Transcript Navigating Your Casio Calculator

Navigating Your Casio Calculator
Dr J Frost ([email protected])
www.drfrostmaths.com
Last modified: 26th November 2013
Click a button.
For details on statistical
calculations, press the
‘Mode’ button.
You didn’t press a button for which information is
provided. Click the button below to go back.
< Return
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.
2 STATS
> 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.
3 TABLE
> Go
Allows you to generate a table of values for a
given function, like the table on the right.
𝑦 = 𝑥 2 − 2𝑥 + 1
𝑥
1
2
3
𝑦
0
1
4
4 VERIF
Allows you to verify whether an equation or inequality is true.
Special Buttons
< Return
SHIFT
If you press a button after pressing SHIFT, it will use
the operation indicated by the gold text above that
button.
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.
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)
< Return
On
On
< Return
Engineers are yet to discover the true nature of this
button, which has eluded mankind for centuries.
But some mathematicians have theorised that
pressing this button turns the calculator on.
Multi-Statements
:
x3
< Return
The semi-colon allows you to write multiple different
expressions, and evaluate them one at a time.
[2] [+] [3]
[ALPHA] [:]
[=]
 5
[=]
 28
[4] [x] [7]
The Absolute Function
Abs
< Return
The absolute/modulus function makes a negative
number positive, and a positive number remains
positive.
5 =5
−7 = 7
On its own it has limited use, but is useful if you want
to plot a table of values, e.g. for
1− 𝑥 2
𝑓 𝑥 =
3
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.
The Reciprocal Function
x-1
From Laws of Indices, you may have learnt that
1
−1
𝑥 =
𝑥
This is known as the ‘reciprocal’ of x.
[6] [x-1] = 1/6
[1/7] [x-1] = 7
< Return
The Factorial Function
x!
x-1
< Return
5! = 5 × 4 × 3 × 2 × 1
3! = 3 × 2 × 1
In general, 𝑥! is the product of 1 to 𝑥.
𝑥! gives the number of ways of arranging 𝑥 objects in
a line. The factorial function tends to also crop up in
Calculus and Number Theory.
The Logarithm Function
log
log
< Return
Just as the ‘square root’ function is the opposite of
‘squaring’, log2 for example is the opposite of finding 2 to
the power of something.
log2 32 = 5, because 25 = 32
log3 81 = 4, because 34 = 81
Use 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.
Fractions


< Return
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:
3.5 + 4.7
0.3
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.
Root Functions
3√
< Return
Use these buttons to get various roots of a number.
e.g.
√
√
25 = 5
3
8=2
4
81 = 3
Recurring Decimals
⎕
< Return
This button allows you to enter recurring decimals.
Your calculator will convert them to fractions.
Recall that 0.354 = 0.3454545 …
Your calculator will convert this to
39
.
110
Powers
x
< Return
Examples:
34 = 3 × 3 × 3 × 3 = 81
25 = 2 × 2 × 2 × 2 × 2 = 32
72 = 7 × 7 = 49
Natural Logarithm
ln
< Return
This finds loge 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.
𝑑
1
ln 𝑥 =
𝑑𝑥
𝑥
Euler’s Constant
e
e
< Return
Euler’s Constant e is equal to 2.71828...
This first button allows you to do e to some power.
Using e1 allows you to see the value of e.
e can also be found above the [× 10𝑥 ] button by using
[ALPHA].
e arises in many different places in maths, notably
𝑑
calculus, where
𝑒𝑥 = 𝑒𝑥
𝑑𝑥
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.
Degrees, Minutes, Seconds
° ′ ′′
< Return
When you have some angle or time as a decimal,
press this key to convert it to degrees, minutes (a 60th
of a degree) and seconds (a 60th of a minute).
4.75 =
[°′ ′′]
→ 4.75
→ 4°45′ 0′′
or…
4.75 [° ′ ′′]
→ 4°45′ 0′′
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.
Factorise
FACT
< Return
This finds the prime factorisation of a number.
You need to enter the number first, then press =.
THEN use the FACT button.
[120] [=] [FACT]

23 x 3 x 5
Hyperbolic Functions
hyp
< Return
After pressing [hyp], use either the sin, cos or tan
button (or inverse sin/cos/tan) to get their
‘hyperbolic’ equivalents: sinh, cosh, tanh.
(cosh 𝜃 , sinh 𝜃) is the parametric form of a hyperbola with Cartesian
equation 𝑥 2 − 𝑦 2 = 1, just as cos 𝜃 , sin 𝜃 is the parametric form of a
circle with equation 𝑥 2 + 𝑦 2 = 1.
These are defined as:
𝑒 𝑥 − 𝑒 −𝑥
sinh 𝑥 =
2
𝑥
𝑒 + 𝑒 −𝑥
cosh 𝑥 =
2
sinh 𝑥
tanh 𝑥 =
cosh 𝑥
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 𝑦 = cosh 𝑥.
Trigonometric Functions
< Return
cos
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.
tan
For example, to solve the following problems...
sin
sin-1
x
4
3
y
60°
x = 3sin60
3
y = cos-1 (3/4)
Brackets
(
)
< Return
Brackets are hugely handy in ensuring operations in
your expression are evaluated in a certain order.
Recall that in ‘BIDMAS’, ‘Brackets’ comes first.
1+1×2

1+1 ×2 
3
4
(because the x is done first)
(using the brackets ensures + is done first)
Storing values in variables
STO
< Return
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.
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] [=]
Engineering Notation

ENG
< Return
Engineering notation is similar to standard form,
except the power of 10 can only be a multiple of 3.
Percentages
%
< Return
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)
Comma
,
< Return
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.
Converting between decimal/surd/fraction
𝑆↔𝐷
< Return
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. 𝜋) into decimal form, and
back again.
[√] [8] []
[𝑆 ↔ 𝐷]
[4] [÷] [9]
[𝑆 ↔ 𝐷]
 2𝜋 2
 8.88576...
4
9

0.4444...
Improper Fractions and Mixed Numbers
𝑏
𝑑
𝑎 ↔
𝑐
𝑐
This allows you to convert between improper
fractions and mixed numbers.
3
2
[24] [÷] [16]

𝑏
[𝑎
𝑐
1
1
2
↔
𝑑
]
𝑐
< Return
Independent Memory
< Return
M+
The independent memory is useful if you’re trying to
keep a running total of calculations.
M-
Once entering an expression, press [M+] instead of [=] to add
your result from the running total.
To subtract the result, use [M-]
M
To display the currently stored total, use [RCL] [M]
(Your value will be preserved when the calculator is turned off.
See the [CLR] button to see how to wipe the value.)
Clear Memory
CLR
< Return
This allows you to delete the values you’ve stored for
variables and in independent memory.
Permutation Function
nPr
< Return
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.
Choose Function
nCr
< Return
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
Polar and Rectangular (Catersian) Coords
Pol
Rec
< Return
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.
y
In Cartesian coordinates:
(√3,1)
(√3,1)
30
In Polar coordinates:
(2, 30)
x
To convert Rectangular to Polar:
[POL] [√][3] [,] [1] [=]
To convert Polar to Rectangular:
[REC] [2] [,] [30] [=]
Statistic
STAT
< Return
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.
Rounding
Rnd
< Return
Rounds a number according to the current accuracy
set on he calculator.
Random Numbers
RAN#
< Return
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
RanInt
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
To get a list of random integers, just put your calculator in TABLE
mode, then use the function 𝑓 𝑋 = 𝑅𝑎𝑛𝐼𝑛𝑡 1,10
Pi
< Return

Pi is typically used in calculations to do with circles.
It is a constant with the value 3.1415...
3
Circumference
Using 𝐶 = 2𝜋𝑟:
[2] [x] [3] [x] [] [=]
Area
Using 𝐴 = 𝜋𝑟 2
[] [x] [3] [x2] [=]
Standard Form
x10x
< Return
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] [× 𝟏𝟎𝒙 ] [5] [=]

320000
The Answer Button
ANS
< Return
This incredibly handy button allows you to use your
previous answer in a subsequent calculation.
[3] [x] [2] [=]
[ANS] [+] [1]
 6
 7
At A Level, it is incredibly useful for iterative formulas:
1
Suppose 𝑥𝑛 = 2 + , and you start with 𝑥1 = 3.
𝑥𝑛
[3] [=]
[2] [+] [1] [/] [ANS]
[=]
[=]




3
2.333...
2.428...
2.411...
As you can see, we can keep hitting the = key to perform further
iterations.
Stats Mode
< Return
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:
Single Variable (X)
Two Variables (X, Y)
Use when you have just one variable, e.g.
height, weight, shoe size.
Use when you have a scatter diagram, e.g.
hours revised against test score.
1 - VAR
A + BX
For your single variable, calculates things like
mean, standard deviation, variance, etc.
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).
y = a + bx
y = a + bx + cx2
> Click to see how to enter your data.
_ + CX2
Assumes y has a quadratic
relationship to x, i.e. Your
points roughly fit onto a
parabola.
ln X
Assumes your data follows
the model y = a ln X + b
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
‘commit’ your table, so that you can now calculate statistics based on it.
> Click to see how you now calculate
statistics based on your table.
AC
Stats Mode – Calculating Statistics
< Back
< Home
Presuming you have just pressed the [AC] button while in Stats mode:
|STAT|
1
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, x2 (useful
when calculating
variance), y, xy, etc.
Var
Reg
MinMax
Allows you to calculate
the mean of x or y, the
number of items n, and
the population or
sample standard
deviations.
Will 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).
Unsurprisingly,
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 [=]
> Practice
< Home
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)
Click to+Reveal
𝑦 = 480.396
14.314𝑥
The Product Moment Correlation Coefficient.
(Use Reg  r. -1 means perfect negative correlation, 0
means no correlation, and 1 mean perfect positive
correlation)
𝑟 = 0.926
Click
to Reveal
The average number of orcs killed
in battle.
(Use Var  𝒚.)
𝑦 = 1317.75
Click
to Reveal
Table Mode
< Back
In some exam questions you’re asked to calculate a table of values for a given function:
f(x) = x2 + 1/2
x
-1
-0.5
0
0.5
f(x)
1.5
0.75
0.5
0.75
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.
> Next
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 [=]
x
-1
-0.5
0
0.5
1
f(x)
1.5
0.75
0.5
0.75 1.5
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.
< Return
Secret Menu!
7
< Return
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.