Posts Tagged ‘geogebra’
Visualisation: Six moving lines
As promised, here is an animated Geogebra visualisation. Click ‘Start’, and six moving lines will appear.
- What behaviours do the lines exhibit?
- What relationships are there between the different lines?
- How are these lines defined?
(Source: movinglines.ggb and movinglines.html.)
Click on ‘Show Values’ and three values will appear — all six of these lines are generated from these three values. If you click on ‘Stop’, and move the sliders, then the lines will be automatically updated to reflect those new values.
- Can you make any conjectures now about how the lines are defined?
- What values would we need to make one (or more) pair of the lines
- perpendicular?
- parallel?
- vertical?
- horizontal?
- Can we generate any line we wish by setting the values appropriately?
I managed to captivate both a lower school class and an A-level class with this visualisation for quite a while last year, and the A-level class made good progress in making and justifying conjectures about what was going on.
Technical Note
One of the few downsides of Geogebra compared to other dynamic geometry software, is that it contains no in-built facility for animation. Thus, to create any animations using Geogebra, you have to create the animations manually in Javascript. It’s not particularly difficult to do, particularly if you know some Javascript, as you can see if you view the source of the .html file above. Putting the ability to animate inside Geogebra itself is, though, my one big wish for future versions.
Tags: animation, geogebra, lines, visualisation
Hinged square dissection
In several recent posts I have referred to ‘linkages’ (which should more properly be called ‘hinged dissections‘. One good recent book on these is Hinged Dissections: Swinging and Twisting, by Greg Frederickson, but there are several classics out there that discuss hinged dissections, including Amusements in Mathematics by H. E. Dudeney.
Dudeney was responsible for one of the most well known hinged dissections, of which this is a simplified example. It converts a square into, well, something else.
The white circles are the hinges — move the coloured circles to move the corresponding parts of the square.
Can you predict what the end result of the transformation is? Can you prove it?
(Source: dudeney-dissection.ggb.)
The Haberdasher’s Puzzle
Dudeney’s classic dissection, published in his ‘Canterbury Puzzles‘ in 1907, is a slightly altered version of this, which allows you to transform a square into an equilateral triangle. You can download a program which will allow you print out a template for this here.
Constructing Dudeney’s dissection takes a touch more effort than the dissection illustrated above, but the process is described incredibly well in this lesson plan, which demonstrates how to make a model of the dissection using foam rubber. I haven’t tried it yet, but it may make a tempting break from lesson planning next week!
Tags: dissection, Dudeney, geogebra, linkage, puzzle
A quickie on quadratics
Fix two points in the plane, and consider all the quadratics which go through those two points.
Find the locus of the stationary points of these quadratics.
The following Geogebra worksheet might help. If you select “Show stationary point” and move the blue point, it will trace out the locus.
(Source: quadratic_stationary_points.ggb.)
Thinking about this question kept me from going to bed until far too late on Thursday. My answer, and other thoughts, will appear tomorrow.
Tags: algebra, geogebra, geometry, puzzle
Visualisation: Yet more rotating squares!
I opened my computer this morning intending to post something new, but soon got caught up in a further exploration of these square-to-square linkages. You soon notice, when creating these, that you are quite constrained in the points you can use to link the squares together. You are also limited in how ‘fast’ you can get the squares to rotate. Trying to use the least different speeds of rotation, I created this:
(Source: rotating_tesselation.ggb -- apologies for the spelling mistake :) )
Moving the slider, you get images like this:
This can easily be extended to a tessellation of the plane by squares and rhombuses.
Question: What does it look like if we do something similar with hexagons? You can see my attempt below. Also, are there any other regular polygons we can do something similar with?
No matter how tempting, I promise I'll move on to something else tomorrow!
(Source: rotating_hexagons.ggb)
Tags: geogebra, Maths, visualisation
Visualisation: More rotating squares
While we’re rotating polygons, here’s another nice visualisation, this time of a linkage of squares that make up a larger square, and rotate around without self-intersecting as you move the slider.
(Source: rotating_square_linkage.ggb)
Just as with the previous visualisation, you can see this just as a source of pretty pictures, like
or you can start asking yourself questions. For example:
- What other ways can we find to connect the small squares together so that they will expand and collapse back into a square without any of the small squares intersecting? How many of these have rotational or reflectional symmetry?
- How can we describe the movement (the locus) and the amount of rotation of the different squares?
- Can we do something similar with other polygons?
- More generally, can we do something like this to move smoothly between different shapes? And what does ’something like this’ mean?
For the second question, we can use the ‘trace’ feature of Geogebra (and all other dynamic geometry programs) to follow the path of particular points, which might give us some idea about how to derive the equations of the points.
On the last point, there are a large category of dissections which involve taking an object, slicing it into a finite number of shapes, and reassembling them into a different shape. There are several in Amusements in Mathematics, a puzzle book by Henry Dudeney from 1917 which I had a part in digitizing for Project Gutenberg. See this section, for example, for a great introduction to dissection puzzles.
Moving further afield, we can do something very similar to this example to demonstrate Pythagoras’ Theorem. That visualisation may appear later this week :).
Tags: Books, dissections, geogebra, visualisations