Tag Archives: quadratic formula

All The Cool Kids Are Guessing and Checking

Last night I landed in the middle of this discussion with Julie:

Until we get Glenn to share his ideas (C’MON GLENN I DON’T WANT TO HEAR YOUR PhD EXCUSES),UPDATE: Glenn has shared part one of his ONE MATHS blog posts here! these are my big three goals for Algebra II this year:

  1. Simplify stuff. Also, I want to check our simplifying.  Let’s plug 7 into (x + 8)(x + 3) and see if we get the same thing when we plug 7 into x^2 + 11x + 24.  Let’s graph it on desmos and see if we get the same graph! Oh, look, we’ll get the same output for any x!  (I think this would also help with the “plug in any number” method for the ACT, which totally blew my mind when I first read about it. “No, wait, it can’t work for any number I pick, can it? How does it know which number I’m going to choose?”)
  2. Solve stuff.  We will do this by legally undoing things. And also check by plugging in the answer and by graphing.
  3. Graph stuff.  90% of which we can graph by using the (h, k) method.

Our department is also planning on using Jonathan Claydon’s layout for Algebra II that hopefully will foster more connections as well!

Back to last night: as things often do in the #MTBoS, the discussion turned to factoring:

So if factoring by grouping or box or slide and divide or airplane or bottoms up is not working for you or your students (i.e. do they all remember the “slide” but forget the “divide” part?), join the cool kids and go back to Guess and Check.

“But wait, Meg, how can you be all hip and cool and use ‘guess and check’?  I mean, that doesn’t sound very mathy at all. I bet you don’t even wear pink on Wednesdays.”

Hey, Mean Girl, it’s not just guess and check….it’s educated guess and check! It’s like finding factors of 111…you could try every number from 1 to 111, but if you’re smart about it, you know not to try 2, 4, 5,… so you can focus on the ones that at least stand a chance!

True story: One year I taught Algebra II both guess and check and slide/divide. Those that had the least math skills chose slide/divide, but then they would get a really big number for ac that had a bajillion factor pairs, and since they weren’t great at arithmetic to start with, they couldn’t choose good number pairs (“hmm, I wonder if 2 and whatever 168 divided by 2 is will add to equal 13? Better break out the calculator and punch. every. button. so. slowly. so. very. very. slowly. Huh, wow, didn’t work. Ok, what about 3 and…”) so it took them way longer to guess and check ac than educated guess and check process. AND THEN THEY STILL FORGOT TO DIVIDE!  Also, in Precal, I have kids that are in love with slide/divide and then slowly but surely, they’ll come in for some extra help….”so could you show me your method again?”

Ok, so here it is:

Factoring 6Factoring 2 Factoring 3And, no, it’s not like our first guess is right every time…Factoring at megcraig.orgbut it usually doesn’t take too long!

Here are the above charts in a handy word doc in case you want to discuss with your department. I actually don’t do a NoteTakerMaker for these, because another big secret is to USE WHITEBOARDS or some other dry erase surface. Unless you want to hear them complain about the guess part of it all day long.

Also, I think next year I’m going to teach a equal and not equal to 1 the same day. If it’s equal to one, awesome, I can lock that in!  Maybe that way they won’t freak out as much when a doesn’t equal 1?  Because full disclosure: yes, mine still complain when a doesn’t equal one.

Let’s round out this post with some more Quadratics (part of my summer goal to get all of my resources online, see more on this page!)

Factoring homework (hint: I sometimes use the first problems as examples in class, then tell them they get to start on #____ or assign just the odds, with evens for optional practice).

Factoring 4File here.

And my first day of factoring review:

Factoring from megcraig.orgFile here. Next year: save grouping for polynomial chapter.

Now let’s solve these puppies!

Factoring from megcraig.orgFile here. (I can’t seem to find my solving by square roots, but I do teach it! Promise!)

And now let’s solve some with complex answers (I usually wait and do complex number operations later–it’s a good “oh, here’s three days before break” section that can really go anywhere in the year)

Factoring from megcraig.org

My favorite annual quote: “So, what’s this backwards j?”

And then the quadratic formula (I save completing the square for converting to vertex form; see it in this post.).  And now let me reveal Jim’s (@mrdardy) awesome quadratic formula manipulation:

Factoring from megcraig.orgIt’s only 1,000 times easier to simplify AND you can still sing “Pop Goes the Weasel” because it is still “all over 2a” AND whoa look at that vertex just pop out!!

And here’s some practice (6 to a page!):

Factoring from megcraig.orgFile here

And the study guide.

Factoring from megcraig.org File here.

Now go forth and spread the news of educated guess and check throughout the land!!!