How I Teach the Introduction to Proofs

Teaching geometry proofs can be intimidating for some teachers.  These tips will help you introduce them to your students and help your students "get it"!  This lesson helps geometry students when they are learning proofs.

Several years ago, when I first found out I would be teaching Geometry, I’ll admit I was a little nervous.  Why?  Proofs.  My last “geometry” class was in high school (I never took a class of only geometry in college).  I knew that I could do proofs, but explaining them would be something totally different.  I have formed my own way of how I like to introduce proofs.  That’s what I’m going to share today.  I’ll write another blog post later about how I actually teach lessons with proofs.

Sometimes at the beginning of the year, I like to teach a lesson about Optical Illusions.  I think it helps lay the groundwork for proofs quite well.  In my curriculum, there is an Introduction to Geometry unit and the next unit is Logic and Proofs.  In the Logic and Proofs unit, I teach Conditional Statements, Biconditional Statements, Laws of Detachment and Syllogism, and the next lesson is Introduction to Proofs.  

Introduction to Proofs Lesson
I usually start class with this powerpoint.  It’s a very short review, because my students have heard me using this vocabulary and talking about patterns before.  I try to stress looking for patterns and making conjectures no matter what lesson I’m teaching.

Introduction to Proofs lesson idea - how to teach the introduction to proofs in geometry

Introduction to Proofs lesson idea - how to teach the introduction to proofs in geometry

Then, I talk about the importance of logical thinking.  I usually explain it in story form.  My little lecture goes something like this:
Okay, so I’m going to tell you a story about what I did last night.  I took a shower and changed clothes.  Oh, but I went to the gym first.  I burned the bread.  My husband and I walked our dog, Zoey.  My husband did not like it.  I went to bed.  Oh, I also cooked dinner for my husband and I.  We watched Netflix together. 
Does that story make a lot of sense?  Could I have told it better?  This is the order that everything happened. 
I went to the gym.  Then, my husband and I walked our dog, Zoey.  I took a shower and changed clothes, then cooked dinner for my husband and I.  I burned the bread and my husband did not like it.  We watched Netflix together.  Then, I went to bed. 
Is that story easier to understand? 
In math, we explain things the same way.  No one likes to have things explained in a jumbled, confusing way.  So, we explain our thinking in a logical order, the same way you would tell a story.

Next, I give out this sheet for the students to practice ordering the story.  They have probably done something like this a million times in English class in elementary or middle school, but I still think it’s good practice.  I honestly have no idea where this worksheet came from (I’ve had it a long time).  If you know who made it, let me know so I can give credit.  I always let them work with partners on this worksheet.  There are a few steps that they will argue over, which I like.

[EDIT: Cherylanne Thyrum figured out where this worksheet originated!  It's from Math Teacher Mambo.  You can find the worksheet under "activities", then "proof story".  Thanks Cherylanne!]

Finally, I start talking about Algebraic Proofs.  I don’t actually talk about the properties of equality until the next day.  On this first day I usually just put a few equations on the board and we solve them.  However, I also write “what we did” next to it (forming a baby two-column proof).  Something as simple as “multiplied” or “subtracted” works for me on this first day.

I typically don’t give homework on this night.  The next day, their warmup is usually to tell me logically, what they did the night before.  I’m looking for a short five or so sentences that make sense (logical and in order)!

Stay tuned:  In a few weeks, I’ll share how I actually teach lessons with proofs.

What do you wish you knew when you first started teaching?

After teaching for awhile, I realize how unprepared I actually was when I first started.  I knew how to plan and present lessons, but dealing with a room full of teenagers everyday and still keeping my life balanced was something I had to figure out for myself.  There are so many things I wish I knew my first year.  So, I reached out to my friends and experienced teachers to ask them…

When I was a first year teacher I was so unprepared!  Here are some tips from experienced teachers to help new teachers at back to school.

What do you wish you knew your first year teaching?

I wish I had known the baggage each student comes in to the classroom with. Before my first year of teaching I thought you simply taught and the students would just receive the information. HA! Now I know the child's well being is so much more important than the information to be taught. Once you build up a child you can start to teach them. 

That it becomes your life! It's not an 8-5 job. You must be passionate!

No need to lecture. After my first day of teaching I cried for about two hours from the stress of standing in front of kids talking for the entire day. I never did it again. Now my students and I happily engage in all manner of discussions and projects, but you'll never find me lecturing for 50 minutes in a row. It just doesn't work for me, and it's not actually the definition of teaching like I originally thought. 

I wish that I knew how useless my teaching classes would actually be.  I can plan a great lesson, but managing group of teenagers is a whole different story.

I wish I had known how hard it was going to be. Not that I would have chosen a different career, but it would have been nice to be more prepared. Plus, I would have given a lot more love to my own teachers.

I wish I knew that this job was going to involve many more hours outside of the classroom.  I'm not talking about the hours spent prepping.  I knew that was going to be part of the job.  I'm talking about the hours you spend worrying over the growth Student A isn't making, or is Student B has somewhere safe to go after school, or what is going on with Student C who hasn't been to school is a few weeks....the emotional wear and tear is daunting.  I do not think I was mentally or emotionally prepared for that.  For some kids, you are their everything:  teacher, coach, caregiver, provider, safe haven, counselor, etc.

How much paper work there was!

To be prepared for any situation at any time!  Make sure you have some sort of lesson extender ready to go, just in case the lesson finishes early (this happened to me also during an observation).  Even if you think you have plenty planned and there's no possible way you'll get it all done in one period/block, have something ready just to be safe!

I wish I had known what I know now about the awesome brains all students bring to every class!

Organization: Creating simpler systems for organizing student work, lessons, classroom stuff etc.
Relationships: How to set boundaries with parents and students.
Warm Demander: How to have high expectations with a warm classroom culture. 

I would never have guessed that even two decades in I'd still have to work evenings and weekends.  I'm not sure that would've deterred me though. One of the reasons I still have a lot to do, other than marking, is that I'm always thinking about new ways to do things.

If you're a first year teacher, I hope that this helps put some things in perspective.  Everyone struggles at first.  If you need help, reach out.  Sometimes you can find help in within your department, sometimes you may need to look in your building or online.  Find someone that will listen and that can help (if needed).  Have a great first year!

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Angles in Triangles Interactive Notebook Page

The first lesson in my Triangle Congruence unit is about triangle theorems.  This is the lesson with the Triangle Sum Theorem, the Exterior Angle Theorem, and the corollaries that follow from those theorems.  

This is the overall page that I would use for this lesson.  This foldable is one that is available in my TpT store.  I would make sure that my students wrote a few reminders off to the side of the foldable.  Basically, I would have them write anything that that particular class period had been struggling with.  Usually that ends up some variation of what I have written in purple.  There is also room for students to write their own reminders or notes.

Angles in Triangles Interactive Student Notebook Page Idea

Each tab has the theorem and an example.  For the Exterior Angle Theorem, I really like color-coding it this way.  After doing this a few times, I can usually prompt students that need help by just circling and putting dots on their paper, no words required.

Angles in Triangles Interactive Student Notebook Page Idea

The bottom tab has all of the corollaries that follow from the top two theorems.  I typically find that my students forget the last one, so I made that the example.  It’s a super helpful shortcut, so I want them to remember it.

Angles in Triangles Interactive Student Notebook Page Idea

I hope this idea is helpful the next time you're teaching this!

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