For our last session of the semester, we had Dr. Giray Okten of FSU’s Mathematics department as our guest speaker! Dr. Okten led an activity involving the creation of secret codes.
We learned about how early civilizations used to encode messages–carrying a secret message in ancient Greece meant having your head shaved! A less objectionable method is the Caesar cipher, which we learned involves shifting the letters in the message forward or backward in the alphabet (e.g., “HI MOM” becomes “IJ NPN” if each letter is shifted forward one).
The students made a cryptography wheel and tried making messages using a Caesar cipher for their parents to decode. Then Dr. Okten had a special message for the students to decode. It revealed to them that they could find candy in the trash can at the back of the room!
And then, as is our habit, we had cake to celebrate the end of the semester!
One last session on logic puzzles! This time, we learned about grid puzzles and deductive reasoning.
For example, four children each bought a different kind of pet at the pet store. Information about what the children liked/disliked and what they didn’t buy helped students deductively conclude what each child did buy. The students did one such puzzle as a group and then worked individually or with one or two others to solve other puzzles ranging in difficulty.
No more crazy islands! …But we did have to get across some rivers.
In this session, we had puzzles ranging in difficulty that asked us to do things like determine the birth order of a set of siblings, or help a farmer get his goat, his cabbage, and his wolf across a river, in a very small boat, without anyone or anything getting eaten. Some of the puzzles were quite a challenge! We used manipulatives to represent ordering and movement in order to find solutions.
This session, we left behind Quint Island and different number systems…but we still ended up on a weird island! On this island, all the inhabitants are either Knights or Knaves. Knights always tell the truth, and Knaves always lie.
We had several examples of conversations with various island dwellers in which they would give us information, and we had to figure out who was telling the truth and who was lying. These puzzles can be tricky! If you’d like to explore more of them, check out this website for a large collection.
We left Small-Digit-Ville in this session, only to find ourselves shipwrecked in another strange place–Quint Island, where the inhabitants use a base-5 money system!
The students learned that the Quint Islanders have red chips worth 1 cent, blue chips worth 5 red chips, green chips worth 5 blue chips, and yellow chips worth 5 green chips. The Quint Islanders expect everyone to use the most efficient means of paying for items–in other words, not using any more chips than necessary–and if a person doesn’t, the islanders know she’s a foreigner and throw her in jail! (Okay, not really.)
Several items were collected that the students could use for selling and buying in order to establish their lives on the island while they waited to rebuild their boat so they could escape the island. The students had a great time with this hands-on application of a different number system!
We continued our exploration of how to reason in base 3 this session. Returning to Small-Digit-Ville, we thought about how to count and represent different numbers. Here’s how one student represented the Small-Digit-Ville number 1202:
Not sure how that equals 1202? Instead of the digits (from right to left) representing 1s, 10s, 100s, and 1000s, in Small-Digit-Ville they represent 1s, 3s, 9s, and 27s. So 1202 means 1 set of 27, 2 sets of 9, 0 sets of 3, and 2 sets of 1. Using the tiles helped the student determine that Small-Digit-Ville’s 1202 is the same as our usual world’s 47.
The older students also considered how we might add and subtract in a base-3 system!
Back from a break between semesters, we started exploring base-n systems. Kelly opened our time with a discussion about chunking–what does it mean to chunk numbers in groups? He discussed how when we write numbers, we naturally chunk ten at a time. But what if we visited an unusual place called Small-Digit-Ville?
The people in Small-Digit-Ville don’t use as many digits as we do. They only use 0, 1, and 2. How can we represent various numbers using only those three digits instead of our usual ten? We explored this and talked about how to represent different amounts of items in Small-Digit-Ville.
We had a special guest speaker this session! Dr. Hector Rosario, of the Chapel Hill Math Circle, came down to Tallahassee to talk about problem solving with our group. He started us off with a puzzler about a bus.
Once we figured out where the bus driver must be, we got on to another type of puzzle: jumping frogs and toads to switch their places. We worked on the problem for a bit individually or with others sitting nearby, and then Dr. Rosario set up some chairs and modeled a simpler version of the problem using students.
When we figured out how to solve the puzzle, we celebrated with some end-of-the-semester cake!
For this session, we continued talking about perimeter and area! The younger students used their understanding about the area of rectangles to figure out how to find area for other shapes, like trapezoids and parallelograms. They used grid paper to think about these ideas.
The older students explored the relationship between perimeter and area – if a shape has a fixed perimeter, what are the possibilities for its area? If a shape has a fixed area, what are the possibilities for its perimeter? They used number tiles to investigate.
It was perimeter and area night at Math Circle! Last time, we considered shapes on a coordinate plane and how we can find the lengths of their sides. This session, we continued that discussion by talking about the perimeter and area of shapes.
The younger students looked at a couple of different ways to determine the perimeter of a rectangle on a coordinate plane, and then talked about area. They used little squares to determine how much space rectangles took up on a plane.
(Credit where credit is due: Some ideas in tonight’s lesson came from this blog entry. Parents, more activities are included there that might be fun to do with your kids in exploring area further!)
The older students investigated different shapes with the same area but different perimeters. They looked at maximum and minimum possible perimeters for rectangles with given areas, and they found a pattern in the answers.
Our next meeting will be on November 29, 5-6 PM. We hope to see you there!