The Magic Math Lanterns: guided visualizations


The game of the magic math lanterns

by Maria Droujkova
Laterna Magica is a seventeenth-century ed tech invention. It projected beautiful transparent pictures on a wall. Let us invoke this object to describe a way of sharing beautiful, transparent math.

The Magic Math Lantern is a storytelling or meditation game. The leader guides participants in visualizing steps in a construction, and they respond describing what they see. As in other Glass Bead Games, the most important game mechanics are: connections among concepts, long pauses for reflection, and the resulting a-ha moments.

A good Magic Math Lantern setup uses elements of the classic story structure. It starts with simple, easy to see elements that set up a stage, such as a horizontal unit interval, or a circle and its diameter. Next, something happens in the story: an adventure that challenges the players, stretches their imagination and invites them on a math quest. They may need to coordinate several triangles mentally, or imagine a 3d rotation, or iterate a transformation, or apply a theorem to adjust lengths in their mental constructions to results of a computation. Finally, when the quest and the mental geometric construction is completed, the players experience an a-ha moment. This culmination of the story restores the harmony at the new level of understanding.

Sometimes the game develops according to the leader's initial plan. Other times, players see and share an alternative construction that is worth investigating. Leaders should expect and embrace the unexpected.

Below, you will find a few Magic Math Lantern scripts. We need to have some live videos to go with them, eventually.

Spiral of Theodorus

by Maria Droujkova
- Imagine a horizontal interval
- OK
- Make it length 1
- Now start at its right end. Imagine a vertical interval, also length one, going up from there.
- Yes
- Connect the ends of the intervals. What do you see?
- A right triangle
- What is the length of its hypotenuse?
- Pythagorean theorem... One plus one is two... It's two!
- Is it?
- Wait no, it's square root of two.
- Look at the hypotenuse closely. Go to its upper end. From there, draw another length one interval, sticking out to up and to the left, perpendicular to the hypotenuse
- I see it
- Connect the end of the hypotenuse and this interval
- Wait, I am lost
(Talk through the previous construction again, with smaller pauses)
- What do you see?
- A new right triangle!
- What is the length of its hypotenuse?
- Two and two is four... Four!
- Uh, no, look again
- Oh, it's two and one - it's square root of three! And now we are doing the same thing again, right? Forever and ever!!!
- Yay! What's the length of the next hypotenuse?
- Square root of four, then square root of five, square root of six.
- How does the whole thing look?
- Hard to see. I think it's a spiral. Yes it is!

external image 400px-Spiral_of_Theodorus.svg.png

Multiplication vs. repeated addition

adapted from a story by Rebecca Hanson

- Imagine... A tree
- Yes
- Imagine... Your tree growing tree times higher.
- Yes
- Imagine... A tree seven meters tall.
- Yes
- What do you see?
(a discussion of how to imagine that particular height)
- Imagine... How you calculate that the seven-meter tree grew three times higher.
- Yes
- What do you see?
- (answers)
- Is it the same picture as the first growing tree?
(divergent discussion)

Picture for the leader:

Star polygons

a math club activity, Fall 2010 Natural Math Club

- Imagine... Six children in a circle.
- Yes
- Imagine... One with a ball.
- Yes
- Imagine... The child counts kids starting from the neighbor - one, two. And passes the ball there.
- Yes
- Imagine... Doing the same again. And again. And again.
- Yes
- Do you see all children playing with the ball, or some?
(divergent discussion - depending on what participants see, and why)

The next visualization can be with five children in a circle. Pictures for the leader: 26208371@N06/5081186838/
external image 5081186838_4a7a4b5bc8_m.jpg StarPolygon.html

Bisectors in a triangle

MariaD and her 12yo, a car game

- Imagine... A triangle.
- Yes
- Imagine... The line splitting the first angle in two.
- Yes (a discussion may happen here about angle splitting)
- Imagine... The line splitting the second angle in two.
- Yes
- Do you see the lines intersect?
- Yes (or discussion if No - what do they see)
- Focus... On the point where lines intersect. Let's call it Point O.
- Yes
- Imagine... The line splitting the third angle in two.
- Yes
- Do you see this line passing through Point O or not?
(divergent discussion, e.g. "Will it always be so?"; this is a setup for investigating the bisector theorem more formally, using paper).

Picture for the leader: http://rchsbowman.wordpress. com/2009/11/03/


Orange topology

by MariaD
We had a very intriguing magic math lantern experience during Apple Math (snack time) yesterday. A couple of weeks ago I asked the kids how to cut an orange into two pieces that have three skins. We discussed the topology of the situation a bit - what counts as a "piece" and what counts as a "skin" and what are allowed cuts. Kids had some good ideas, such as to cut skins separately, which we had to discuss. As an aside, it's respectful to acknowledge that kids did solve a problem AS STATED and to offer "the next one" with modified conditions when they find loopholes in your conditions.

Anyway, yesterday we had a bushel of excellent red oranges for snack, and I attempted to demonstrate how the solution works. The trick is to cut the orange through. Here is the result of my fifth or so attempt, which came out right: 26208371@N06/5169487343/

The "hole" comes out in one piece with two skins on either end, and the rest is one piece with one skin. Here Ava is looking through it.

Kids all wanted to eat the super-special "hole" piece with two skins. I asked them how to solve the problem (my suggestion of "Fight to the death for it" did not get approved). The solution was to cut the piece into four, lengthwise. I asked kids to imagine how many pieces AND skins this will make.

Now, some of these children have been solving fruit cutting puzzles of various sorts for a couple of years. Still, nobody could predict what will happen to the magic orange.

Which tells me it's a rich problem setting worth investigating more. I worked with this formula (mx+b, in this case 2x+1) in growth patterns, though with older kids.

Here we are counting the results of cutting: 26208371@N06/5170107682/

Origami by phone

from British Origami Society (more there)

The art of verbal instructions, sometimes known as "over the phone" folding, makes you realise (in my case anyway) what a poor grasp of your native language you have! Phrases like "fold to here" or "this way up" have no meaning. Some years ago I taught a group of blind students who encouraged me (none too gently!) to explain myself clearly. Have a try for yourself - sit back to back with another folder (it's cheaper and more friendly than phoning) and teach them a simple fold.

traditional - text Mark Gilcrist
Find the center of a 8.5x11" sheet of paper and fold each long side to the middle.
  1. Fold back 1" off the top of the paper. Crease to the back side.
  2. Bring top corners together to the middle of paper, meeting about 3/4" from top edge. (This makes the shirt collar)
  3. Bring bottom edge of paper up to fit snuggly under collar at the top. Crease the fold.
  4. Re-open the flap just folded. Fold open flaps made from middle crease to lower corners. (These flaps become the sleeves)
  5. Leaving new flaps open, refold lower edge to attach under collar. Sleeves are now present.
  6. Add stamped patterns to shirt and maybe buttons or a tie.

Proof without words

from MAA, submitted by Garrett Eastman

This can be a good source of guided meditations!

Shapes out of shapes - introducing the magic lantern

via Kirby Urner; game by Maria Droujkova

There is an activity we do with little kids... We take a 3d object (a favorite toy or your hand or something else interesting) and decompose it into simple shapes and then draw the shapes. I ask kids to trace the shapes over the object first. If they don't understand the game, I place the object behind a glass window and they trace 2D shapes on the window, with a marker. The drawings come out amazingly fun.

I think I will use this collection to introduce the idea. And this activity overall can introduce the magic lantern game.

How many cubes fit into a sphere?

Imagine a sphere of radius 1 (2, 3...). Imagine stacking cubes inside. How many fit?

Future ideas


Sine function

The arrow rotates around and around... Picture for the leader:

Convex shapes

LinkedIn Math, Math Education discussion (members only)
  • Imagine a convex shape
  • Imagine another convex shape
  • Imagine them intersect
  • Is the intersection you see convex?

sin(45°) and doubling the area of squares after Socrates

From Math Future discussions (1, 2) and Math Game Design discussion
  1. Imagine a unit square
  2. Imagine its diagonals
  3. Imagine the triangles formed by the diagonals unfolding, making a larger shape.
  4. What is this shape?
  5. What is the area of this larger shape?
  6. What is the length of its side?
  7. (look up and compute numbers in decimals) sqrt(2)=1.414 and sin(45°)=0.707
  8. From the diagonal of the smaller square, sin(45°)=1/sqrt(2) but from numbers it looks like it's sqrt(2)/2 - why the "coincidence"?


based on work by Linda Stojanovska
  1. Imagine an interval.
  2. What color is it?
  3. Call it something! (Let's say listener calls it "a")
  4. Attach another interval at the end of a, on the same line.
  5. What color is it?
  6. Call it something! (Let's say it got called "b")
  7. Are you seeing the interval a+b?
  8. Imagine the point (a+b)/2
  9. Where is it? (middle)
  10. Now imagine the point (a+b)/3
  11. Where is it?
  12. ...(a+b)/N - repeat until the pattern of approaching the starting point is seen
  13. Now do it with coordinates of points A and B
  14. Where is (A+B)/2?
  15. Where is (A+B)/3?
  16. Wow - it's not on AB

Miscellaneous good ideas