Inequalities Foldable

This week, I'll finally finish systems in my Math 8 class.  I say "finally" because we started systems the first week of February.  We have had so many snow days, delays, and testing interruptions that it's not surprising we're still working on that topic after two months.  My kids are pretty good with knowing what to do when but their lack of fluency with integers is still killing their overall accuracy.  We've broken out the calculators 100% of the time, but still they make errors. 

Systems is also arguably the hardest unit we have to cover this year because it is one of the most complex with the most steps.  I'm hoping that our next unit makes students feel more successful.  We're going to spend time on inequalities next.  This should be a partial review for them.  I put together what might just be my favorite foldable to date and I'm sharing it below.

When I was looking for inspiration on inequalities, I found Sarah's post from the fall about "flipping the sign."  I took her idea and ran with it in a slightly different direction (pun intended).  I included both instances of flipping the sign on the flip-flops and oriented them in the opposite direction so it would be easy to write the notes on them. 

Here are the files.  Please leave me a comment if you try them out; I'd love to hear about it.  I've changed the fonts to ones I think will be friendly, things in the Arial family.  Not as cute, but hopefully you won't have to reformat anything before you print!


What's usually the hardest topic in your course for students?  

Mathematically yours,
Miss B

The Quadratic Formula

A couple of weeks ago, one of my students casually mentioned that they'd learned the quadratic formula last year when I was using the ± sign.  "Great," I thought, "there goes the point of teaching the other methods of solving quadratics if the last teacher just went straight to the formula."  Given that quadratics are in no way part of the curriculum for their last class, I was perplexed.  I finally got a little more background today.  Apparently they did it "one day" and were told they "didn't have to memorize the formula," though one of my girls did have it memorized!

All that is to say that last week we studied solving quadratic equations by factoring and by square roots.  Those methods went pretty well.  Then we did completing the square.  If you haven't ever introduced it as a puzzle using Algebra tiles, you are missing the hook you need.  The intro to completing the square goes something like this:

Alright everyone, we're going to play a little game.  I need you to please take out an x squared tile and 2 x tiles.  You may add ones tiles if you want.  Please make a square.  

The kids make a square by adding one.

Great.  Let's notice the area of the square and the side lengths and note that on our recording sheet. (Use graph paper here to make the diagrams better).

Next, try the same kind of puzzle with one x squared tile and 4 x tiles.  Kids will add on 4 ones.

Repeat as needed with other even numbers of x until you have success (or you run out of tiles!)

Would you build a square from one x squared tile, 5 x tiles, and ones tiles?   Disaster will strike.  Kids will build you rectangles, convinced they are squares until you suggest they measure.  Other kids will claim it's not possible.  Still others will add negative x tiles in an attempt to make it work.  Whining will most likely ensue. 

So, why isn't this one working?  Let's see what you have tried.  Build some unsuccessful models, talk about why they fail.  Did you start with this model?  Build rectangle (x+2)(x+3).  Most kids will say "yes."  Why didn't it work?  They'll answer that the sides were different lengths.  So if this side is longer, could I move a tile to the opposite side?  No?  That makes the same thing?  Hmm...if we start building this with the idea of putting equal x tiles on each of two sides of the x squared tile, what will happen?  We'll have one left over.  How could we share it equally?  There might be trouble here if your tiles are plastic like mine.  They were stuck on the idea that they couldn't physically destroy my tiles.  One of the kids saw the idea of dividing that tile.  We drew the (messy) model of (x+2.5)^2.  We made the connection (easily, even) that the b value divided by two and squared provided the c value. 

Once we got that far, kids understood the "what should we add to both sides to make a perfect square" aspect of completing the square, which is the only part that is really different from what they already know.  My kids have turned out to be pretty confident at completing the square.  Today, some of them got to the end of the kite relay.  The last problem was completing the square of the standard form of a quadratic equation.  I actually had three or four groups get it.  The hardest part for almost every group: adding fractions and remembering to get a common denominator.  Oh, how we forget the foundational things! 

We wrote it up formally in our notes after lots of messy whiteboarding.  The kids' first reaction: "You make it look so easy."  Sorry, friends!  Their second reaction: sheer joy when they realized we'd derived the quadratic equation by completing the square.  It was cute. 

What's your best tip for teaching quadratics?

Mathematically yours,
Miss B

My Students' Favorite New Activity- Coloring Relay

Come March, my students have seen most of the tricks up my sleeve.  We've written ON our desks (the scandal!), we've played "Shark Tank," we've done Kagan structures, we've written on whiteboards, we've played "Mountain Climber," we've done some boring worksheets, we've made foldables, we've worked through some tough tasks, we've coached each other in "Problem Master," we've modeled with manipulatives, we've used Desmos.com to visualize data, we've done stations...you get the idea.  So, it's nice to be able to do something new in the spring that feels fresh, especially when the kids are getting antsy!

Recently, I had all of my classes do a "Pot of Gold" activity.  Every time a group got a question correct, they were able to color part of a picture.  I thought it might be a good way to get the kids out of their seats somewhat and keep everyone working.  Boy, was I right.  The kids liked the activity so much that we did it again today.  We started learning completing the square yesterday and today they had to solve a series of questions in their teams to get to color a kite coloring page I downloaded.

Kite image from PrimaryGames.com

Here's how to make it a success.  Choose a simplistic picture so that your students aren't spending lots of time coloring.  I chose a kite (see above).  Number areas on the picture so that they know where they can color after each problem.  Post the coloring sheets in the front of the room with some colored pencils and let the kids get started on the math.  Despite my insistence that it was not, in fact, a competition, they took it that way.  It was, however, largely healthy.  No one got left behind.  I told the groups that everyone had to have the correct work on their whiteboards to get credit for the problem and be able to color and to move on to the next question.  They could not raise their hand for me to check their work until everyone at the table agreed.  I said, "Don't call me over while two people on your team are finishing because you want to be done first.  You have to reach consensus."  They did an amazing job coaching each other, discussing how to simplify their answers, and helping each other find mistakes on their boards.  I only had one student who I had to speak with regarding being on task (and this was not a surprise, unfortunately).  55 out of 56, I'd call that awesome for engagement.  Even some of my reticent children were talkative because they needed their group members to agree.  In my earlier class, we literally were so engrossed in what we were doing that one of my students had to point out that it was time to go and that there were already other students in the hall.  Whoops!  No teams finished today, but they begged to finish tomorrow.  :)  

So, print off some coloring sheets, chop up that boring worksheet into task cards, and watch how quickly and accurately your students will finish their work so they can color!  

What motivates your students to work well with a team?

Mathematically yours,
Miss B
       

I need some advice, please!

Like the title says, I'm looking for some advice.  I have a new student joining my class tomorrow (or really, probably not until next Wednesday because the student had the bad timing of registering in the middle of state testing and he will have to take the tests for the next four school days).  In any case, he's the first new student I've gotten all year and with this being my first year of implementing Interactive Notebooks, I have no idea how I want to handle it.  I see the following options:

1. Give the student a composition book, have a place for a table of contents, guidelines, and rubrics, and then start his pages with whatever number we're on immediately thereafter. 

2. Give the student a notebook that I spend hours assembling with the papers that I dig out of files so that his notebook is identical to those of students who have been with me since August. 

3. Give the student a notebook with selected pages that I think would be most helpful (without actually knowing anything about this student's academic progress) and then pick up with the rest of the class.  This would probably include a few reference pages on overarching topics and the most recent things we've worked on.

If we were back in September or maybe even October, option 1 would be my first choice.  Given that we're something like 55 pages into the notebook, I cannot see how it would be practical to do #2 but when I say, "Refer to page 25 for slope-intercept form," I can't very well have this student without a resource. 

What would you do?  I would love to hear your ideas, even if you haven't gone through this particular situation before.  I know what I'm leaning toward, but I'm also more than a bit conflicted. 

Mathematically yours,
Miss B

My idea of Professional Development

Earlier tonight, I answered a series of tweets by @jybuell concerning teachers' content knowledge.  It got me thinking about the direction professional development needs to take.  At my own school, I see a range of comfort levels from teachers when it comes to the new content in their curriculum.  I'm a pretty savvy mathematician and end up fielding a lot of content questions from colleagues, but even that didn't prepare me for tape diagrams and double number lines; someone had to teach me how to use them.  Then there's this picture that has been bouncing around Facebook for the past few days.  Who among us didn't need a few moments to decipher the thought process taking place at the bottom of the page?  All of this is to say that most if not all teachers need some support with content knowledge as we adopt Common Core.  That support may be in grasping the "deep conceptual understanding" we want our students to have or it may be in understanding the finer points of new methods were asked to use to get our students to that understanding. 

Where does that leave us?  With a precious few inservice days built into the calendars and plenty of initiatives around every corner, we need to have professional development (now "professional learning," according to some) that addresses the needs of the individual teachers.  As teachers, we are asked to differentiate lessons for our students and I think it's time that administration look at doing the same thing for their teachers. 

One thing I would love to see happen is local conferences.  My idea is relatively simple: have each teacher work individually or in a small group of not more than three to select something to present to other teachers.  It could be anything from a method of classroom management, strategies for cooperative learning, web-based resources useful in a variety of disciplines, interactive notebooks, information gleaned from a conference or class, or interesting math tasks.  Whatever it is, teachers would be asked to plan and deliver one session on an inservice day and to attend other sessions of their choice given that day.  In larger districts, several schools could combine at one location to offer more sessions.  My school pretty much always has the same few people present inservice and in the past three years, it's always been about Common Core, PARCC, or some combination of those two things.  I know there are people now implementing aspects of Common Core that are worthy of being shared and I would like to learn from them.  In many ways, I think a local conference would be more practical than a national or regional conference.  I attended NCTM regional in the fall and I realized that so many things were not useful to me- either they didn't apply to my population (hello, no technology at home), were things that my district had done previously and moved away from because we found them ineffective, or were just sales pitches disguised as sessions.  One notable exception: @Mathalicious.  I totally need to spend my entire curriculum budget on that next year.  Second runner up: Dinah Zike's Foldables.  Yes, a sales pitch, but my kids love the books I learned to make and I didn't spend a cent on the books. 

I realize my idea involves lots of logistics and teacher buy-in but I think it would be a valuable way to share best practices.  As I say frequently, "One day when I run the world..."

In what ways should professional development change to make it more effective for you?

Mathematically yours,
Miss B

When snow days and state testing collide...

This year we've had ten snow days.  That's an awful lot more than normal and I think it might be a record for my lifetime, including when I was in school.  Unfortunately, they've been scattered throughout the past two months, so we have had a very hard time getting into any sort of routine since Christmas break ended nearly two months ago.  Yesterday and today were snow days and they've already announced a delay for tomorrow.  The biggest problem with that is that we were slated to start our six days of reading and math testing today so now our schedule will be even stranger than before.  I just keep trying to roll with the punches, but it's getting harder and harder to believe I'll have time to finish my curriculum if things keep going this way.

I'm going to see just two of my classes for 45 minutes each tomorrow (when we usually have 90 minutes), so I wanted something light and active for my poor kids who will have spent the morning taking a 99-minute math test and then have to come back to me for even more math!

Enter the Pot of Gold activity developed by KFouss.  I used her completing the square version last school year and it was a hit.  I made a small change by including some coloring.  Is it necessary?  Absolutely not.  Will it help my kids have some fun and be motivated after testing?  I think so.  It might also breed a little competition among the groups when they see certain groups making faster progress.

The files are below.  If you don't have the font KG Begin Again, go download it now.  I'm pretty sure these files will be formatted terribly without that font. 

First up, systems:

Next, factoring polynomials:

How do you keep kids working when it's testing season?

Mathematically yours,
Miss B