Day 61: Pass the Proof

Proofs take time to learn. They take an even longer time to get good at. I have been trying to tackle proofs from every direction I can possibly think of. The latest of which is called Pass the Proof.

Here is a quick rundown:

  • students break into groups of four
  • each group gets a sheet that has 3 proofs on it, each looks something like this

61.2

or this

61.1

  • each student has three options; fill in a statement, fill in a reason, or erase one of those two.
  • After doing so they pass the sheet to the next person in the group.
  • Repeat till Q.E.D.

I made into a game. The first correct group received 3 points, second – two points, third – one point. We briefly talked about each proof upon completion.

I really enjoyed this activity!

** Update ** Here are the proofs I used.

Day 60: Slope

For the past two days algebra has been diving into the idea of slope. Fawn Nguyen has an awesome activity and post on using stairs to get students thinking about slope as steepness. This can be found right here.

All my classes pretty much reached the same point as her’s on the first day. I am not going to re-create an identical post to hers so go check her’s out… Seriously!

On the second day however I moved the class in a little different direction…

I lost quite a bit of student engagement when I pushed students to think about what we could do with the different bases/heights we measured. We went through all the operations and I asked students “What operation would be the best for COMPARING the base and the height?”

We settled on division. Some classes looked at the base/height some looked at the height/base. We talked about what a large base and small height would look like as a stair case and vise-versa. After that each group measured a the base and height of a particular case and we threw all those measurements into a spread sheet and ranked them.

There was some great conversation on what ranking the numbers from smallest to largest related to in terms of least/most steep.

I held on ever further in introducing the word slope.

After I had students measure the base and height of an individual step, we talked this and how measuring something in millimeters is a whole lot more precise than using inches. We also talked about how the measurements of each step are proportional to the measurements of the overall height and base.

Day60

Then we dove into this:

Day60.1

This activity came from James Cleveland over at The Roots of the Equation. I love this because it drives home the idea of slope as a ratio. I threw student’s rankings up on the board and then we quickly calculated the height/base of each.

Only at this point did I introduce the word slope, we talked a little more about what it measured and found the slope between a few points.

A great couple days of classroom action, students really seemed to enjoy the openness of these activities… even though they were complaining a little.

Day 59: Observation

An administrator came into Algebra today and I have my first of two formal evaluations for the school year.

I used Fawn’s Staircase and Steepness for the second year in a row. As always there were some hiccups along the way (like rulers turning into helicopter rotors and reaching maximum velocity). I want to write about the lesson tomorrow and use today to reflect on the positives and negatives of the lesson.

Positives

  • Students were challenged in multiple ways (use appropriate tools, make sense of calculations, put thinking into words).
  • The activity made math social.
  • I had student buy-in across all ability levels.
  • The students who had learned slope last year, had a tough time recalling the information (this says to me that there is value in the activity across all ability levels).

Negative

  • I did not spend enough time pre-loading students with expectations for the day. Behaviors became an issues because students were uncomfortable with their thinking being challenged + loss of interest.
  • All around time management could have been better.

STRUGGLE FOR THE YEAR (pretty much on a daily basis): I try and present math in a way which removes the mindset of memorizing equations/learning through repetition. When trying to do so, I have a hard time balancing that mindset and finding a form of classroom management that pairs well with it. I want my classroom to be social, which in turn is noisy. But students tend to push to the point where it becomes unproductive on a daily basis.

GAHHHHHHHHH….. IT IS SO HARD TO FIND THE RIGHT BALANCE.

Day 57: Fence Problem Part II

Geometry spent their second day on the The Fence Problem. I went over the opener (in yesterday’s post) and it was interesting to see that a majority of students created a right triangle.

Day56.1

To create an area of 15 u^2 students needed some x value of 6. After I put the right triangle on the board I paused for a bit and just let students process the situation. A couple minutes passed and a student raised their hand and said (6,-10) will also work. We drew the triangle then I started getting other responses; (6, 2.5), (6, -5), (6,-100) ect.

I asked students to take out the fence problem again and used this new knowledge to help them.

They struggled still in proving a general case. I am OK with this.

We eventually worked through the solution as a class.

Here are a few of my thoughts after this activity:

  • I need to do more problems where I introduce a key idea on the second day and give students an entire period to struggle.
  • I am still too helpful.
  • These are students on the advanced track; they probably have never had a problem that takes them more than a day to solve.
  • Generalizing is a huge idea in geometry (proofs) I need to do a better job of tying everything together.

I encourage everyone out there to try this activity; it is worth the time and struggle for students!

Day 56: The Fence Problem

My Geometry students lost it today over this problem.

I gave it to them and asked that they work quietly on it for 10 minutes. The automatically went for rulers and protractors. A lot of them came up with some pretty cool solutions for the drawing provided.

Then I told them it wasn’t drawn to scale.

They still have a rough time going from actual numbers to variables to represent all different cases. The most popular solution was to cut the land in half because it looked like all the little pieces would add up. When I asked them to prove to me that the areas were they same they ran into some big problems.

I ended the day by throwing them a lifeline which I have mixed feelings about.

Day56

For most of them, this is the first time they have ever come out of a math class with little to no progress on a problem. This is good. Struggle is good.

Tomorrow we are going to start class with this

Day56.1

I am excited to see how this goes!

Day two can be found here.