games,  homeschooling,  mathematics

Math and Graph Games.

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Yesterday the children and I taped sheets of scrap paper to the floor to create the Land of Many Ponds. Basically it involves a lot of “ponds” connected by little streams, represented in the map by pieces of paper with masking tape trails for the streams.
The basic idea of the game is to have three people moving around the map at different rates. One person moves from one “pond” to the next. The second person always flies over a pond and lands on the second one from him, and the third person lands only on every third pond. The three people must somehow meet up.
(The long-since defunct webpage I originally got the idea from had a complicated story about a monster who moves from one pond to the next, a swan who can fly to the second pond and a superhero who jumps over to the third pond. The monster wants to be friends but the swan is too scared to do so unless the superhero arrives at the same pond at the same time to let her know that the monster isn’t going to hurt her.)
Each pond is connected to three other ponds and everything forms loops. No pond is further than three steps away from any other pond, but because your taking steps according to specific multiples you have to plan your route carefully. We had friends over and four children playing, so we had two people moving three ponds. The children were very excited and not much strategy was followed, so I left the map taped to the floor and decided we’d play again later when the kids were calmer.
Our later attempts worked easier. The kids had exhausted themselves jumping from one pond to the next and were ready to play by the rules. Also, I think the game was too hard for the kids until after they had a bit more experience moving around the board. If I do it again with other children, I’ll start everyone off playing a simpler game, like having everyone choose a pond and then putting a marker in another pond, and challenging everyone to get there by going over six other ponds. Children need practice planning how to get to an unmoving location before they start trying to meet at moving ones.

Other alternative games are as follows:

  1. Place differently colored blocks randomly on four or five of the ponds. Two players start at alternative ends of the “rainbow.” (So if your using four blocks, one person might start at whatever color is closest to the top of the rainbow and another at whichever color is closest to the bottom.) The players choose or are assigned a how many steps they are allowed per turn. I recommend using 2, 3, 4, or 5 although we did experiment as well with 8. Each player has to then try to move through, stopping at each of the colored blocks in order going either up or down the rainbow. (We start at alternative sides so that no one can just copy the other person’s path.)
  2. Play tag. Whomever is it can move three spaces. Everyone else moves just two. A person is tagged only if It can end his turn on the same space that person is on. Alternate It taking a turn and everyone else taking a turn. After a while, switch it up. Whomever is it can only move two spaces and everyone else moves three. How does that change the game?
  3. Play guessing games. Place an assortment of colored blocks on every pond. Have someone call out a hint, and everyone has to run to land on a pond that matches the clues. In this game people don’t have to follow the paths to move but the hints should be related to the paths. Hints can be things like: Find a pond with a blue cube. Or find a pond that is two steps away from a pond with a blue cube. Or find a pond that is not a neighbour to a pond with a red block.
  4.  Stand on a pond and challenge everyone to move to a different pond as far away as possible. (Distance is measured by the number of edges connecting them, not by physical size.)
In addition to playing these games we have been exploring Euler’s (pronounced “oiler”) Curcuit . Euler’s Curcuits are connections of dots/vertices and lines/edges that are connected in such a way that a person can trace the whole pattern without lifting the pencil, ending on the same location one started and not going over the same line more than once.  We made simple ones at first and then more complicated ones, drawing them first in pencil on a sheet of paper and then placing the sheet in a clear plastic folder so that we could use dry erase markers to trace it over, erasing and redoing it as many times as we want. The basic rule is that as long as every dot has an even number of lines connected to it, the pattern will be a Euler’s Curcuit. If two of the dots have an odd number of lines, then the graph will be a Euler’s path instead, and can be traced starting at one odd vertice and ending at the other odd one.

I presented the logic to my oldest by asking him how many times he’s gone through the closet door. He hides there frequently, but of course he hasn’t counted. Then I asked if he’s been through the door an even or an odd number of times. The smile grew on his face as he understood that it must have been an even number of times, because if it was an odd number then he’d still be inside the closet! So we talked about the lines of an Euler’s path being like doors that can only be gone through once. If you go through it, your going to be stuck in that room, unless there’s another door/line. If there is, then you can go through. You keep following the open doors around and around. The only ways to “lose” on a Euler’s path is to start at the wrong location or go to the ending location too soon, and the only way to “lose” on a Euler’s cycle is to return to your starting place too soon.

After using the Euler’s cycles and the Land of Many Ponds, I asked my oldest son to compare the Land of Ponds with an Euler’s cycle, and explain how we could turn the one into an Euler’s cycle.

I also played a game walking them through an imaginary cave. I had drawn out a map (which the boys cannot see) and then another copy with just part of the map for them (just the “rooms”) with little squares to show many tunnels there are leading out. I made up the hints as I went along, but the map on the side is what it ended up being like. Number 1, I decided, was a room where whomever lived in the tunnel had bunkbeds and I threw in an age appropriate math question about the number of people who could sleep in the room. I also had the kids “find” a clue that told them that they could visit the monster once, but he’d eat them if they visited him twice. The monster lived in room 2, and they could enter his room once but had to leave by the same door they came. That left them worried about where to go, because it meant that three of the tunnels were too unsafe to explore – any of the three could lead to the monster. Only when they found a clue elsewhere saying that the monsters home was seperated from the entranceway (topmost elipse) by three rooms could they figure out where was safe to go and where wasn’t. Number three was where they found the treasure, but only after the monster gave them the key to the treasure chest. In amongst everything else they had some simple math multiplication questions pertaining to things like the number of rocks they had to move to clear a caved-in path, the number of treasure in the chest, the number of bunkbeds, etc. Since my oldest is learning the letters of the Greek alphabet I threw in a couple of puzzles matching letters to keys to open doors.
File:Rock paper scissors lizard spock.png
So far we’ve been using these graphs/maps/cycles for games, but on Monday my plan is to move onto showing my oldest other uses of graphs. I’ll try posing some of the simpler challenges from for him to solve. I also want to talk to him about directed vs. undirected graphs (using our Land of Many Ponds as an example of an undirected one, and the rules graph of Rock, Paper, Sissors, Lizard, Spock as an example of a directed one.) We might or might not go on to talk about finding the best path in challenges where the edges have different weights/distances/costs marked on them. Or about how a graph where the vertices are connected by single line (no loops) is called a tree.

I wonder sometimes why I do all this. We have a pretty decent math curriculum to work through and none of this other stuff is, strictly speaking, necessary. And yet I love trying to understand things, so I like the challenge for myself of learning about this kind of thing, to share with my children. I like definitions and classifications and I think the kids do too. My oldest doesn’t necessarily like memorizing names or technical terms so I try to minimize those, but his face brightens up when he understands some new connection.
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