Casio Classpad - Haese Mathematics

Transcription

Casio Classpad
Replacement calculator screens for the Core material
Chapter 1: Univariate data
Page 30
Using the calculator to find the mean, median and
mode
Use the calculator to find the mean and median of the following data :
2, 3, 5, 4, 3, 6, 5, 7, 3, 8, 1, 7, 5, 5, 9
Tap on the Statistics icon I in the applications menu.
Use list 1; check that the cursor is in the first position
of list 1.
Type 2 into the first position; this value will appear at
the bottom of the screen as [ 1] = 2.
Key E to move to the next position in the list.
Type the remaining data values into list 1 keying
E after each entry.
Note: There are 15 data values in the set so the
screen shows all the data is entered in list 1.
To find the descriptive statistics for the data
Tap Calc in the menu bar to get you into the menus for
finding descriptive statistics.
We are dealing with only one variable
so tap the option One-Variable
The Set Calculation screen allows you to select the
list that contains the relevant data.
Tap OK
All the available descriptive statistics for this variable
appear on the screen :On this screen the first statistic, • ,• is the mean.
The mean of the data is 4.867 (to 3 decimal places)
The second statistic: ¦ x = 73 means that the sum of all
the data values as 73; the Greek symbol 6 meaning
‘sum of’.
The next three statistics we will consider in section 1E.
‘n = 15’ refers to the fact that there are 15 data values in the
set.
Scroll down to access other statistics
Med = 5 means the median is 5
Mode = 5 gives the mode for this data set
The other statistics on this part of the screen
are the statistics of the five-number summary
which is covered in section 1F
1
Page 34
Using the calculator to find the range and inter-quartile range:
Key the data into a list. The data does not have to be ordered.
Tap Calc in the menu bar to get you into the
menus for
finding descriptive statistics.
list that contains the relevant data.
Tap OK
The screens at right show all the statistics for the
data. Scrolling down will reveal the lower part of the
screen.
The range is maxX – minX = 14 – 3 = 11
The IQR = Q3 – Q1
=9–5
=4
Page 38
Using the calculator to find the standard deviation
Key the data into a list.
Tap Calc in the menu bar to get you into
the menus for finding descriptive statistics.
The Set Calculation screen allows you to
select the list that contains the relevant data.
Tap OK
2
The sample standard deviation that we use is
The sample standard deviation is 2.5166.....
Note: The variance is not given on the screen. The variance can be
found by squaring the standard deviation.
Page 38
Standard deviation for grouped data
Key the data into the lists using list 1 for the data values and list 2 for the
frequency.
Tap Calc in the menu bar to get you into the
menus for finding descriptive statistics.
lists that contains the data (list 1) and the frequency
(list 2)
Tap OK
The sample standard deviation that we use is
The sample mean is 3.38 and the sample standard
deviation is 1.3536...
3
Page 47
Using the graphing calculator to find descriptive statistics and construct a
boxplot.
Using the data from the example above: 2, 3, 5, 4, 3, 6, 5, 7, 3, 8, 1, 7,
5, 5, 9
Key the data into a list.
T
Tap on
Tap on SetGraph in the menu bar and tap Setting...
to access the Set StatGraphs screen
Turn Draw On
The type of graph is a MedBox
The data is in list 1
Tick the Show Outliers box.
Tap Set to get back to the lists screen
Tap the graph icon y to display the boxplot in
the bottom half of the screen
.
Tap Analysis in the menu bar and select Trace
then use the cursor keys to see the values of the
five number summary displayed on the graph.
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Page 50
Example 20
Using the calculator:
Page 51
Using the calculator to draw the boxplot in Example 20.
Tap on SetGraph and tap Setting...
Turn Draw on
Tap Set to get back to the lists screen
Tap the graph icon y to display the boxplot in
the bottom half of the screen
Tap Analysis in the menu bar and choose Trace then the cursor keys to see
the values of the five number summary displayed on the graph.
.
Page 55
Using the random number generator on the calculator:
Go to the Main menu Jand press the
Keyboard k .
Access the catalogue by tapping on cat.
Locate rand(
Tap twice for rand( to appear on the screen.
Type in the range of integers that we are
considering 1,45) and then tap EXE
Other random numbers in the same range can be
obtained by tapping EXE repeatedly.
5
To obtain a list of six random numbers in the range 1 to 45 inclusive
i. Locate randList in the catalogue. Tap twice to
for randList( to appear on the screen.
ii. Type in 6,1,45) ; the number of random numbers
required and the range of numbers, all separated by
commas. Close the bracket and press EXE
A list of six random numbers in the range 1 to 45 will
be generated. Other sets of random numbers in the
same range can be found by tapping repeatedly EXE
Page 56
Example 21
Core Chapter 2 : Bivariate data
Page 63
Using the calculator to graph parallel boxplots.
Enter the data into the lists:
Set-up the first boxplot:
Turn Draw on
Set-up the second boxplot:
Tap on the 2 for the second graph
The data for the second graph is in list 2.
Similarly set-up the third graph ; the data for the third graph is in list 3.
After setting-up all three graphs tap on Set
6
Tap on the graph icon y to produce the parallel
boxplots.
Tap the Resize option at the bottom of the screen to get
a full screen graph.
Page 69
Using the calculator to construct a scatterplot
Enter the data into the lists; the independent variable (height) in list 1 and the
dependent variable (weight) in list 2
Set-up the graph as shown, tapping Set to get back
to the lists screen.
Tap on the graph icon to produce the graph.
Tapping Analysis, then Trace and then using the
cursor keys allows you to identify the points.
Page 75
Using the calculator to find Pearson’s correlation coefficient
Enter
Enter the data into the lists; the independent variable in list 1 and the dependent
variable in list 2
Produce a scatterplot for the data; it will reveal any
errors made in entering the data and any outliers. It
will also indicate whether the data is linear.
The scatterplot indicates strong, positive, linear
correlation between the variables.
7
Tap on Calc and then on Linear Reg
This means that we are fitting a linear model or linear regression of the
form y = ax + b to the data. Regression will be discussed in greater detail in
Chapter 3
Complete the Set Calculation screen as shown,
tapping OK when finished. The appropriate
Stat Calculation screen will appear.
The linear regression screen for this data is shown
at right. Pearson’s correlation coefficient r is
0.9130....for this example.
This r value indicates a strong, positive correlation
as indicated by the scatterplot.
Page 78
Calculation of the coefficient of determination.
r2 is found on the linear regression screen of the calculator.
r2 = 0.8336....
Alternatively, if the value of r is known, then this can
simply be squared.
Core Chapter 3 : Linear regression
Page 88
Using the calculator to find the three median regression line.
Enter the data into the lists and draw a scatterplot to make sure the data is entered
correctly.
Tap Calc from the menu bar and select MedMed Line.
Complete the Set Calculations
screen as shown at right:
8
Tap OK and the coefficients for the equation of the three-median
regression line are shown.
The equation in this case is
Tap OK and the regression line appears on the
scatterplot.
Page 91
Using the calculator to find the least squares regression line.
Enter the data into the lists and draw a scatterplot to make sure the data is
entered correctly.
Tap Calc in the menu bar and select Linear Reg
Complete the Set Calculations
screen as shown at right:
Tap OK
The figures for Linear Regression appear on the screen.
The values a and b are the coefficients for the equation of the least-squares
regression line. a is the slope of the line and b is the intercept.
The equation is
Note: Least squares regression lines are often written in the form
which in this case would be
The slope is always the number multiplying the independent variable x.
Tap OK and the least squares regression line
appears on the scatterplot.
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Page 98
Using the calculator to find and plot the residuals.
Using the data from Page 91:
Tap Calc to access the Set Calculations screen. Choose Linear
Regression.
Change Copy Residual from Off to list 3. This will paste the residual
values into list 3. Tap OK
Tap SetGraph and go into Settings...
Turn Off graph 1
Tap on graph 2 and complete the settings as
shown.
Tap on Set and then the graph icon y.
The residual plot is shown on the lower half of the screen.
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Page 104
a.
Example 4
Enter the data into the lists and construct a scatterplot .
The data does not appear to be linear.
Fit a least-squares regression line to the data and construct a
residual plot:
From the menu bar tap Calc and then
Settings...
Complete the
Complete the Set Calculations screen as
shown. Tap OK.
The Linear regression screen appears. Tap OK.
The least squares regression line
appears on the scatterplot.
Tap the label of list 3 to fill the column
with the residual values.
Set-up graph 2 for the residual plot (turn graph 1 Off). Tap Set.
The residual plot shows a pattern suggesting that the
linear model is not the best fit.
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b.
We can apply the x2 transformation by adding an extra row
containing x2 values in the table. To do this on the table go to the Cal
location for list 4 and, using the keyboard, type in list1^2.
Tap on the label of list 4 and the column will
fill with the values from list 1, squared.
Go to SetGraph in the menu bar and choose
Settings... to construct a scatterplot of
y(list 2) versus x2(list 4).
Tap Set and then the graph icon y.
The scatterplot of y versus x2 appears to be
linear.
Go to SetGraph and Settings... and complete the Set Calculation screen
so that the residuals for this transformed graph are in list 5.
Tap OK to see the Linear Regression screen
(Tap OK) to fill list 5 with the residuals.
and
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Tap SetGraph and Setting... and complete the screen for graph 3 as shown (make
sure the other graphs are turned Off)
Tap Set and the graph icon y to construct the
residuals plot for the transformation.
The residuals plot confirms that the relationship between
y and x2 is linear.
c.
From the linear regression screen above we can write the equation. The slope, a, is 0.4186...
and the intercept, b, is 1.0781... so the equation connecting y and x2 is
Page 109
a.
Example 5
Enter the data and construct a scatterplot.
The data is clearly not linear and
has the general shape.
audience size
day
We therefore consider the four transformations
where x represents the day and y represents the audience size.
For each of the transformations we add the appropriate column to the data in the lists, construct a
scatterplot including a least squares regression line, and a residual plot.
Transformation logx
13
Transformation logy
Transformation
Transformation
See the text for the remainder of Example 5.
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Core Chapter 4 : Time series
Page 136
Example 6
To graph the deseasonalised data open the Statistics application and enter the time period 1 to 48 in
list 1 and the deseasonalised data in list 2.
The time period data, 1 to 48, can be entered into list1 using a sequence:
Tap on Cal for list1 and then the soft keyboard and cat to select seq(
[alternatively you could type seq( ] then specify the sequence
as sequence, variable, starting value, ending value, step)
These values are shown at right. Tap EXE and list1 will fill
with the values 1 to 48.
Enter the deseasonalised data into list2.
Tap SetGraph in the menu bar and choose Setting... to get to the Set
StatGraphs screen. Choose xyLine for a time-series graph. Tap Set.
Tap Calc and choose Linear Reg. Complete as shown.
Tap OK and the slope (a) and
intercept (b) for the least-squares
regression line will be displayed.
Tap OK and the graph will be drawn along with the least squares
regression line. (Shown as a full screen here)
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Casio ClassPad
Replacement calculator screens for Module 1 : Number Patterns
Chapter 5 : Arithmetic and Geometric Sequences
Page 157
Finding terms of a sequence using the calculator.
Method 1
Tap on the Graph and Table: W application. Complete the screen as
shown
Tap on the table icon # in the toolbar. A table of
values appears in the lower screen. Scroll down to
find the 25th term.
Note: You may need to choose the maximum value for x in the table.
Tap on the 8 icon to enter the Table Input screen.
Change the End value to at least 25. Tap OK
Method 2
On the Menu screen tap on the Sequence application, H.
Choose Explicit as the type of sequence and type in the sequence using the n
available on the toolbar.
To create a table for the sequence tap 8 to display the
Sequence TableInput box. Enter the required values and
tap OK.
Tap on the table icon # in the
toolbar.
A table of values appears in the
lower screen. Scroll down to find
the 25th term.
1
Page 158-159 Displaying a sequence using a calculator
This section does not have an equivalent on the Casio CassPad.
Page 159
Example 2
On the menu screen tap on the Main application J.
a.
Type
given as a fraction.
and tap E and the answer
will be
b.
Type in the two subtractions and it is seen that the answers, given
as fractions, are not the same value so the sequence is not arithmetic.
Page 160
Example 3
On the Menu screen tap on the Sequence application, H.
Choose Explicit as the type of sequence and type in the sequence,
,using the n available on the toolbar.
To create a table for the sequence tap 8 to display the Sequence TableInput
box. Enter the required values (End value must be at least 25) and tap OK.
Tap on the table icon #
Note :
The appearance of the formula may change when you create the table.
A table of values appears in the lower screen. Scroll down to find the 25th term.
Page 162
Example 5
,using the n
box. Enter the required values ( The End value must be fairly large in this case;
you may have to change the value to find the term that is equal to 927) and tap
OK.
Tap on the table icon #
Scroll down the table until you come to the value 927
for anE. This means that term 186 has the value 927.
2
Page 162
Example 6
0, using the n
box. Enter the required values ( The End value must be fairly large in this
case; you may have to change the value to find the term that is more than
1000) and tap OK.
Tap on the table icon #.
Scroll down the table until you come to the first value
of an that is greater than 1000. Term 24 is the first
term that is greater than 1000.
Page 164
Graphing Arithmetic Sequences.
, using the n available on the toolbar.
Tap on the table icon #.
To plot the sequence the table window must be active.
Tap on Graph in the top menu bar and choose
G-plot. This will plot the sequence values as
disconnected points.
Tap on the graph icon $
Tap Analysis and choose Trace then use the
cursor keys to trace the points on the graph.
to
Page 167
Example 9
It is possible to include a column in the table showing the sum of n terms.
Tap on the diamond ♦in the menu bar
Choose ∑display and turn On
Tap on the table icon # and
three columns will fill; the third
column gives the sum of n terms
of the sequence.
Scroll down to find that the sum
of 25 terms of the sequence is
1975.
3
Page 168
Example 12
Tap on the table icon # and the three columns will fill; the third column,
∑anE, contains the sum of n terms of the sequence.
Scrolling down this column will show that the sum of 26 terms will first exceed
50,000.
Page 174
Example 15
Tap on the table icon # and scroll down to find the value of the 10th
term is 1536.
Page 175
Example 16
b.
Tap on the table icon # and scroll down to see that the
value of term six is 81.
Tap on Graph and choose
G-Plot. The graph should
appear.
The window for the graph may not be correct . Tap on the window icon6 and set
it up as shown.
Note : the xmin (n) value is 1 and the xmax value is 6.
The corresponding terms(an values) have values from 1/3 to 81 so ymin is set at 0
and ymax is set at 90 to cover these two values.
4
c.
Scrolling down the table will show that term
13 is the first term that has a value greater than 100,000
Page 176
Example 17
Tap the table icon #and then the Graph. Choose G-plot and the graph
should appear.
If the window is not correct then tap Zoom and choose
Auto or Quick Initialise.
The graph shown at right is Zoom Auto.
Alternatively values can be
chosen for the window by
tapping 6
The maximum y (anE) value is
16 and the minimum is -8.
Page 177
Example 19
Tap on main menu J and then the keyboard k key. Select the 2D option
and tap on the nth root icon
Type in 4, arrow right, and type 5.0625.
Tap EXE and the answer 1.5 appears.
5
Page 179
Example 21
The last term is 1/729, which, as a decimal, is 1.3717...×10-3 or 0.0013717...
Enter the sequence and tap on the table icon.
Scrolling down reveals that term 7 has the value
1.3717...×10-3 so we are looking for the sum of 7
terms.
From the table the sum of
7 terms is 1.49931....
Alternatively, the sum of the 7 terms can be calculated by substituting in
the formula. The answer will be given as a fraction. Highlight the
fraction answer and tap on u to give the answer as a decimal.
Page 180
Example 22
The first term is 5 and the common ratio is 2 so the formula for the nth term
is 5×2n-1
Type in the sequence and tap on the table icon#. Scrolling down the
table shows that the sum of 11 terms is the minimum number of terms to
give a sum of at least 10,000.
Note: The sum column is On in this case. Tap on the diamond ♦in the menu
bar. Choose ∑display and turn On.
6
Page 185
Example 24
Enter the sequence as shown and tap on the table icon#.
Scroll down the table until you come to a value that is greater than 20,000.
Since Vn is defined as the amount after n years then we can say that after 12
years the value of the investment is at least $20,000.
Note: By highlighting the amount a12, the value to10 decimal places appears
below, and it can be seen that, rounded to the nearest dollar, this is $21291.
Page 185
Example 25
Scroll down the table until you come to a value that is greater than 6,000.
The population will reach 6000 twelve years after 1990; in the year 2002.
Page 186
Example 26
d.
Scroll down the table until you come to a value that is greater than 2000.
They will first be making an investment of more than $2000 on Maria’s 25th
birthday.
e.
Scrolling down the ‘sum’ table to n = 21
shows the total sum of $28 676.49 has been made
up to and including Maria’s 21st birthday.
Note : To include the ‘sum’ column
Page 189
Example 28
e.
Make sure that you have ∑display turned On
Scrolling down the ‘sum’ table to a sum of $12.83 show that this occurs
when n = 20. A phonecall of 20 minutes costs $12.83.
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Page 193
Example 30
d.
Enter both the sequences and tap the table icon #.
Scroll down the table until you come to the values where bnE > anE
This first occurs at the beginning of the eighth
year.
To draw the graph :
Tap on the window icon 6 and complete as
shown. The ‘y’ values can be seen on the table :
the minimum value is $1000 and the values for
term 10 are approximately $2000
Tap OK to bring up the graph:
Tap Analysis, choose Trace and use the cursor
help distinguish which points belong to which
keys to
sequence.
Chapter 6 : Difference Equations
Page 203
Generating a sequence on the calculator.
Difference equations are known as recursive sequences so we use this option when entering the rules
for difference equations. The variable an for this type of sequence is found on the menu bar.
Page 203
Example 4
This sequence is of the form an+1 = 2an -1; a1 = 5 so tap on the p next
to ^. Choose&
Enter the sequence as shown and tap the table icon #.
Tap on the window icon 6 and set up the
window as shown. The value of term one is 5
and the value of term seven is 257 so y- values
are set to cover these.
Tap OK and the graph
will appear.
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Page 213
Example 10
c.
Enter the sequence as shown; make sure the Sequence Table Input
8 screen starts at 0.
Tap on the window icon 6 and set up the window as
shown. The value of term one is 500 and the value of
term nine is 84 so y- values are set to cover these.
Tap on OK to bring up the
graph.
d.
From the table it can be seen that after week 9 the size of the
colony will not be sufficient to provide 120 mice for research purposes.
e.
Enter the sequence as shown and tap on the table icon #. From
the table it can be seen that the population is now increasing as the weeks
increase.
The window will need to be changed to graph
this sequence:
Tap OK to show
the graph.
9
f.
Enter the sequence as shown and tap on the table icon #. From
the table it can be seen that the population is remaining constant at 500 as
the weeks increase.
Tap on Graph and choose G-Plot. Tap on the
graph icon $.
The graph shows that the population is stable at
500.
Page 218
Using the calculator for second-order difference equations.
To enter a second-order difference equation firstly the type of equation
needs to be chosen.
Tap on& and choose ( which means that the equation is of the type
Complete the screen as shown accessing the variables
from the top menu.
Tap on the table icon, #.
To graph the first seven terms first choose an appropriate window
observing the required values from the table.
Tap OK to show the graph.
Tap Analysis in the menu bar and choose
Trace to track the sequence on the graph
the cursor keys.
using
10
Casio ClassPad
Replacement calculator screens for Module 2 : Geometry and Trigonometry.
Chapter 6: Geometry. No replacement screens.
Chapter 7 : Trigonometry
Page 267
Example 1
Tap on the main menuJ and press the Keyboard buttonk. Tap on
TRIG.
Note: The default option for measuring angle is Radians but for the
calculations in Further Maths you will be working in Degrees. Tap on Rad
(for radians) on the bottom menu and it should change to Deg (for degrees).
The calculator will now assume all angles are in degrees and will give answers
as degrees.
To give all answers as decimals tap on Standard in the bottom menu and
change this to Decimal
Tap on sin then 46) Tap EXE.
Sin (46o) = 0.7193 correct to 4 decimal places.
Page 267
Example 2
Tap on the main menuJ and press the Keyboard buttonk. Tap on TRIG and check that the
calculator is in Degree mode.
Tap on cos-1 then 0.8649) Tap EXE and the answer is given as a decimal
degree.
cos (30.1287o) = 0.8649
To change the answer into degrees, minutes and
seconds (DMS):
Tap on Interactive in the top menu, choose
Transformation and then the bottom option toDMS
Tap OK and the answer will be given in degrees,
minutes and seconds ie. 30 degrees, 7 minutes and 43
seconds.
Page 270
Example 5
Tap on the main menuJ and press the Keyboard buttonk. Tap on
TRIG and check that the calculator is in Degree mode.
Tap on cos-1 then 26 ÷ 38) Tap EXE and the answer is given as a decimal
degree.
cos (46.8264..o) =
To change the answer into degrees, minutes and seconds (DMS):
Tap on Interactive in the top menu, choose Transformation and then the
bottom option toDMS
Tap OK and the answer will be given as 46 degrees, 49 minutes and 35
seconds.
1
Page 272
Example 6
Follow the same instructions as for Example 5 above.
Page 272
Example 7
Follow the same instructions as for Example 5 above to find the angle then tap 12 ÷
cos (ans) and tap EXE.
Alternatively
You can use the calculator to solve the equation
On the Main menu highlight the answer 30.9637... by running the stylus over the
number. Go to Edit in the tool bar and select copy.
Go to Menu and select the solve equation N option. Tap
k and TRIG.
Tap cos , tap Edit from the tool bar and tap Paste. This
should paste the value from the Main application into the
equation so that you have cos(30.9637...
To complete the equation tap ) = 12 ÷ x
Tap solve 1 and the calculator will
solve the equation for you. This
means that you do not have to transform
your equations to solve them.
Page 275
Example 8
There are three methods for doing this and other examples on the calculator. It is worthwhile trying all
the methods so that you can see which one suits you.
Method 1
The Interactive menu from the menu bar will enable you to access the ‘solve equation’ facility on
this calculator which means that you do not have to transpose equations to solve them.
a.
Go to Main J menu. Tap k and TRIG.
Tap Interactive from the menu bar and select Equation/Inequality and solve.
Select solve numerically and complete the screen as shown typing the
equation sin(x)/15=sin(30)/9 into the equation line. Tap OK and OK on the
Warning box (the warning box appears because this
example is an example of the ambiguous case of the
sine rule).
2
The calculator will solve the
Enter 180 – x – 30.
Tap on EXE to find angle A;
equation for you; angle B is 56.4426...o
angle A is 93.5573..o
b.
Tap Interactive from the menu bar and select Equation/Inequality
and solve.
Enter the equation
, using y as the variable
in this case and ans to use the previous answer (angle A). Tap OK to give the
answer; y = 17.9653...
Method 2
The whole calculation can be done on the Main
application however, the equations will have to be
transposed as in the text.
Method 3
Note: Example 8 is an example of the ambiguous case of the sine rule which means there are two
possible solutions (not examined in Further Maths). The sequence for drawing the triangle in this
case needs to be angle first then the side lengths.
Tap on Geometry Gin the menu and then on the triangle O in the toolbar. Use the stylus to
construct a triangle.
Tap on two of the sides and then the arrow on the
rightp. The ‘measure angle’ Qsymbol will be on
the left. The angle between the highlighted sides in this
case is measured as 74.47589o. Highlight this figure
and change it to 30. Highlight the tick box to the right.
Tap on the screen and then tap on the side AC
Tap on the arrow on the right of the toolbar and select the icon that measures
lengthx.
The length of the highlighted side in the triangle drawn will be shown. Highlight
this figure and change it to 15 then highlight the tick box.
Touch the screen again to get back to the triangle.
Tap on the screen and then tap on the side AB
T
T
3
Tap on the arrow on the right of the toolbar and select the icon that measures lengthx. Highlight
the length and change it to 9 units. Tick the box on the right.
The diagram should now look like the one in the text.
a.
Tap on the sides AB and BC. The angle B will be measured as
56.44269o
Tap on the sides AB and AC. The angle A will be
measured as 93.55731o
b.
Tap on the side BC and the length will be
calculated for you and appear in the toolbar. The length
is 17.96532 cm.
Note: Drawing the triangles requires some practice but is worthwhile as
then there are no calculations to be made.
The area of the triangle above can also be found by highlighting all the
sides and then the area icon E will appear on the left and the area
(67.36994 cm2) is given in the toolbar.
Page 277
Example 10
Using the main Japplication from the icon panel, pressing
k and selecting the TRIG option. Make sure that you have selected
Decimal and Degree on the status bar.
Calculations are typed as shown followed by EXE for each calculation.
Alternatively i. You can draw the diagram, entering the lengths and angles as
shown above in Example 8, and use the diagram to solve for AD
ii.
4
Page 280
Example 11
Use the main J application from the icon panel and pressing
k
Enter the calculation as shown using the x2 option and going to the TRIG
window for cos (use I to get back to the mth option)
ans) EXE to find the square root of the previous answer.
Tap
Alternatively
Tap on Geometry Gin the menu and then on the triangle O in the toolbar. Use the stylus to
construct a triangle.
Tap on the side AB. Tap on the arrow on the right of the toolbar and select
the icon that measures lengthx.
The length of the highlighted side in the triangle drawn will be shown.
Highlight this figure and change it to 16 then highlight the tick box.
Repeat this process for side AC changing the length to 23 units. (You may
need to zoom out if the triangle extends off the screen; Select View and
Zoom Out from the tool bar)
Tap on the sides AB and AC and change the angle between them to 40o The
triangle is now set-up as for the example.
Tap on side BC and its length (14.8725 cm) is shown
on the tool bar
.
Page 281
Example 12
Construct the triangle in the Geometry application (See example 11 above.)
Tap on the side marked x; AC in this case, to find that the length is
12.6965.... m
5
Page 281
Example 13
Tap on Geometry Gin the application menu and then on the triangle O in the toolbar. Use the
stylus to construct a triangle.
Tap on the side AB. Tap on the arrow on the right of the toolbar and select the icon that measures
lengthx.
The length of the highlighted side in the triangle drawn will be shown. Highlight this figure and
change it to 8 then highlight the tick box.
Repeat this process for sides AC and BC changing the lengths to 9 and 11 units respectively.
By tapping on two sides
the magnitude of the
angle appears in the
toolbar above and all the
angles of the triangle are
found.
Page 283
Example 14
Tap on Geometry Gin the application menu and then on the parallelogram o in the toolbar. Use
the stylus to construct a parallelogram.
Tap on the side AB. Tap on the arrow on the right of the toolbar and select the
icon that measures lengthx.
The length of the highlighted side in the triangle drawn will be shown. Highlight
this figure and change it to 15 then highlight the tick box.
Repeat the process for side AD changing its length to 21 units.
Highlight both sides AB and AD to change the angle
between them to 65o.
6
Go back to the toolbar and select the line segment icon y. Use the stylus to draw a line from B to D
and from A to C.
Tap on G to stop the line
drawing function.
Tap on BD and go to the length
toolbar. BD is 19.99376 cm long.
Tap on the screen and then AC to
find its length is 30.53276 cm.
Page 287
Example 15
a.
Tap on Geometry Gin the application menu and then on the triangle O in the toolbar.
Use the stylus to construct a triangle.
Tap on the side AB. Tap on the arrow on the right of the toolbar and select the icon that measures
lengthx.
The length of the highlighted side in the triangle drawn will be shown. Highlight this figure and
change it to 25 then highlight the tick box. Repeat the process for side
CB changing it to 18.
Highlight both sides AC and CB to change the angle between them to
105o.
Tap on all three sides and the area icon E
should appear. The area of the triangle is
217.333 cm2
b.
The area of the triangle can best be found using
as in the text but to check you could construct
a triangle as described above changing the lengths of the sides to 16, 12
and 20 cm.
Tap on all three sides and the area icon E should appear.
The area of the triangle is 96 cm2.
Page 288
Example 16
Tap on Geometry Gin the application menu and then on the triangle O
in the toolbar. Use the stylus to construct a triangle.
Tap on the side BC. Tap on the arrow on the right of the toolbar and select the
Highlight this figure and change it to 30 then highlight the tick box. Repeat
the process for side AB changing it to 24.
Highlight both sides AB and AC to change the angle between them to 44o.
Highlight all three sides and the area icon E should appear.
7
The area of the triangle is 351.8176 cm2.
Chapter 9
Page 294
Applications of Geometry and Trigonometry
Example 2
Tap on Geometry Gin the application menu and then on the triangle O in the toolbar or the line
segment icon y. Use the stylus to construct a triangle.
Tap on the side AB. Tap on the arrow on the right of the toolbar and select the
Highlight this figure and change it to 7.5 then highlight the tick box. Repeat
the process for side AC changing it to 5.
Highlight both these sides to change the angle between them to 68o.
Highlight CB to find its length; 7.2907... km
b.
Highlight both BC and AC to find that the
angle between these sides is 39.48427o.
The bearing is found by subtracting this angle and 68o
from 360o.
The angle can be highlighted,
copied and pasted into the main
J screen.
Page 299
Example 5
menu and then on the line
Tap on Geometry Gin the
segment icon y. Use the
stylus to construct the
diagram.
Tap on AB to change it to 50 and pairs of sides to establish the angles between them.
a.
Tap on the side AD
(length 65.57 m)
b.
Tap on the side BC
(length 59.16 m)
8
c.
Tap on the side AC (length 96.91 m)
Tap on the points D and C
(length 111.19 m)
d.
Construct a line joining D to C and highlight the triangles ADC and ABC
to find the areas.
(Areas 3159.9 m2 and 1211.424 m2 resp.)
These values can be copied and
pasted into the main J
screen to calculate the total
area.
9
Casio ClassPad
Replacement screens for Module 3 : Graphs and relations
Chapter 10: Construction and interpretation of graphs.
Page 339
Example 8
Choose Graph and Table W from the application menu.
Enter 5x – 3 next to y1 and tick the box next to this equation.
Tap the graph icon$ in the toolbar.
Tap Zoom in the menu bar and choose one of the zoom options that will
show the features of the graph. In this case I have chosen Box then used
the stylus on the graph to constructed a box around the graph enclosing the
intercepts with the axes.
To find the intercepts with the axes: Tap on the right arrowu in the tool
bar.
The first icon = on the left finds the y-intercept.
Tap on this to show that the y-intercept is -3.
The second icon Y on the
left finds the x-intercept.
Tap on this to show that the
x-intercept is 0.6
Page 347
Example 14
Choose Graph and Table W from the application menu.
Enter -2x + 7 next to y1 and tick the box next to this equation.
Enter 4x - 5 next to y2 and tick the box next to this equation.
Tap the graph icon$ in the toolbar.
Tap Zoom in the menu bar and change the window if the point of intersection is not on the screen.
Tap Analysis then G-Solve then
intersection
(2, 3) appears on the graph screen.
Intersect and the point of
1
Page 349
Using the calculator to solve simultaneous equations.
Tap on the Main J application and press the keyboard buttonk. Choose
the 2D option) and choose
Enter the first equation in the top box and the second
equation in the lower box.
The two variables x,y are typed into the outer box.
Tap EXE w to get the answer.
Page 371
Using the calculator for Example 28
Tap on the Statistics applicationI and enter the data in the
lists.
Tap on G in the toolbar to reach the Set StatGraphs screen,
tap Set and then the graph icon y.
To test the x2 tranformation.:
Go to the Cal row of List 3. Tap on the soft keyboard
k and select the qwerty keyboard abc and type
list1^2. Tap on the label of list3 and the list will fill with
the squares of list1.
Graph x2 versus y; set-up as shown
on the screen at right.
Tap Set and then the graph y icon and the graph will appear: the points
appear to be linear.
2
Repeating the process for the x3 transformation:
The points on the graph do not appear to be
linear.
If it is known that the relationship between x and y is of the form
then the calculator can be
used to find the values of k and n.
Go to Calc in the menu bar, choose Power Reg and the Set Calculation screen
will appear. Set-up this screen as shown and then tap
OK. The Stat Calculation screen will show that the
value of k (a) is 3.19 and the value of n (b) is 2 so the
equation of the relationship between x and y is
The graph shows the original
points with the regression line
fitted.
Chapter 11: Linear programming
There are no replacement screens for this chapter however it is possible to graph inequalities on the
calculator and solving simultaneous equations is simplified by using the calculator.
Page 385
Example 3
Go to the Graph and Table menu W
You will need to transpose any equations that do not have y as the subject.
: Tap on the y = icon in the toolbar choose Z and type 0.
Constraint
Tick the box next to this relation.
Constraint
Tap on the y = icon in the toolbar choose ' and type 0.
Tick the box next to this relation.
Constraint
Tap on the y = icon in the
toolbar, choose X
type
and tick the box next to this relation.
Constraint
Type
and tick the box next to this relation.
3
Tap on the graph icon $ and the feasible region will be shaded.
The point of intersection of y3 and y4 can be found:
Tap on the Main menu J and press the keyboard buttonk.
Choose the 2D option) and choose
Enter the first equation in the top box and the
second equation in the lower
box.
The two variables x,y are
typed into the outer box.
Tap EXE w to get the
answer.
4
Casio ClassPad
Replacement calculator screens for Module 4: Business Related Mathematics
Chapter 12 : Financial transactions and asset values.
Page 437
Example 13
Tap on the Statistics I application and enter the numbers 0 to 7 in list1. Go
.
to the Cal row of list2 and type
Tap on the label of list2 to fill the list with the inflated
values for each year.
Tap on the Set StatGraphs icon
G and set-up the scatterplot.
Tap Setand then tap the graph
icon y.
Page 443
Example 18
Tap on the NumSolve applicationN. Type in the equation
Tap on Solve and the value x = 5.36118... is given.
Rounding this value up we can say that after 6 years the value
will be less than $4000.
Chapter 13: Loans and Investment
Page 452
Example 1
Tap on the Financial icon in the application menu. The initial screen, at right, will
appear. Tap on Simple Interest to access the simple interest calculation screen.
Complete the screen as shown entering the term of 4 years
as 4 × 365 days and the amount invested (an amount ‘going
out’) as -2460.
Tap on the button SI next to the box to calculate the simple
interest and the button SFV to calculate the simple future
value (principal + interest).
Note : Make sure the Format for Days in Year is on 365
1
Page 452
Example 2
Tap on the Financial application and choose Simple Interest.
Enter the time period as
and the loan value as 18000 (an amount
‘coming in’, so positive) Tap the buttons next to the last two boxes to find the
interest paid and the total amount owing.
Page 452
Example 3
Tap on the Financial application and choose Simple Interest.
Enter the time period as 185 days and the loan value as 2500 (an amount
‘coming in’ so positive) Tap the buttons next to the last two boxes to find the
interest paid and the total amount owing.
Page 454
Example 4
Tap on the Financial application.
a.
On the initial screen there is a Day Count calculator that will
calculate the number of days between two dates.
Entering the two dates as month, day, year and
tapping on the Days button will give 174 days
from day 1 to day 2. If both days are to be
included then you will need to add one day.
b. and c.
Tap on the Calculations
and choose Simple
Interest.
Note : The simple interest calculator will only calculate the interest and the (principal + interest). For
examples where you are required to calculate the principal, the time period or the interest rate you will
need to use the transposed formula and substitution.
2
Page 463
Example 11
From the initial Financial application screen choose Compound Interest.
N is the number of instalment periods; 5 years in this example.
I% is the Annual interest rate; 5.5% in this example.
PV is the present value or initial investment; $10 000 in this example.
PMT is the amount paid each period; $0 in this example.
FV is the future value; the amount that needs to be
calculated in this example.
In this example there is 1 instalment period per year,
P/Y, and the interest is calculated once per year (C/Y)
a.
Tap the FV box and the amount
of the investment after 5 years is
calculated.
b.
Highlight the FV value
and tap the copy box x in the
toolbar. Tap on the Main menu
Jand paste the value on the
screen.
Type – 10000 and tap EXE to
calculate the interest.
Page 464
Example 12
From the Financial application choose Compound Interest.
a.
b. Enter the N value as 5×4
Page 465
c. Enter the N value as 5×365
Example 13
a. Interest is 893.09
b.
$14 883.62 is in the account after 3 years (36 months)
3
c.
Copy the FV value from part b. by tapping on the copy icon x in the
toolbar. Tap on the Main applicationJ and paste this value.
Complete the calculation as shown.
Page 467-8
Finding times and periods for compound interest.
The value of N can be calculated easily by the calculator by entering the other
values in the calculation and then tapping on the N box.
Note the signs for PV and FV.
Page 469
Example 14
The value of N can be calculated by the calculator by entering the other values
in the calculation and then tapping on the N box.
a.
the minimum number of time periods is 41
b.
41 time periods (months) is 3 years, 5 months.
Page 470
Example 15
The values of N can be calculated by the calculator by entering the other
values in the calculation and then tapping on the N box.
It will take 321 days for the investment to grow from $1700 to $1800.
4
Page 471- 473
Using the calculator to plot interest graphs.
Tap on the Statistics I application and enter the numbers 0 to 10 in list1.
.
Go to the Cal row of list2 and type
Tap on the label of list2 to fill the list with the compound interest values for
each year.
Go to the Cal row of list3 and type
.
Tap on the label of list3 to fill the list with the (principal + simple interest)
values for each year.
Tap on the Set StatGraphs icon G and set-up the
scatterplots.
Tap the graph icon y.
Tap Analysis then Trace and use
the cursor keys to find the value
particular points on the graph.
Page 477
of
Example 17
a.
From the initial Financial application screen choose Compound
Interest.
N is the number of instalment periods; 5×12 months in this example.
I% is the Annual interest rate; 6% in this example.
PV is the present value or initial investment; -$1000 in this example
(outgoing amount).
PMT is the amount paid each period; -$250 in this example (outgoing
amount).
FV is the future value; the amount that needs to be calculated in this example.
P/Y and C/Y are both 12 as there are 12 payments per year and 12
compounding period per year.
Tap on the FV box to get the amount accumulated.($18 791.36)
b.
Alternatively this calculation can be done on the Amortization
screen. The sum of the interest can be accessed on this screen.
Amortization lets you calculate the interest and principal portions of a
payment or payments.
PM1 is the number of the first instalment period in the interval under
consideration; 1 in this example
PM2 is the number of the last instalment period in the interval under
consideration; 60 in this example.
Note: The initial investment and regular payment can be entered as positive
amounts on this screen and the balance and sum of interest payments are also
positive. If they are entered as negative amounts then the balance and sum of
interest will be negative.
The sum of the interest for the 5 years is $2791.26
5
Page 478
Example 18
Using the Compound Interest screen from the Financial application:
Set-up as shown.
Tap on the N box to find the number of months
save $10 000.
It will take 25 months to save $10 000.
Page 485
to
Examples 22 and 23
Calculations of the amount owing using the annuities formula can be done on the Main application.
Page 488
Example 25
From the initial Financial application screen choose Compound Interest.
$10000 is borrowed (incoming amount, so positive) for 5 years and interest
is charged at 7% p.a. compounding yearly. A repayment of $2000 (outgoing
amount, so negative) is made each year.
Tap on the FV box to find the amount still owing
after 5 years: $2524.04. This is shown as a
negative value indicating that it is still owing
(outgoing).
Page 489
Example 26
a.
Enter the instalment period (N) as 4.5 × 12
There are 12 repayments and compounding periods per year.
The amount still owing is $9619.15
6
b.
Enter the instalment period (N) as 4.5 × 4
There are 4 repayment and compounding periods per
The amount still owing is $9793.21
Page 489
year.
Example 27
The repayment amount (PMT) is required in this
example. If the loan is ‘paid out’ this means that the
future value is zero.
The monthly repayment is $492.02 to pay out the loan in 5 years.
Page 490
Example 28
The loan is for 20 years so there are 20 × 12 =
240 instalment periods (N)
The repayments (outgoing amount, so negative)
are $1250 per month.
The loan is to be paid out in the 20 years so FV
is zero.
Tap on PV to find the amount that can be
borrowed; $158,760.54
Page 491
Example 29
Calculate the yearly repayment amount for the loan then tap on the spread
sheet icon in the toolbar
Tap on Edit and choose Fill Sequence. You will need to enter the annuities
formula
in the
Expr. box. Complete the remainder of the screen as
shown. Tap OK.
Highlight the 25 values in column A and tap on the
graph icon D and the
owing each year will be
graph of the amount
drawn.
7
Page 491
Example 30
Enter the values for I%, PV, PMT, FV, P/Y and C/Y as shown.
Tap on N to find the number of instalment periods; 28.004 will need to be
rounded-up to 29.
Page 492
Solving for the principal, P
Enter N as 20 × 12 months and the other values as shown.
Tap on PV for the amount that can be borrowed; $158
760.54
Page 492
Example 31
Enter N as 10 × 26 fortnights and the other values as
shown.
Tap on I% to give an interest rate of 6.10%
Page 496
Example 32
Solving for N, for a loan of $20 000 at 7.5% interest and repayments of $120,
brings an Error message. This occurs because $120 is less than the interest on
the loan for a month so the loan will be increasing in value and will never reach
zero.
8
Page 496
Example 32
a.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and
C/Y as shown. N = 158, rounding-up.
Tap on Calculations in the menu bar and choose Amortization.
The values for I%, PV, PMT, FV, P/Y and C/Y will
carry over to this screen.
Complete the PM1 and PM2
values as 1 and 158
respectively then scroll down
and tap on ΣInt
$11 484.67 is paid in interest
for this loan.
b.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and C/Y as
shown. N = 61, rounding-up.
Tap on Calculations in the menu bar and choose
Amortization.
values as 1 and 61 respectively
then scroll down
and tap on ΣInt
$4055.39 is paid in interest
for this loan.
c.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and C/Y
as shown. N = 38, rounding-up.
Amortization.
The values for I%, PV, PMT,
FV, P/Y and C/Y will carry over
to this screen.
values as 1 and 38 respectively
then scroll down and tap on ΣInt
for this loan.
9
Page 498
Example 33
a.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and C/Y as
shown. N = 14, rounding-up.
Amortization.
Complete the PM1 and PM2 values as 1 and 14
and tap on ΣInt
for this loan.
b.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and C/Y
as shown. N = 53, rounding-up.
The values for I%, PV, PMT, FV, P/Y and C/Y will carry over to this screen.
and tap on ΣInt
for this loan.
c.
Solve for N entering the values for I%, PV, PMT, FV, P/Y and C/Y as shown. N = 158,
rounding-up.
and tap on ΣInt
$11 484.67 is paid in interest
for this loan.
10
Page 499
Example 34
a.
Calculate the repayment amount (PMT) per month; $757.39
respectively then scroll down and tap on ΣInt
The total cost of the loan is interest + fees =
$81 774.62 + 240 × $8 = $83 694.62
b.
Calculate the repayment amount (PMT) per month; $769.31
The values for I%, PV, PMT, FV, P/Y and C/Y will carry over to this
screen.
respectively then scroll down and tap on ΣInt
The total cost of the loan is (interest only)
$84 633.88
Option A is a cheaper loan.
Page 500
Example 35
a.
Calculate the repayment amount; $1075.56 per month for option A
Go to the Simple Interest screen to calculate the amount that is to be repaid
per month for option B:
The total amount that is to repaid is $114 000 so
the monthly repayments are $114000/96 =
$1187.50
11
b.
The total amount of interest paid for option A is $28 253.96 and for
option B it is $39 000. The difference is $10746.04; more interest paid for
option B.
c.
An equivalent reducing balance loan rate to option B can be found
by using the repayment amount from option B and tapping on I%.
An equivalent rate is 11.23%
Page 503
Example 36
The Simple Interest screen can be used to calculate the total amount that has to be
repaid; $1296
Monthly instalments will be $1296/6 = $216
Page 504
Effective rate of interest
Nick is making six monthly payments of $150 on a loan of $800. This has an
equivalent reducing balance interest rate of 41.67%.
12
Casio ClassPad
Replacement screens for Module 5: Networks and Decision Mathematics
Chapter 14: Undirected graphs and networks
No replacement screens for this chapter.
Chapter 15: Directed graphs and networks
Page 564
Finding connectivity matrices on a calculator.
Go to the Main application J then k. Go to the 2D keyboard and choose
the CALC option
Tap on the 2 × 2 matrix symbol 8 and a 2×2 matrix
will appear on the screen.
In this case we want a 5 × 5
matrix so tap 3 times on 7
and this will produce three more
rows.
Tap on 6 3 times to produce three more columns and so we end with a
5 × 5 matrix.T
Tap on the first element and tap 0 to enter the first element.. Use the arrows
to go to the next element and continue entering the elements.
Some of the elements in the matrix are entered in the
screen at right
When all the element are entered
in the matrix tap outside the
matrix and save W the matrix as
a variable, (tap VAR) A, in this
case.
Tap w
1
It is now possible to find powers of the matrix A.
Tap A ^ 2 w
Tap A + A^2 w
2
Casio ClassPad
Replacement screens for Module 6: Matrices
Chapter 16 Matrix representation and its application.
Page 625
Using a calculator for matrix operations.
Go to the Main J application thenk. Go to the 2D keyboard and choose
the CALC option
Tap on the 2×2 matrix symbol 8 and a 2×2 matrix
will appear on the screen.
In this case we want a 3×3 matrix
so tap once on 7 and this will
produce another row.
Tap on 6 once to produce another columns and so we end with a
3×3 matrix.T
Tap on the first element and tap 1 to enter the first element. Use the arrows to
go to the next element and continue entering the elements.
All the elements in the matrix are entered on the screen at
right.
To save the matrix tap outside the
matrix and save W the matrix as
a variable, (tap VAR) A, in this
case.
Tap wand the matrix is saved
as A.
Establish the other matrix as B
1
To calculate A × B : type A × B (make sure you are using variables for A and
B) and then w
To calculate 2A + B : type 2A + B (make sure you are
using variables for A and B) and then w
Note : To clear the matrices from the screen tap Edit
and then Clear All.
To clear the matrices from the variable names tap Edit
and then Clear All Variables.
Page 630
Powers of matrices
To find A2 : Type A^2 w
To find A3 : Type A^3 w
Page 637
Using the calculator to find inverses.
Set-up matrix A
Tap A ^ -1 w to find the inverse of A.
Check that A × A-1 = I
2
Page 638
Example 10
Set-up matrix B
Tap on the 2×2 icon, 8. In this case we want a 4×4 matrix so tap twice
on 7 and this will produce two more rows.
Tap on 6 twice to produce two more columns and so we end with a 4×4
matrix.T
Enter the elements into the matrix and save it as B.
Tap on ans^-1wto find the inverse with the elements
as decimals.
Highlight the inverse matrix
and tap the decimal-fraction
icon . in the toolbar. The
elements in the matrix will
now be presented as fractions.
Page 640
Example 12
Set-up matrix A: Tap8and then 6 and then 7 to establish a 3×3
matrix.
Enter the elements and then save the matrix as matrix A
Set-up matrix B: Tap7 twice to establish a 3×1
matrix. Enter the elements and save as matrix B.
To solve the simultaneous
equations we need to
calculate A-1×B
Type A-1×Bw
3
Chapter 17 Transition matrices
Page 648
Example 1
Establish the transition matrix A and the initial state
matrix B.
c.
The state matrix for Tuesday
is AB
The state matrix for Friday is
A4×B
d.
To see the progression to steady state, A5,
10
A and A15 have been calculated.
The steady state proportion of students buying their
lunch at the canteen is 0.375 or 37.5%.
Summary
Page 657
Using the calculator for matrix operations.
Establish matrices A and B.
• Addition: Tap A + B w
•
Multiplication by a scalar:
Tap 3A w
Tap 4A – 2B w.
4
Multiplication
Tap AB w
•
Powers of a square matrix
Tap A^2w
Tap A^5w
•
Determinant of a matrix
Go to Interactive in the menu bar and choose Matrix-Calculation
Tap det then tap A and OK
Alternatively type det(A)
The determinant of
matrix A is 10
Similarly the determinant
of matrix B is 5.
•
The inverse of a matrix
Establish the matrix B. Check that the determinant is not zero.
To find the inverse: Tap B^-1w and the inverse is
given with the elements as decimals.
Highlight the matrix and tap the
‘decimal to fraction’ icon .
and the elements of the matrix
will be given as fractions.
5

Casio Classpad - Haese Mathematics

Transcription

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