Day 139: Factor Dice

Today is Friday; one strike.

I also introduced factoring trinomials; two strikes.

It ended up being one of the best days of the year.

I was frantically digging around the MTBoS looking for activities to introduce factoring. I came across a whole lot of crazy techniques; X method, diamond method, bottoms up…. Honestly, none of them made sense.

Then I came across this gem over at Dan Meyer’s Blog.

I pulled out a whole lot of dice from the department closet and after our opener I rolled a couple in front of the class. I told ’em that my dice added to 9, paused for about 15 seconds, them told ’em that the two multiplied to 18.

No sweat.

We did a couple more, then I broke the class into groups of two. Each group had two dice; a red and green and a folder to shield their dice. One person rolled the dice, calculated their sum and product then the other guessed.

They messed around with those for a while, this got old for some of the groups, so I gave ’em two more dice.

They added the red dice to create a number and did the same with the green dice. Then one student calculated the sum and product while the other student guessed the value of the dice.

For some groups, this wasn’t even enough. So… I kept increasing the number of dice.

Somewhere in between one, two, and three pairs of dice I upped the difficulty even more:

Green = Positive

Red = Negative

So now the sum is -2, and the product is -80.

This activity lasted about 15 minutes. I had 100% engagement across the board in 4 periods of algebra. No joke.

Students went back to their seats and I put this up on the board.

Last year I had students expand out binomial products till they picked up on the pattern. This year they looked at why A and C wouldn’t work here. We talked about how FOIL works, which I have actually been referring to as F (O+I) L this year.

Then things clicked.

Students understood where all that work with the dice came into play.

After a couple more multiple choice sort of problems, they were on their own.

I am fully aware that the dice activity only serves as a stepping stone in developing factoring skills. However, I threw up a couple of trinomials with a = 2 and students took them down. They understand that factoring these things is a guess and check process, and with more practice things will get easier.

Monday will be the true test to see how well they absorbed everything after a weekend passes.

Day 26: Emperor’s Banquet

It is always a good week when I have back to back lessons from Fawn’s site. Geometry presented and wrapped up Mrs. Murphy’s Laundry with group presentations. As the period went on, the presentations continued to develop, which was great to see.

In Algebra, students worked through The Emperor’s Banquet. I put the problem up on the projector and had students read it then work by themselves for five minutes to make sense of the problem. After, they discussed their progress with their neighbors.

I saw a lot of students taking this direction. A few went up and shared how they decided where to sit if there were 5 guests at the banquet.

In my eyes this image shows a great example of first step in developing algebraic thinking on the student’s end. After this point students slowly added mathematical framework to the problem. It was extremely difficult for me to step back and just let things happened. But after a while numbers started appearing.

What happened in green was awesome; a huge shift in structure, which makes communication much easier on the student’s end.

More patience on my end paid off and tables started emerging; another big step, this time into abstraction.

Conjectures started coming out and we created a table on the board. After about 10 minutes the table reached 17 guests and most of the conjectures had counterexamples. So we started searching for patterns different ways (in green).

At this point I felt very stretched in my ability to provide further scaffolding. In a few periods, students noticed linear patterns between 3, 6, and 12 guests. Which was extended to answer almost every case (excluding primes) ie. where should you sit if there are 18 guests? For 9 guests the last seat is 2, so double both of those, 18 guests, the last seat is 4. One co-worker suggested going in the direction of piecewise, and helping students see that certain rules work for certain number of guests; interesting!

I went into today’s lesson with the hope that students would look deeper into patterns, but what came out of today was much richer on both my end and the student’s.

**This problem is also great for Algebra 2 and even pre-calc… I believe it can be modeled by a logarithmic composed with a floor function **

Day 25: Mrs. Murphy’s Laundry

Geometry worked through Mrs. Murphy’s Laundry today, which is from from Fawn’s 180. For the first time this year I heard students talking about the problem on their way out the door and in the hall an hour after.

I started out by giving students 5 minutes to work on the problem by themselves. Then broke them into groups using IsntantClassroom. From here each group had around 40 minutes to agree on whom to arrest.

This is the first time Geometry worked on whiteboards; here’s a quick preview of how things unfolded.

This week we have been talking about inductive/deductive reasoning, during the lesson the only thing I stressed was that each group needed to be able to clearly communicate their reasoning for arresting someone.

At one point two groups felt they had eliminated all suspects… So they had a large group discussion and were able to figure something out.

Tomorrow groups will present their cases in front of the class. Here are a few boards of groups that were able to reach a conclusion.

I am looking forward to hearing students present their cases tomorrow.

In Algebra students worked through a few sets of problems involving proportions and unit rates.

Day 20: If… Then…

I spent a lot of time today talking about college and telling stories about my crazy roommates. It is important for students to see that I am not just a mean-lean math machine. Plus it was college day so every teacher was sharing out.

Algebra spent the rest of the time practicing absolute value and multi-step equations.

In Geometry we picked back up on Sam’s lesson. Students created big and little posters, we will gallery walk tomorrow.

A few take-aways:

• I did not emphasize that the black statement card is true; therefore their first statement needs to be an If Then version of it.
• Students started off slow but quickly picked up on what was happening.
• I am really excited for the gallery walk and giving students time tomorrow to work through and discuss which statements are logically equivalent.
• Some students mixed up the order of the statements, this will also be interesting.

Day 11: HWD (How We Do) – Homework

In Geometry we played around with formulas and put some visual patterns into the mix as well. However, today we calculated perimeter instead.

In Algebra we wrapped up Crossing The River and talked about the equation 25 = 4A + 1. I spent some class time going through a few examples of one and two step equations introducing some vocab along the way.

After we fished up I gave students some practice problems for the last 20 minutes of class.

I want to use today to write about how homework/classwork goes in my classes.

I have students who come from 6 or 7 different schools so there is a wide range of abilities. I don’t believe that giving ever student a single assignment  of 2 – 50 evens reaches students the best way. So here is how classwork/homework goes.

Two different assignments for two different ability levels. Students pick one and get to work. I explain that the /homework part of this only comes into play if they choose not to use their time in class productively. I do not ask students to take the problems they didn’t complete home and work on them there. I am more interested in the quality of student’s work over the quantity.

With two minutes left in class I throw up something along these lines:

Student pick one of the two questions and bring it in complete the next day.