I would say my most developed unit in geometry was my logic unit. I took it as an opportunity to introduce geometry in a refreshing way for student who may have been turned off to math prior to taking my class. For the first several weeks, we focused on logic puzzles that we not "mathematical" in the traditional sense. This encouraged thinking and problem solving, but it also engaged all of my students, as all of the puzzles were approachable and had an accessible solution. The aim of the unit is to get students to write up the solutions to the puzzles in a two-column, "statement-justification" format.
I grouped thee puzzles into "important stuff", "neat stuff" and "hard stuff", in the style of PCMI math camp, although I think for HS I would reduce the grouping to "important stuff" and "fun stuff". The "important stuff" was the bare essentials for teaching basics: converse, contrapositive, law of syllogism, etc. All of that stuff was taught via simple logic grid puzzles that are easy to find online. I also taught proof by contradiction because, while it is not a required standard in NY, it was an invaluable tool as we went on throughout the year.
The "fun stuff" or "hard stuff" was more engaging and really got the students asking questions. Much of that I will post here.
Since many people on Twitter or talking about approaching their Geometry units right now, I feel like I had best get my stuff uploaded as fast as possible and if I try to do it all at once, I will get bogged down by the volume of my material. I've decided to upload documents incrementally, so check back soon or follow me on Twitter, @solutionsforx Write a comment below if you have any questions or requests.
See an overview of my 2017-18 geometry school year with links to all lessons
This puzzle is my own creation. It is very successful and keeps the students busy. Find the treasure through process of elimination. Yes, there is ONE answer to this puzzle.
All-Time Greatest Classroom Puzzles
I collected these puzzles from the Internet. They are the best for engaging students and eliciting active classroom discussion.
There is an island filled with grass and trees and plants. The only inhabitants are 100 lions and 1 sheep. The lions are special:
1) They are infinitely logical, smart, and completely aware of their surroundings.
2) They can survive by just eating grass (and there is an infinite amount of grass on the island).
3) They prefer of course to eat sheep.
4) Their only food options are grass or sheep.
Now, here's the kicker:
5) If a lion eats a sheep he TURNS into a sheep (and could then be eaten by other lions).
6) A lion would rather eat grass all his life than be eaten by another lion (after he turned into a sheep).
1) Assume that one lion is closest to the sheep and will get to it before all others. Assume that there is never an issue with who gets to the sheep first. The issue is whether the first lion will get eaten by other lions afterwards or not.
2) The sheep cannot get away from the lion if the lion decides to eat it.
3) Do not assume anything that hasn't been stated above.
So now the question:
Will that one sheep get eaten or not and why?
A friend of mine shared this problem with me and I love it.
There is an island populated with 1000 logicians. Of the logicians, 300 have blue eyes and 700 have brown eyes. On this island, having blue eyes is considered the worst possible fate. If a logician ever discovers that he or she has blue eyes, he or she is to eat their final meal that evening and jump off a cliff the very next morning. However, while every logician can plainly observe the eye color of every other logician, there are no mirrors, no photos and no reflections. Furthermore, logicians never discuss eye color. That is, no logician ever tells any other logician whether or not his or her eyes are blue or brown, nor do they discuss the eye color of other logicians. On the other hand, logicians are aware of all of the facts of the island, including the laws that govern the behavior of their entire population and the fact that there is a total population of 1000 logicians, as well as the eye color of every other logician except his or her own.
One day, a prophet shows up on the island and declares before everyone: "At least one of you has blue eyes." The prophet leaves the island, answering no questions and never to return again, leaving the logicians to figure out the next logical move.
The warden meets with 23 new prisoners when they arrive. He tells them, "You may meet today and plan a strategy. But after today, you will be in isolated cells and will have no communication with one another.
"In the prison there is a switch room which contains two light switches labeled A and B, each of which can be in either the 'on' or the 'off' position. Both are originally in the 'off' position. The switches are not connected to anything.
"After today, from time to time whenever I feel so inclined, I will select one prisoner at random and escort him to the switch room. This prisoner will select one of the two switches and reverse its position. He must move one, but only one of the switches. He can't move both but he can't move none either. Then he'll be led back to his cell."
"No one else will enter the switch room until I lead the next prisoner there, and he'll be instructed to do the same thing. I'm going to choose prisoners at random. I may choose the same guy three times in a row, or I may jump around and come back."
"But, given enough time, everyone will eventually visit the switch room as many times as everyone else. At any time any one of you may declare to me, 'We have all visited the switch room.'
"If it is true, then you will all be set free. If it is false, and somebody has not yet visited the switch room, you will be fed to the alligators."
What strategy can the prisoners use to survive and be set free?
SELF-REFERENTIAL APTITUDE TEST
I have not given this to an entire class of students, but I have given it to a few students who show interest in going above and beyond the rest of the content in this unit. If you haven't seen these before, they are a tone of fun. Check it out here.
This puzzle is a great warm up and fully exercises negations and conditional statements. NOTE: This puzzle has been personalized and is true to me. I recommend you edit it to be true to yourself.
I have three younger brothers, so when we were growing up, we would sometimes bicker in the car during long drives. On a particularly long ride, we were passing a drawing I had made between us and were arguing over the desireable qualities of these six characters. (I called the characters 'moocrums' (I don't know why!))
Tommy: If a moocrum has arms, then it should have one eye.
Daniel: If a moocrum has a mouth, then it doesn't need (and should not have) arms.
William: Well I think if a moocrum has hair, it should have one eye.
Young Mr. Nockles: Well I think if a moocrum does NOT have a mouth, it should have two eyes.
Okay, there isn't much special about this assignment except that it demonstrates the goal of the unit. This, by the way, is the only assessment I give students in the logic unit. I feel it is important to limit assessment in the first unit and keep it simple because geometry students are often freshman or sophomores, nervous about math class, and a simple first assessment helps them relax into the course. I have provided multiple versions of this quiz because everyone who know me knows I allow students to retake exams. As you can see, it would not be difficult to make your own version of this quiz. In my own course, after at least two attempts, everyone got an 'A' on this quiz.
IMPORTANT NOTE: I highly recommend teaching proof by contradiction. Students have no preconceived idea of what that could be so they are quick to observe and follow the conventions of a proof by contradiction. It sets them up for direct proof and two-column proof. For assessing them, I make these expectations abundantly clear: 1. State what you are trying to prove and what you will assume (You must correctly write the negation of what you are trying to prove). 2. Correctly identify every possible alternative case and where a contradiction lies. 3. Conclude with a statement like this, "Therefore, by contradiction, it must be true that ___."
This activity is designed to demonstrate to students why geometry is important. Spoiler alert: in geometry, we learn how to write logical arguments to explain ourselves. In this activity, students write "proofs" for "real life" applications involving pharmaceuticals and law. To be clear, in my own curriculum, I don't assign this task until unit three (1. Logic, 2. Construction, 3. Proof) but to understand how the aim of this unit feeds into the later Proof unit, I think it is helpful to look at this assignment.