Thursday, January 11, 2018

Homeschool Math: Resources for Art of Problem Solving's Introduction to Algebra: Chapter 1

Will's eighth-grade math spine is Art of Problem Solving's Introduction to Algebra. To her, it feels like a big step up from Math Mammoth 7, which she pretty much breezed through with little outside assistance, and she is working through the text quite slowly, much more slowly than she'd do in an organized class.

But of course, I'm ensuring that she actually deeply understands the concepts as she goes, which is far more important than zipping through at a steady pace.

Part of that process is providing a lot more scaffolding to the concepts than AOPS provides. By that, I mean that if a kid doesn't understand a concept when it's explained one way, then I find a couple more ways to explain it. I find a visual way to explain it. I find a hands-on way to work through it. I find drill problems to practice and cement it. That gets harder as the math gets more advanced--not because there aren't visual or hands-on ways to explain any math concept, because there always are, but because visual and hands-on learning is often neglected for older kids.

Here, then, are my hard-won extra resources for Art of Problem Solving's Introduction to Algebra chapter 1, which covers the order of operations, distribution and factoring, an introduction to equations, exponents, fractional exponents, and radicals:

Order of Operations

  • Order of Operations notebook pages. You'll find that I use a LOT of material from Math Equals Love, and that's because it's excellent, relevant, and just the kind of visual, hands-on exploration that makes math make sense. Here, I used the practice problems with Will so that she would have examples to put in her math notebook. We also discovered right away that her biggest issue with algebra is going to be writing out solutions step by freaking step. I still do not understand why she is balking at this, but she will actually erase a previous step and write the new one in its place rather than writing out the solutions. It's maddening. I've tried explaining to her every way I know how why writing out the solutions is important, but honestly, I think she's just decided to be stubborn about it. I now employ natural consequences in that if a problem is incorrect and she's written out the solution for it, I will mark exactly where she made her mistake and sometimes give her a hint about what to do next. If she's not written out the solution, I just mark it incorrect to try again, full stop.
  • Order of Operations worksheets. Drill problems, if you need them!

Distribution and Factoring

Introduction to Equations

Exponents, Fractional Exponents, and Radicals

  • The same Order of Operations notebooking pages, above, have a section on Negatives and Exponents that I think is necessary to review before beginning to learn exponent rules. I saved that section for Will to do here.
  • Exponent Properties. THIS handout was the game changer for Will's understanding of exponent rules. Before we completed these handouts (she and I both worked them, then compared our answers and discussed), she did not understand exponent rules and could not remember them. After we completed these handouts and discussed them, she understood them, and they were easy for her to memorize. Here's a little of my work in progress with the handouts:

See how working out the solution means that the math rule makes sense? Math rules. Make. Sense. If you don't understand WHY a math rule works, then you better figure it out, because there's no point in memorizing it otherwise.
  • Exponents Game. I don't usually make games and manipulatives anymore, as often they don't get enough use to justify the work and materials. I made this game, however, so that Will would have some practice without having to write and write and write.
  • Here the Exponent Rules are broken down more quickly, as a review. I'm holding onto this to present at another time, if Will seems like she needs to explore the concept again.
  • Exponent Rule Mistakes. We also didn't use these pages, but only because Will was ready to move on. They're still a possibility if she needs to review the concepts again later.
  • Radicals. Will started off absolutely baffled by radicals, so we used every single one of the resources here, other than the ones on rationalizing the denominator. Explicitly working through these resources on factoring radicals, adding and subtracting radicals, and multiplying radicals is the only way that Will was then able to understand the AOPS section on them. We also both completed the entire prime factorization chart in one evening, because Will was enjoying it (!!!). Previously, both kids have also memorized all of the prime numbers under 100, and I will tell you that has made life incalculably easier for both of them.
  • Two Methods of Prime Factorization. I taught Will both of these methods, but we both tend to prefer the factor tree, I think because we have the primes under 100 memorized.
  • Exponent worksheets. Drill problems, if you need them!
  • Radical Expressions worksheets.

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