Who: J1 and J2
When: while not hacking minecraft on raspberry pi
where: in my home office
Recently, we found a couple of nice games and puzzles built around multiplication that I wanted to share with y'all. The important point is that these aren't drill-in-disguise where the primary objective is reciting multiplication facts, they are games with their own goal that is facilitated by multiplication.
BojagiIn this puzzle, you have to cover a grid with rectangles. The trick is that the grid has numbers sprinkled throughout and every rectangle you draw has to contain exactly one number that is equal to the area of the rectangle.
|Solving (left) and make-your-own (right)|
Here's the game and a collection of puzzles: http://bojagi-gotmath.rhcloud.com/
For credit, the game was built by David Radcliffe (@daveinstpaul) and I originally read about this game on Moebius Noodles.
Why do I love it?
(1) Fun and challenging puzzle worth doing on its own
(2) Great reinforcement of the area model for multiplication
(3) Tremendous scope for further investigations
(4) YOU CAN MAKE YOUR OWN PUZZLES!
The last two points are related. Creating a puzzle helps stimulate thoughts about the structure of the puzzle.
Examples of things you might want to explore
a. Is there always a single solution or could there be several?
b. How many ways are there to partition a rectangle into sub-rectangles?
c. If there can sometimes be multiple solutions, can we recognize this or recognize that the solution will be unique in advance (before we solve the puzzle)?
d. If you put numbers into the square grid, will they always form a puzzle that can be solved? If not, what conditions are necessary? What conditions are sufficient?
e. Is there an algorithm that will always find the solution (when one exists)?
f. Are any game versions fun (and what mathematical structure do they have?
This game comes from Calculation Nation. They have several good games, so it is worth taking a look at their collection. Other than this one, I particularly enjoy Nextu.
The objective here is to get 4 in a row before your opponent does. On each turn, you move one of the sliders below the playing square and capture the square that is the product of those two.
Why do I love it?
(1) The game is fun and challenging (this is the sine qua non of games, no?)
(2) Players have to use multiplication and division when planning their next move
(3) The interaction with your opponent is not straightforward
(4) J1, J2 and I had a delightful conversation about which numbers are in the playing board, which are missing, and why.
In this case, I don't see as much scope for further investigation, but you might have more ideas than I do. At least I will leave you with one:
Why should we have expected that the playing board would be a square, thus justifying the name?
Some other things related to multiplication
A beautiful multiplication table emphasizing prime factorization, from Math4Love:
This is linked with their game Prime Climb (which I don't have, so insert unhappy smileys here!)
As the name suggests, a game related to factorization. Rules here. I will blog about this when we play the game at home or with one of the school classes. Also, I noticed a pencilcode user that started building a program related to implement this game (Introbot's FactorGame).
An alternative algorithm
A video (here) and picture (below) are getting pushed around the web. My take:
thoughtlessly teaching/learning/applying any algorithm isn't very useful, but playfully investigating and thoughtfully considering why it works is always worthwhile.