate

” (note that

in binary by comparing each digit)[DEC] [13] [SHIFT 3] [and] [10] [=] Slide53

Unit Conversions

< Return

Your calculator can convert between different units.

These are all listed on your calculator case.

Convert 13km/h to m/s

[13] [SHIFT] [CONV] [19] [=]

will give 3.6111 m/sSlide54

Scientific Constants

< Return

Your calculator has a number of scientific constants, mostly used in Physics. None are used in A Level or GCSE maths exams.

These are all listed on your calculator case.

Exam Warning:

The value of

(gravitational acceleration on Earth) is very accurate on your calculator, where M1 exams require you to use the less accurate 9.8ms

-2

.

Calculate 3g

[3] [SHIFT] [CONST] [35] [=]

will give 29.41995 Slide55

Matrices

< Return

Matrices are rectangular grids of numbers which can be used to represent linear transformations (such as rotations and enlargements) as well as to solve simultaneous equations.

Your calculator can add, subtract and multiply matrices, as well as find the determinant or inverse of a matrix.

Entering matrices

Similarly to the STATS mode, you enter your data first, before pressing [AC] to advance to the calculation mode.

When you first enter

matrix mode, you will be asked to define ‘

MatA

’. Once entering your data, press [AC], and your matrix will be saved to the variable ‘

MatA

’. You can similarly define further matrices by using [SHIFT 4], selecting ‘

Data

’ then say ‘

MatB

’, entering your matrix before pressing [AC] again to return to the calculation mode.

Using [SHIFT 4] -> [MatA] will use the matrix MatA in a calculation, NOT set its data: use ‘Data’ for this. Note that to modify the size of an existing matrix variable, use [SHIFT 4] [Dim].

Entering

[MODE] [6]. Follow instructions to enter

MatA

. Use [=] after entering each number. Then press [AC] to return to calculation mode.

[SHIFT 4] [Select 2:Data] Enter matrix B and press [AC].Calculate .[SHIFT 4] [MatA] [] [SHIFT 4] [MatB] [=] Calculate [SHIFT 4] [det] [SHIFT 4] [MatA] [)] [=]Calculate [SHIFT 4] [MatA] [] [=]Calculate [SHIFT 4] [MatA] [] [=]Note that the button sadly does not work, so you can’t find any arbitrary power of a matrix, only and using the specific power buttons. Slide56

Summation

< Return

You can evaluate expressions of the form

For example

.

Very useful for FP1 at A Level!

“Determine

”

Press

button (using Shift).

Use the directional buttons to move between the parts of the expression.

In order to enter

(note your variable needs to be

), use [ALPHA] [X].

 Slide57

Differentiation and Integration

< Return

Note first that your calculator can’t do algebraic differentiation or integration, that is, it wouldn’t be able to determine that

.

However, it can find the gradient at a particular point on the curve, or do

definite

integration (i.e. the area under a curve).

“Determine the gradient of

at the point

”

Press [

].

Enter function

(using [ALPHA] [X] to get

). Set value of

to 4 using directional arrows. Answer is 47.75.

“Calculate the area under the curve

between

and

.”Press [ and use arrows to enter (using [ALPHA] [X] to get )Answer is 0.386.Important note: Your calculator uses the trapezium rule with tiny strips in order to get the answer. It thus does not give exact results. Use only to verify your answer in exams. Slide58

Solving Polynomials/Simultaneous

Eqns

< Return

While the ‘SOLVE’ button could allow you to solve a quadratic equation, it would only give a single solution and not give an exact answer. The [MODE]

[EQN] mode however overcomes these problems. It can also find roots when they are complex numbers (involving

).

(Note however it still can’t express roots of

cubics

exactly)

“Solve

”

[MODE] [EQN] [

]

[1] [=] [-1] [=] [3] [=] [-4] [=]

Use down arrows to scroll through solutions.

“Solve the simultaneous equations:

[MODE] [EQN] [

]

[2] [=] [-1] [=] [4] [=]

[3] [=] [2] [=] [5] [=]Use arrow keys to scroll between and solution.Note that you can also solve 3 simultaneous equations involving 3 unknowns.