Category Archives: Algebra

2 is better than 1 (sometimes)

I’m a fan of visual context over simple memorization of formulas.  Another pet peeve of mine is the requirement that denominators be rationalized without reason.  A good conversation on rationalizing the denominator takes place over at “Why am I teaching this?”.  One place I’ve see rationalized denominators lose their context is in a unit circle.  I get that it’s a standard.  I get that the angle radian measure makes most sense with a radius equal to one.

A friend of mine, Jen Silverman, makes some great protractors that help with this radian angle measure as well.  I think my frustration comes from the idea that the unit circle is a beautiful overlay that simplifies so many interactive relationships between the trigonometry ratios.  Too often students never come to this realization and instead resort to tricks.

Here’s another post using the same trick.

I used to try to get students to think of the Unit Circle key segments as just a small set of lengths that interact in different ways.

Some applets to help in relating the unit circle to the cartesian form of basic trigonometric graphs:

While others simply go back to memorizing the rationalized denominator forms.

Still, I feel the abstraction away from Trigonometric Ratios loses the relationships within the right triangles created by these points along the unit circle.

So my proposal is: Introduce Circle Trigonometry with a radius of 2.  This would double all the segment lengths.  Trigonometry ratios would relate more directly, and students could redraw and label the pieces more intuitively [it’s pretty hard to imagine what Sqrt(2)/2 looks like compared to Sqrt(3)/3)].  I haven’t built the applet for this, yet.  Expect it to show up in a future revision of this post soon.  If you’ve ventured this way before, or you’re interested in encouraging/discouraging me from doing so, please comment below.

Sum it Up, Angle Edition: Part 1

360 Wheel

Somewhere, I don’t know where for sure, a student learned that one common way to measure a full rotation is 360o.  I would even say this generation is more aware of this concept with the popularity of sports that involve rotation(s).  I could mention the number 1080 in a class and many students would have at least a minimal intuition that it had something to do with revolutions/rotations.  Imagine you are that student and start from there:

The Basics (prior knowledge)

  1. Full Turn =  360o.
  2. Half Turn = 180o (call it straight angle)
  3. Quarter Turn = 90o (call it a right angle)

Number Sense (and decomposition)

“How many ways do you think you can fill in these boxes?”

90 = [?] + [?]
90 = [?] + 41
90 = 29 + [?]
180 = [?] + [?]
180 = [89] + [?]
180 = [?] + 27
180 = [?] + [?] + [?]
180 = 90 + 80 + [?]
180 = 70 + [?] + [?]
 Supplementary Supplementary 3 piece Complementary


After 1-3 minutes of getting out their Chromebooks, loading the applet and playing, I give the students a specific angle measure to move to, like angle BAE = 40o.  Next a simple task: find all the measures.

“Mr Butler, I don’t know what to do?”  “Mr. Butler, but there’s not enough information.”  and more of this kind of thing for about 30 seconds.

Then one student elbows the other, “ow, why’d you do that?!”

“Check this out, I can find this one.”

“Oh oh, I can find the next one look here it’s a quarter turn, we just need the other piece and go backwards like we did a few minutes ago”

10 minutes later we share.  And there are soooooo many different methods used, a good amount of them correct.  In the end it was easy to check though.  The total angles made a full revolution, the one measure everyone knew coming in.  For those that were incorrect we talk about how the error took place, (was it arithmetic? or reasoning?)

We repeat this process with two or three more different starting measures for the angle controlled by the slider (let the students choose the angle for at least one of those).


Oddly enough, vocabulary seems to stick better once the students have a context to put it in.  These quarter turns, that make a CCCCorner are call CCCComplementary, these half turns, that make a SSSSStraight line (or Straight angle) are called SSSSSupplementary.  Somewhere in this process students start to notice the relationship in angles across from each other at an intersection are equal.  We draw the criss-cross X and get the term Vertical.  At this point we are only about 3/4 through 1 class period.


Move the angle to _____ and  prove to me (with addition or subtraction) which angles are:

  • Complementary
  • Supplementary
  • Vertical

(There’s two for each)

Mr. Butler adds, “Hey class, oh and by the way, we just reviewed Algebra and solving equations.”

“You’re sneaky Mr. Butler”

Then the bell rings…

… to be continued…

Say what you mean, the importance of language in a math class

“Why?” asked a student on some day in that one class.  The teacher replied, “Because.”  Often this conversation goes on without even a word spoken aloud.  Teachers develop routines in the classroom, and students learn to follow them.  The cliche at home of “Do as I say, not as I do,” doesn’t always fit into the classroom structure of I do, we do, you do (which may not be the best method either).  It might have transformed into more of a modeling-focused perspective of “Do as I do [, not as I say].”  Words in the classroom can get confusing, leading many teachers to struggle using proper language with students and instead depending on visual aids and demonstrations.  These conversations can get even more complicated when the students jump in.

Students are true artists when it comes to words and interpreting directions.  They are the best peer-editing service, often exposing any fault or loophole left open in an activity.  A skilled teacher can anticipate these attempts for learners to go off script and plan accordingly.  An even more skilled teacher can say less with more, only giving a few cleverly designed guidelines that allow students to explore and still arrive at some type of expected outcome.  For example, an assignment may have required students to write 1-2 page research paper on the topic of proportional relationships and its applications in scaling up or expanding a business.  This guideline is specific but has too many holes.  Through various interpretations of format, the students can complete the task without demonstrating any understanding.  Some think that more detail in the directions is the remedy for this situation.  I’m on the other side, preferring to give directions that would say something like, “In enough words and/or pictures, show me how proportional reasoning can help expand a business.”

Many view math as an objective and direct subject, being so reasonable and having one answer (or at least a best answer).  Some also say that one method is the best method and should be the only one that any true mathematician should use.  There is a strong movement to have a more open middle lead by dynamic educators such as Dan Meyer.  Math still has structure, and such structure should be highlighted.  I think it’s just foolish of us to attempt to summarize that structure into discrete, specific statements and expect the statements to transfer the understanding directly and objectively.  Math is just as much abstract as it is concrete.

This entire preface is to address one major issue with the most common prompt given in a math activity: “Solve it.”  First of all, we might even need clarification on the it that needs to be solved.  More importantly what does it mean (in the context of a math activity) to solve.  I ask this question to my students every year, sometimes multiple times a year, and their most common responses often include:

  • work it out
  • break it down
  • simplify it
  • find the solution
  • or everyone’s favorite Nike slogan (yeah, that one you’re thinking of right now)
So we wait, and see if students can come up with anything else.  Usually it gets really quiet; maybe because students aren’t quite sure why we are talking about words in a math class.  I start the scaffolding by reminding them that the meaning of a word can be found in its root.  I highlight the root, solve, and then ask the students to brainstorm other words with the same root.  This list is often populated by:
  • solution
  • solved
  • solver
  • solving
  • and sometimes an unrelated word like sold
I try to edit the list and ask the students to think of a word that’s only four letters that is pronounced the same, spelled the same, and has the same meaning in both english and spanish.  At this point students have sometimes guessed the word soul, or they forgot that I said four letters and go back to their originals from the first round of guessing.  The tension is building, both for myself and for the students.  The students are still confused but at least interested.  The gears are turning but not really going anywhere for most.  Usually one or a few that listened carefully to the clues asks in a half certain tone, “solo?”
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