Recently, I **read** on Twitter some teachers’ frustration with students who just want to know the quick procedures to do the math at hand and don’t have much interest in the meaning of the underlying concepts. I often come across this dilemma in my one-on-one work with students; in this tutoring role I especially feel the pressure to teach the “how-to” for an upcoming test and don’t always have the time to explore the “why” with the student. I wrote a bit about this tension before in **this post**.

Another **conversation** on Twitter was specific to Algebra 2, about how to build on past knowledge even when some/all of the students seem not to remember that past knowledge. How might we deepen students’ understanding and not simply retread the procedures?

I was faced with these dual dilemmas when I worked with a student this week reviewing Complex Numbers and Quadratic Equations for an upcoming test. My approach: __pay attention to mathematical structure__.

**A. Fractions involving imaginary numbers:**

These three examples, examined together, allowed us to explore how to handle a negative value in the radicand (“inside the house”) and also how to handle a two-part numerator* with a one-part denominator. Once the imaginary ** i ** was extracted and the radical simplified as much as possible, we took a look at when we could and couldn’t “simplify” the denominator.

I wanted to help my student avoid the common mistake of trying to “cancel”** when you can’t. We used structure to explain. When there is a two-part numerator and a one-part denominator, you can do one of 3 things:

“Distribute the denominator” to make two separate fractions. I find this is the most reliable routine to avoid mistakes.

Divide “all parts” by a common factor.

Factor a common factor (if any) in the number, then simplify with the denominator.

{*I wasn’t sure if the numerators qualify as “binomials” since they are numeric values, but my student and I discussed how they have two terms on top and one term on bottom, which can be challenging to simplify. This structure will be encountered later when solving equations using the quadratic formula.}

{**I have avoided the use of “cancel” since I became more familiar with the “**Nix the Tricks**” philosophy of using precise mathematical language and avoiding tricks and “rules that expire”. See resources below for more on this.}

**B. Operations with complex numbers:**

Again, we looked at a set of three problems to examine structure, which leads us to the appropriate procedures:

What is the same and different about #10 and #11? What operation is needed in each? Which is easier for you?

What is the same and different about #11 and #12? What do you call these expressions: 4 – 5*i* and 4 + 5*i* ? If you notice this structure, how does the problem become easier?

This led to a fruitful discussion of combining “like terms”, what is a “conjugate”, and whether it mattered if the multiplication was done in any particular order. My student had been taught to always list answers for polynomials in order of decreasing degree, as in *x ^{2} – 2x + 1*, so he was writing any

**terms first. This isn’t wrong, but the rearranging of the order of the multiplication could have caused a mistake, so we talked about whether**

*i*^{2}**is a variable or not, and when might it be helpful to treat it like a variable.**

*i*By noticing the structure of conjugates and why they are used, we got away from merely memorizing math terminology and instead added to conceptual understanding.

**C. Using the discriminant**

The discriminant is one of my favorite parts of the quadratic equations unit! Students must pay attention to the structure of a quadratic equation (is it in the standard form *ax ^{2} + bx + c = 0* ?) before using the discriminant to give clues about the number and type of solutions.

Rather than memorize what the discriminant means, look at where it “lives” in the quadratic formula. It is in the radicand, which is why a positive value yields real solutions and a negative value does not. The radical follows the ± , which is why nonzero discriminants give two solutions (either a real pair or a complex pair). And the two solutions are conjugates of each other, something that I hadn’t really thought about when I got real solutions using the quadratic formula. (And there is a nice surprise when you examine the two parts of the “numerical conjugates” and relate them to the graph of the quadratic. See note below for more on this.)

I recently read **this post** about one teacher’s success having students evaluate the discriminant first, then tackle the rest of the quadratic formula. Her strategy integrates the use of the discriminant with quadratic formula solving, instead of making it a stand-alone procedure.

**Calculator Note**: when evaluating the Quadratic formula on the TI-84+ family of calculators, use the fraction template to make the calculator input match the written arithmetic. Press ALPHA then Y= for the fraction template, or get it from the MATH menu. Then edit the previous entry for the second solution (use the UP arrow to highlight the previous entry and press ENTER to edit). Here is #26 from above:

** Another Calculator Note**: the TI-84+ family in

**mode can handle the addition and multiplication questions #10-12, so if you are assessing student proficiency on these skills, have them do it without using the calculator. The color TI-84 Plus CE can operate with an imaginary number within the fraction template, such as questions #4-6 and anything with the quadratic formula. You can use either the**

*a+bi***symbol (found above the decimal point) or a square root of a negative number.**

*i*However the B&W TI-84+ can’t use an imaginary number within the fraction template. Use a set of parentheses and the division “slash” for this to work:

Notes and Resources:

Nix the Tricks website and book: **nixthetricks.com**. And lest you think that “Distribute the denominator” is yet another trick, consider this: The fraction bar is a type of grouping symbol (like **parentheses**) and it indicates division. Dividing is equivalent to multiplying by the reciprocal. So the “distribute the denominator” work for #4 above is also this:

Three articles about “Rules That Expire” have been published in the NCTM journals. Currently all three are available as “FREE PREVIEWS” on the website.

- Circumventing High School Rules That Expire, Barbara J. Dougherty, Sarah B. Bush, & Karen S. Karp (2017). The Mathematics Teacher, Vol. 111, No. 2.
- 12 Math Rules That Expire in the Middle Grades, Karen S. Karp, Sarah B. Bush, Barbara J. Dougherty (2015). Mathematics Teaching in the Middle School, Vol. 21, No. 4.
- 13 Rules That Expire, Karen S. Karp, Sarah B. Bush, Barbara J. Dougherty (2014). Teaching Children Mathematics, Vol. 21, No. 1.

“Look for and make use of structure” is one of the Standards for Mathematical Practice (SMP #7) in the Common Core State Standards found here: http://www.corestandards.org/Math/Practice/

The “nice surprise” about the solutions to a quadratic equation written as “numerical conjugates” and their relationship to the quadratic graph was pointed out to me by Marc Garneau. His **post here** gives more detail and a student activity to go with it.

Thanks as always to the #MTBoS and #iTeachMath community on Twitter for great conversations!