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Le Plat Diviseur

August 31st, 2006

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A plat diviseur is a plate designed to make portioning cakes easy. Click here to see one. It works by putting dots at appropriate angles around the rim of the plate, so that when someone wants to cut, for example, five portions they merely cut from the center of the plate to wherever there is a five. For neatness one cut is always common, and tends to have just the number 0!

Displayed on screen this image could be used to motivate an activity in which students practice finding angles by creating their own plat diviseurs. It can be quite a fun activity, especially if you buy paper plates for them to use (though I recommend that they practice on paper first!).

They could also be asked which numbers are absent; they hopefully notice that 2,4 and 8 are missing. Discussing why can get into thinking about halving and fractions.

The reciprocity of 3 and 1/3, 5 and 1/5 can also be brought up, by discussing how often the image of a divided circle is often produced as a visualisation of proper fractions.

Since 5 and 7 are not factors of 360 this is also an interesting discussion point. While discussing factors, they should note that points for 3,6 and 9 are coincident at two points, and that if 2,4 and 8 are added what other numbers would have coincident points. I remember asking whether 10 and 12 would have coincident points if we had them as divisions too, which caused some thought.

Note that there are very few good images of plat diviseurs on the internet. I originally got the idea from Problem Pictures.

Domino Problem

August 31st, 2006

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How many spots are there in a complete set of dominoes?

It’s a lovely problem that offers the opportunity to discuss methods of calculation, and methods for avoiding mistakes. It links to combinations and permutations, as well as triangle numbers, and is a really simple question to pose. An intermediate question is to ask how many dominoes there are in a set, and establish that solution before continuing to the original question.

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Flickr – A Free Problem Pictures

August 30th, 2006

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At University I was introduced to a really interesting CD called Problem Pictures. It had some fascinating images that all had some mathematical significance. I printed out about fifty and they made the walls of my classroom bright and appealing. It also had questions related to each photo, but I found that I rarely used them, preferring to create my own questions and ideas.

Enter Flickr, a great online photo sharing tool. Though you will not get questions related to each photo, the database is huge. Typing into the Flickr search engine ‘geometry’ turns up 10,810 photos at the time of writing. Most of them are really relevant; some of them are utterly spectacular. Here is a random sample:

Top Hit in Flickr Geometry Search at time of post

 

#flickr_badge_source_txt {padding:0; font: 11px Arial, Helvetica, Sans serif; color:#666666;} #flickr_badge_icon {display:block !important; margin:0 !important; border: 1px solid rgb(0, 0, 0) !important;} #flickr_icon_td {padding:0 5px 0 0 !important;} .flickr_badge_image {text-align:center !important;} .flickr_badge_image img {border: 1px solid black !important;} #flickr_www {display:block; text-align:left; padding:0 10px 0 10px !important; font: 11px Arial, Helvetica, Sans serif !important; color:#3993ff !important;} #flickr_badge_uber_wrapper a:hover, #flickr_badge_uber_wrapper a:link, #flickr_badge_uber_wrapper a:active, #flickr_badge_uber_wrapper a:visited {text-decoration:none !important; background:inherit !important;color:#3993ff;} #flickr_badge_wrapper {} #flickr_badge_source {padding:0 !important; font: 11px Arial, Helvetica, Sans serif !important; color:#666666 !important;}

I have found that putting an image full screen on an interactive whiteboard when students come into the room is a great way of capturing the students’ attention straight away.

Bracket Expansion

August 29th, 2006

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This activity is for a first introduction to expansion of brackets or for a review activity.

Bracket Expansion Excel Spreadsheet

Imagine telling a class that 4(x+3) = 4x+12 and them asking you “why?”. What if instead you presented the information to them in a way that they generated the idea that they were equal and the explanations for their ideas themselves?

This sheet generates pairs of equations (for just expressions, read on)*, one with the bracket in place, one with it expanded. It is a very simple spreadsheet. It contains simple macros that allow either equation to be hidden. Though you can use it however you like, my suggestion follows…

I would use the sheet with both equations or expressions* visible and change the values a few times. I would ask students to try to explain the relationship between the two expressions.

Most likely the first thing that they notice is the methodological relationship the product of the two numbers in the first expression is the last number in the second expression. A brief glance at some other variations will demonstrate that this is the case.

However, they are unlikely to have grasped the numerical relationship; that they each have the same value, irrespective of the value of x. This will probably require some prompting. You could ask each student to choose a value of x and substitute it into the two values. What do they notice. Choose random numbers a few more times. Repeat the activity. Once they have established that they are always getting the same numbers then you can move on to the big one: the conceptual relationship.

The concept we’re aiming at is surely for the students to grasp that the link between the two expressions they first noticed is necessary for them to have the same value. It will be difficult for this activity alone to convince them of that, but I would at this point urge them strongly to consider whether the two things they’ve noticed are related. Since they’re expecting a relationship, they may trust that linkage. I haven’t linked it for them however, at least not in their minds.

I would then remove the expanded expression and ask them to construct it themselves, making sure that for whatever number they make x the two expressions match. This is a crucial check that should be encouraged early.

* The ‘y=’ on both the equations can be easily removed to just give the expressions. To be honest, that’s my preferred usage, because of the problems students have understanding when we’re using equality as a question signifier and when we’re using it as a binary relation between the two expressions on either side. To get rid of the ‘y=’ click on the ‘control’ tab and delete G4 and G5, where the ‘y=’ text is seen. Go back to the ‘display’ tab and they will have gone.

Also note that you must have macro security set to medium and allow macros for the buttons to function. Please disable macros and view the macro code if you are unsure whether this is safe, or if you do not understand code, ask someone who does.

Countdown

August 29th, 2006

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Countdown Flash Executable

There may be more clever Countdown applications out there, but I like mine with all its wobbly buttons. The anagrams and conundrums are included in the game because they are included on TV. In my opinion, a little bit of letter work is good logical reasoning and therefore constitues just as much a maths class warmup as the number problems. It also hooks non-mathematical students into the game, especially if you keep scores in some way!

I appreciate the concerns you may have in downloading and running an executable file (or if you don’t have those concerns, you should!) You download the file at your own risk of course, but it is simply a .swf file wrapped in a Flash Projector. I do it this way because some schools don’t have the latest Flash player installed on their systems. I am as certain as I can be that it is virus free.

Mathematics and Aesthetics

August 28th, 2006

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Teaching Theory of Knowledge you teach a lot of students who actively dislike mathematics and don’t want to engage in any mathematical thought. The idea that mathematics could be a beautiful thing, or that beauty may have a mathematical aspect is often surprising.

This activity was one I concocted for my students and it has worked well on numerous occassions. Having downloaded and printed in colour the Mathematics and Beauty Cards PDF you can give a set to a group of four to six students with the straightforward instruction that they order the cards from most beautiful to least beautiful without any equivocation. This will likely take them twenty minutes. If you have an interactive whiteboard you may then also find useful the Mathematics and Beauty Flash Application which offers a very simple mechanism for reviewing the decisions students make.

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Marx & Engels – The German Ideology

August 28th, 2006

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My dad’s great guide to Marx & Engel’s famous work deserves to be used by Philosophy students:

<h2 style="text-align: center;"The German Ideology – A Student’s Guide

You are welcome to use this file in whole or in part for any educational purpose so long as the author is recognised.