September 13th, 2006
I’ve lost the origin of this puzzle, but it has a unique solution. It’s a good puzzle for bright students at KS3 & KS4 to frown about
A census taker comes to a man’s home. Upon inquiring about the ages of his three children, he receives the following reply: “If you multiply their ages, you get 72. Furthermore, the sum of their ages is the house number”, which the census taker knows.
The census taker considers the situation for a moment, then says: “Okay. I need one more piece of information.”
“Sure thing”, replies the man. “My oldest child loves chocolate.” The census taker smiles, thanks the man, and walks away.
How old are the man’s three children?
Posted in puzzle •
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September 13th, 2006
Geometer’s Sketchpad is one of my favourite programs but extremely annoying to author in. Here I have created a zipped archive of some of my past Sketchpad efforts for anyone who wants them. I have included a brief description of each file below.
If you don’t have Geometer’s Sketchpad then these will not be very useful to you. You can get hold of a copy from The Curriculum Press in the UK.
ConstructingAnEllipse
An extremely simple file exhibiting the construction of an ellipse defined by two centers and a point. The problem is that the “Locus Constructions” that are revealled don’t tell the whole story; how did I arrive at choosing those two circles to construct the circle with? This requires thought about the definition of an ellipse and the nature of construction.
ConeFromCircle
If you cut a sector from a circle (the bright-red sector) you can join the two radial sides together to form a cone. What is that cone’s volume? How do you maximise the volume of the cone? This sketch allows an exploration of the structure of the problem prior to doing the maths on it. It is a good exploration for construction of graphs at lower levels, or using calculus at higher levels.
Projectile Motion
A useful sketch for teaching calculus, start with the initial position of the projectile on the ground. Some of the ‘features’ on this sketch are odd and may require some exploration before they make sense!
ProjectiveCuboid
An example of how to construct a cube using two-point perspective
DoQuadrilateralsTessellate
I believe that this sketch comprises a proof that all quadrilaterals do in fact tessellate. It is an interesting question to whether it does in fact constitute a proof (what is a proof anyway?!) Choose your quadrilateral and click the buttons in order to follow the argument steps.
HexagonalSlideTessellation & HexagonalRotationalTessellation
Dynamic examples of how to distort tessellating hexagons in such a way to ensure that the result also tessellates.
Egg
An interesting construction of an egg using four arcs of four circles, it questions students to consider the relationship between mathematics and aesthetics.
Posted in KS4 (GCSE), KS5 (VI Form), geometry •
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September 8th, 2006
To motivate students to think about the nature of language and language acquisition, the following activity is great fun. It requires a lot of patience and careful management. The activity is to communicate only through clapping in order to move a student to a particular position in the room.
Setup
I ask one game student to leave the room so that he cannot hear the class instructions. He will be invited back to a very strange environment, and that he shouldn’t be embarrassed or worried about the activity, but just do what he thought we wanted him to do.
Once he leaves the room, I tell the students that they may not talk under any circumstances from now on. They should not discuss with one another the activity until I say they can. I will bring the volunteer back into the room, and they are to make him move to the front of the room standing exactly where I am now (move to some unlikely position first). They can do this only by clapping. They can use body language if they must, but can’t use pointing, but or try to mouth words to the subject. I then ask one student to go and get the subject and bring him into the room.
The likely result is disaster! Allow the disaster to run for a while and then stop the class. Ask the subject to leave. Ask a couple of students to to review what happened, allow students one minute to discuss with one another strategies. In one minute, stop the discussion, and get the subject back in.
Repeat this a few times. It may result in total disaster, but more likely they will eventually settle on a hot/cold strategy whereby the clapping intensifies as the subject moves to the room.
A New Subject
If possible, then ask another teacher to come to the room - this may need to be setup ahead of time. The class should by this point be quite coordinated in their efforts, and hopefully the teacher will move to the correct position.
Discussion
Read the rest of this »
Posted in KS5 (VI Form), philosophy •
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September 8th, 2006
With this sheet cleared and on the whiteboard I ask a student to give me any two numbers. Whatever the total is, I suggest to her that her numbers are no good because the last box doesn’t equal the target, and ask if any other student would like a go. Slowly, in their own time, I allow students to guess and get things wrong repeatedly. This can continue as long as need be. It is important to allow them to discuss the answers with one another.
When a student eventually gets a correct combination in the first and second box I click the record button and it is kept underneath (macros need to be enabled for this - check them first please to ensure that I’m not trying to give you a virus - I am not, of course!).
Slowly a pattern should emerge which should lead to a raft of solutions. At this point the students will hopefully start to talk about the patterns. You may want them to write their explanations of patterns down or discuss them in groups to report back to the class.
Depending on the level of the class there are then several options that the students could be asked to work on.
- Taking the first box on the x-axis and the second box on the y-axis, drawing a graph. How should the graph be described?
- Labelling the boxes with a and b, what should the next box be labelled (”no, not c” I hear you explain to numerous students!)
- Have we got all the solutions? This can lead on to thoughts about different sets of numbers, as they will ordinarily not wish to get out of positive integers.
This idea is an extrapolation of an idea I once read in a 1970s-1980s copy of Micromath or Mathematics Teacher from the ATM, though the spreadsheet was made by me (it’s very simple). If you have any reference for this idea, please let me know or add a comment so that the author of the original idea can be recognised.
