Saturday, January 21, 2012

Some Resources on Patterns and Systems of Equations

Here are some resources I'd like to share (yes, I'm blogging a lot, but that's because various things have been on my mind and I have been too busy doing semester grades to post anything).

First: Some fun visual patterns from NCTM. You can use this for your elementary students simply as pattern recognition exercises, OR expand them into algebraic exercises / modeling writeups as I am going to do for my middle-schoolers.

Secondly, I realized that when I shared my shapes puzzles for teaching systems of equations a while ago, I didn't share the surrounding lessons that then reinforced their conceptual understanding and eased the student transition to symbolic manipulations. I'm reusing these lessons this year and they are simply working magically for me. The kids are doing substitutions in their heads for entire expressions. When they look at 2n + 5p = 44 and n + p = 10, they can quickly say to me: "You can replace the smaller equation into the larger one twice, with 3p left over, so that means 10 + 10 + 3p = 44, so 3p = 24 and p = 8." And they completely understand why algebraic "elimination" is simply a shortcut that comes out of the substitution concept. (Some of them are even mad at me for giving it another name, because in their heads elimination and substitution are exactly same methods with a couple of steps skipped.)

So, let me try and do this:
Lesson 0: Savings Race was the first lesson I used after getting back from vacation, to get the kids thinking about linear functions and to briefly preview the idea of break-even points. It primes the kids for some of the concepts that will come along soon.

Lesson 1: Shapes puzzles introduces the idea of solving for unknowns with multiple requirements and visualizing variables/equations as composed of visual shapes. Also, the kids start to concretely develop an understanding of substitution and "scaling down" a value. No mini-lesson teaching necessary. Just go around and facilitate if they get stuck with the puzzles, but there was that eerie silence for much of the period when the kids were just thinking and they didn't want my help. I introduced the term "substitution" at the end of class using a pair of shapes equations.

Lesson 2: Line segments reinforces the same concepts from Lesson 1, plus it asks the kids to write algebraic descriptions of the relationships so that I can go around and facilitate how they could have substituted using symbols instead of using pictures. Again, no mini-lesson teaching necessary. I re-introduced the term "substitution" at the end of class using a pair of equations they can visualize easily in their heads.

Lesson 3: Transition to Algebra asked the kids to solve various pairs of equations. I asked the kids to do mostly algebraic manipulation at this point, or if they need to, draw out a few shapes but still write equivalent algebraic symbols next to them to show a transition to symbolic thinking. I went around and facilitated, but there was no mini lesson. THE KIDS TALKED A LOT TO EACH OTHER DURING THIS CLASS WHILE MAKING JOINT DISCOVERIES! At the end of this assignment, I went over the different terminology of "substitution" versus "elimination" methods using examples from this worksheet. Since most of them had been doing elimination in their heads, I asked them to start writing down -( SMALL EQUATION ) to show they are mentally subtracting the smaller equation from the larger one.

Lesson 4: Drilling elimination method asked the kids to do everything using elimination. I did not tell the kids how to decide if it's addition or subtraction between the two equations, but I told them that the ones where they don't want to subtract in their heads are going to "feel a little different" when they get to them. More often than not, kids grabbed me when they thought subtraction would not work for that system, and we discussed how addition would cancel out opposite terms. This day, I also gave the kids computers and asked them to check their answers using Wolfram Alpha instead of turning to me or checking with their partners.

Lesson 5: Graphical solution and meaning of systems helps the kids see one type of situation where systems are used and gets them to practice some basic linear skills and graph-reading skills. No mini lesson necessary, although I did chat with individual kids to tie this to the graphs they saw the day before on Wolfram Alpha.

Lesson 6: Group project lets the kids practice analyzing break-even points. Both lessons 5 and 6 are easy transitions, because they followed from Lesson 0 above.

...We're not done with the unit yet (I intend on going all the way through quadratic-linear systems), but I thought instead of sharing the whole unit, I'd share just the bits above on how I developed the most fundamental concepts of systems. I hope this is useful to you! Like I said, these are lesson I dug up from the dusty digital filing cabinet, but they're working like magic for me with no rote teaching, so I am linking to them here in hopes that parts of them can be used in more than just my classroom!


  1. This is some great stuff! Different from rote practice practice practice. I really like the visual diagrams and such. We just finished substitution and elimination, but kind of hit a wall with student frustration/apathy to continually find x and y algebraically. May look at working in once a week shapes, line segments lessons.

    Do you have any ideas for exponents?

    Thanks again!

  2. I don't have anything exciting for exponents, beyond making kids write them out and cancel x's manually (for subtraction rules) or count x's manually (for addition/multiplication rules). Once you explain the rules and they have a conceptual understanding, you can reinforce by doing bingo games:

    Have the kids fill out a bingo grid with numbers -12 through 12, and then play a game "This is the same as X raised to what power?" and you can make up questions on the spot. Put the questions on the board and don't go over the answers until about 10 problems in or unless kids have a question. They should all be quietly doing it in their heads.

    I think exponents is understanding coupled with drill, drill, drill. Hopefully the bingo game helps with the boring drilling part. You can also make your own exponent math cards and have kids play games like War or Concentration with them.

  3. I love this! I can't wait until we get to our unit on systems of equations now! Thank you!