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Autograph Newsletter 12 – Don Steward Puzzles

May 26, 2014 - Autograph, Autograph Newsletters
Welcome
Welcome to the twelfth Autograph Newsletter! Each jam-packed edition looks at a specific topic in mathematics and how Autograph can help engage students and enable them to understand the key concepts better.
Contents
Introduction
Diagnostic Question
Free Online Autograph Activity
Further Puzzles
Video Tutorials
Handy Autograph Tip
Introduction
One of my all-time favourite mathematics blogs is Median, created by Don Steward (donsteward.blogspot.co.uk). The range of activities and the levels of depth, challenge and imagination are simply outstanding. The vast majority of my ideas for lessons and activities these days come from this blog and the equally outstanding NRICH. Many of the rich, intriguing activities that Don creates lend themselves very well to being analysed on Autograph. I am always keen to ensure that students first tackle the problems on pen and paper, and then Autograph is used to test theories, answer questions, and embark on wonderful extensions. Below you will find a selection of my favourite puzzles from Don’s website, with accompanying Autograph files. I hope you and your students enjoy them, and thank you to Don for creating such a wonderful resource.
Diagnostic Question
Usually we have a Diagnostic Question in this spot, but for this special edition of the newsletter we are instead going to have… a puzzle! Project this on the board, encourage your students to develop a systematic way of solving it, and challenge them to explain their method to their peers. Encourage them to compare each other’s answers and methods, and describe the merits and flaws with the different techniques. For a full discussion of the problem, grids to print and the solution, click here.
How many triangles are there
Free Online Autograph Activity
Leap Frog
Leap over the three points on paper first. Once you have noticed something, try to explain it. Then turn to Autograph to look at the puzzle in more depth.
Leap Frog
These Autograph activities do not require the full version of Autograph to run them. You just need to install the free Autograph Player (you will be guided through how to do this), which means you can use these activities in the classroom or set them for your students to do at home.
Further Puzzles
The following ideas for puzzles are also taken from Don Steward’s website. Try them on paper first and then turn to Autograph to look at them in more depth. Click on the image to download the individual Autograph files.
Idea 1 – Extending Leap Frog
Download  1. Leap Frog Extended.agg
One way of extending the Leap Frog puzzle above is to think about what happens if you leap half way towards each point instead of twice as far.
What do you think will happen?
Try this out on paper first, with each member of the classes starting with their points in different positions
What seems to be happening?
Which students have similar outcomes? Which have different?
Get your students to keep the three points in the same position but try another starting point
When you have some predictions, use the Autograph file
 Drag the three points and the starting point around and observe the effect on the path and position of the leaps
What is happening here? Why?
You can add more leaps if you like by creating vectors (see the Handy Tip at the end of this newsletter)
What will happen if you leap ¾ of the distance?
How about 1.5 times the distance?
Idea 2 – Four Lines
Download  2. Four Lines.agg
This classic geometry puzzle works particularly well on Autograph, and Don has added a second pattern that I had never seen before!
Can you use four connected straight lines to go through each of the 9 dots in the first pattern?
 Drag the points on the green dashed line around to experiment
When you have cracked this, try the same for the heptagonal pattern… and when you figure out how to join these up with four connected lines, please let me know how! :-)
Finally, can your students create a similar challenge?
Idea 3 – Quadrilateral Puzzle
Download  3. Quadrilateral Puzzle.agg
Don Steward produces another lovely little problem
On squared paper, get everyone in the class to draw any four points that form a quadrialteral
Find the mid-points of the sides of this quadrialteral
Join up the mid-points to form a new quadrilateral
Compare the new quadrilaterals that the class have formed
What have they got in common?
Can you explain this?
Look at the Autograph file
Drag the corners of the quadrilateral around the experiment
When is the new quadrilateral a rectangle?
When it is a square?
Can you explain this?
How about experimenting with triangles and their midpoints?
Idea 4 – Pizza Cutting
Download  4. Pizza Cutting.agg
Another classic problem that is easier to analyse on Autograph
Print your students out a set of circles to represent pizzas
Challenge them to create the most number of slices pizza with 1 straight cut
How about 2 straight cuts?
Things get a bit trickier when you try 3, 4, 5 and 6 cuts
Use the Autograph file to help analyse this further
Drag down the page to see more pizzas
Drag the corners of the quadrilateral around the experiment
Look at the maximum number of slices for each circle
Can your students spot a pattern?
Can they explain the pattern?
Can they use it to predict how many slices you could divide a pizza up into using 7 cuts?
How about 100 cuts?
Would it make a difference if the pizzas were a different shape? Square? Triangular?
Video Tutorials
The following video takes you on a short trip around Don Steward’s wonderful blog.
Handy Autograph Tip
In order to create the Leap Frog activity, I had to create vectors between two points, and then create multiples of these vectors. Here is how I did it:
Open Autograph in Standard Mode
Make sure you are in Whiteboard Mode
Place two points anywhere on the page fairly close together
Click on an unoccupied part of the graph area to deselect everything
Select the two points in turn, with the one you want to be at the foot of the vector selected first
Right-click and choose Create Vector from the menu.
Click on an unoccupied part of the graph area to deselect everything
Select the point at the foot of the vector at the vector itself
Right-click and choose Multiply Vector from the menu
Choose a Multiply Factor (e.g. 0.5)
It may now be useful to hide the original vector
Ensure that only the vector itself is selected, right-click and choose Hide Object from the menu
Dragging the points around to confirm that the vector has been created successfully
Challenge: Can you use this tip to construct the Leap Frog activity?

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