Teaching Geometry - Practice Right Tri Trig #VNPS

I knew I had a lot in yesterday's lesson and I had today's lesson as a day of processing and practicing.  We were back at the board to do special right triangle trig with special right triangles and points on a plane.  I think it went okay.  I think they are getting the ratios now.

Here is the powerpoint lesson: Practice Right Triangle Lesson

Here are the board problems: Practice Trig Board

Here I asked them to draw and label a 30:60:90 triangle.  Then, I asked them to find sines and tangents:



Here was same triangle, different student, different questions: 
(These students wanted to be in the picture.)


Then, I switched to the 45:45:90 triangle:  We also looked at writing inverse:




Here we looked at both special triangles together:


We were running out of time, but we graphed the magnetic graph and put a point on the coordinate plane to introduce finding the ratios given a point and the idea of a reference angle:


Tomorrow: Law of Sines

This is a HEAVY unit!

Teaching Geometry - Right Triangle Trigonometry #VNPS

I am figuring out three things about teaching Geometry:
1.) Geometry is when they really are introduced to a lot of things for the first time.  Therefore, it is great for discovery type lessons.  Versus when I have been teaching Algebra 2 and I am saying, you have already learned this, let's make it more challenging.
2.) Discovering at the boards with VNPS is exhausting.  I am doing 3 lessons a day at the board with classes of 28 kids all at the board.  Then, sometimes, add my Alg 1 and 2 classes at the board.
3.) Learning is messy and productive struggle is uncomfortable for both the kids and the teacher.  I want to jump in and help, but I can see them building their understanding one piece at a time and I don't jump in and save them.  I let them build.

Lesson in point - yesterday's lesson about Right Triangle Trig.  The lesson was ambitious and I knew that going in.  I was supposed to teach them all 6 trig functions, including how to solve for the ratio or the angle on the calculator plus introduce angles of elevation and depression.

I did my best to really stress that the trig functions are just side ratios.  They were able to stumble through my board problems.  I did introduce SOHCAHTOA as a when of remembering them.  I introduced the three reciprocals.  I got to one problem introducing angle of elevation and they were able to do it.  I did not get to angle of depression.

Here is the powerpoint lesson: Right Triangle Trig Lesson

Here are the board problems: Right Tri Trig Board

Here is some student work:

I read somewhere online to start them off relating trig to a ramp.  I asked them to draw the ground and a ramp that had a slope of 3/4.  This is what I got:


So, I asked them to erase it and label the horizontal and vertical instead:



We learned about theta and from theta we labeled opposite, adjacent, and hypotenuse.  I told them the 3/4 was a tangent ratio.  I asked them to come up with 5 more combinations of side ratios using opp, adj, and hyp. Then, we named them:



I left each lesson not feeling done.  They left feeling confused.

The thought came to me that I fed them all a huge Thanksgiving dinner and then I kicked them out without giving them time to digest.

They tried the homework.  I went over it today.  I went over each problem.  They were starting to see the connections for each side ratio.  Some of the word problems were challenging because they were overlapping triangles.

I did discover one student's misunderstanding.  She thought COSECANT was another acronym and couldn't figure out what it stood for.  She kept asked what it was. I kept saying it was the reciprocal of Sine.  Then, she finally said, but what does it stand for.  I think I corrected her.

Next lesson...practice.

Teaching Geometry - Special Right Triangles #VNPS

Today's lesson: Special Right Triangles - good for drawing and checking vocabulary

We were going to the boards to discover both special right triangles, using squares and equilateral triangles. 

We started with 4 squares.  (I forgot to take square pictures). We had 4 different sizes and I asked them to find the diagonals.  Some started the Pythagorean Theorem and then they saw the pattern!

Then, we did 4 equilateral triangles:


Then we drew altitudes.  I asked them to label their angles and find the altitude. What did they notice?



I already taught the area of a regular hexagon using equilateral triangles and the Pythagorean Theorem, so they caught on to this one pretty quickly.



Here is the google slide: Special Right Triangle Lesson

Here are the board problem: Special Right Triangle Board Problems.

Teaching Geometry - Pythagorean Theorem #VNPS

My students are very confident with the Pythagorean Theorem.  They learned it prior to my class and we have used it in class without a formal introduction.  Today's lesson was to include Pythagorean Triples (new), more difficult problems with sides measuring square roots or binomials as well as Pythagorean Inequalities.

Again, we were up at the boards.  I gave them this list of triples without telling them what they were.  I asked them to write what they noticed.



They had so many different noticings I wanted them to share, on the spot I decided to have them do a gallery walk.  In their groups, I asked them to circle the classroom and read each other's remarks.  When they got back to their spot.  I asked them to share out something they noticed about what someone else wrote.  I liked this.  One group in the first class noticed the last four.  So, in my other two classes, I asked them to notice about the last four as well.  Some noticed it added 3, 4, and 5 when you went down the list.  Some kids noticed it was 3*7, 4*7, and 5*7, etc.

 







After this, we did these board problems.



And, Google Slide Lesson: Pythagorean Theorem

They did well with the regular Pythagorean.  No one tripped up when I gave them a hypotenuse instead.  We struggled with squaring the binomial - only one group per class got it wrong.  We learned from their mistakes.

I finally, we drew a triangle from the triple 5, 12, 13.  I asked if it was a right triangle and how they knew that and then I asked what would happen if we changed the 13 to a 14.  They quickly grasped that it would now be obtuse and that was our inequalities.

Student work

Back to desks to recap and practice. 

Teaching Geometry - Geometric Mean #VNPS

I taught geometric mean years ago.  I did it with the different ratios within the triangle.  I did it with two numbers.  I wanted to have my kids discover it at the white boards and understand how to set up the ratios, so I started researched.

It was our first lesson as a new class with new students in the new term, so I wanted to get them used to my memes, so I showed this as my title of my slides and I got them up to the boards.



In the process of researching the lesson, I learned two things:
1.) You can find the geometric mean of more than two numbers.  You just need to take that root.
2.) The geometric mean is the middle number in a geometric progression.

Here is the google slide lesson: Geometric Mean

I guess I just never really thought about #2 before but it made sense and I could relate it back to Algebra 1 when the students learned about geometric sequences.

When I introduce a new subject, I like to connect it to something they already know.  Naturally, that would be arithmetic mean.  We started with this:



I asked what they thought Geometric Mean might do.  We worked our way to this:



They did come up with the idea of multiplication.  I gave them two numbers and they had it.  They could calculate the geometric mean.  Then, I explained we would use proportions in similar triangles and use the geo mean to help us find other sides of the triangles.

Most of my whiteboarding (#VNPS) is oral, but I gave them this picture.

Here is my script for the board work.



I asked them to classify it to orientate them with the picture.  I asked what "h" was and it took a bit but we got to the altitude.

I had them pull apart the three triangles so the right angle was in the left corner.  That was tough, but we got there.

Then, I asked them to write the ratio of short leg to longer leg.

I forgot to take a picture of that one, but here is a picture of student work for hypotenuse to short leg:


And...voila...there it is staring at them - that diagonal geometric mean.  At least one group in the class had that aha moment and found the geometric mean.

We moved onto the second ratio and set it up.  And, a student taught me even more about the Geo Mean:
3.) The three geometric means are the three altitudes of the triangle!!!

I also thought of them as the altitude and the two legs, but yes, they are all altitudes.  How cool!

They set up the other ratios and found the geometric means within the proportions.  We came back to our desks to recap with notes and did a practice sheet.

I was happy with their work and their understanding.

January...it's the new September (#mtbosblog18 post 1)

My 1st blog post in the #mtbosblog18 challenge.  I challenge you to blog on the 18th of each month for the year 2018!

Our school does semesterization.  Some of the courses are only one semester long and it works for them.  For math, we are a full year long.  This means on Tuesday, Jan 23rd after we finish our midterms, we start fresh with term 2 and this means new classes, some new kids, and new periods.  So, if a student had me first term, they might leave and go to a new teacher or they might stay with me but have new classmates and a new class period.  This new term I will have 117 students - 53 returning and 64 new students for 55% new.  That's a lot!

In September I do a lot of getting to know the students from day 1.  Unfortunately, I don't have that precious time to repeat it.  I will hand out an index card and collect the new students' birthdays so I can add them to my weekly birthday board.  I will do another escrapbook like I did last year.  I make a scrapbook style page about me - pictures and my hobbies and then I share it with the class to make their own slides.  I love this!  It is a great way to get to know the new students but to get a chance to learn about the students who have had me all along.

Here is my page this year:


I am a To-Do list person and I love to check things off and feel accomplished.
Keep in mind I teach 3 different classes, Algebra 1, Algebra 2, and Geometry.

Here is my list:

• Update my canvas pages
• Update flippity.net for my random groupings at my VNPS
• Make new Jan/February seating charts (I change each month but both are short, so 1 will do)
• Make new email groups - this means looking through each kids' demographics to see if their email includes a middle initial (this takes a LOT of time!)
• Print class lists and keep printing until day of because they might change
• Copy my Assignment Sheet 11 for each class (Long term assignments due each cycle)
• Make and copy my new calendar and syllabus for the new unit for each class
• Make a new poster for the 2018 challenge
• Update my Deltamath groups (I use this site for practice problems every now and then)
• Put my old lessons away in binders
• Make my escrapbook page and make the classes google slide show and share (once I get email lists made)
• Organize new student birthdays
• Replenish classroom supplies - new board markers, new paper, etc
• Wash my whiteboards
• Clean my computer desktop
• Prepare 1st day lesson for each class, hopefully 1st week of lessons
• I administer the AMC math test on Feb 7th, so I need to make sure I send an email to the PTA to get parent volunteers to help with that.
• We have a math team competition on Jan 24th, so I need to make the teams for that and book the bus.
• Figure out Equatio (first I have to find it. I know I have it, can't find it though)

Keep in mind this all has to be done while midterms are happening which includes grading all of my exams, analyzing results, and recording results, plus providing extra help sessions to students. 

Teaching Geometry - Expected Value

I am teaching a new prep this year - Accelerated Geometry, along with Acc Alg 1 and Acc Alg 2.  It is a lot.  I have taught a lower level Geometry course, but a while ago.  I have a colleague who also teaches it, so I have the quizzes (I make different versions). I have the homework.  (I still do each problem).  I have the Do Nows, but have to do them.  And, then of course there is the presentation - the lessons, those are mine and how I run the class.  I have the what and when, but I get to create the how and I enjoy the creation part.

This is where expected value comes in.  I have not taught it before.  I'm not even sure I learned it before.  So, I immersed myself in it. I went down the EV rabbit hole, reaching, researching, reading definitions, real life uses for it, formulas, worksheets, examples.  We have a target example in the book I knew I was going to use but I thought it might not be a good starting point. 



It has the whole area of a circle and the probability of hitting each ring.  I was teaching probability so it should be okay, but I didn't want them to enter at that point and get hung up.  I am all for starting with a challenging problem, but today's lesson was not it.

I also am doing a lot of VNPS - having the kids work at the whiteboards to discover Geometry.  I knew I would be teaching this lesson on the first day back from winter break, so I wanted to have them up and working and discussing, but I just couldn't do it.  I didn't want to force it.  It just didn't fit with that kind of lesson and that is okay.  I felt this topic had to be a whole class, step by step procedure.  That doesn't mean it had to be boring.  I just wanted to slow it down a little and take it one piece at a time.  I also decided to start at a carnival with a card game.  A standard deck of cards because we were familiar with the probability in a deck of cards.

I started like this:



I got them interested.  Some had questions - what do we win?  (I was going for 3Act style, not giving away all the information.)

I gave them some more information and polled the class.  We had some gamblers ready to play.



It went pretty smoothly and seemed to make sense.  I built up to the target.  Now they have seen the idea of EV.  I created a table to organize my work.  I had to consider all possibilities - winning and losing.  And, then it is about multiplying the "prize" by the probability.  Not so bad.  We ended up with an EV of -0.73 and we talked about what that meant.  Did they see how the carnival could rig their games?  Yes, they did.

When I introduced the target problem, I started by saying we couldn't just jump straight to the EV chart.  We had to figure out the probabilities first and that took some time, but again,  they were okay with it. We left things in terms of pi and conveniently pi/pi simplified to 1.  We walked through the problem step by step and they found success.  Then we did a couple more on their own for practice.  I liked the way the lesson went and was happy with it. 

Here is the slideshow if you are interested.