Source: http://www.zazzle.ca/satan_alphabet_in_math_tee_shirt-235801256845953337
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.
Balancing Scales
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?"
Act 1 - Penny balance from Timon Piccini on Vimeo.
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:
To see a copy of what this looks like just click on the link below.
- Student views visual and solves.
- Student develops visual from words and solves.
- Student views equation, develops visual (if necessary), and solve.
To see a copy of what this looks like just click on the link below.
| one_step_equations_-_addition.docx |
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!
To really hammer home the connection between them we play lots and lots of Algebranary!
Downfalls
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.
It is not perfect, but this has all been incredibly valuable for student understanding the basics of solving linear equations.
- 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.
Moving Further
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.
The Goods
| penny_equations_-_student_work.zip |
| penny_equations_-_mess_of_all_my_media.zip |
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.
| linear_equations_with_pennies.pptx |
