Slides from Chicago’s Common Core Conference

In February, a conference was held at the University of Illinois at Chicago on the Common Core State Standards.  Initially planned for 150-200 attendees, the conference capped at 900 attendees with a long waiting list.

CME Project lead author Al Cuoco gave a talk about the curriculum and its connections to the Common Core, referencing both the Standards for Mathematical Practice and Standards for Mathematical Content.  Slides are available at

http://www.lsri.uic.edu/ccss/ccss_cuoco.pdf

Slides, videos, and photos of all speakers are available at

http://www.lsri.uic.edu/ccss

Of particular interest to me was the talk by Phil Daro, one of the three lead authors of the standards, who led off the conference with a lot of great information about CCSS and its principles.

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Factoring Non-Monic Quadratics

Regardless of the course, non-monic factoring was always a thorny issue in my teaching.  My students never seemed to get “good” at it, even though they seemed alright at monic factoring (“monic” just means the first coefficient is 1, like x^2 + 14x + 48).  This topic made me really question why I was teaching it, for several reasons:

  • The very next thing was the quadratic formula, and for most of the things non-monic factoring could be useful for, I felt the quadratic formula would be just as good.
  • I couldn’t find many places later in my curriculum where non-monic factoring was being used, so it seemed like a topic taught for a single purpose.
  • The methods I saw for factoring non-monic quadratics had little or nothing to do with the methods for factoring monic quadratics.
  • The method I learned and first taught amounted to trial-and-error.

And maybe you know this method too: to factor 6x^2 + 31x + 35, you write down all the factors of 6, separately write down all the factors of 35, and start making pairs.  Eventually you either find the pair that works, or you run out of pairs:

(x+1)(6x + 35) = 6x^2 + 41x + 35, nope

(x+5)(6x + 7) = 6x^2 + 37x + 35, nope

(x+7)(6x + 5) = 6x^2 + 47x + 35, nope, keep trying…

And I was polite in picking 6 and 35 here, two numbers with only two prime factors each!  I feel this method is a mathematical nightmare.  Keep testing, keep checking.  And don’t give the ones that aren’t factorable, since the only way to know it’s unfactorable is to test all the possibilities, and that’s just mean.

A year or two later, I learned and taught the “key number method”: multiply the coefficients of “a” and “c” (6 \cdot 35 = 210).  Then you break up the middle term (31x) into two pieces whose coefficients multiply to 210:

6x^2 + 31x + 35 = 6x^2 + 10x + 21x + 35

Then “group” in pairs and a miracle occurs:

(6x^2 + 10x) + (21x + 35) = 2x(3x + 5) + 7(3x + 5)

= (3x + 5)(2x + 7)

This worked a lot better for my students, by which I mean they got correct answers faster and with greater accuracy.  But the core of this method is the “miracle” that splitting the 31x in this exact, specific way will do great things.  It works because it works.  (There are better explanations, but my students just memorized what to do.)

One advantage of the key number method is it can be applied to monics, too, visualizing the “sum and product” concept:

x^2 + 14x + 48 = x^2 + 6x + 8x + 48

= x(x+6) + 8(x+6)

= (x+6)(x+8)

But this generally comes after the fact: I didn’t teach students to factor monics in this way.

While working on CME Project, I learned (through Al Cuoco and Jeremy Kahan, a field test teacher) about a “scaling” method that uses monic factoring as the core of non-monic factoring.  It feels a lot more natural, cements monic factoring, and fits tightly with Mathematical Practice #7, “Look for and make use of structure.”  It starts with specific non-monics like this one:

25x^2 + 70x + 48

Try factoring that for a second using either of the methods presented above.  It’s messy!  But, would you believe this is actually a monic quadratic?  It’s just got a different variable: 5x.

25x^2 + 70x + 48 = (5x)^2 + 14(5x) + 48

Now cover your finger over each 5x: it reads finger^2 + 14 \cdot finger + 48.  Doesn’t matter what’s under the finger: it factors!

F^2 + 14F + 48 = (F+6)(F+8)

And you’re done when you lift your finger, remembering that F = 5x.  In teaching, I used capital letters for these substitutions, to remind students that there was more work to be done later.

(5x+6)(5x+8)

How fast was that?  And understandable, too!  The core concept of a replacement of variable (the book calls this “chunking”) plays forward deeply into later topics and courses: when I say I used capital letters for substitutions, I generally was doing that with Precalculus or Calculus students, but the concept can be seen much, much earlier.  By using it frequently, it becomes a tool students actively look to use when they see something complicated.

But I fudged the example: it’s got 25x^2.  How about that original one, 6x^2 + 31x + 35?  It doesn’t have a perfect square term, but … wishful thinking … we can make one by multiplying through by 6, then paying it back later.

6(6x^2 + 31x + 35) = (6x)^2 + 31(6x) + 210

\mathbf{= F^2 + 31F + 210}

\mathbf{= (F+10)(F+21)}

Note that this method includes the step that was part of the “key number method”: the 210 is produced by multiplying the coefficients of “a” and “c”, but this time there is a more mathematical reason for doing so.  And the payoff is the same, since we then need two numbers that add to 31 and multiply to 210 — but we use the monic factoring method to perform that step.  This cements monic factoring skills, as it becomes part of the process in the later topic.  And now the miracle, as we have common factors in the two right-side terms:

6(6x^2 + 31x + 35) = (6x+10)(6x+21)

= 2(3x+5) \cdot 3(2x + 7)

= 6(3x+5)(2x+7)

And now you zap the 6 from each side and you’re done:

6x^2 + 31x + 35 = (3x+5)(2x+7)

It was shocking to me that this method works at all, and especially shocking that it always works: any factorable non-monic quadratic can be dealt with using this method.  And variable substitution is a natural method used in other places: completing the square is a variable replacement using \left(x - \frac b 2\right) as the variable … x^4 - 1 is a difference of squares … circles and ellipses all relate to the unit circle x^2 + y^2 = 1… trigonometric equations are just regular equations when you cover your hand over the “\sin x” part … a z-score is a linear substitution … and others.

The biggest benefit of presenting substitution methods as early as possible is that students learn a general-purpose tool they can apply repeatedly across grades and topics.  I also think it makes quadratic factoring easier and faster to teach.  What do you think?

Next: how this method can be used to develop the quadratic formula…

(If you know how to better display equations easily in places like this, let me know.  The LaTeX equations look pretty bad in the vertical alignment category, and I had to force a white background on each equation.  As long as it’s readable, I guess, but somehow I think it could be better.  Thanks to Mark Betnel for the pointer to the LaTeX commands available.)

NCTM 2011: Mathematics of Game Shows slides

At NCTM 2011, attendees learned some mathematics while vying to win a selection of fine cheeses and a copy of Slumdog Millionaire.  This talk addressed the math problems that arise around game shows, especially those of interest to the producers of the show and not just the contestants.

Here are the slides for the talk.  What games and problems should be part of the next one?

Mathematics of Game Shows @ NCTM 2011

General-Purpose Tools

High school math can be filled with specific tools for one purpose only.  Use this box to solve a word problem about people painting houses, but this other box for rate-time-distance problems, and there are plenty more where those came from. Use FOIL to expand a binomial multiplied by another binomial, but don’t try it on a trinomial!

In reflecting on my own practice I realized I taught inequalities to my Algebra I students in one way, then to my Precalculus and AP Calculus students in a completely different way!  Why did I do this?? It makes no sense, and contributes to students’ feelings that mathematics is a giant toolbox you either know or don’t know how to use.

Here are some ideas about how to throw away some of the special-purpose tools…

  • For word problems, encourage students to test answers to the word problem, not to guess the answer, but to define the process used to test an answer.  This process (called “guess-check-generalize”) works for all word problem “types”.  This means you don’t have to talk about word problem types, just give problems.  This method (testing numbers and looking for generality) is a useful skill in plenty of other places, and is basically what is described in Common Core’s practice #8, “Look for and express regularity in repeated reasoning”.
  • Instead of FOIL, use an “expansion box”: you can introduce it when working with numbers.  Expansion boxes act like the area model for multiplication seen in elementary  and middle schools: a grid that works equally well when multiplying 23 by 17 … 2 1/2 by 3 1/3… (x-1) by (x^2+x+1)… (3+i) by (3-i).  Oh, and it also works for the product of two binomials.  Phil Daro, one of Common Core’s lead authors, specifically calls out FOIL as an artifact of high school mathematics that has no larger utility.
  • For inequalities, teach the Precalculus / Calculus method earlier. The solution to -3x + 7 < 16 is closely tied to the solution to -3x + 7 = 16. Determine the solution(s) to the equation, then graph them on a number line to determine test zones.  Test somewhere in each zone to see if the zone makes the inequality true or false, then shade the ones that are true.

When teaching Algebra 1, my students learned to solve -3x + 7 < 16 by the “basic moves of inequalities” but those moves were very difficult to justify to my students. What I didn’t think about is that those basic moves don’t carry beyond linear inequalities. And I shouldn’t have been too surprised that they didn’t remember these rules in the long run: there was nothing to tie it all together except memorization. Similarly, I can’t tell you how many times I bashed the wall trying to get my students to solve |x-3| < 5. And even after breaking through that wall, the same wall was rebuilt weeks later as if nothing had happened.

By using general-purpose tools, there is less clutter in the curriculum and more “hooks” for students to connect topics and concepts together.  The methods we teach students should work for more than just today’s lesson — if it’s just for today, what’s the real purpose of teaching it to our students?

What’s your opinion? What other “special-purpose tools” are out there, and what are the alternatives?

What is MPI?

EDC’s Mathematical Practices Institute (MPI) offers teacher professional development, curriculum support, and technical assistance for schools and districts across the country. Through its programs, MPI provides opportunities and tools for teachers and school administrators to effectively implement the Common Core State Standards for mathematical practice.

MPI professional development offers ways to teach “bread-and-butter” mathematics topics such as factoring, graphing lines, and solving word problems while supporting the larger goals of school mathematics.

With this blog, we hope to join the positive conversations happening online around mathematics and education, to listen to others’ ideas, and to share our own.

NCSM/NCTM Schedule

Come talk to us next week in Indianapolis!

NCSM

Monday, April 11, 12:15-1:15: Ryota Matsuura, Al Cuoco, Glenn Stevens, Sarah Sword. “Mathematical Habits of Mind for Teaching: Assessing Mathematical Knowledge for Teaching at the Secondary Level”

Monday, April 11, 1:30-2:30: Katherine Schwinden, June Mark. “Implementing New Curriculum Well: What Does It Take?”

Tuesday, April 12, 3:00-4:00: Ryota Matsuura, Al Cuoco, Sarah Sword, Glenn Stevens. “Mathematical Habits of Mind for Teaching”

NCTM

Thursday, April 14, 8:00-9:00: Paul Goldenberg, Cindy Carter.  “What Babies (and Brain Science) Say About Elementary School”

Thursday, April 14, 11:00-12:00: Sarah Sword, Al Cuoco, Bowen Kerins.  “Geometric Optimization”

Thursday, April 14, 12:30-1:30: Paul Goldenberg, June Mark, Al Cuoco.  “Algebra Grades K-12: A Perspective from Three Programs”

Thursday, April 14, 4:00-5:00: Bowen Kerins.  “CME Project – Get to the Core”

Friday, April 15, 8:00-9:00: Bowen Kerins, Kevin Waterman.  “Mathematics of Game Shows”

Friday, April 15, 11:00-12:00: Kevin Waterman, Stephanie Ragucci.  “Linear Algebra and Geometry Together”

Friday, April 15, 12:30-1:30: Mary Wedow.  “Habits of Mind Approach to Eighth-Grade Algebra”