Math = Love

Tuesday, October 6, 2015

Reviewing Integer Operations + Order of Operations in Algebra 2

My Algebra 2 classes were participants in my action research project over remediating integer operations.  I blogged about that more here.  Based on what group students were randomly assigned to, they received a certain set of notes.

Group 1:

Group 2:

This second group of students used a number line and bingo chip to work out the questions.

Then, we took notes over the Order of Operations.  I re-sized my order of operations hop scotch this year.  Link to download files is at the bottom of the post, as always.

Usually, I find that students don't really listen to reviewing the order of operations because they are convinced that they already know how to do it.  After all, they've been doing the order of operations since third grade or so.

To (hopefully) get their attention, I told them that I was going to tell them how their third grade teacher got the order of operations wrong.  We were going to Algebra 2-itize the order of operations.  Of course, a student told me Algebra 2-itize wasn't a word.  #sad

Here's the Cliff Notes version:

Parentheses - Your third grade teacher taught you that you should always work out the parentheses first.  He or she should have taught you that grouping symbols always need to be done first.  These could be parentheses.  They could be fancy brackets.  Did you know that there are invisible parentheses around the numerator and denominator of every fraction???  [Later, I decided I should have also said that there are invisible parentheses inside any radical!]  We should cut your third grade teacher a bit of slack, though, because when you were at that age, the only grouping symbols you ever saw were parentheses.

Exponents - Your third grade teacher taught you that exponents should always come next in the order of operations.  This year in Algebra 2, we're going to learn a cool way to rewrite any radical symbol as an exponent that is a fraction.  This means that radicals (or roots) have to be done during the exponent stage of the order of operations.

Multiplication and Division from Left to Right - This one is pretty self-explanatory.  Except, students have to be reminded of the left to right aspect.  Many students admitted that they had a teacher tell them that they ALWAYS have to do all the multiplication before the division.  And, I believe this.  It's one of the things about PEMDAS that drives me crazy.  

Addition and Subtraction from Left to Right - Again, I emphasized the left to right aspect.

Then, I threw this problem up on the board:

I asked, "What does the Order of Operations say we should do first?"  Half of the class was convinced that we do 4(5) first.  The other half of the class said we needed to take care of the exponent.  This led to a discussion of another way our third grade teachers misled us.  When we were taught to always do parentheses first, parentheses were only ever used as grouping symbols.  We were still using the x to mean multiply.  It wasn't until we got to middle school that we really started using parentheses to mean multiplication.

I made them write a note about how when we see parentheses in a problem, we need to check and see if they are grouping parentheses or multiplication parentheses.

We did a few practice problems together in our notebooks.  Here's a few:

Foldables and graphic organizers can be downloaded here.

Solving Equations using the Three Essential Rules of Math Foldable

When reviewing solving equations in Algebra 2, I borrowed Glenn's Three Essential Rules of Math.

I typed them up as a foldable to reduce the amount of writing my kiddos had to do.  Here's the outside:

And, examples on the inside:

We did a few practice problems.  I made them draw in the 1's and 0's that they made, but I guess I forgot to do it when I wrote out my own solutions in my notebook.  Oops...

And, for the record, we did more than 3 practice problems.  But, I only copied down three.

I can't decide how well I like this method.  The ones and zeros didn't help my students as much as I'd hoped.  I think if they were taught this way from the beginning, it might work much better.  Once they get to Algebra 2, they seem pretty set in their equation solving ways whether they are doing it correctly or incorrectly.  

Download the file here!  The font is HVD Comic Serif Pro.  

Monday, October 5, 2015

Algebra 2 Exponent Rule Review

Every year, I keep trying a different way to review exponent rules.  For my Algebra 2 students, I gave them a list of exponent rules.

We wrote a word summary of what type of problem each exponent rule would help us with.  My marker choice was not good for photographing...  Sorry!

They were really struggling with applying these to the problems we were simplifying.  They kept claiming that they just didn't know where to start.

Eventually, I broke down and gave them an order of steps to follow.  This helped them a lot.  Though, I wish they could solve these problems without me writing out step by step directions.

We were working on simplifying expressions like this:

I guess where I struggle with this is that it *should* be review in Algebra 2.  We don't have time to derive all of the rules from scratch.  But, they act like they've never seen anything like this before.  My main motivation for doing this as a skill is to prepare for dealing with negative and zero exponents when we work with rational exponents and logarithms.

I also drilled into their heads that (2xy)^2 = 4x^2y^2.  For the past few years, most students have said 2x^2y^2 which was driving me bonkers.  We did lots and lots of practice problems with that, and I don't think I've seen anyone make that mistake at all lately.  Yay!

If anyone has ideas about how to review this without reteaching it from the beginning, I'd LOVE to hear it!

Exponent Rules file is here.

Sunday, October 4, 2015

Factoring out the GCF of a Polynomial Foldable

I've always taught factoring out the GCF of a polynomial the way I was taught to do it.  And, historically, my students have always struggled with this.  I think they usually get the gist of what we're doing, but they usually struggle with the GREATEST part of GCF.

Last year, I saw Jan Lichtenberger's post about using the upside down division method to make factoring the GCF (and a host of other concepts as well) easier.  It's like my beloved Birthday Cake Method, but upside down.

Also, if you haven't checked out the rest of her posts at her Equation Freak blog, you're missing out.  So many great ideas there!

I modified her instructions the tiniest bit to make them fit with what I'd just covered with my students.  I gave them this instructions on the outside of a booklet foldable, but they really learned from just walking through the four examples we did as a class.

Seriously.  This took less than half of a class period.  And my students rocked it.

Even though they didn't really use the steps, I hope they might reference them if they forget at anytime.

Here are the four practice problems we did together:

Some notes about how this went:

* It's a lot of writing.  A lot more writing than the standard way of solving this type of problem.  I'm believing more and more that making students write problems out the long way is a good thing.  It makes them start thinking about shortcuts.  Did you hear that?  It makes THEM start thinking about shortcuts.  Students thinking in math class is always a good thing.

* If you don't factor out the highest number that goes into each term the first time, it's not the end of the world.  In my example, I factored out a 2 three times.  In one class, we actually factored out a four and then a two.  In my other class, someone saw right away that we could factor out an 8 and save us some writing.  By breaking this down step-by-step, students were actually thinking "Can I factor anything more out?" instead of just getting an answer and thinking they must be done like in the past.
* Student also caught on really easily to the fact that all the terms had to have it in common to divide it out.  I used to tell them this in the past, but it never really stuck.  On the second problem, a student wanted us to factor out a z.  A couple of other students chimed in why this couldn't happen.  In that same problem, another student suggested we factor out a five.  Again, other students chimed in.  I actually didn't do that much talking in this lesson.  It was beautiful.

* With the third problem, I wrote it out as taking x out twice.  Both my classes realized that we could just take an x squared out and be done with it.  Yay!

* Then, my students shocked me yet again when they didn't run away screaming from the last problem.  This is usually the type of problem that students just skip without trying which makes me sad.  Instead, they walked through it step by step.  My first Algebra 2 class of the day took out the x^2, y^2, and z^2 just as I wrote it above.  My second Algebra 2 class of the day insisted on us doing it a bit differently.  They factored out the 2 first.  Then, they asked if they could take the x^2, y^2, and z^2 out ALL at the same time.  I tried to talk them out of it.  Wouldn't that be confusing?  But, they insisted.  And, they did it perfectly.  They were so proud of themselves for doing it in two steps instead of the four steps I wanted them to take.  What they didn't know was that the way they solved this problem was almost exactly like how I'd tried to teach students to solve it before and it had gone horribly wrong.

*  I guess what I like best about this method is that students can stick with it for the rest of their mathematical careers, or they can internalize the process and factor out the GCF without this structure.  Either way, kids are doing algebra!  I'm starting to think of this method as training wheels without the stigma (hopefully!) of training wheels.

*  I told my class how my students in the past have struggled with this.  They couldn't believe it.  Now, I'm hoping that they'll be more likely to factor out the GCF before beginning to fact otherwise since they think it's so easy.  *Fingers Crossed*

Download the file here.  If you download the editable Publisher file, you'll need to download this font: Londrina Solid.  I've also uploaded it as a non-editable PDF.  

Saturday, October 3, 2015

Calculator Hospital

Guys, I'm not the most organized person.  I'm really not that organized at all.

My desk regularly looks like this:

Do you see those two calculators in the center of the picture?  Students brought them to me because the screens were doing something wonky.  I need to try and reset them to fix this.  Or maybe they just need new batteries.

This happens all the time.  I usually tell students to just give me the calculator and grab a different one.  I set the calculator on my desk with the best of intentions of fixing it.  A few days go by, and I start to forget why that calculator is sitting on my desk.  In a desk cleaning spree, I will put it back in a calculator pocket.

My desk looks cleaner, but soon a student tries to use it again and the whole process starts all over again.

Not any more.

I took an empty plastic tub and labeled it "Calculator Hospital."  Dead/sick/dying calculators now have a designated place to go.

Now, why didn't I think of this sooner???

Oh, and don't worry.  My desk looks much cleaner now.  At least for a little bit...

Friday, October 2, 2015

HOYVUX Posters

Yesterday, I mentioned some HOYVUX notes we took in Algebra 2 as a review.  Here they are again, for reference.

A month ago, Mary Williams posted a link on twitter to this video sneak peak of her classroom.  I got really excited when it revealed HOYVUX posters.  As soon as I saw them, I decided these HAD to go on my to do list.  

I finally put together this version:

Laminating + flourescent lights doesn't make for the best pictures.  Sorry about that.

Here's a better pic of HOY:

And VUX:

I've made them to print two to a page, but you could easily edit the Publisher file to print 1 to a page to make them bigger.  I wanted to hang them above my coordinate plane, so I needed to keep them small to fit.  

To edit the Publisher file, you'll need to download these free fonts: La Unica and GrutchShaded.

Publisher file and PDF file have been uploaded here.

Thursday, October 1, 2015

Graphing Ordered Pairs and Linear Functions Interactive Notebook Review

One of my review topics for Algebra 2 was graphing ordered pairs and linear functions.  We covered this thoroughly in Algebra 1, but I know they needed to be reminded after two summers plus a year of geometry.

I typically use this foldable with my students in Algebra 1, so I wanted something on the quicker and easier side that they hadn't seen before.  I whipped up a quick graphic organizer over graphing ordered pairs.  

Then, I gave my students a coordinate graphing picture from  My picture looked a bit different from one of my students which makes me think that I might have accidentally skipped a few points.  I did this over the course of a day in between helping students.  I should have marked off each point as I graphed it to make sure I didn't miss any with all my starting and stopping.

I ended up giving the same picture to my Algebra 1 students later, so once I've done another for that notebook, I'll try and update this post with a proper kangaroo.  

Kids asked if I chose a kangaroo on purpose since I spent my summer in Australia.  Duh.  

Plus, kangaroos are cute.  

I now interrupt this math-y post with some cute kangaroo pics from this summer.  

You're very welcome.

Okay.  Enough cuteness.  Back to the math.  

I typed up a quick set of steps to save writing time while reviewing graphing linear equations.  

We talked through HOYVUX again since it had been a while since they'd seen it.  Some of my students had a different Algebra 1 teacher, so we also had to rewatch Slope Dude.  The other students insisted.  

Before class, I quickly put together a poof book of practice equations to graph.

I found these awesome instructions (via Pinterest) that show how to easily fold the poof book.

There was only one problem with this poof book that students quickly pointed out.  With the exception of the vertical line, all of the other lines ended up having a negative slope.

Oops!  I'll definitely fix this next time I use this with a group of students.  Don't worry, I'll upload the editable Publisher file so you can change the equations to your heart's content!

All the files are here.  (And, remember - you can always access every single file on my blog here!)