Tag Archives: proof

Geometry – Logic Stuff

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I’m totally reducing the amount of time I spend on this next year. But I’m blogging out what I did just in case you teach it as well.

Things I liked (in order that I taught it):

Inductive and Deductive reasoning sort – I made a card sort with inductive and deductive reasoning examples.  I had the kids sort the cards with zero information, then tell me what they felt was different about the two sets of cards.  They did a great job!  I’m not including my cards because I need to change them next year.  I will only do words on both types of cards, and no patterns or numbers.  A few students took inductive to mean “numbers” or patterns only, and deductive to mean “words”.

I need to spend more time turning statements into conditional statements.

Conditional / Converse / Inverse / Contrapositive Activity from Sam and the great folks at PCMI.  Wow – what a great activity!!  TIPS:

  • It takes a few days but it is worth it!  All students were very engaged the entire time.
  • DON’T LET THE STUDENTS WRITE THEIR OWN STATEMENTS.  Use the ones Sam etc. wrote.  You will thank me later.
  • It is “paper intensive” so make group packets before they come in to preserve your sanity.
  • I put the directions in a picture frame so students could see it more easily as it is a “directions intensive” activity.  They did well and I love when they practice following directions.
  • After the posters were all finished, I had students do a “gallery walk” with post-its so they could make comments on other posters in case a student got the truth value incorrect.  This was great!

Symbolic Logic Proofs – I made cards for them to refer to.  They used them often.  We made “index card pockets” in their graph books to hold them.  I loved these things!

Proof strips – I laminated card stock and we wrote on these to re-arrange the statements.

Post-it flags – I also let them use post-it flags to write the givens on so they could be easily rearranged.

I also combined this with Speed Dating for the win.  I don’t think I can ever thank Kate enough for Speed Dating.

I did another sorting activity thanks to Pam Wilson’s intro to logic post.  This went well!   I let them sort on the floor if they wanted and then check their answers with another group.

Engaging Students in Parallel Proofs

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I am really starting to like, even love, teaching Geometry.  There, I said it.  Hopefully I won’t lose my math card.

I am finding that Geometry problems are just one big puzzle.  And I love puzzles!  I am also inspired to see some students who were maybe not great Algebra students have success with Geometry.  I have identified those students, and I plan to work with them one on one this year to strengthen their Algebra skills so they can keep their newfound engagement with mathematics going strong!  It is so exciting to see these student’s perception of their own mathematics ability improving, and their attitudes about math changing!

Parallel proofs were a blast on Friday thanks to the amazing Bowman and his Global Math Whiteboarding talk Tuesday night.  If you missed it, you can watch it here.  You can even take a look at the terrific whiteboard brainstorming sheet that we all worked on together during the meeting.  I have a set of giant whiteboards, but I often forget about them.  I have been wanting to use them, and this was an amazing opportunity.

I assigned two students per giant whiteboard with two different colored markers and one eraser.  The two different colored markers were a fabulous idea.  The students can color code things (so helpful in Geometry), and I could make sure both partners were equally participating.  I’m not sure who gave this tip at Global Math but THANK YOU.  As a bonus, the colored markers I found in my cabinet were scented.  I didn’t realize how much this would thrill high school students.  Some things that I think they will love they could care less about, but break out scented Expo markers and WATCH OUT because you will have some really happy teenagers.  Who knew??

Back to parallel line proofs.  I gave them the proof, but told them not to prove it yet.  I told them to just draw the picture and label the GIVENS only.  Then, I had them discuss with their partner how they knew the “prove” was true.  For example, given that m<1 is 100 degrees, how did they know that m<8 was 80 degrees?  Once they talked about it, I had them write down their thinking (steps) on their whiteboard.  Then, they shared their giant whiteboards with their steps with the rest of the class.

They discovered:

  • There are many ways to do a single proof
  • Some ways can take many more steps
  • Some ways are equally as efficient as others, there is not one most efficient way
  • How to carefully explain their steps to the class so everyone would understand what they were doing to “prove” they knew it.
  • They do not like it when other people talk while they are presenting. This is great training for when I am talking!  lol

After they shared, I modeled how to write down the proof steps, explaining that they have to SHOW why m<2 = 80 degrees, if m<1 is 100 degrees using substitution and angle subtraction.

After that, I gave them another proof, but let them do the steps and work on the proof before sharing with each other.  The sharing is crucial, as it helped students see the difference ways each group solved the proof.  My favorite part was how mad a student became when she realized she had taken TWO extra steps than another group to prove her statement.  Now that is engagement!  In fact, this activity produced some of the highest levels of engagement I have seen all year!  Thanks again Bowman!!

Proof Stations – Geometry

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After a smashingly successful first day doing proofs in Geometry I was excited to prepare for day two!  Our Geometry classes are not tracked (leveled).  On day 1 I noticed that I had a wide range of proof writing ability.

I still wanted the students to have time to think and process the proofs as they did the day before, but I did not want to go too slowly for some students, or two quickly for others.  I wanted everyone to be able to think and to learn at their own pace.  Enter Stations.

I am a huge fan of stations for review, but I really haven’t done much at the beginning of a unit.  This worked beautifully!

image-3I gave my students six proofs in (of course) an INB foldable.  All they had were the givens and the prove.  I had them work on them silently for 10 minutes.  After the 10 minutes, they were able to work with a friend, and move around to the stations.

At each station, I had a copy of the same proofs that were on their papers, with some “hints” added.  The “hints” were the “fill in the blank” that can help students along if they get stuck.  On the back of the picture frame I had the proof fully worked out.

I encouraged the students to try the proofs FIRST, then if they got stuck or needed a hint they could go to the station.  Or, if they finished the proof they could check their answer with the proof answer on the back of the station frame.  Many students did not WANT the hints.  I agree with them, I really hate reading how someone else has solved a proof before I have time to think about it and try it on my own.  But, I did want students to check to make sure their proofs were similar to mine.  Since they are just beginning proofs, they often leave steps out.

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This activity worked so well!  Students had time to think and try to work out the proofs on their own.  They were able to move around at their own pace.  They could get a hint if needed, and check their answers.  Another great thing came out of this as there are usually a few ways to solve each proof.  I loved it when a student proved something in a different way and came to ask me about it.

We still have a long way to go, but I feel this gave them a great foundation!

** I did not provide the proofs I used here as I “borrowed” them from many sources and am not sure where they all even came from.  🙂

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Beginning Geometry Proofs

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After a couple weeks of p, q logic proofs, we started proofs “for reals” in Geometry this week.  First, we did algebraic proofs.  The kids were not happy that they had to show all of their steps.  “But I can do this in my head!”  Yeah, right.  At least it was a pretty easy introduction to proofs.  Anywho…

After learning all about segment and angle addition postulates, we started beginning baby proofs.  The first day went really well.  I took a page from the amazing Fawn’s book and modeled my first proof lesson after her problem solving lessons.  After all, proofs are just giant problems you have to figure out.  I gave them a proof.  We wrote down our “given” and marked up our diagram.  Then, I had them “think” silently for a few minutes.  I encouraged them to write what they were thinking or to take notes, but they didn’t have to do any “work”.  I just wanted them to think about what they knew.

After those few minutes, I gave them a few minutes to talk to their group about their observations and what they were thinking.  I walked around and heard amazing things.  “Could we just add these up?”,  “Aren’t these equal?” and “I think we could use the segment addition postulate here.”  Then, we pulled it all together.  I let the students tell me (and each other) what I should do next.  The students explained their thinking to their peers and I also jumped in occasionally when needed.

Learning how to “do” proofs (how to prove something) is problem solving.  This takes time and effort.  This takes a person looking at something on their own and really trying to figure it out.  You can give a student strategies, but I don’t think you can “teach” someone how to do a proof.  They need time to figure it out on their own.  The time spent in class “thinking” and “talking” to each other before actually doing this together as a class was well worth the time.  First of all, I have freshmen, so a few of them are not doing their homework as consistently as I would like.  This means that the in-class time is the only dedicated time they will get.  Also, many of my students get frustrated on homework at home and will just quit.  I wanted them to have support when they were starting out.

The best part was when we worked as a class.  Since I gave them time to think silenty and work together, some of my students “figured” out the approach they would take.  They were excited to explain it to their peers.  Their peers are much tougher on them than I am!  Students often had to explain their reasoning in a couple of different ways.  The listeners wanted to know how and WHY their classmates are doing a certain step.  And voila, there are the proof “reasons”.

I was worried about teaching proofs, but it has been amazing so far!

Day 2 – Proof Stations!