Logarithmic Math Date

Almost every day I write the date on the board using a mathematical expression. Today I was lazy and just wrote February 12, 2014. Sometimes I think my kids don't like the date being written in math language, so I just put the regular date. During my last class of the day, just as I was finished doing a proof at the board and we were about to play Mastermind, one of my students asked me to come back so she could ask me something. When I got there, she told me the date on the board was bothering her and and wondered if I could write the date as log(base 2)4096 = x. I said probably no one else would understand it, but ok. So I went up and wrote it on the board, then we went on to play Mastermind. At the end of the hour as students were leaving, another student came up to me and said that he thinks that x doesn't equal 12. I grabbed my calculator and showed him how to rewrite the equation so that 2^12 = 4096. The first student also argued that the equation was right. I couldn't believe I was having this conversation! These are two freshmen who are in both geometry and algebra 2 who are thinking about logarithms outside of their algebra 2 class. It was so cool! I also had to share it with the algebra 2 teacher because I thought it was so great.

Fractions

The internet was down at school so we couldn't do our normal Estimation 180 to start my 8th grade math classes. However, the state exam is coming up, so I thought I would refresh their memory about fractions. I put three problems on the board - one each of addition, multiplication, and division. Students got to work and very few had to ask me how to do the problems. As I walked around I was impressed with how many students were getting correct answers, and quickly too! I was expecting more than half the class to have forgotten what to do.

I credit it to the month we spent doing fractions every day at the beginning of class. I know this is an area where kids struggle, so I made sure we practiced a ton. Honestly, I didn't spend much time on the conceptual aspect of fraction operations; most of the work we did was drill work. Even after doing fractions almost every day for a month, there were still students who struggled. After that month, I let fractions go and focused on other things. I wondered how much all that practice would stick with them. I read somewhere on the MTBoS about a teacher's frustrating experience teaching a concept. He spent extra time on it, and even then felt that his students didn't get it. However, later in the year when it came up again, they all knew the concept. I hoped that this would happen with my students and fractions.

Today before we went over the problems together, I told my students how proud I was of them for remembering how to do fraction operations. One of the students said to me - That's because you're obsessed with fractions. I laughed and took the comment with happiness. I said - That's the point! I made you do it every day because I know how hard fractions are and I don't want you to leave here not knowing how to do them. So go ahead and think of me as being obsessed with fractions.

Mastermind

I read this post about using Mastermind to teach proofs and logical thinking. I've been meaning to introduce the game for a while in my high school geometry classes for a Mastermind Monday (along with Tangram Tuesday), but didn't get around to it until yesterday. As I introduced the game, my students paid close attention, which is not always the case when I am talking. I tried to use the language that Avery describes on his blog post, but I had a hard time with it. Instead, I asked students after their gave their guess, "What was your strategy?" or "What were you thinking when you made that guess?" If I heard students say something like, "There's not a 1 in this game" or some other conjecture, I would ask them to repeat it for everyone to hear. There were individual arguments over what to guess next more than whole class discussions. I need to work on making sure the class is all involved equally in the game, as well as work on having them make conjectures rather than just describe a strategy. Otherwise, the game was fantastic! The first time we played, every students was engaged. Their minds just latched on. In every one of my four geometry classes they asked to play it again today, and we did. This game is absolutely becoming part of my regular arsenal of activities.

Video for Class

I'm have to go to a training session at school tomorrow, which means I'll be missing one period of the day. Today's a snow day, so we're already missing a day this week. We have lots to get done in class and I don't want to waste tomorrow. So I had the genius idea (!) of making a video for them. But then I had to figure out how to produce it. I thought of using screen-cast-o-matic - that thing is amazing. It's like a screenshot, but it does videos. However, for my video I have to write math stuff and draw triangles, which isn't so easy with a trackpad or even a mouse. So no scree-cast-o-matic. Instead, here's what I came up with. I feel so low tech.

Negative Sign vs. Subtraction

I had an amazing conversation with my 2^nd hour 8^th graders. They were very confused about why you would subtract the 9 in an inequality like this: 9 – 3x > 42.

When I said you had to look at the sign in front of the 9, not all of them understood. So I showed them that if you add 9, then you have 9 + 9 on the left, which results in 18. But we want 0. So you have to do whatever it takes to get to 0.

Then you bring the -3x down. This is where it got interesting. I said negative three, and they said to me – it’s not negative, that’s a subtraction sign! So I said that I know they think there’s a difference between a subtraction sign and a negative sign, but really they’re the same thing. Think about changing a subtraction sign into “plus a negative.” That’s saying that they’re the same.

One student in particular was tenacious with her questions. She knew exactly what she didn’t understand and she stuck with it. Another seemed stuck in the mindset of “But you didn’t teach this to us” (especially the part about subtracting 9 instead of adding it).

So I put the following problems on the board.

5 + 3x = 24

5 – 3x = 24

3x + 5 = 24

3x – 5 = 24

We talked about the first two and why you subtract the 5.

The one student started to see patterns between them and she tried to say what they were. She said that when there’s addition, you always subtract the number. But then I showed her a new problem where that’s not always true.

-5 + 3x = 24

We also talked about how in the second problem on the board, 5 – 3x = 24, you have to bring the negative with the 3x. I know it looks like a subtraction sign, and it is, but it’s also a negative 3x. They could think of it as changing the subtraction sign to plus a negative, and then voila, it’s negative.

What I loved about this conversation is that it’s a real mind-bender for students. It shows their conceptual understanding maturing from what they learned in elementary school, where a negative is different than subtraction. I told them this – that sometimes learning is confusing. And I had seen them making many mistakes on this and was very glad to be having this conversation.

This conversation worked because they were ready for it. It reminds me of when I tried to teach them about dividing fractions in a conceptual way, rather than the procedure, and they were so confused and didn’t want to hear it. Then later one student was actually curious about it. Why does dividing by a decimal (or a fraction) result in a bigger number? Usually dividing results in a smaller number. His curiosity about this made him ready to talk about it.

I also got to tell them that it doesn’t matter to me at what point they understand this stuff. If they get it yesterday, great; if they get it today, that’s fine too. They can retake if they need to. I don’t want to limit when they can learn it.