Pop-up dodecahedron model: How to

I've made one of these from a cereal box and rubber bands, and it's delightful! Here is the model:





Here's how to do it:

Cut out two copies of the "pentagon flower shape" from a cereal box or other similar cardboard. Make each model about 10 cm across. Crease the join lines so that the five pentagons bend freely from the centre one.

Place the two models on top of one another as in the centre diagram. Lace an elastic band alternately over and under the "petals" as shown in the right-hand diagram, holding the model down flat.

Then let go and -- with any luck -- up pops a 3D dodecahedron! Do try this at home before you bring it to class. It is quite wonderful when the 3D figure suddenly appears, and making the model helps students remember the composition of this polyhedron.

Adapted from Ian Stewart (2009). Professor Stewart's Cabinet of Mathematical Curiosities. NY: Basic Books, p. 7. (A highly recommended resource book for your classroom as well.)

Paulus Gerdes' books at Lulu.com --many available as free pdf downloads!

Very sad to say, Paulus Gerdes passed away this winter at a relatively young age, and the world has lost a great ethnomathematician.

Gerdes' books are all available at this Lulu.com site, many of them as inexpensive or free pdf downloads. Here is an example of one of them:

Explorations in ethnomathematics and ethnoscience in Mozambique

One of the ways to work against neocolonialism and ethnocentrism is to work reflexively -- that is, to view one's own culture as the subject of anthropological or other study, inspired by a sense of the importance of all cultures.

Reading chapters 1, 2, 3 or 5 from this book by Gerdes with your group, can you think of things (or activities) in your own cultural world (and/or that of your students) that might lend themselves to mathematical study, exploration or exemplification? Would you consider using these in your teaching? Why or why not? How, when, with what focus?

Readings for our Feb. 18 class

Hi everyone. Next week is UBC's Reading Week, and we will be meeting off campus at Terra Breads Café in Olympic Village: 1605 Manitoba Street, on Songbird Square (with the huge bird sculptures), about 1 block west of Science World. The nearest Skytrain station is Main Street Science World.

Here are our readings for next week's class, on the topic of Proof:

1) Gila Hanna & Ed Barbeau on proofs as bearers of mathematical knowledge

2) Vicki Zack on Grade 5 students' counterarguments and proofs

3) John Borwein on experimental computerized mathematics, the pleasure of discovery and the role of proof

Readings for our February 11 class: Ethnomathematics

Hi all! Here are our readings on ethnomathematics:



1) Paulus Gerdes on "frozen mathematics" and geometry

2) Carraher, Carraher & Schliemann on street mathematics and school mathematics 

3) Marcia Ascher on Marshall Island wave maps as an example of ethnomathematics

'Something to think about': Mathematical problems from Mason, Burton & Stacey, Thinking Mathematically (2nd Edition)

It helps to be doing mathematical problem solving to get a sense of how it works for both learners and teachers.

As you work on these puzzles, note with your 'teacher bird' awareness how your 'learner bird' is engaging with the problem solving!

1) Thirty-one:
Two players alternately name a number from 1, 2, 3, 4 or 5. The first player to bring the combined total of all the numbers announced to 31 wins. what is the best number to announce if you go first?

- Try playing it!
-What do you need to record?
-What totals enable you to win in one move? Generalize!
-Can you find a strategy for playing that will guarantee you to win?

extend:
•What if 31 is changed to some other number?
•What if the permitted numbers are 1, 2, 3, 4, 5 and 6?
•What if there are 3 players?
•What if the permitted numbers are 1, 3, 5 or 2, 3, 7?

2) Sticky Angles:
Given a supply of sticks, all the same length, and a supply of angles all the same size, can you join the sticks together end to end at the given angle to make a closed ring?

-Have you tried it physically?
-Introduce a way of making a supply of angles!
-Have you stayed in the plane?
-Have you tried folding a strip of paper appropriately to mimic many sticks joined at the correct angle?
-Will your method always work, or is your angle special in some way?

extend:
•What is the shortest such sequence when it is possible?
•What length sequences are possible?
•Does it help to have more than one angle available, particularly when you are confined to the plane?

3) Sequence:
Write down a sequence of 0s and 1s. Underneath each consecutive pair write a 0 if they are the same and a 1 if not. Repeat this process until you are left with a single digit. Can you predict what the final digit will be?

-Specialize systematically.
-Allow your system to alter as you begin to see what is going on.
-Try to be systematic about patterns and not about lengths of sequences.
-Try working backwards from the final digit.
-Find a convincing argument to support your conjecture.

extend:
•Write down a sequence of 0s and 1s in a circle and proceed as before.
•Set your result in a more general context by using 0, 1 and 2 with some appropriate rule.

4) Milk crate
A certain square milk crate can hold 36 bottles of milk. Can you arrange 14 bottles in the crate so that each row and column has an even number of bottles?

-Depict the crate. Find a way to be able to manipulate bottle substitutes.
-Specialize to crates of other sizes
-How many bottles might there be in each row and column?
-Specializing to larger crates might help.
-Eventually you will deal with the 36-bottle crate, but what about square crates in general?
-Can you build up new arrangements from old ones?

extend:
•Try to place other numbers of bottles.
•Try rectangular crates.
•What is the largest/ smallest number of bottles that can be suitably arranged in a given crate?
•How many ways are there to place the bottles?

Some real-life workplace mathematical problem solving from the 90s (from Colours of Infinity)

In this older but nonetheless fascinating film on fractals, Colours of Infinity (featuring Benoit Mandelbrot, Arthur C. Clarke and Ian Stewart), Michael Barnsley talks about how he made a breakthrough in fractal image compression for video and computer images.

35:30 - 42:00.

YouTube video: Watch kids solve a word problem -- "How old is the shepherd?"

This is a replication of the famous "what is the name of the captain?" word problem -- a problem with misleading or insufficient information, and which many kids attempt to solve nonetheless.

"How old is the shepherd?" video

What do you think of this?