Linear vs Non-Linear Functions

In Algebra I, we are working on determining whether a function is linear or non-linear.  We have been using visualpatterns.org as Thursday bellringer problems each week, so I started this lesson by showing the students 3 patterns (2 linear, 1 non-linear) and asked them to write an equation for each pattern and graph the points using my version of Sheri Walker's handout from visual patterns.org.

The first two were pretty easy for them since they looked like the bellringer patterns (nice and linear).  The third one was a little tricky.  With some assistance the students were able to figure out the pattern.  We then talked about what made the third pattern different than the first two.  A lot of students picked up on the fact that the rate of change was different, some even mentioned that the equation had an exponent.  We compared the graphs and discovered that we couldn't connect the points with a line.  This opened up the discussion about how a constant rate of change (here's the INB page for rate of change from @mathequalslove) is what determines a linear pattern.  We practiced with a few examples and that was a wrap for day 1.

Today we started by reviewing with Plickers.  First time I've used them...LOVE!!!  I gave them four patterns and they had to determine if they were linear or non-linear.  Plickers are seriously awesome.  I will definitely be using them more often.  Great formative assessment.

Then we built our own non-linear and linear patterns with pattern blocks.  This was a great opportunity for my co-teacher and I to walk around, get a feel for who was still struggling and offer more one-on-one instruction.  I asked students to write their names on post-its and put them on their desks.  Then each student figured out the equation for their own linear pattern.  I had the students switch a few times, record whose pattern they were working on and figure out the equation.  I made sure I got a picture of each student's desk with name so that I could do some error checking.

 

 

 




Overall, this was a great activity.  I feel like students are really developing an understanding of rate of change and the fact that constant rate of change determines a linear function.

Next we will be delving deeper into linear functions and start making the connection between rate of change and slope.  I'm planning on doing a lot of experiments and investigations to help develop really solid understanding of linear functions.  I LOVE Andrew Busch's resources and will probably be using several of them.

Algebra Party!!

We are moving at a snail's pace in Honors Geometry this year... New 8 period, shorter class, day (up from 7), new textbook & curriculum, and ever changing daily schedules have made it hard to find my grove to get through material!  That being said, we are working on solving problems involving segments and angles using algebra.  I didn't want this week to just be a boring week where I show lots of examples in class and students do lots of examples for homework, so I decided to throw a big ole Algebra Party.

It all started with some angle bisector algebra problems.  I used a carousel approach where I gave each group of 4 students 4 different problems and a big whiteboard.  There were 4 steps to each problem:  Draw and label a picture, write an equation, solve the equation, and solve for the requested segment.  Each student did step 1 for their own problem, then rotated to the next problem to check and do step 2, etc. until all four problems were complete.  The students said that it really helped them to understand the different types of problems better.

Then we had a half day with 15 minute classes on Friday.  To ramp up the excitement for a week of lots of Algebra, I decided to have the students make party hats.  They are really very simple hats, but they loved it!  Our school motto this year is "212 it - go the extra degree" and one student made a 212 hat:  It says:  STEAMY MATH 44,944 / 212 = 212 

The second party game was clock partners.  I had the students sign up for partners for each hour of the clock and just called out random hours for three problems.  They got to work with different partners and do some problems with vertical and supplementary angles. 

To conclude the week of party games I will be doing two more party games.  The first will be another carousel activity, but randomly assigning students to groups to practice complementary and supplementary angles.  The second will be a scavenger hunt with acute and obtuse inequalities.  

Who said math class had to be boring?

Average Velocity vs Instantaneous Velocity

In Calculus, I wanted to start my unit on Derivatives by getting the students to conceptualize the limit definition of the derivative.  I decided to use an idea from @ThinkThankThunk that he blogged about here.  Now, I didn't start by doing this on day 1, but I really liked the idea of somehow demonstrating the difference between average and instantaneous velocity.

Day 1 & 2:  I went outside and we had one of the students run/walk/jog at varying speeds for 200 ft.  I timed her first for the 200 ft, then every, 50 ft, then every 25 ft.  We discussed the difference between average and instantaneous velocity and how we could go about determining instantaneous velocity.

Day 3, 4, & 5:  We used the CBR2 to do a ball bounce activity to help develop the limit definition of the derivative.  Handouts are here, here (Activities 1.3, 1.8, and 1.9), and here. And a little Desmos demo thrown in.

Overall, I felt like the lesson went well.  Yes, it took quite a few days, but the understanding of what a derivative is, what it is measuring, and how it measures that will be invaluable as we move on to more difficult topics.