# «Exercises demonstrating how to use Autograph to visualise mathematical concepts in the classroom. Simon Woodhead, Alan Catley, Craig Barton and ...»

2012

Autograph Training Material

Exercises demonstrating how

to use Autograph to visualise

mathematical concepts in the

classroom.

Simon Woodhead, Alan Catley,

Craig Barton and Douglas Butler

Eastmond Publishing Ltd

PO Box 46, Oundle, PE8 4EJ, UK

Tel: +44 (0) 1832 273040

Fax: +44 (0) 1832 273529

simon@autograph-maths.com

Autograph Training Material

Table of Contents

Getting Going

Euler’s Nine-Point Circle

Best Practice

Whiteboard Mode and the Onscreen Keyboard

Technology in Secondary Mathematics

Graphing Quadratic Equations

Linear Programming

Cubic Investigation

Iteration

Parametric Equations

Binomial Theorem

Trigonometry

Conic Sections

Geometry Sound Mirrors

Transformation Geometry

Transformations in Three Dimensions

Angle at Centre Theorem

Vectors

Lines and Planes from Vectors

Creating Pin Boards

Statistics Baby Weights

Scatter Diagrams

Poisson and Normal approximations to the Binomial

The Central Limit Theorem

www.autograph-maths.com 2 Autograph Training Material Calculus Introducing Differentiation

Differentiating Trigonometric Functions

Finding the Area under a Curve

A Goat Grazing Half a Square Field

Volume of Revolution

The Exponential Function

Mechanics The Human Cannonball

Terminal Velocity

Euler’s Nine-Point Circle In this starter activity you will be introduced to Object Selection and the Right-click Menu, which are used in most Autograph files.

Select any two points, then right-click and choose Line Segment.

**There are three ways of selecting more than one object:**

1. Hold down Shift whilst clicking on each object.

2. If the objects are points, click and drag a selection rectangle around them.

3. Use Whiteboard Mode which allows you to select more than one object without holding down the Shift key.

Repeat for the other two pairs of points to complete the triangle.

Select all three lines and make them the same colour.

Select any two points, then right-click and select Mid-Point. Repeat for the other pairs.

Select any vertex and the opposite side of the triangle, then right-click and choose Perpendicular Line. This is called an Altitude. Repeat for the other vertices.

Select all three altitudes and make them the same colour.

In Point Mode hold down the Ctrl key and move the cursor to the intersection of an altitude and a side of the triangle. When the cursor changes to a small circle left-click to attach a point to the intersection. Repeat for the other two altitudes.

In the same way attach a point to the intersection of the three altitudes.

Select the intersection of the three altitudes and a vertex, then right-click and choose Mid-point. Repeat for the other two vertices.

**The Nine Points of Euler’s Circle:**

1. The three mid-points the sides of the triangle.

2. The three intersections of the sides and the altitudes.

3. The three mid-points between the vertices and intersection of the altitudes.

Select any three of these points, then right-click and choose Circle (3 pts).

Select one of the vertices of the triangle and move it.

If you follow these instructions Autograph will plot a perfect quadratic. However the students have not had the opportunity to predict where the curve crosses the axes, what happens to the curve for large negative and large positive values of x, where the maximums and minimums are, the general shape of the curve, etc. When asked in future what the shape of the curve y = x(x – 1) is they may recall it or know how to enter it in Autograph, but it is unlikely they will understand why it is the shape it is. What will they answer when asked about y = x(x – 1)(x – 2)?

The Three Step Rule

**An Autograph activity following best practice consists of the following three steps:**

In Slow Plot Mode graphs are plotted slowly from left to right, and plotting can be paused to allow for predictions to be made. Slow Plot Mode will stay on until it is turned off again. You do not need to turn it on again every time you open a new page.

Prediction Students can use the Scribble Tool to mark their predictions.

Students can use the Scribble Tool to mark the points where the curve crosses the axes, what happens to the curve at large negative and large positive values of x, where the maximums and minimums are, the general shape of the curve, etc.

Show Click Pause Plotting again. The graph will now plot slowly from left to right, hopefully passing through the points the students have marked with the Scribble Tool.

About these Autograph Activities When you see this icon you should give your students the opportunity to predict what will happen next before continuing.

We have not given detailed instructions for each prediction step as we have aimed to keep the activities in this training material concise.

Suppose you now want to use Autograph from your interactive whiteboard.

Go to View Preferences Whiteboard and make sure all four options are ticked.

At this point you should notice that lines are thicker, text is enlarged and the Onscreen Keyboard has opened.

Click Text on the Onscreen Keyboard to show more keys.

Attach a point to the curve y = x², right-click and choose Vector. Enter ( ).

Click Text on the Onscreen Keyboard to return to the minimum configuration.

Use the left and right arrow keys on the Onscreen Keyboard to move the point along the

In Whiteboard Mode to select more than one object you simply click on the objects in turn.

You must remember to deselect everything before starting a new selection. You can deselect everything by clicking in an unoccupied part of the graph area or by pressing Esc on the Onscreen Keyboard.

Click Esc on the Onscreen Keyboard to deselect everything.

Click on the point at (0, 0) and the curve y = (x – a)² + b, so they are both selected. Then right-click and choose Move to Next Intersection.

Select the area between the curves and click Text Box. Use the Onscreen Keyboard to change the word “Area” to “Area Between two Quadratics”.

Click and drag the yellow diamond so it is pointing at the area between the curves.

Use the Constant Controller to change the values of a and b.

The Onscreen Keyboard can be used to type mathematics in other applications, e.g. emails.

Use the Onscreen Keyboard to type: ∫ 1/√(1 + x²) dx = sin x + C The character will not display correctly because it is not in most fonts. Go to Format Font and select Arial for Autograph Uni.

The Arial of Autograph Uni font was specially commissioned for Autograph to support certain mathematical symbols that are not available in normal fonts.

Technology in Secondary Mathematics More Autograph Resources www.tsm-resources.com/autograph

**Visit the Technology in Secondary Mathematics (TSM) Autograph page for resources, including:**

TSM Workshop 2012 www.tsm-resources.com/tsm-2012 The annual TSM workshop has now been running for 10 years and provides the opportunity for teachers of mathematics to learn how to use Autograph and other software in the classroom.

The workshop is a full three days giving participants time to learn Autograph at their own pace.

For those wishing to go a bit further there is the chance to qualify as an Autograph Certified Trainer.

To enter x² type xx or press Alt-2.

Click Pause Plotting immediately.

Use the Scribble Tool to mark the points where the graph y = x² + x – 2 crosses the axes, any minimums or maximums, etc.

Use the Constant Controller to change the values of a, b and c. Observe the relationship between the vertical line and the quadratic.

Choose a parameter in the Constant Controller using the drop-down list. Change the value of the parameter using the up and down arrows and change the step size using the left and right arrows.

Select the vertical line, then right-click and choose Delete Object.

Enter a point with coordinates (b² − 4ac, 0).

Select the point, then right-click and choose Circle (Radius). Enter 0.4 and click OK.

Select the point and add a Text Box, tick Show Detailed Object Text.

Use the Constant Controller to change the values of a, b and c.

Take careful note of the values of b² − 4ac and the corresponding position of the graph y = ax² + bx + c for various values of a, b and c. Hint: Look at how many times the graph hits the x-axis and compare this to when b² − 4ac 0, b² − 4ac = 0 or b² − 4ac 0.

Make a conjecture about how you can tell the number of roots there are to ax² + bx + c = 0 by calculating the value of b² − 4ac.

Linear Programming Problem A group of students is planning a day trip to London to raise money for charity. They have priced tickets at £10 for adults and £5 for children.

Constraint 1: The minibus they have hired can only seat 14 people.

Constraint 2: The event will only run if there are 10 or more people.

Constraint 3: There must be at least as many children as adults.

Solution Let x be the number of children and y be the number of adults, then the three constraints can

**be expressed as follows:**

Constraint 1: x + y ≤ 14 Constraint 2: x + y ≥ 10 Constraint 3: x ≥ y

Type = to get ≤ and = to get ≥.

The students want to make as much money as possible for charity, in mathematical terms they want to maximise 5x + 10y. We call this the Objective Function.

Use the Constant Controller to find the maximum value of 5x + 10y (and the respective values of x and y) such that the objective function remains in the feasible region.

Cubic Investigation In this investigation students would normally first be introduced to a special case, for example y = (x – 2)(x + 3)(x + 4), and then asked to look at this more general case.

Select the point, then right-click and choose Tangent.

What do you notice about where the tangent crosses the x-axis?

Select the tangent, right-click and choose Edit Draw Options. Change the Dash Style to Dashed.

Use the Constant Controller to change the values of a, b and c. What happens when two roots are equal? Can you make two of the tangents parallel?

Now that you have seen this result can you prove it mathematically? Assuming one of the roots is 0 will make the mathematics a little easier.

Iteration Many equations cannot be solved using conventional methods, for example 2 = x³. In such cases we need to use numerical methods to find solutions.

The character is available through the Arial for Autograph Uni Font.

By eye what do you think the x coordinate is at the intersection of these curves?

Show that you can rearrange 2 = x³ as x = (2 )^(1/3). Therefore the x coordinate at the intersection of the graphs y = x and y = (2 )^(1/3), is the same as the x coordinate at the intersection of y = 2 and y = x³.

We are going to use the iterative formula xn+1 = (2xn)^(1/3) to find the solution to this equation.

Select the point and the curve y = (2 )^(1/3), then right-click and choose x=g(x) iteration.

Click on the right arrow to step through the iteration. What appears to be happening on the graph page and in the dialog?

Zoom in to inspect more closely what is happening.

Parametric Equations Many different types of equation can be entered in Autograph: cartesian, trigonometric, exponential, hyperbolic, implicit, conics, polar, parametric, piecewise and differential. In this activity we look at a parametric form of the Lissajous equation.

Open the Constant Controller and set the step size to 1 so we are only considering integer values of b. Investigate the family of curves by varying b.

What can you say about the ranges of x and y for those curves? What can you say about the value of b if the curve is closed? Is there a relationship between the value of b and the number of regions enclosed by the curve?

Notice how the graph is traced and retraced. Try different values of b. Is the graph always retraced? Make a conjecture about the value of b and the number of times the graph is traced.

Change the step size to 0.1 and repeat the investigation to see if this offers any insight into what happens between odd and even values of b.

Open Help (F1) and navigate to Resources Example Equations: 2D Parametric Equations and try some of the examples there.

Binomial Theorem The Binomial approximation is often used for approximating powers of numbers close to 1, but how close to 1 do we need to be in order for the approximation to be any good? For reference

**Newton’s generalised Binomial theorem is:**