Tag Archives: Proofs

Day 63: Class Goals

Coming off of Thanksgiving break I feel ready to go. In Algebra, we are just starting to work with slope. I am excited for tomorrow’s lesson… stay tuned for that.

I graded Geometry’s concept tests over the break, and a majority of the class bombed Triangle Congruency… Hard.

The fact that 7/8 of the classes scores didn’t raise tells me I messed up somewhere along the way. We spent about 40 of the 49 minutes in the period reviewing different problems on whiteboards. I think it helped them quite a bit.

For the last 9 minutes I threw this up on the board:

Day63

They were giggling and having a great time. I thought it was interesting that none of my 20 students tried to make the side at a slant… Taking the easy way out I guess.

After about two minutes I revealed the bottom of the slide:

Day63.1

It is really interesting to me that after about 3 seconds, most of the class decided it would be too much work and checked out.

Did I ask the wrong question? Were students not perplexed by this? Is there a lower entry point into coordinate geometry that would have been more effective? Was a bad idea to give this problem with 7 minutes left?

Maybe.

I don’t regret asking students to find a place to start on this for homework. I set the bar high… Students should see that I want them to struggle and be challenged.

This summer I went to one of Dan’s workshops. One idea that stuck with me was you can always add to a problem, but you can never take away. I believe that if I led students through a coordinate geometry problem before hand, it would take away from the magic and struggle that makes math so great. Working at a problem for a longggg time then finally getting it, ya know?

I had parent teacher conferences a few weeks back, the only talking point I had planned was to project my class goals up on the board for parents to see. These were on my syllabus this year and I refer to them constantly to make sure I am excited and passionate about what I am teaching…

Here they are:

  • Encourage the development of mathematical reasoning by:
    1. Incorporating multimedia into lessons.
    2. Presenting students with low entry and high exit problems.
    3. Focusing on student work.
  • Develop patient problem solving skills.
  • Make math social.
  • Provide the appropriate level of mathematical rigor for each student.

I believe there is an appropriate place for guided examples, notes and repetition; after the struggle. After students discover and apply the mathematical tools that make their lives easier.

Tomorrow I am planning on having students talk about the problem for a few minutes, creating a plan of attack with their groups, then trying it again. I will work in examples along the way according to where they are at in the solution process.

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 30: Proofs?

I have a sub tomorrow. Last year it would have been no big deal. Probably because I was so focused on surviving. Now I am kinda disappointed I don’t get the spend the day with my students and frustrated at myself for giving them all the same worksheet when they really need to keep moving forward.

Today in Algebra I was planning on taking the easy road and giving students a day to practice. I wasn’t feeling great about it because that would mean back-to-back days of practice on a concept they have almost mastered already. What is awesome about my students is they would have done this without any thought. But just because they are willing to work for me without question, doesn’t mean I should take advantage of it.

Instead I did a sort of speed dating activity. We rearranged the room so there were rows of desks facing each other. Students had two minutes to work on a problem with their partner then one partner rotated. I threw other questions in there like state flags and capitals of states; they loved it.

Afterwards I asked them for feedback; didn’t want to change much, just maybe a little more than 2 minutes on some problems (that doesn’t mean there isn’t any room for structuring it a little better on my end though!).

In Geometry we started proofs. (Actually, first there was an awesome visual pattern found here)

And I was worried about how today would go.

At a workshop this summer with Dan he talked about how you can always add more, but after all the mathematical structure has been added you can’t take it away.

I went with it and put this up on the board:

Day30.2

I asked students what they could conclude from this.

There wasn’t any complaining or distractions or groaning. They jumped right on it.

Some threw numbers into the mix;

Day30.1

And eventually we were able to reach the conclusion. I talked about how this was a specific case and how 6.1 and 83.9 is another. If we want this to be true for every case, we would be spending a lot of time plugging in numbers making sure they work.

After I asked students for to write out a game plan for how they could use the provided information to reach the conclusion we wanted.

They did it.

And it was awesome.

I guess all my emphasis on how the process is so much more important than the solution is paying off. Those whose game plan fell short didn’t care; they learned from it and hearing about other’s approaches. I wrapped the discussion up by writing a formal proof using THEIR rules.

For the rest of the period students worked on Justin’s Formal System Proofs. (Provided by a co-worker) This is a great way to introduce proofs and reinforce logic in general.

Day30Here is the direct link to the goods.

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Featured comment:

Joshua provides two great suggestions for fostering student interest:

…encourage your students to have opinions about proofs. Which ones do they like, which ones do they find most convincing, which ones do they find easiest to understand, for two proofs of the same result, how do they compare, etc…

Prove or Disprove and Salvage if Possible (=PODASIP, inspired by the PROMYS Program in Boston.) Give them at least a couple cases where they’ve been asked to prove something that isn’t true. This ambiguity is common for working mathematicians and many non-mathematical situations…

Day 29: Why? Why? Why?

In Algebra I used the textbooks exploration in solving one step inequalities. The goal was for students to realize the same properties of equality apply to inequalities.

This went ok.

It is had to differentiate and account for different student ability levels when giving every student the same worksheet. But at times I am feel like I am out of other options, especially when dealing with inequalities. These are my weak spot.

I am throwing around a few ideas of how to have students ask the question for the triangle inequality but that idea is still n the works. If anyone has any activities/ideas for inequalites let me know! I think the hardest thing for me is the similarity between solving a multi-step inequality and solving a multi-step equation.

In Geometry I introduced algebraic proofs by throwing this up on the board:

Day29

Solving for the variable was a piece of cake, justifying each step was brutal.

I really enjoyed asking students WHY they could subtract 8 from both sides. There was a flurry of reasons; they understood that it has something to do with the equal sign but couldn’t quite construct a viable argument as to the exact justification.

This is my first year teaching Geometry so these lessons are definitely at a still-in-the-works point, but I am excited to see how the year develops!