Mazes, Maps and Moebius Strips 2 – More Islands

p1030974This Thursday was the second meeting of our Mazes… math circle. Fortunately, everyone was well and we had a full house – RB, L, and E were back and A and V joined us. A’s younger brother, M stayed as well.

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To begin with we did a bit of a review – looked at the collection of shapes, discussed what they shared in common. Then I gave them a more difficult island puzzle. Interestingly, everyone approached it by either tracing the lines with their fingers, coloring in the ocean or “walking” the map with their fingers sounding off “land, water, land, water, land, water”. Neither RB, L or E mentioned evenness or oddness that they discovered at the last meeting.

p1030981My plan called for a more complicated island puzzle that all of them would solve together. But I really wanted to have time to start working on maps, so decided to skip it. But here it is in case you’d like to work on it.

Instead, we moved to another land-water puzzle. I showed it to the kids and told them that this is only a part of a map of an island. The rest of the map was lost. The only markings on the remaining piece were the word “Ocean” at the bottom and a few dots where treasure might be hidden. Since I only had one copy of the map, they were to work on it as a team.

p1030977The next 20 minutes were wonderful. I just stepped back and observed. Some of the kids used the same strategies as with the previous puzzles and applied them to this one. But tracing and coloring didn’t work very well because this was not a whole map. They refined their approach and drew a straight line from the “ocean” to one of the dots, sounding off “land, water” every time it would cross one  of the island’s contour lines.

RB kept disagreeing. He said that he believes all the dots are on land. He explained it this way:

Logically, we know we only have a piece of a map. It was torn out. So before that happened, there were other pieces of the map, other pieces of the island, around it except at the bottom. It is ocean there. And so around this piece, we know there was more island which is land. And we don’t know if the island had any rivers or lagoons or lakes. And it probably had just a solid shoreline, so ocean wouldn’t get into the island. And so then everything we see here is land.

A., in a very mature way that was absolutely amazing to observe, countered:

I think I understand what you’re saying. But if all of this is land, then what are the lines for?

The puzzle, even though it appeared to be identical to all the previous island puzzles, was different. The island wasn’t made out of a single closed curve. Instead, it had simple closed curves inside simple closed curves inside simple closed curves. An island on a lake on an island on a lake on an island on a lake

It’s like a matryoshka doll! – V. observed

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L. decided to write down the reasoning behind his solution.

Back to the very important question of “what are the lines for”. M suggested that an island might be a mountain. “But even without any inlets, it might still have pools of water from rains or from underground sources“, A. suggested. “Or ocean waves running over the island in a hurricane.” – V added.

I felt this was a good time to step in. First, I thanked them all for their insights and also for a mature and thoughtful way they handled the discussion. “It feels like a real-life adult meeting!” – V said.

I continued: “As you can see, some of you think this dot is in the water and some think it’s on land. Why do you think that is?”

A: “Because these lines, they separate land from water. Why else would there be lines on a map?

Me: “Have you seen maps of mountains? They have these lines called elevation lines. Sometimes lines mean shorelines and sometimes – elevation or something else. Some of you assumed that the lines on this map mean the same as on all the other island map puzzles you’ve seen so far. Some of you, however, worked from a different assumption. So you all used great strategies and logic, but arrived at different answers. That’s because you started from different starting points.”

They discussed the problem some more and settled on “this is water if all the lines are shorelines. It is land if all the lines are elevation lines.

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We had just 10 minutes or so left, so I didn’t have as much time to introduce the map coloring puzzles. Before the circle I drew a few maps on 3×5 cards and colored one of them as an example. I used it to explain the coloring rules – if two countries share a border, you cannot color them the same color. But if they share a point, you can. The goal is to figure out the least number of colors needed to color each map. Then I passed around the cards and they got busy coloring.

Everyone, from 5-year old M to 10-year old L used too many colors at first. For example, L guessed that his map required 6 colors (it was a 2-color map) and started coloring it accordingly. I asked him if we could get by with fewer colors and still follow the rules. His first answer was “no”. But as he worked, he realized that he could use fewer colors. And the final result did use only 2 colors. The scenario was pretty much the same for all the kids.

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E. got a beautiful space-themed coloring map. He did it justice using at least 5 different colors, working with utmost focus to color even the tiniest stars. It looked beautiful! I really didn’t want to interrupt his work. Once he was done and we had time to admire the results together, I drew a copy of the same map and suggested that it might be interesting to see what this map would look like when colored with the least number of colors. He said he would take it home and work on it for the next time.

Soon I ran out of prepared maps. So I gave them blank 3×5 cards and asked to draw map puzzles and trade with each other. They were happy with the idea. At this point the older boys – RB, L and A started to come up with ideas for avoiding the hard work of coloring the entire map. Unfortunately, we ran out of time. So we’ll pick up here next Thursday.

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