At some point kids learned that algebra is hard, confusing, or some other description preceded usually by an expletive, and followed by a sigh.  As a lover of Math, I never understood that algebra was difficult, and it became stunningly obvious the first time I tried to teach basic equations that I was not "normal."  I stood in front of the board, and told my grade eights "Just do the opposite operation, that's it!  If you see adding just subtract."  It seemed so obvious to me.  No stress, just tell them.  We see it all the time. It just didn't work for my students, and it didn't work for me.  So I thought long and hard, and this is what I came up with.

With my mind on 3 acts so often, I wanted to boil down my problems to a simple question.  I began to ask my students "How many pennies are in that cup?"

If you ask students how many pennies were in that first cup (assuming I divided the pennies equally), they will have the answer before the video is finished.

The thing that arose as the difficiency of this method, is that the problems became so easy in this case, that students couldn't see the connection between the algebra and the cups.  I worked for a while on a basic set of work that would bridge the gap between the visual and the abstract.  My process for my students went as follows:

1. Student views visual and solves.
2. Student develops visual from words and solves.
3. Student views equation, develops visual (if necessary), and solve.

To see a copy of what this looks like just click on the link below.

I hummed and hawed about how to bridge two and three, but I found by the time students had finished step 2, they could complete step three without any hesitation.  It was fantastic! As soon as students get a bit stuck on the equation, I tell them, "Let's draw it on this lovely whiteboard."

To really hammer home the connection between them we play lots and lots of Algebranary!

There are a number of downfalls with this method, that I am still trying to work through, and I would love any critique that you can give.

• When kids can do it in their head, they feel no need to draw anything; when the numbers are too large drawing is cumbersome, and inefficient (the necessary breaking of visual models to get to the abstract I guess).
• Visually subtraction is impossible (very hard?) to show in a static image, so in my slides you will see I resort to number lines (I save the visual of that for last), since students will have a hard time drawing on their own.
• Though students are able to UNDERSTAND the steps of solving equations, this set up does not create the desire FOR algebraic knowledge like some of the awesome projects we often see on the blogosphere.

It is not perfect, but this has all been incredibly valuable for student understanding the basics of solving linear equations.

I have always imagined this would also work great for inequalities (unbalanced scales and ask "What is the minimum number of pennies to do this?"). Dig through the slides, and the pictures, and the videos, take what you like, tell me where you would go, and hopefully enjoy.

I started to organize the video files, then I got lazy.  If you want to deal with the raw files however, you can download this mess of files. Otherwise see how I organized them in my PowerPoint.

This lesson is not a feat of great technology.  This is not a three act post. This is however something that I am proud of because it was simple, clear, and really really fun.  I am in my third year of teaching, and every year I notice misconceptions that I want to get out of students heads.  I also want to introduce them to natural understandings of mathematics that transitions students easily to higher level thinking, and higher level mathematics.  I love this lesson because I feel as though it did that.

My struggle has been to get kids to understand that a variable is a number.  They see x and they think that it is a letter, but I want them to know that it is a number.  However, I want them to know that it is any number.  Depending on the number we put in, we get a certain outcome.  Rather than tell them this, I start with a favourite past time of mine, mad libs.

Students in my class have a vague concept of variable but I am really driving home that a variable can be anything, so I begin with variables in English.  I prepared this spread sheet and mail merge document so that they can see clearly how variables work.

The paragraph first looks like this:

• Once upon a/an «Noun_1», there were three little pigs. The first pig was very «Adjective_1», and he built a house for himself out of«Pl_Noun». The second little pig was «Adjective_2», and he built a house out of «Pl_Noun_2».
  

And wonderful grade eights turn it into this beauty:

• Once upon a fire breathing llama, there were three little pigs. The first pig was very colourful, and he built a house for himself out of cats. The second little pig was scary, and he built a house out of apes. 
  

I talk a bit about programming here and say, "This is what computers do, they take information and insert it where it is supposed to be.  Here wherever I have 'Noun 1' what information is stored there gets displayed." I then talk about we get a certain outcome based on the variables we put in.  I ask about the first variable 'noun 1', "What if I put in this variable the word 'time'? How would that change the outcome of the sentence?"

"It wouldn't be funny!" Most students say. At which point we move on to our next session...

I tell students here that we are going to use ________________________ as a variable.  I give them a chart, and I want them to fill in the chart with variables that produce a factual sentence, a nonsensical sentence, and a funny sentence.  Here are a few examples....

At this point you can't help but have wonder and awe at the buy in factor! It goes with out saying that you have to remind students to be appropriate (the cat carcass one is weird, but it made me laugh!), but kids love thinking of these and sharing them.

At this point I tell students that they have used variables for ever in math.  They have all done 3+__ = 5. The __ is a variable, you can put any number in there (although only one number makes that statement true!).  So in this example we do the same thing, but instead of _____________ we now say X.  We discuss again what is factual what is not.  We can even begin to talk about one to one correspondence (is "speedo" factual or funny? Hint: the answer is yes).

The jump then becomes ridiculously easy. I have students do the same thing with numbers.  There tables now read even and odd (instead of funny and factual), and they insert values of x that produce even and odd. They do three functions: x+3, 3x, and 2x+1.

Now all you wiley math teachers out there are saying 

Becausewith 2x+1 you get some interesting results.  Kids start to X out the even section.  Here's where the messing with your kids in a totally awesome way part comes in. Ask them, "you can't find a value that produces an even number?" They say "no it is impossible." You say, "Yes it is." I let the class know that there is a number that works, they just have to 'open their mind' (but not in a hippy way). They rack their brains, and finally asking, "can we use decimals?" to which you respond "is it a number?" because remember a variable means any number.

Why stop at odd and even?  This I feel is the homerun to this lesson.  I gave them first 4x+1>5 and then 2x+11=7. They fill in a true and false table for each of these, and what they notice is fantastic!  In the first example students ask, "What about when x=1?" To which I reply, "Is it greater than 5?" "No" "Then is it true?"  I had students writing down 1- (-infinity) in the false section.  This wasn't 1 minus negative infinity, this was 1 TO negative infinity!  In one lesson I had students figuring out inequalities, variables functions, and equations.

At that point all we needed to do was show a little notation, and they were totally on board for inequalities and equations.  Awesome! I am going to keep giving them these right up until I formally teach them equations (inequalities aren't even touched on in grade 8, but hey, get them ahead!).  Because once they get bored of filling these in, we can say, "How can we speed this process up?"

This is the worksheet I whipped up.  Take it and do WHATEVER you want with it!