Intro to Proofs in Geometry

I wanted to blog about this a looooong loooong time ago but the school year got in the way along with moving across the country twice because of family health dramas (NY to California in the fall, now California to Oregon.  I know Cali to Oregon doesn't seem that far, but it is over 1,000 miles from San Diego to Portland.  California is freekishly big.)  So though I know posting lesson plans in the summer is kind of silly, I want to get it out of my system before I forget what I did.

The school in which I taught this past year was a high poverty school where 30% of our students had IEPs.  It is a charter school so it's pretty small meaning I got to work very closely with my students, colleagues and parents but we did lack funding and our students were weak in a lot of basic skills.  In fact, most of my geometry students this past year hadn't even passed algebra 1 yet.  The previous algebra teacher found them so lacking in basic skills that she gave the entire algebra 1 class "incompletes" because they didn't finish the algebra 1 curriculum.  My principal decided to enroll all these students in geometry anyway because she figured (rightly I think) that they needed a bit of a break from algebra and if they saw some algebra in a geometrical context it might make going back to algebra more meaningful (which it did.  Every time algebra popped up in geometry the students were actually excited because it was familiar and wasn't too difficult.  They really mastered equation solving, writing expressions and equations of lines by studying these topics through geometry.)  All of this meant that when proofs came up I was super freaked out.  I've always struggled with teaching them and I feel like I've done a very poor job in the past.  I put a lot of thought into how to build proofs into our curriculum this past year and I feel like what I did was relatively successful.  My students weren't scared of proofs for the first time in my teaching career.  When they came up, the students knew at least where to start and always attempted them.  So I want to lay down what I did just so I don't forget.

So here is the description of how the unit flowed.  It's definitely a more traditional approach to proofs and I stuck to two-column proofs.  I tried to transition the students to paragraph proofs, but their skills and confidence were too low; they liked the organized nature of two-column proofs.  First, I didn't nix the logic unit.  Even though logic is not in Common Core anymore, I think that the reasoning done in the logic unit helps prepare students for proofs.  

Logic Unit Lesson 1: Intro to conditional statements.

• First I did Sam Shah's lesson introducing conditional statements.  I did the drawing activity and posted all their pictures on the wall.  It went really well- I was surprised at how much trouble some students had following the directions precisely.  A lot of them didn't know the geometric vocabulary (like what an isosceles triangle is) or were hesitant drawing so it was a great activity to do towards the beginning of the class.
• Then I just did a mini-lecture on the notation of conditional statements, Euler diagrams and what a negation is.
• Then I gave them this assignment: Logic Unit Lesson 1: Conditional Statements Intro 

Logic Unit Lesson 2: Manipulations of Conditional statements

• I did the second part of Sam Shah's lesson on conditional statements
• I did a mini lecture on the vocab: converse, inverse, contrapositive and negatin
• Finally I gave them this assignment: Logic Unit Lesson 2: Manipulations of Conditional Statements.  

Logic Unit Lesson 3: Word Proofs

• First we did a syllogism activity  where I just cut the cards apart and had them put the syllogism in the correct order and a mini lecture on syllogisms
• Then we did word proofs.  This is one of the most successful lessons I've ever taught on proofs.  I totally stole it from another blogger, and of course forgot to save their name in the name of the file I downloaded like I normally do.  When I figure out who made it I'll update this post.  I reformatted the file I stole from that other blogger and did the lesson in kind of a workshop style.  I did one or two of the word proofs on the board to demonstrate how to do it, then I gave them time to work on their own and then we compared answers.  I had them come up and show different solutions they'd discovered and we talked about the fact that there's more than one way to do a proof correctly.  Like I mentioned before, this was actually a pretty bright class but one lacking in discipline and both basic math and study skills.  I had to hold the kids back.  They were chomping at the bit to do more and more and more puzzles.  I made the last two pages of the lesson optional and almost all students did them anyway.  Whoever designed this lesson was brilliant because it really hooked students who are usually disengaged with math.  You can't not want to solve one of these puzzles when they're presented to you.  
• There are two pages in the lesson to have the students make their own puzzles and switch papers with each other.  We had to skip this part because we ran out of time.
• Here's the lesson: Logic Unit Lesson 3: Word Proofs

Logic Unit Lesson 4: Angle Proofs

• This lesson was a little less fun than the last one, but it was very effective.  First we went over basic angle terminology: complementary and supplementary angles, vertical angles and linear pairs (they'd learned these before), and we also talked about the algebra properties of equality, transitivity and the substitution property.  
• Then I had them do the lesson below in pairs.  It's structured exactly like the word proofs from the last lesson with 4 or so "rules" and space for them to use the rules to go from the given to the prove.  The students were able to stumble their way through these proofs without me doing any examples on the board based on what they did in the last lesson.  
• Here's the lesson: Logic Unit Lesson 4: Angle Proofs
• Finally we concluded with a Scrambled Proofs Activity

Here's where I made a mistake.  I did parallel lines and transversals as the next unit because I liked flowing from points, to lines, to parallel lines to triangles to polygons.  It seemed like the logical way to structure the course.  Also, congruent triangle proofs are so much richer and more interesting if the students already know their parallel lines and transversals angle relationships.  But if I were to do it again, I would do congruent triangles after the logic unit and then do parallel lines and transversals.  The proofs for parallel lines and transversals are a little more abstract and involve more vocabulary than congruent triangle proofs so trying to launch from the intro to proofs unit straight into parallel lines and transversals was too big a jump.  So here is how I built up proof using congruent triangles after I failed at teaching them proofs through parallel lines and transversal relationships.  

Congruent Triangles Lesson 1: Intro to Congruence

• For this lesson I just did a standard lecture over what congruence is, the notation for congruence and examples of using congruence to find missing parts.  Nothing really exciting.  The only thing about this lesson that I like is my warm-up.  We define congruence as "identical in every way" and then I ask students if the two identical twins are congruent which leads to a great discussion of what "corresponding parts" means: 
  
• Also, I always get a laugh out of my student by choosing another teacher who's the same height as me (at this school I used the principal) and I use that teacher to discuss the difference between the "equal" symbol and the "congruence" symbol.  Our heights can be equated, but if you accidentally use the congruence symbol you're saying I and this other teacher are identical in every way.  
• Here's the assignment for that lesson: Congruent Triangle Lesson 1: Intro to Congruence

Congruent Triangles Lesson 2: Triangle Congruence Theorems

• I used this cool illuminations app on congruent triangle theorems along with this worksheet that I wrote to introduce SSS, SAS, ASA and AAS.  I've done this activity twice, the first time I let the students pair up and use their own computers.  A lot of the students wouldn't or couldn't follow the directions so I spent the whole period frantically running around trouble shooting.  The more motivated students in the class were able to make the connections I wanted but the other students left class mostly confused.  So the second time, we did each activity first as a whole class with computers closed, then I let them open their laptops and play with the simulation to confirm the results for themselves. This worked much better.  All the students were successful on the homework without need for more instruction.  
• Then I totally stole a few worksheets here and here from Math Teacher Mambo for the students to work on independently.  Here's my mash-up of her brilliance: Congruent Triangle Lesson 2: Congruent Triangle Theorems

Congruent Triangles Lesson 3: Using Theorems in Proofs

• This is a lesson I created that I'm exceptionally proud of.  It was super boring though.  I realized that students weren't really getting what theorems were for or how to use them.  They were still struggling with problems like: if angle A and angle B are a linear pair and angle A measures 40 degrees, what is the measure of angle B.  They weren't thinking about what "linear pair" means and how to connect that meaning to the problem.  Even if they got this far they didn't understand that in a "proving" situation they needed to state how they know that angle A and angle B add to 180 (that they need to say: by def of linear pair or state the theorem).
• So I made this worksheet: Congruent Triangles Lesson 3: Using Theorems to Make Deductions.  
• In pairs I had them go through their notes and fill in the blanks for all the theorems we've covered.  This was the boring part.  I gave them a time limit though which helped keep them focused and this activity also reinforced the importance of taking notes.  I refused to tell them any answers.  If they didn't have it in their notes they needed to find someone who did.  
• The "classwork" part of the lesson is where it all really paid off.  I did the first few examples with them- how to cite the correct theorem that had been used in each situation.  As soon as I started doing these problems on the board, a bunch of "ohhhh so that's why we needed those stupid theorems" exclamations went through the room.  It was especially rewarding to watch them do the last page where they have to think backwards- find the theorem that applies to the situation then figure out what deduction can be made.  
• Students guarded the list of theorems they made as the first part of the lesson fiercely and insisted that I do a similar "fill in the blank theorem review" at the end of every unit.  

Congruent Triangles Unit Lesson 4: Proving Triangles Congruent.

• This lesson went quite well although it's a very traditional lesson.  I just did a few example proofs with them and talked about how to set up a two-column proof table (I know... but these students liked structure.  I tried to show them a paragraph proof and their eyes all crossed and they started throwing paper.)  
• I gave them this packet of problems from letspracticegeometry.com (there are a lot of typos in this worksheet though.  I would like to rewrite it) but without the first two pages.  
• Then I gave them this proof template worksheet thing that I created.  It has the students choose which proofs they want to try, the harder proofs being worth more points.  They have to reach a certain number of points to get full credit.  
• This small spin on a worksheet created a night and day difference in students' attitudes about proofs.  Every other time I've taught proofs students have been super whinny about them and would give up quickly.  But when I handed out the above assignment I saw at least half the students immediately turning to the last page to do the harder proofs.  A lot of them struggled on the proofs through a good chunk of the period without finishing and I kept suggesting they just go do more of the easier proofs, or build up to the harder ones but they said that they wanted to do hardest ones.  The fact that there was a choice between easy and hard involved made them want to prove to themselves that they could do the hard.  Students with less confidence started with the easy ones and were able to advance to the harder ones pretty smoothly.  Everyone was engaged and no one was complaining that I was making them do proofs.  

Congruent Triangles Unit Lesson 5: CPCTC theorem proofs

• I taught this lesson the same as the last one.  Examples then a "choose your own problems" proof worksheet.  Here's the packet from  letspracticegeometry.com and here's the proof template worksheet I gave them.
• I've had trouble in the past with students using CPCTC inappropriately so I put the following message up on the projector in giant letters and made them recite it in unison a few times.  I kept it up as they worked on the proofs and I didn't have students misusing CPCTC!  

To use CPCTC you MUST

FIRST: Prove triangles congruent

THEN: Say parts are congruent with CPCTC

CPCTC says that:

IF two triangles are congruent THEN their corresponding parts are congruent.

Prove the "IF" first, Only then can you use the "THEN"

• Again, through this class period students were working on proofs without complaint and without giving up.  If they started to have trouble they could persevere or choose a new problem and this flexibility eliminated a lot of the griping I've experienced in the past with proof practice.  Boring but effective.  

WHEW.  That's all.  I just wanted to catalog what I'd done because this was my most successful proof teaching experience so far.  It still needs a lot of work though.  And I know that under Common Core, it may not even be relevant anymore because I didn't work in any proving congruence with transformations.  

Pedagogy vs. Compassion

This year has been the most tumultuous of my life which is why my posts this year have been so infrequent but this summer, my first summer off in my 5 years of full time teaching, I hope to spend some time reflecting on my teaching career so far.  I also want to record my experiences in the school I most recently taught at in San Diego California before they grow cobwebs.  I have a lot of lesson plans I want to share but first I want to think about how teaching at a disadvantaged, high poverty, high IEP percentage school was different from teaching at my relatively privileged charter school in Oregon and VERY privileged private school in New York.  It wasn't that different.

I did have to change how I taught.  I used inquiry based approaches at the other two schools in which I've worked.  The students had good study skills, were well organized and cared about their education so getting them engaged in self-discovery lessons wasn't that difficult.  They knew how to accept challenges and persevere even if they didn't know how to do something.  Boy did this bomb in my school in San Diego.  In geometry class, if I gave the students rulers they were immediately put to use as either projectiles or weapons.  If I asked them to spend 10 minutes completing an activity on their own, all the cell phones came out or hands went up asking for help.  No one had the initiative to even attempt an activity on their own.  Games descended into chaos.  I quickly learned that these students needed a very firm hand and they would only behave under direct instruction.  Maybe I should have persevered with inquiry based approaches and over time they would have gotten better but the standards hanging over my head made me too nervous to spend too much time on this classroom chaos.

Their study skills were so weak that most didn't take notes, bring paper or pencils to class, and many didn't know their multiplication tables.  I spent a lot of my time teaching them how to listen in class, how to take notes, how to use their notes effectively, how to show work and how to care.  I did use a lot of questioning in my direct instruction lessons- I never actually completed a problem myself on the board, always asking for student input- but it was still direct instruction.  At the end of the year though, as I was grading their final project and their final tests, I was astonished to realize that they'd mastered as much content as the students in my relatively privileged Oregon school and also exhibited the same enthusiasm for math that my Oregon students exhibited.  Here's an excerpt from an e-mail a student sent me at the end of this school year- it's almost identical to letters I received from my Oregon students:

I'm not sure why, but it just recently dawned upon me that you will be leaving after this year and I'll probably never see you again, so I decided to write you a farewell letter. I've never really been compelled to write one to a teacher before so you'll have to bear with me here. I wanted to start off by thanking you for everything you've done, I can honestly say you're the best teacher I've ever had in my entire life. That being said, the support you've given me and the mentality of perseverance you have instilled in the classroom has really inspired me to work even harder and I wanted you to know you have made a big impact on my life. I want you to know that you'll always have a special place in my heart, even years from now, I'm sure I'll look back and be able to confidently say you helped me achieve my goals.

Without inquiry based learning, you-tube videos, gimmicks, games or technology my students in San Diego reached a similar mastery of content and a similar changing of attitudes about math that my students in Oregon attained.

My pedagogy didn't matter.  Or rather, I used the methods that I thought would work for my students.  Method mattered much less than I would have thought.

I don't want this post to sound boastful- I had the same number of failures and frustrations as other teachers but I did feel successful at the end of the year.  I am left questioning the amount of time I've poured into thinking about my method- feeling guilty over not using more inquiry based approaches or not doing enough projects or relying too much on direct instruction or not letting learners of different styles shine since direct instruction caters to auditory and visual learners.  Certainly method was important but it wasn't a question of "is direct instruction or inquiry instruction the correct way to teach," it was a question of "is direct instruction or inquiry instruction the correct way to teach for my students."  Would my Oregon students have learned as well as they did had I used direct instruction on them?  I don't know.  Or was the method of instruction really not that important at all?

I got numerous notes from students at the end of this school year and all of them cited my ability to listen to their difficulties, to work with them after school, and my stubborn refusal to let them give up that helped them succeed.  (There were a few students that I failed though, don't get me wrong.  And I felt like giving up on a lot of them sometimes.)  None of them mentioned my lectures as being boring and a lot of them thanked me for teaching them how to listen and take notes because it helped them in their other classes.  Where does this leave me in the pedagogy wars? I don't know... but maybe it's time I directed my guilt away from my methods of instruction and try to hone what does make me feel successful- treating each student with compassion and trying to be flexible in finding what works for them, regardless of what methods are fashionable in the larger ed-community.

Absenteeism

Right now, I'm making lesson plans for my first period algebra 1 class.  Here is the first slide I decided to add to my presentation for tomorrow.

"Right now...

• 4 of you have As
• 2 of you have Cs
• 9 of you have Fs

It's really easy for me to tell though without looking at my gradebook who is passing and who is not: those who are here everyday are passing.  Those of you who are absent two or more times a week are failing."

What do I do?  I can't teach students who don't show up.  When I make this announcement, most likely at least half my class will be absent and won't even hear the message I'm trying to convey.

Update: 3/8
The Monday after writing the above post I decided to try a new grading system in my class to see if that could help with attendance. I had been assigning homework and calling it homework, but I had been giving the students time to complete it in class.  Only if they didn't finish it in class would they need to do it at home.  I did this because I didn't want students to feel pressured to get the work done quickly- when I assign only classwork the slower and more careful students tend to get stressed out.  The problem though was that students were not using class time well.  When I asked them to work they said they would finish it at home, and then of course it (and the student) never came back.

I thought that maybe if I made their grade entirely based on them showing up and using class time well, then I would have more luck with both attendance and with comprehension.  Miraculously all the students did show up on Monday and I told my students that attendance was our biggest problem.  That those who were failing were failing because they weren't here.  I explained that I was going to make their grade based entirely on if they came to class, took notes and if they completed the work asked of them during class.  Immediately, I saw relief wash through the classroom.  I think because for the first time all year, they realized that they could pass.  That they could do what I was asking them to do.  The late homework, missed lessons and poor classwork completion had been weighing on them and had been causing them to avoid class.  It was easy for them to not show up because this class is first period and at our school, only freshmen and sophomores have to come to first period.  So my freshmen were hanging out with their Junior and Senior friends instead of coming to class.

They want to do well and only their guilt and lack of confidence had been keeping them away from class.  They constantly tell me that they like me as a teacher which is why I was so baffled by their poor attendance.  Maybe the fact that they do seem to like me contributed to them not wanting to face me when they thought they'd let me down.

Since changing my grading system two weeks ago, my attendance has sky rocketed.  They're all completing class work, asking questions and performing well on quizzes.  They still definitely lack initiative.  Since I require them to turn in an exit ticket to receive credit for the day's work, the end of class has gotten awfully chaotic as students frantically try to get my help because they don't trust their own abilities.  But they're trying and showing up now.  We can work on initiative later.  

I am torn about this no- homework system.  I have been following the homework vs. no homework debate and I'm more on the side of assigning homework because I've seen students grow so much from wresting with problems when they have no one around to help them.  They take better notes, ask better questions, and demonstrate much more mastery over the material than when I don't assign homework.  This experiment has reinforced my belief that homework does significantly contribute to learning because my other algebra 1 class to whom I still assign homework are demonstrating much more confidence with the material and are growing more rapidly.  Both my first and fifth period Algebra 1 classes are composed of low-income students who have failed algebra at least once before.  But my fifth period class has time earlier in the day (usually during lunch) to complete their homework so their homework turn in rate is good, their attendance is good and their learning is evident.  But clearly when students can't do homework and the not doing it wears down their self confidence and causes them to avoid class, the homework needs to be nixed because it's doing much more harm than good.

I guess this just reinforces my belief that there are no absolutes in education.  Every thing about teaching needs to be modified depending on the composition of students sitting in your classroom.  When students do homework it's good for them, but when they can't do it and are still expected to do it, it's bad for them.

Asking for help

I seek help on-line constantly when it comes to lesson planning.  I've grown used to the idea that anything I can think of, someone out there in the blogosphere has probably already perfected and I love that I can see kernels of lessons I've just dreamed come alive in others' hands.  This doesn't even include the gazillions of ideas I've never thought of that are about a hundred times better than anything I can dream.

But when it comes to actually teaching- implementing the lessons, getting my kids excited, supporting their growth, encouraging them to persevere, I've never received much help (administrators never pop in.  I've been formally observed only once and that was by a coworker) and I feel like at this stage, I don't need much help.  I have a thriving community of students coming during lunch to do math because they enjoy it, and I've watched the most recalcitrant math students slowly gain confidence and enthusiasm and I feel like this is what I'm good at.  I'm good at patiently coaxing students into learning that they can learn math and over time, that they enjoy learning it.

But this semester I have the most stubbornly anti math student I've ever taught.  For three weeks, she was an angel in my advisory and a demon in my math classroom.  She refuses to accept help saying that she doesn't need it, she'll do it at home.  Then she proceeds to do nothing at all through the whole 80 minute block.  When I try to help her she slides under her desk, covers her paper, refuses to look at the problem, gets up and walks away, or starts ranting about the uselessness of math.  She's a wonderful student in advisory so I know she's bright and capable, but she refuses to cooperate in math (especially whenever division becomes involved.  She says she never learned it and she never wants to learn it.)  She slept through all of my math classes two weeks ago and refused to stir when I tried to rouse her.  She got a 30% on her first test and even though I discussed with her the consequences of her actions through all of advisory that day she slept through math again the next day. I asked her if she wants to fail? It means she'll have to do it all again next year.  She replied she doesn't but she can BS her way through the other tests.  I said that learning to read is tedious, but once you do learn, it's magical what you can discover and that math is the same way.  She replied that reading is vital but math is superfluous.  I said that everyone needs help to learn math because it's several thousand years of accumulated knowledge that we're trying to impart in a few short years and that all I would like is for her to let me help her.  Right now I don't even care about notes or homework or tests.  I would just like her to allow me to talk to her about math without arguing.  She wouldn't budge.

I thought that I'd have to just wait her out.  I'd need to stop nagging her and let her come around on her own.  Maybe over time she'd start to feel left out.  Or she'd realize that she couldn't BS her way on her own and she didn't want to fail.  She was so obstinate that maybe just the fact that I was pushing was making her push against me and if I stopped pushing she'd stop fighting.  I was worried though that she would get so far behind by the time she came round that it would be too late to learn what she needed to learn since she was already so far behind.

So I turned to our vice principle, explained what was going on and what I'd tried and he said he'd talk to her.  The next day she took notes, completed her homework and asked for help.  I asked him what he said and he told me he'd talked about how many thousands of years of knowledge we were trying to teach her in a tiny span of time and that she could not learn without my help.  He said that this will be maybe the only time in her life where she had a teacher who was willing to give her extra time, extra help and who really cared about her and if she waited, she would never get the help she needed.  It was almost exactly the same logic I'd tried on her.  Her efforts have continued through the week.

I guess this just reinforces my belief that if I ever get to a place where I think I've figured it out- that means I've grown too complacent.  Teaching will always and forever be something I'll need help with and that's the way it's supposed to be because it's a collaborative endeavor.  I hope I'm always humble enough to ask for the help I need.

Angleatron Failure and Distance Formula Game Success

I taught the lesson on angleatrons that I previously posted about and it was not very successful at all.  I'm reluctant to write about my failures because I'm already the type of person who doubts everything I do and even my most successful lessons leave me feeling like I'm not the teacher I wish I was.  This is also why I'm a terrible blogger.  In my most insecure moments, I can't help but compare my teaching to these fantastic teachers I so admire and aspire to me more like.  I hope the fact that I am constantly striving to be better makes me a better teacher, but it also makes me very uncomfortable in my own skin much of the time.

The lesson was unsuccessful for several reasons beyond my control.  My speakers broke partway through showing the video so the students couldn't hear Vi Hart's narrations.  I then tried to paraphrase what she was doing with paper folding but students grew bored watching a soundless video.  This made me rush through the video to move on to the activity, but then students were confused about how to do the paper folding.  Their confusion reinforced my reasoning behind doing the activity because if students couldn't grasp the idea that the corner of their paper can be used as a 90 degree angle, then they really did need to practice basic angle drawings.  About half the class did take off doing drawings and folding angles.  A couple of them produced really beautiful designs and I think all of them did grasp what 90 degree and 45 degree angles are supposed to look like.  The other half of the class adamantly refused to draw, or refused to draw precisely (sloppily drawing 90 degree angles that looked more like they were 100 degrees because they refused to use the corner of their papers to guide their drawings.)  Their reluctance and difficulty only convinced me that they did need to practice, but the activity didn't work for the students who needed the practice.

I did try some other games this past week and they were much more successful.  For me, the simpler the game, the easier it is for me to pull off because I have a very minimalist classroom (I have to buy all my own supplies, the students have tiny desks and we don't have a white board, only a smart board which allows only one student to write on it at a time.)  I came up with a game to practice the distance formula which worked beautifully mostly because it was so simple.  First, I had to bribe the students to play because playing games involves more thinking than taking notes and they actually wanted me to keep lecturing so that they could passively copy/ sleep.  Then I asked them to group into threes and told them they were competing against their group members to convince them to work with people other than their best friends.  Finally, I just displayed four numbers on the smart board.  The students could rearrange the numbers into two ordered pairs however they wanted and could add negatives if they wanted.  The person in their group that was able to organize the ordered pairs in such a way as to maximize distance won and earned a candy.  I started with 0,0, 4, 12.  Then gave them 2,3,4,5.  Then started giving them bigger numbers.  At first the students just paired the first two digits and the last two digits and used the distance formula.  But after a round or two they started figuring out how to add negatives and rearrange the bigger numbers with smaller numbers to get larger distances.  They also were doing a good job of checking each other's work because they only earned candy if they did the calculations correctly.  By the end of the game, every student had figured out how to maximize distance and they were all tying and I was going bankrupt on Jolly Ranchers.  My favorite part was when one person in a group announced their largest distance was 13.2 and students from a different group came over and clustered around asking the person from the first group how they'd gotten such a big distance. I think the game worked very nicely because it was simple, strategic, competitive but not so competitive that students who were "losing" became disheartened.  By the end everyone was winning.

I didn't like the game because I don't like the distance formula.  I would much rather students use the Pythagorean theorem enough that they could then extrapolate the distance formula by picturing triangles on the coordinate plane without needing to graph.  Unfortunately I just didn't have the time to reinforce this method of calculating distance so I caved and taught them the distance formula (but at least I did show them how it came from the Pythagorean theorem, though half my class fell asleep or glazed over when I tried to show the derivation to them.  I've tried having them do the derivation themselves but their algebra skills are too weak.)  At least though, they did do some critical thinking in terms of figuring out how to maximize distance.  That was the saving grace of this game.

Angleatrons

I inherited a geometry class last semester that was already two months into the curriculum and it was very frustrating that their basic sense of shape hadn't been strengthened.  I was supposed to start with congruent triangles, but many of them didn't even know what a right angle was supposed to look like.  It was too late to go back and work on basic drawing skills but I've been thinking about how to help students with little practical drawing experience succeed in geometry.  Gone are the days when all students had formal art classes and without these classes, their visualization and drawing skills are so weak that geometry can be really challenging and frustrating.

With the new semester, I'm starting over with a new class and I'm working on building more drawing and visualization into my curriculum.  I've just written up a lesson tied to Vi Hart's angleatron video. I want my students to be able to do rough sketches of all the basic angles so that their drawings, when we get to triangles and polygons can be at least a little bit accurate.

First I'll show my students the video and have them try to explain how the different angleatrons were formed.

Then I'll have the students make the different angleatrons, name their vertices, sides and the angles themselves.

Then I want my students to try making 3 different geometric patterns using their angleatrons like Vi Hart did.
Finally, they'll each pick the pattern they like the best and we'll make a class quilt out of their different patterns.

I'm a little nervous because a lot of my students really hate drawing, but I hope the structure of this activity and Vi Hart's beautiful examples will help.

Here's the lesson sheet I'm planning to use:

Angleatrons Lab