Today was the last day of classes before spring break, I gave concept tests in both algebra and geometry.
After the tests students played Cross Nim. This has become my favorite game; it can be as simple or complex as students choose. A majority of students were thinking 3 to 4 steps ahead, which is great in developing a sense of reasoning!
I am off to a foreign land for spring break, see y’all in a week!
This morning mrdardy dropped by:
This is one of my all-time favorite problems. I usually set this up with 500 lockers. This kind of forces students to use the important problem-solving principle of reducing a complex problem to a much simpler one. Students typically offer 10, 15 or 20 lockers as a manageable set that they can actually simulate by hand. I have used this problem with Geometry kids, Algebra II kids and with Precalcuus kids and it is always an interesting conversation.
I had planned to give Geometry the locker problem today and took his suggestion of upping the number of initial lockers to 500.
However, geometry wasn’t too happy with me today; yesterday I handed out four lunch detentions and ripped into a couple kids. The word spread and they kind of had a mini revolt of taking an extreme and not talking at all. I talked to them about how they are honors students and next year they will be a minority in Algebra 2; a class filled with mostly juniors and seniors. Lately, some of their behaviors haven’t been progression towards that level and yesterday they pushed a little too hard.
Anyways, they were overwhelmed by the initial idea of going through 500 lockers. I didn’t say anything and after a couple minutes students were testing smaller cases. Eventually, they were victorious and tackled the problem.
Spring break is next week for us. Algebra finished up multiplying polynomials and after the break we will start factoring. Today, students worked through one of my favorite problems The Locker Problem, which is out of Fostering Algebraic Thinking by Mark Driscoll.
The Locker Problem
There are 20 lockers in one hallway of the King School. At the start of the school year, the janitor closed all the lockers and put a fresh coat of paint on the doors, which are numbered from 1 to 20.
When the 20 students from Mrs. Mahoney’s class returned from summer vacation, they decided to burn off some energy. They came up with a plan: The first student ran down the row of lockers and opened every door. The second student started with locker #2 and closed every second door. The third student started with locker #3 and changed the state of every third locker door. The fourth student started with locker #4 and changed the state of every fourth locker, and so on, until all 20 students had passed by the lockers.
Which lockers are open after the twentieth student is finished? Which locker or lockers changed the most?
I like this problem so much because it has a very low point of entry and investment for students. I had each group work on the problem by themselves for about 10 minutes then they broke into groups.
The problem also has various exit points depending on the direction each group decided to take. I had some students looking at differences of squares while others were using pictures, and another group was looking at factors. Either way, all students were able to engaged in the problem.
Algebra explored what the side lengths are for a rectangle with area x² – y² today. The paper folding required to get there was a lot of work and I am not sure if many of them totally understood what was going on.
Either way it is good for them to see a visual representation of where the heck these things come from.
Geometry practiced Law of Sines and Cosines, which we spent yesterday deriving. I talked to them quite a bit about how parallel these are to calculus in that they are using geometry to set up the problem then algebra to solve for either the side or angle. To me, that seems a lot like how calculus is; use an integral or something to set up them problem but after that it is all algebra to reach a solution.
Algebra explored the expansion of (a + b)² today. Across 100 students, none of them were able to correctly expand (x + 3)² coming into the day. We plugged x = 2 into different expansions. Most students had 13, some had 25. Everyone agreed that 25 was spot on, but couldn’t come up with an explanation as to what (x + 3)² expands to and gives 25 out.
We first talked about the area of a square with side length z and how the are is z². I asked ’em what area meant, they told me length times width. I asked ’em what area meant without using an equation… it took bit but we got here.
We folded the paper and labeled like so
We talked about how the new side length is a + b and to find the area of this square take (a + b)²
They found the length of each piece which makes up the area…
This is where they struggled:
The area of the square calculated from the formula is (a + b)²
But the area is also defied by these squares and rectangles added together.
(a + b)² = a² + 2ab + b²
We used this to expand (x + 3)² to x² + 6x + 9, which gave 25 out for x = 2!!
After a couple more examples students had a good hang of things, I saw a good mix of multiplying out and visual representations.
Tomorrow they are exploring (a + b)(a – b)… it should be fun!
A week ago I came across Kate Nowak’s awesome post on Trig. I was already into the heat of things, but pulled her activity on finding the height of some really tall things. Geometry had a lot of fun messing around with inclinometer apps. In turn we had some great conversations on real error. Students were getting heights of over 500 meters then 2 meters for the same object.
Next week we are diving into Law of Sines and Cosines, should be fun!
Yesterday algebra started multiplying polynomials. A lot of the problems were products of monomials with polynomials, something like:
x(3x² + 8x – 14)
For the entire year I have been looking for a way to get students to remember to distribute the x to each term. A lot of students call it Santa hat, I didn’t introduce that, who knows…?
I was looking through Youtube earlier in the week and came across this video.
I played it for my classes after they worked on the problem above; blank stares, they were not sure how it connected. I told ’em that x was a new car and asked “According to Oprah, who gets a new car?”
EVERYBODY GETS A CAR
EVERYBODY GETS AN X
Then we moved onto products of binomials
(x + 3)(x + 4)
EVERYBODY GETS AN X
EVERYBODY GETS A 3
Today, I talked about FOIL. I was really excited for once to get a bunch of blank stares again, after showing them this:
They didn’t need FOIL to remember to multiply terms, or make a face by connecting terms. They just did it. Their future is looking hopeful when multiplying binomial by polynomial or polynomial by polynomial.
Students even started using the word “distribute” instead of “everybody gets an x”, without any prompting on my end.
This content isn’t the most exciting, but my students are excited that they are having a ton of success in multiplying these things.