Resolving Dissonance

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Noticing and Wondering

This past Tuesday at the Global Math Department Max presented a conference on noticing and wondering.  I’m glad I was there for it!  I learned a lot about tying in what students already know to what we are trying to learn next.  I’ve heard of and used KWL’s before (know, want to know, learned), but noticing and wondering worked so much better!!

My 9th grade class really struggles with math.  Many of them get very frustrated and shut down when anything seems even slightly challenging.  Unless it’s interesting too.  Or is presented in an excessively non-threatening way.  Asking what they notice instead of what they know did that for them this week.  We actually got through what I wanted to for the day plus some extra things.

We are learning about features of functions (maximum, minimum, increasing, decreasing, intercepts, etc.) and how that all looks with graphs, tables, equations, and in interval notation.

I started with asking students what they noticed about this graph:

ScreenHunter_01 Feb. 15 14.02

They noticed that:
– the lines intersect
– one starts higher than the other one
– the lines are straight
– the lines have different slopes
– the lines have different labels

Next I gave them a context:

Aly and Dayne work at a water park and have to drain the water at the end of each month for the ride they supervise. Each uses a pump to remove the water from the small pool at the bottom of their ride. The graph represents the amount of water in Aly’s pool, a(x), and Dayne’s pool, d(x),over time.

I asked them again what they noticed now:

– Aly’s pool drains faster
– Aly’s pool is bigger
– the pools drain at a consistent rate, even as the water gets low (we had a problem a while ago where that wasn’t the case)

Then I gave them some more information.  Dayne’s pool started at 24,000 gallons and finished draining after 24 minutes.  We also talked about what it means when they cross and how that can be expresses as a(x)=d(x)

They wondered:
– When do they cross
– How much water did Aly start with
– When was Aly’s pool done draining

When talking about those we also talked about:
– what a(x) means, and what a(5), a(6), etc mean.
– when is a(x) > d(x)
– what does d(x) = 2000 mean and what would that x value be?

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How I teach graphing a system of equations

English: Revision of File:FuncionLineal02.svg

 (Photo credit: Wikipedia)

By request from @jreulbach at http://ispeakmath.wordpress.com/ – this is how I teach graphing.  On twitter she mentioned that her students struggle with graphing, but could solve a system using substitution.  Very ironic because mine get graphing but took forever to get substitution!

A lot of how I teach graphing came from the curriculum my school uses for 9th grade.  Also, with any subject I teach I try to make as many connections to anything else they know as I can.  For my ninth grade class very first we started of with what an intercept is.  I introduced that with football.  When someone intercepts the ball, their path and the path of the ball cross.

To talk about slope I showed them this cartoon.  They get confused when graphing because for slope you go up first, then over next.  For graphing a coordinate you go over first, then up/down.  We had a graphing competition that seemed to resolve that.  They were in teams and had to find the correct point if I gave a point, and the correct slope if I gave a line.  First team to get it won that round.

Those two things didn’t take very long, about one day each.  It was stuff they’ve learned before, just needed reminded of.

As we went through the rest of the unit we used the organizer found here. but instead of printing this out to give to them, I had a blank grid we filled in as we went along.  Only the honors students did the matrix, everyone did graphing, elimination, and substitution.  We learned graphing first, then the other two.  I also used this method for notes.  We only filled in the organizer once they had it down.

I also had them write slope-intercept, standard, point-slope, and recursive formulas for lines we were working on.  and practice going back and forth between the different forms, and graphing from the different forms.

Once we know how to graph one line very well, we get into two lines.  I start by introducing it as another kind of intercept.  But instead of looking at the x or y axis, we are going to use another line.  I go back to football on this and explain it like someone tackling another player.  The two have to cross paths for that to happen.

After we graph the two lines I have them pick the point where it looks like the lines cross, plug that into both equations, and see if they are both true at that point.  If they are both true, that one point works for both lines.  So that has to be where they cross.  If it doesn’t work, that’s not the right point.  I’m REALLY emphasizing checking work with all my students and that seems to help also.  It helps them get better at catching their own mistakes before they practice too much the wrong way.

For the younger grades, that aren’t into systems yet – they are just learning graphing – I still use the same slope and intercept instructions, and we still practice the confusing slope vs. graphing a point problem with the game.  That seems like the biggest confusion for them.  You start vertical for finding slope, and horizontal for graphing.

With the younger ones I also have them create pictures with graphs, then give someone else the instructions for making their picture.  Here’s an example of how to do pictures with graphing, but instead of giving my students one I have them create their own.  Depending on how long we have to work on it I’ll have them do a picture with anywhere from 10-20 points as the minimum.

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Linear Equations Card Matching Game

graf of linear equation

graph of linear equation (Photo credit: Wikipedia)

One of the standards for my ninth graders is that when given a context, graph, equation, or a context they can use that to create the other three. I have one standard for linear equations and one for exponential equations.

They mostly get the linear one. They still REALLY have a hard time with exponential. Mostly with coming up with a context. If I give them the context, then can usually work from that to create the other three parts (graph, equation, and table). But if I give them one of the other three, only one of them has been able to give me a context that works.

I saw a card matching game for linear functions on another website where students match the graph, the equation, a description (slope, intercept), and the table. That’s almost what I need! So I downloaded the files that blogger provided then made contexts to match them.

I’ve put off writing this blog because I REALLY want to say where I got the cards from! But I now can’t find it… thought it was on my Pinterest board, but the one on there was a different teacher that didn’t have downloads available. (edit: after some searching through my browser history, I found the original download!)

There are two equations I couldn’t come up with a real context for. And some of the other ones are stretching things a bit. I’ve read a little recently about trying to make math in school more authentic. Definitely something I’m aiming toward as I have time to work on it, and I do include some more realistic things for other lessons.

To make the matching more authentic, at some point I’ll need to do the context cards first – then create the rest.  This time however… all the other work was already done and I’m in a time crunch, so used what was available.

Here are my files for the linear matching game:

Equations
Graphs
Tables
Context
Descriptions

Coming soon I’ll have pictures posted of the game.  Right now they are headed for my “parent basket” for a volunteer to copy, cut, and laminate them.

You can find my links to the matching game for exponential equations here.  These ones I like better than the linear!  I started with context first, then made the table, graph, and equation.  It was tons easier to make realistic context doing it that way.

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