Arc Length vs. Arc Measure

Students always seem to get confused about the difference between the length of an arc and the measure of an arc.  When I teach this concept, I try to relate it to sports.  The vast majority of the kids in my class have run around the track at some point.  So I start asking them questions. :)

If you have to run one lap around the track which lane do you want to run in?
The inside one!

Why?
It's shorter!

Why?
The circle is smaller.

Interesting...  You have still run one lap though.  All the way around the circle would be 360 degrees.  So, it should be the same distance.  If you run the outside lane, that's the same distance as the inside lane.
[At this point, controlled chaos ensues.  Most of the athletes become indignant.]

Then, I tell them that the measure of an arc is the degree measure.  The length of the arc is the actual distance you would have to run.  Most of them get it by this point.

I made a handy little "cheat sheet" for students to glue in their notebooks.

2 comments free resource, general teaching tip, geometry, interactive notebook

Coordinate Quadrilaterals INB Pages

Instead of teaching coordinate quadrilaterals throughout the quadrilaterals unit, like I normally do, I taught it all together.  It worked.  I don’t know if it was better or worse than doing it throughout.  

The first day, I only taught about proving a quadrilateral is a parallelogram.  Before I started the lesson, I gave all of my students a sticky note and had them find the formulas for midpoint, slope, and distance and write them down.  They stuck it on their desk and referred to it during the lesson.

I felt super fancy with the foldable I used for this lesson… partly because I used an “I can…” statement on the foldable, and partly because I used a shutter fold, but kept one flap up.  I also feel like the way I have a flap for each formula takes any guess work out for the students.  Under each flap, we worked the example, using that method.  So, when my students had their homework they could mimic the work under the flap.

On the facing page, I gave students a scrap of graph paper.  I gave them the four points in the margin and had them draw the graph.  The worked with their partners to prove it was a parallelogram using all three methods.  I honestly didn’t have to use the graph, but I think it helps them visualize things better.

The second day, I taught students how to determine the type of quadrilateral, using coordinate geometry.  Some of these problems take up so much room (if you want things to be organized), so I used a giant hamburger book.  There were three problems in this book.  It took almost the whole class period to work the three problems - that could be because my students worked slow.

The third day, I had my students do a stations maze to practice their coordinate geometry.  They worked on it the whole class period.

I haven’t decided if I will teach it all together again next year, or if I will do a little at a time throughout the unit.  Good thing I have time to decide!

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Systems of Equations INB Pages

This year, I taught systems of equations right after Christmas break.  My students STRUGGLED.  I think I need to do a two day review of graphing equations before jumping in next year.

First, we talked about what a "system" of equations actually is.  We also talked about the three types of solutions they could have and what they looked like graphically.  At the bottom of the page, I included a magic book that substitute points into a system to determine if they were a solution.  The file was corrupted, so I will get to remake that for next year.  Yay.

Next, we solved systems of equations by graphing.  I gave the students a step by step list for their notebooks that had one example included.  Next year, I think I will include a "drill and kill" practice sheet on the facing page.

The next day, we solved systems of equations by substitution.  I used the same step by step list with one example format as the previous day.  This one worked well and I liked the way I highlighted what we were substituting into each equation.

On the next page, I had a practice sheet.  Each time I substituted into an equation, I highlighted the substitution.  Showing that helped my visual students SO MUCH.  I also started throwing in word problems.  They hated me a little for that. :)

I spent the next day reviewing and showing them what the special cases looked like algebraically.  No cool foldable for this page.

I continued following the same format for solving systems of equations by elimination.  I had a step by step list with one example for their notebooks.  This time, I highlighted the "stacked" variables.  I was trying to make them notice that they needed to make sure they were combining the same variables.

The example page came next.  I continued highlighted the "stacked" variables.  Also, more word problems :)

We had several days of review and when I was sure that my students had it down, I pulled out the big guns...MORE WORD PROBLEMS.  I used legal paper to make a giant hamburger book.  It gave us plenty of room to work all of the problems.  My students whined loudly that they were going to die.  No word problem related deaths... yet.


Intro to Systems and Solving by Graphing INB Page
Solving Systems by Substitution and Elimination INB Page
Substitution and Elimination Practice Pages

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Factoring INB Pages

I teach factoring a little bit differently than most people, I think.  I only teach one method and it works every time.  There's no need for boxes or fancy flow charts to tell students what to do when.  You can read my post about How I Teach Factoring.

For this unit, my interactive notebook pages are kind of bland.  However, they got the job done!

First, I taught factoring by grouping.  I listed the steps to factor by grouping, then had an example below.  Inside the hamburger book, I have lots of examples.

Next, I taught factoring trinomials.  I had the steps listed and an example below (again).  Inside the giant hamburger book there are a bunch of examples.

Lastly, I taught all those special factoring patterns...sort of.  There really isn't a need for students to know all the factoring patterns, until they notice the pattern for themselves.  They can still factor everything (that's required in my Algebra 1 standards) with the splitting the middle method I've taught them. This hamburger book lays it all out for them.

Yes, this unit had only three lessons.  Our textbook has an entire chapter for this unit!   We did a lot of practice, so we spent more than three days working on these skills, but it wasn't necessary to teach a million special cases.

I hope this helps you maybe re-think your factoring unit!

Factoring by Grouping Foldable
Factoring Trinomials Foldable
Special Factoring Patterns Foldable

4 comments algebra, free resource, interactive notebook