The
two simultaneous equations y + 14x = 7 and 3y – 7x
= 7 solve to (2/7, 3). First discuss and
construct the 2 lines manually on the white board,
noting the large negative slope (– 14). THEN
have Autograph plot them.
Next, select the 2 lines and solve for the intersection, first at 4 d.p.
(the default setting), then at 10 d.p. (in the ‘Page’ => ‘Settings’ menu),
and discuss the recurring nature of in decimal form.
As well as offering full analysis of bivariate data (scatter
diagrams and line of best fit), Autograph has a significant
section for single variable statistics and probability.
Data
can be entered directly or copied from a spreadsheet.
Click anywhere on the web site table and ‘Select
all’ (Ctrl-A).
Click anywhere on the web site table
and ‘Select
all’ (Ctrl-A).
In Excel, paste (Ctrl-V). [Excel XP
has a useful facility for downloading web-data directly.]Use
the split-screen facility to view the top and bottom of
the data. To construct a histogram of the pressure variation
over
Birmingham, select the ‘Pressure’ data
(click on the top value, then Shift-click on the last).
Paste into the Raw Data entry
box in Autograph. Sort. Notice a rogue value at the start.
Discuss, and delete.Before plotting anything, note the
range and variation and discuss the likely distribution.
Set up appropriate ‘Min’, ‘Max’ and ‘Width’ for
the class intervals, and enter a name for the data set
Plot the Histogram, (use autoscale
and horizontal zoom to obtain a good fit on the page).
Use the ‘Table
of Statistics’ option in Autograph
to create a summary in the ‘Results Box’. This
can be copied directly into Word – you will however
need to select it all and change the font to ‘Autograph’ for
the symbols.
To tidy up the columns, select the tabular text and use
the option ‘Table’ => ‘Text to columns’ and
finally the default ‘Table Autoformat‘ option
to obtain the layout illustrated. Any Autograph screen can
also be copied and pasted to Word.
The object-based environment
of Autograph provides an ideal basis for the
study of transformations. It is exceptionally easy to
enter shapes, lines or points, and to create dependent
transformations, which can be animated dynamically.
In
Autograph a shape can be created by ‘grouping’ a
number of points ('cursors'). This shape can now be dragged
around as required.
With
the shape selected the right-click menu lists all the transformations that
are possible without the additional selection of other
objects. Of those listed, Shear and Stretch can
be animated by varying the appropriate factor.
To
set up Enlargement, Rotation or Reflection,
it is important to select the appropriate objects first:
2.
Enlargement: need first to select a grouped object AND
the centre of enlargement. Factor can be animated
3.
Rotation: need first to select a grouped object AND the
centre of rotation. Angle of rotation can be animated.
4.
Reflection: reflection line can be constructed from coordinate
points, or entered as an equation.
Matrices and Transformations
Transformations
can also be defined in Autograph by Matrices (e.g.
for the IGCSE). There is a long list of preset matrices
for standard transformations. Matrix
elements can include constants which can be varied dynamically.
This is a great way to visualise the association of each
element.
Translation:
this is achieved by creating a Vector and
selecting both the vector and the grouped object. All transformations
can be ‘chained’ indefinitely in Autograph.
There are many ways to generate functions in Autograph:
1.
Selecting a number of points on the
screen: here 3 points to create a quadratic. The points
can be dragged around, e.g. to explore what happens when
two of the points are coincident on the x-axis
2.
and 3. Entering the equation y = x n.
When an entry includes constants, these can be varied
dynamically, or a family plotted. In this case, a comma-separated
list to generate the family, and zooming in allows an
exploration of the behaviour of xn for large n.
4.
A look at the transformation of y = x2 in
the form y – b = (x – a)2,
when the constants ‘a’ and ‘b’ can
be varied.
5. Functions can be defined as f(x) = …, and g(x) = …
giving, for example, an opportunity to explore transformations of sinx and
the same transformations on, say, y = x2
