Teaching so many grades at once has made me re-think (because I have to!) the sequence of topics that we teach in school. This is a work in progress, but here is a rough list of topics in sequence, and what I personally think is ABSOLUTELY NECESSARY for the kids to understand regarding each topic:
* We absolutely need to introduce the idea of linear patterns as soon as possible. Something like:
_____, ______, 3, _______, _______, ________, 47, ________
The kids should be able to do this starting in early 7th grade (or before), well before they even consider what x's and y's are. They should be able to reason it out like this: "In 4 steps, the value goes up 44, so in each step, how much does it go up by??"
This proportional/linear reasoning underlies so much algebra but can be introduced completely without!! (And notice that you can use it to reinforce integer operations and also fractional understanding.)
_______, 0, ______, _______, 1, _______, _______, _______
* Instead of grade 7 being a re-teach of grade 6 skills, we need to focus on estimation skills in grade 7. Can kids plot points on a graph when scale is 3 or bigger?? Can kids plot points on a graph when they have mixed number types? Can kids mentally estimate 5% and 20% and 15% and 90% QUICKLY because they understand how to benchmark against 10%? Can kids figure out which 2 integers 23/9 falls between, AND which of those integers it is closer to??
Estimation needs to be the focus in Grade 7, to help kids see the "bigger picture" of values and less of the rote algorithms.
* Work in Grade 7 on helping kids set up equations. Help them figure out that "x" is the quantity that's being asked. Help them list out all the givens. Help them figure out that most equations are written like "part + part + part = TOTAL" or "part * part = TOTAL" so that they can always try to look for the total in a question. Help them figure out that "each" and "every" represent rates, which represent multiplications somewhere.
* Let kids figure out how to go backwards to find a Mystery Number. Don't try to teach rote algebra. Instead, draw arrow diagrams that represent the steps of evaluation, and kids'll figure out on their own how to work backwards.
* When kids are flying through easy equation setups, give them a twist. Make them figure out that if two quantities add to 20, then they are x and 20-x. Help them figure out why it's not x and x-20. Help them figure out that if two quantities multiply to be 70, then they are x and 70/x, not x and x/70. Because they'll need that flexible understanding down the road.
* Then, proportions. Can kids figure out WHICH situations are proportions and which are NOT?! How about proportions in the coordinate plane - what do they tell us? What about part-to-part ratios vs. part-to-whole ratios?
It sounds silly, but making tables works wonders in reinforcing proportional relationships. Kids will naturally observe that proportional quantities always start at (0, 0) and grow "proportionately" (in lack of another word), and they'll want to develop "shortcuts" like figuring out the scale factor immediately.
* Introduce bivariate linear patterns and introduce non-linearity alongside it. Before we teach any algebra, help kids learn that: x is the cause and y is the effect. Almost ALWAYS. For goodness' sake, don't tell them which variable is x and which is y, and DON'T always put the x on the left side of the table. Let them figure it out.
* Visually introduce linear patterns. Let them play with the idea of building equations and then making predictions, forwards and backwards, with the simple equation. Emphasize the utter importance of thinking about what X and Y each represents in the problem, in order to figure out where to plug in the given starting values. They'll need this understanding down the road in higher algebra, to keep track of the meaning of their computed results.
* Tie slope to UNIT rates. Yes, rise is a rate. But rise/run gives us the UNIT rate, which is more useful more often. Let them graph while THINKING about the meaning of unit rate and initial value. Don't feed them graphing algorithms!
* Then, introduce quadratic patterns and the idea of second differences. DON'T introduce quadratic skills before kids learn to visually recognize quadratic patterns from a table!!
* Help kids learn to find quadratic equations. Even if it's not the easiest way. (I do the Legrange Method and I think my kids hate me for it. Down the road I'll teach them to solve for a, b, c, using a system of equations.)
* Now, can the kid make predictions with a quadratic equation? Forwards and BACKWARDS. Don't let them start on factoring until they see the value of going backwards. If you need to, give them a graphing calculator for now, and tell them that soon we're going to figure out how to "not cheat" with the calculator and how to do that backwards solving-for-x by hand.
* Factor with the box method, and MAKE SURE THAT KIDS UNDERSTAND WHY factoring helps us solve the problem. (Here, insert joke about product of hair counts at NY Yankee Stadium being 0. Why?? It only takes 1 bald man. Start referring to your "zeroed factor" as a bald person.) Then introduce geometric problems that force kids to set up quadratic equations with x's on both sides, to solve them, and then to eliminate nonsensible solutions. The whole shebang. Focus on the idea that not all algebraic solutions always make sense.
* Embed linearity and quadratics in visuals, applications, word problems. Take your time. Make sure kids can viscerally connect to the idea of a slope being a rate and a y-intercept being an initial value; make them go backwards from equations to filling in blanks in a story. Make sure kids graph EVERYTHING and they understand always how the graph and table and equation are all interchangeable.
* Secretly introduce the idea of linear systems through applications like a savings race. Help kids approach the idea of break-even points from a graphical perspective, and use visual puzzles to get them thinking about equivalent quantities and substitution. Finally, wrap it up in a neat algebra bow by tying those methods to traditional algebra symbols and operations.
Make sure your kids can explain how to properly check a system!
* Get back to quadratics and now ask the kids to find quadratic equations fitting tables, using a systems of equations approach. Integrate different concepts into one seamless application for them. Don't let them walk away from this unit not knowing how to find a quadratic equation given ANY quadratic table. Re-emphasize now why we need exactly 3 points to find a quadratic equation. Use Dan Meyer's basketball pictures to let it sink in.
* Introduce square roots and combining square roots through Tangrams and other composite shapes made of right triangles. Tie this into combining like terms in general, and get them to think about decomposing shapes into smaller parts...
* Make sure kids understand dimensions. THEN introduce scaling in the coordinate plane.
* Lastly, exponents. I don't have good ideas for reinforcing exponent rules, except making kids write out all the terms being multiplied. Do you??
I feel like in the end, I'm still missing some topics, but I would be VERY HAPPY if every middle schooler I have the pleasure of teaching can leave with all of these concepts integrated together under their mathematical belt. That'd give us so much to build off of in high school, when the algebra gets trickier. What is your list of middle school must-knows??