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Preface

the scientific calculator has evolved from being a computational device for scientists and engineers to becoming an important educational tool

powerful and flexible environment for students and their teachers to explore mathematical ideas and relationships

together with advice from experienced teachers,

have culminated in the advanced scientific calculator,

- the CASIO fx-991 ES PLUS,

with substantial mathematical capabilities and extensive use of natural displays of mathematical notation

This publication comprises a series of modules to help make best use of the opportunities for mathematics education afforded by these developments

The focus of the modules is on the use of the calculator in the development of students’ understanding of mathematical concepts and relationships,

as an integral part of the development of mathematical meaning for the students

The calculator is not only a device to be used to undertake or to check computations,

once the mathematics has been understood

The mathematics involved in the modules spans a wide range from the early years of secondary school to the early undergraduate years,

and we leave it to the reader to decide which modules suit their purposes

we are confident that the mathematical ideas included in the modules will be of interest to mathematics teachers and their students across international boundaries

The modules are intended for use by both students and teachers

focusing on calculator skills relevant to the mathematics associated with the module

In addition,

a set of exploratory Activities is provided for each module,

to illustrate some of the ways in which the calculator can be used to explore mathematical ideas through the use of the calculator

these are not intended to be exhaustive,

and we expect that teachers will develop further activities of these kinds to suit their students

as well as some advice about the classroom use of the activities (including answers where appropriate)

Permission is given for the reproduction of any of the materials for educational purposes

We are grateful to CASIO for supporting the development of these materials,

and appreciate in particular the assistance of Mr Yoshino throughout the developmental process

We were also pleased to receive feedback from mathematics teachers in several countries during the writing,

which helped to shape the materials

We hope that users of these materials enjoy working with the calculator as much as we have enjoyed developing the materials and we wish both teachers and their students a productive engagement with mathematics through the use of the calculator

Table of Contents

Table of Contents ································································································································ 1 Module 1: Introduction to the calculator························································································ 5 Entering and editing commands·································································································· 5 Mathematical commands ············································································································ 6 Recalling commands ··················································································································· 8 Scientific and engineering notation····························································································· 9 Calculator modes······················································································································· 10 SET UP ····································································································································· 11 Memories ·································································································································· 14 Initialising the calculator··········································································································· 15 Exercises and Notes for teachers ······························································································ 16 Module 2: Representing numbers ································································································· 18 Representing decimals ·············································································································· 18 Representing fractions··············································································································· 19 Representing percentages·········································································································· 20 Recurring decimals ··················································································································· 21 Powers······································································································································· 22 Scientific notation ····················································································································· 24 Roots ········································································································································· 25 Reciprocals································································································································ 26 Exercises,

Activities and Notes for teachers············································································· 27 Module 3: Functions ······················································································································· 30 Evaluating expressions and functions ······················································································· 30 Table of values ·························································································································· 31 Linear and quadratic functions·································································································· 32 Cubic functions ························································································································· 34 Reciprocal functions ················································································································· 35 Maximum and minimum values ······························································································· 36 Intersection of two graphs········································································································· 37 Exercises,

Activities and Notes for teachers············································································· 39

Module 4: Equations······················································································································· 42 Equations and tables·················································································································· 42 Automatic solving ····················································································································· 43 Solving quadratic equations ······································································································ 44 Systems of linear equations······································································································· 45 Using the solver ························································································································ 47 Exercises,

Activities and Notes for teachers············································································· 50 Module 5: Trigonometry ················································································································ 53 Trigonometry and right triangles ······························································································ 53 Tables of values ························································································································ 55 Exact values ······························································································································ 56 Radian measure ························································································································· 57 Gradian measure ······················································································································· 57 Solving triangles with the Sine Rule························································································· 58 Solving triangles with the Cosine Rule····················································································· 59 The Pythagorean Identity ·········································································································· 60 Coordinate systems ··················································································································· 61 Exercises,

Activities and Notes for teachers············································································· 62 Module 6: Exponential and logarithmic functions······································································· 65 Exponents and roots ·················································································································· 65 Exploring exponential functions ······························································································· 66 Using exponential models········································································································· 68 Logarithms ································································································································ 69 Properties of logarithms ············································································································ 70 Logarithms to other bases ········································································································· 71 e and natural logarithms ··········································································································· 73 Exercises,

Activities and Notes for teachers············································································· 75 Module 7: Matrices ························································································································· 78 Defining matrices······················································································································ 78 Matrix arithmetic······················································································································· 79 Matrix inversion························································································································ 80 Transformation matrices ··········································································································· 82 Matrices and equations·············································································································· 83 Exercises,

Activities and Notes for teachers············································································· 84

Module 8: Vectors ··························································································································· 87 Representing vectors ················································································································· 87 Vector magnitude and direction································································································ 88 Vector arithmetic······················································································································· 90 An example from sailing··········································································································· 92 Dot product ······························································································································· 93 Three-dimensional vectors········································································································ 94 Cross product ···························································································································· 95 Exercises,

Activities and Notes for teachers············································································· 96 Module 9: Further numbers··········································································································· 99 Scientific constants ··················································································································· 99 Measurement conversions······································································································· 100 Numbers to other bases ··········································································································· 100 Binary logical operations ········································································································ 102 Complex numbers ··················································································································· 103 Argand diagrams ····················································································································· 104 Polar form ······························································································································· 105 Powers and roots of complex numbers ··················································································· 105 Exercises,

Activities and Notes for teachers··········································································· 107 Module 10: Univariate statistics ·································································································· 110 Getting started with statistics ·································································································· 110 Entering,

editing and checking data························································································ 110 Retrieving statistics ················································································································· 112 Determining normal probabilities ··························································································· 113 Frequency data ························································································································ 115 Inferential statistics ················································································································· 117 Exercises,

Activities and Notes for teachers··········································································· 119 Module 11: Bivariate statistics····································································································· 122 Getting started with bivariate statistics ··················································································· 122 Entering,

editing and checking data························································································ 122 Retrieving statistics ················································································································· 123 Using a linear model ··············································································································· 124 Other regression models·········································································································· 126 An exponential model ············································································································· 129 A note about curve fitting ······································································································· 130 Exercises,

Activities and Notes for teachers··········································································· 131

Module 12: Probability················································································································· 134 Probability of an event ············································································································ 134 Simulating events···················································································································· 134 Simulating integers ················································································································· 136 Combinatorics ························································································································· 137 The Normal probability distribution ······················································································· 139 Exercises,

Activities and Notes for teachers··········································································· 142 Module 13: Recursion,

sequences and series ·············································································· 145 Recursion ································································································································ 145 Clever counting ······················································································································· 146 Recursion and multiplication ·································································································· 147 Sequences································································································································ 148 Arithmetic sequences ·············································································································· 150 Geometric sequences··············································································································· 151 Series······································································································································· 153 Exercises,

Activities and Notes for teachers··········································································· 154 Module 14: Calculus ····················································································································· 157 Continuity and discontinuity··································································································· 157 Exploring the gradient of a curve···························································································· 158 The derivative of a function ···································································································· 159 Properties of derivatives·········································································································· 160 Two special derivatives··········································································································· 161 Exploring limits······················································································································· 162 Integration as area under a curve ···························································································· 164 Exercises,

Activities and Notes for teachers··········································································· 167

Barry Kissane & Marian Kemp

Module 1 Introduction to the calculator The CASIO fx-991 ES PLUS calculator has many capabilities helpful for doing and learning mathematics

In this module,

the general operations of the calculator will be explained and illustrated,

to help you to become an efficient user of the calculator

Entering and editing commands We will start with some computations

tap w1 to enter COMPutation mode

- as shown below

even before you enter any calculations

The D'symbol shows you that the calculator assumes that angles are measured in degrees

The Math symbol shows you that the calculator has been set to accept and display calculations in natural mathematical notation

- as we will explain below

An alternative is to tap q before tapping =,

which will produce a decimal result immediately

Notice that no unnecessary decimal places are shown: the result is shown as 5

- 4 and not 5

which students sometimes obtain with hand methods of calculation

You should use the z key to enter negative numbers,

- as in the screen above

it is still acceptable for entry,

- as the screens below show

As you can see,

the calculator automatically shows arrows on the display when a command is longer than the display

If you need to check or edit what has already been entered,

you can move backwards and forwards with the two cursor keys

! and $ (These keys are on opposite sides of the large oval REPLAY key at the top of the keyboard

- it will jump to the left end
- similarly,
- if you tap

! when the cursor is at the left end of the display,

- it will jump to the right end

it is not necessary to return the cursor to the end of the display before tapping =

although the clear arrow indicates that there is more to be seen

you can erase it and start again using the C key or you can edit it using the o key

You can then add another character by entering it from the keyboard

Characters can also be inserted using qo,

but it is generally not necessary to do this

Try this for yourself by entering the above command and then editing it to replace the 8 with a 3 before you tap =

Mathematical commands Many special mathematical operations are available on the calculator

so only a brief look at a few of the keyboard commands is provided here

you can usually enter a mathematical command in the same way in which you would write it,

as the calculator uses natural display

In some cases,

the keyboard command is the first one you need to use,

- while in other cases,

a command is entered after the number concerned

The square root key s'is discussed in more detail in the next module

q= was used to get a numerical approximation in the form of a decimal

Notice that the sine command j has been completed with a right parenthesis

although this is not strictly necessary here,

as the calculator will compute the value without it

It is a good practice to close parentheses,

rather than trust the calculator to do it for you

the command gives an approximation to sin 52o

- q and c,

and is represented on the calculator keyboard with Abs (written above the c'key)

ignoring the direction or sign of the difference

Tapping the d'key after entering a number will give the square of the number

- in the example shown above,
- 123 x 123 = 15 129

as shown to find the fourth power of 137 above

which is 12 x 11 x 10 x … x 1

This is the number of different orders in which twelve things can be arranged in line: 479 001 600

and so is discussed in detail in the Probability module

In general,

when using natural display mode,

you should enter these commands in the calculator in the same way in which you would write them by hand

The first example above shows a fraction being entered

using the cursor keys such as R and E to move between these,

or you can start by entering 26 and then tap the a key

The second example above shows the use of the calculator to find the logarithm to base 2 of 32: that is,

the power of 2 that is needed to obtain 32

Tap the i key first and then enter both 2 and 32 as shown,

- using the $ and
- ! keys to move between these

represented in mathematics by 52C5

then use the nCr command with qP and then enter the 5

The result in this case shows that there are 2 598 960 different five-card hands available from a complete deck of 52 playing cards

Calculations of these kinds are discussed in more detail in Module 12

Module 1: Introduction to the calculator

When using mathematical commands in calculations,

it is often necessary to use the cursor key $ to exit from a command before continuing

study the following two screens carefully

In the first case,

- after tapping s,

we entered 9+16 and then tapped =

- after entering s9,

we tapped $ to move the cursor out of the square root before tapping +s16 to complete the command and then = to get the result

- using fractions:

$ was used to exit the fraction before adding 4

it is always possible to use parentheses to clarify meanings in a mathematical expression,

but it is not always necessary

- in the three screens below,

the parentheses are necessary in the first case,

- but not in the second,

as the third screen makes clear

it is a good idea to develop expertise in constructing expressions without them,

- when possible

Recalling commands You may have noticed a small upward arrow (next to Math) at the top of your calculator display

When you are at the top of the list,

the arrow points downwards to show this

both up and down arrows will show,

to indicate that you can recall commands in either direction

These three possibilities are shown below

If you move the cursor key to a previous command,

you can enter the command again by tapping =,

or can edit it first (using o) and then tap =

without having to enter each complete command again

there is an even easier approach

! key and then edit the command directly

For example,

the following screen shows an estimate of almost 113 million for the 2020 population of the Philippines,

which had a Learning Mathematics with ES PLUS Series Scientific Calculator

Barry Kissane & Marian Kemp

population of 92 337 852 in the 2010 census,

and a population growth rate of 2% per annum

The easiest way to obtain an estimate for later years,

assuming the annual population growth rate stays the same,

- is to tap

! and edit the command to change the exponent of 10 to a different number each time

The screens below show the results for 2030 and 2040

The very high growth rate of the Philippines in 2010 will lead to a population of more than 167 million in 2040

The same calculator process could be used to predict the population if the growth rate was assumed to be reduced drastically from 2% to 1%,

- as shown below,

where the number of years as well as the growth rate have both been edited

if the growth rate were to be reduced to 1%,

a figure around 10 million fewer than predicted for a growth rate of 2%

Successive predictions of these kinds can be made efficiently in this way,

without needing to enter long and complicated expressions more then once

or change modes (as described below) but will not be erased when you tap the C key,

so it is wise to keep the calculator in the same mode and switched on if you think it likely that you will need some of the same sorts of calculations repeatedly

Scientific and engineering notation Scientific notation When numbers become too large or too small to fit the screen,

they will automatically be described in scientific notation,

which involves a number between 1 and 10 and a power of 10

which is described later in this module

the screen below shows two powers of 2 that require scientific notation to be expressed

which does not fit on the screen,

so it has been approximated using scientific notation

the second result has been approximated from 0

- 000000000931322574615479… to fit the screen

Start with the number between 1 and 10,

tap the K key and then immediately enter the power of 10

For example,

the average distance from the Earth to the Sun is 1

- 495978875 x 108 km,

which can be entered in scientific notation as shown on the screen below

Notice that the exponent of 8 is not shown as raised on the screen,

although it is interpreted by the calculator as a power

notice that he calculator does not regard this number as large enough to require scientific notation,

and so it is represented as a number,

indicating that the sun is on average about 149 597 887

- 5 km from the earth – an average distance of almost 150 million kilometres

if you use the K key to enter a number in scientific notation incorrectly (i

using a number that is not between 1 and 10),

the calculator will represent it correctly,

- as the screen below shows

Engineering notation Engineering notation is a different way of interpreting large and small numbers,

using scientific notation with powers of 10 that are multiples of 3

since units often have different names with such powers

For example,

a distance of 56 789 m can be interpreted as 56

- 789 km or as 56 789 000 mm,
- as shown on the screen below,

by converting numbers through successively tapping the b key

The number itself is not changed by these steps,

but its representation is changed to make it easier to interpret

we have used the calculator only for computations

the calculator can be used to explore many other aspects of mathematics,

which are accessed in various modes

To see the choices,

- tap the w key,
- to get the screen shown below

For now,

note the following brief overview of the other modes

Barry Kissane & Marian Kemp

- used in advanced mathematics

An example of a complex number is i = -1

the number i can be entered into the calculator and used for calculations via qb,

but this command will have no effect in other modes

special complex number operations are also available with q2(CMPLX)

both univariate and bivariate,

which are dealt with in the two respective Statistics modules

In this mode,

- q1 (STAT) provides various statistics calculations

Mode 4: Base N mode allows you to undertake computations in different number bases as well as the usual decimal number base

Both the calculator keyboard and the q3 (BASE) menu provide suitable commands for converting numbers between binary,

hexadecimal and decimal number bases

These are especially important for computer science,

as these bases are commonly used in computers

In particular,

both quadratic and cubic equations can be solved,

as well as systems of either two or three linear equations

the matrix menu in q4 (MATRIX) provides access to various matrix operations

which is useful for various purposes including sketching graphs and solving equations

Mode 8: Vector mode allows you to define and use up to three vectors with dimensions 2 or 3

this is indicated in the display

For example,

the two screens below show the calculator in Statistics and Matrix modes respectively

but you will not have the benefits of natural display notation on the screen,

so it is better to use Computation mode if you intend to do some calculations

we suggest that you keep your calculator in COMP mode

Notice that the calculator screen memory is cleared whenever you shift modes

SET UP We suggest that you now change back to Computation mode by tapping w1

In any mode,

the calculator can be set up in various ways by accessing T (via qw)

When you do this,

you will notice that there are two screens in the SET UP menu,

and you can move from one to the other using the R and E cursor keys

Here are the two screens:

Display format The calculator display can be set up to either natural display (Math) mode or single line (Line) mode by tapping 1 or 2 respectively

and it is usually better to use this

exact results will not usually be shown and the symbols will look a little different

It will also be more difficult to enter commands

these two screens show the same information,

the first in Math mode and the second in Line mode:

it is slightly more difficult to enter the fractions in Line mode,

as the numbers need to be entered in precisely the same order as they are written

you will be presented with the choice of Output mode

people like results of some calculations in Math mode to be represented as decimal approximations rather than exact numbers (although this is always possible using q= or the n key)

If you prefer to do this,

you should choose the Line out (LineO) format for results after choosing Math Input and Output (MathIO),

- as shown on the screen below

the screens below are both in Math mode,

but the first shows the result as an exact number with Math output and the second as an approximate number with Line output:

- radians or gradians,

and the choice made in T is shown in the screen display with a small D,

Your choices can always be over-ridden in practice using qM (i

which is also explained in the Module 4

Most people leave their calculator in degrees if they are generally concerned with practical problems or radians if they are generally concerned with theoretical problems

there are a few choices for the way that numbers are displayed as decimals

- select Scientific notation,

It can sometimes be a useful idea to choose Fix or Sci (e

to ensure that all results are given in similar ways,

especially if all results are money values),

but we think it is generally Learning Mathematics with ES PLUS Series Scientific Calculator

best to choose Normal decimal formats,

allowing the calculator to display as many decimal places as are appropriate

When Normal is chosen,

there are two choices available,

- called Norm1 and Norm2

These are almost the same,

except that using Norm1 will result in scientific notation being used routinely for small numbers before Norm2 will do so

two screens below show the same calculation as a decimal after selecting Norm1 and Norm2 respectively,

and using q= to force a decimal result

but you should decide this for yourself,

as it is mostly a matter of personal preference and also depends on the kinds of calculations you generally wish to complete

Here is the same number (134÷5) represented in the three formats Fix,

Sci and Norm respectively

Notice that both Fix and Sci are shown in the display when they have been set up as the chosen format

you need also to select the number of decimal places to be used (five are shown in the first screen above)

you need also to select the number of digits to be displayed (five are shown in the screen above,

so there are four decimal places showing and one digit to the left of the decimal point)

Fraction format The second Set Up screen shown below shows a choice of two ways of giving fraction results: as mixed fractions (using 1 ab/c) or proper fractions (using 2 d/c)

As shown in the next module,

results can easily be converted with N from one of these to the other (via qn),

so the decision is not very important

the same calculation has been completed in each of these two formats above

Decimal point display You can select 5 Disp to choose between a dot or a comma for a decimal point in the calculator display

Here are the two choices:

Contrast You can select 6CONT to adjust the contrast of the screen to suit your lighting conditions

Hold down,

- or tap repeatedly,

! or $ cursor keys until the contrast is suitable

but this is also a personal preference

This is convenient for recording values that you wish to use several times or for intermediate results

Both variable memories (labelled A to F as well as X and Y) and an independent memory (labelled M) are available

- tap qJ(STO)

tap the memory key for the variable concerned,

shown with pink letters above the keys on the keyboard

the memory key for B is x and that for X is )

The screens below show the process of storing a value of 7 into memory B

Notice that neither the Q key nor the = key is used here

- with a present value of 7

- tap the Q key,
- followed by the variable key

Variables are used on the calculator in the same way that they are in algebra,

- as shown below,

after storing a value of 8 to memory A

To change the value of a memory variable,

you need to store a different number into the memory,

as storing replaces any existing value

but it is not necessary to do so,

since storing a number replaces the existing number

although you can use it as a variable memory if you wish

- using m or qm (M-)

while the third screen shows the result being recalled

Notice that whenever M contains a non-zero number,

the screen display shows an M to alert you to this

which recalls the most recent calculator result

You might have seen this appearing when doing a succession of calculations

the first screen below shows the calculator being used to find 7

When +5

- 1= is then pressed,

the calculator assumes that the value of 5

- 1 is to be added to the previous result,
- which it refers to as Ans,

since there is no number before the + sign

(Ans was not entered by the user

When a previous result is not to be used immediately,

- as it is in the above case,

then the Ans memory can be recalled with M,

as shown below to find 265 – (7

- 3) after first calculating the value in parentheses:

- where it is especially useful

while it is not necessary to initialise the calculator before use,

this is the easiest way to reset a number of settings at once

After turning the calculator on with the W key,

tap q and 9 to show the Clear menu,

shown in the first screen below

The middle screen above shows the resulting message,

while the third screen shows that the default settings involve Math Set Up and Degrees for angle measures

As the screen above shows,

you can choose to clear only the Set Up or the memories,

- if you wish

© 2013 CASIO COMPUTER CO

You should get a result of 1137

- changing the 760 to 76,

and check that the resulting sum is now 453

Find cos 52o

The hypotenuse of a right triangle with shorter sides 7 and 11 can be found by calculating 7 2 + 112

22 14 ÷

Use the calculator to evaluate

- there are nC2 handshakes

Evaluate 38

which is the number of different orders in which the students in the previous question could line up outside their classroom

Find the absolute value of 3

4 – 7

which is represented in standard mathematical notation as 3

Use the calculator to evaluate 2 + 3

the population of the Philippines was 92 337 852 in the 2010 census

use the calculator to find out approximately when the population will reach 150 million

- (Hint: To do this,

enter a command and edit it successively until you get the desired result

Change the Mode of the calculator to use natural display but to give answers always as decimals in Line mode

4 x 1017

B and C the values of 7,

- 8 and 9 respectively

but do not write down the result

Notes for teachers This module is important for new users of the calculator,

as it deals with many aspects of calculator use that are assumed (and so are not repeated) in other modules

The text of the module is intended to be read by students and will h