Tag Archives: functions

Alg II Files: Functions & Graphs

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TMC has got me all mathy-feeling, so here’s another unit! Or at least, the first part of it!

I didn’t really change much to this chapter as from previous years, but I’ll go ahead and post everything so it’s all in one place. As usual, find more files and FAQs on my Algebra II Files page.

Let’s begin with what is and is not a function:

(file here) I took the NAGS from Sarah at mathequalslove and I think the rabbits came from Shireen at MathTeacherMambo. (Definitely 2 of my top five math blogs).

Here’s part II, but according to my calendar, I did part 1 and 2 the same day.

(file) I’m just going to warn you if it’s the first time teaching Algebra II, the struggle is real when trying to find function values from a table or graph. Just be prepared.

Also the magic parentheses for evaluating a function = amazing. We took the parenthetical promise (h/t mathequalslove again) in Unit 1 that said every time we substitute in a value, we put it in parentheses. And we’re going to be substituting for x, so let’s go ahead and put the parentheses first like:  3(       ) + 2 then INPUT the INPUT INTO the parentheses!

Here’s some homework:

Unit 3 1

(file) Then it’s the ever-popular graph stories!

Unit 3 3Unit 3 4

(file) Then…it’s time for….domain and range!!!!  Y’all, I just totally had a genius idea: have them figure out what the scale/domain/range should be for the graph story graphs first! One of you try it out and let me know how it goes. But since I didn’t think of that until just now, here’s what I used:

Unit 3 2

(file) Since we’re doing this before we did inequalities, it’s domain and range PLUS learning interval notation. Note the color with a purpose! I have them “box in” the graph before they think about writing the interval. I also use Sam’s domain and range meter, sometimes breaking out the spaghetti to use if I feel like picking a million tiny pieces of spaghetti off the floor.

Here’s some slightly lagging homework for the chapter:

Unit 3 5

(file) Then it’s a graphing line “review.”

(file) As you can see, I always go back and forth on graphing standard by converting or by x and y intercepts. Here’s the boring homework file.

Now here’s some exciting stuff! This is a pretty magical activity that is a really good introduction to the (h, k) form. Just read question 1 and let that awesomeness just sink into your brain.

(file) And some practice:

Unit 3 6

(file) Then it’s time for a study guide:

Unit 3 7

(file) And of course a study guide video!

Off to make another post!

Category: Alg II | Tags: , , ,

Precal Files: Function Transformations, Compositions, and Inverses

See more precal files and FAQs here!

As you may have guess from my TMC presentation, I LOVE function transformations. LOVE LOVE LOVE. So let’s get started with a foldable of parent functions:

Function Files from megcraig.org Function Files from megcraig.org(File with instructions and these pictures here)

Homework for the next 3+ days of transformations: (Could someone tell me if that second part is from your blog?!?!)

Function Files from megcraig.org(File here). After the first day they have a quiz of sketching the parent functions. I think I may add writing the t-table out as well.

Then let’s start transforming!

Function Files from megcraig.org(File here) Also see a more in-depth explanation in this post. And a great post from Shelley! And a great Geogebra app from Jed!  SO MUCH AWESOMENESS!

Here’s a practice worksheet:

Function Files from megcraig.org I actually had students ASK to make a table like the day before because they could see the transformation easier. I also added these type of questions this year:

Function Files from megcraig.org(similar file here)

There is also a GREAT activity I used that is a bit copyrighted. If you are part of a NMSI/LTF school, look for the “Graphing Transformations” activity. Basically it gave the students a graph in the first quadrant. Then it asked them how the domain/range/max/min/x-values of max/min/x-intercepts/y-intercepts/AROC/area under the curve change based on different transformations. (They told them what the area under the curve was.) It would be really easy to recreate and there was a lot of great thinking and previewing of Calculus in it.

Also STAY TUNED TO THIS BLOG for another great activity to practice writing equations of transformations.

Next up, let’s do transform our parent functions!

Function Files from megcraig.org

(File here) Read more about this method at the end of this post. The big idea is that we move the ORIGIN (not the “vertex” since not every graph has a vertex) and count our stretched/shrunk graph from our new origin. So easy and beautiful! Works great for conics and trig functions, too!

We did some speed dating practice with it:

Function Files from megcraig.org Function Files from megcraig.org(file here) The first pages are the questions, the second set are the answers. I may change some of them up to make the difficulty more equitable. Some people had really quick graphs and others took a bit longer. Maybe making it so there’s just one hard one, but two easy ones? I’ll let y’all sort that out and get back to me.

So after what seems like forever (yet not enough time), we move onto function compositions:

Function Files from megcraig.org(file here) Things to notice: I write the outside function first, putting (            ) wherever there is an x. Then plug in the inside function into those parentheses, leaving a (       ) wherever there’s an x in that function. Then plug in the value. This seemed to go a lot smoother than finding g(5), then plugging that into f, especially if you have a composition of more than 2 functions, or if you have 2 x’s in the “outside” function.  Also, notice that cool way of simplifying the complex fractions on #4. Read more about it here.

Homework: Function Files from megcraig.org(file here)

Then some inverses. I want to do A LOT more with them this year and start talking about them WAY EARLIER (See my flowchart epiphany here). But here’s what I did last year:

Function Files from megcraig.org(file here)

Function Files from megcraig.org(file here) (yes, even though it says 1.7 instead of 1.8 at the top. Numbers are hard.)

And a really good in-class sheet with some practice Free Response Questions:

Function Files from megcraig.org Function Files from megcraig.org(file here)

And then it’s study guide day!

Function Files from megcraig.org(file here)

Now go forth and transform.

Algebra II Files: Functions & Radicals

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News flash: I secretly love making math powerpoints. I need to find a job that is just making them (and NoteTakerMakers) all day. Or maybe half a day because, ok, it would probably get old after a while. But for now, enjoy the bounty of my obsession.

Our textbook starts the radicals chapter by doing composition and inverse of functions, so that’s where I start as well:

Function Files from megcraig.org

(file here).  I have to spotlight my two favorite slides:

Function Files from megcraig.org

Yes, “pig squared” gets a laugh every time.

Function Files from megcraig.orgFunny story: On one of my student’s review of a Vi Hart video, the student said that Vi talked about doing some operation with dolphins. The student said that didn’t bother her because “my teacher does math with corgis and unicorns.” Awesome!

And yes, there is an NTM to go with it:

Function Files from megcraig.org(file here) For the past [redacted] years, I’ve always worked from the inside out on functions, then I had an epiphany last year…try working from the outside in!  For example, on that first problem, it’s j(h(1/2)).  Let’s start with j, which is 6x, but we know we’re going to replace x, so we’ll write 6(              ). What are we filling that with? Oh, h! So now we have 6(2(     )+5) and what do we want to put in there? oh, 1/2! 6(2(1/2)+5)!  I found it really helpful for when there’s more than one x that you have to plug in for, like #4.  Anyway, just thought I’d mention it since it’d hard to tell what order I’m doing things on the key. We need magical time-telling paper. Get on that, people.

Here’s the homework (I found that finding function values from a chart or graph is something that Precal students struggled with, so I tried to add some practice)

function files from megcraig.org(file here)

Ok, now inverses!

Function Files from megcraig.org(file here) To find out the 5 things to know about inverses, you’ll have to view the powerpoint (clickbait!):

Function Files from megcraig.org(file here)  Let’s zoom in on my favorite question from the homework that was posted above:

Function Files from megcraig.org(although I guess I should make it a 1:1 function?) Discussing this problem the next day is a great way to reinforce the idea of inverses!

Then it’s time to graph some radicals:

Radical files from megcraig.org(file here) and review for a quiz:

Radical files from megcraig.org(file here)

Then it’s time for the phrase that strikes fear in teachers, students, puppies and unicorns: EXPONENT RULES.

Radical Files from megcraig.org(file here) To clarify some stuff, the PMA/RDS at the top is from someone in the MTBoS. Exponent rules follow the pattern of doing operation “below” it: power means you multiply, multiply means you add, and you can’t do anything with addition since there is not a function lower than it.  Then the same thing is true for roots/division/subtraction. I really wish I could find the original post because that person explained it a lot better than I can right now.

If you’re not aware of the Dead Puppy Theorem, go visit Bowman immediately!  I made my own corollary which is “Every time you say a negative exponent makes the number negative, a unicorn dies.”  “But Ms Craig, there’s not any unicorns left!”  “EXACTLY.  That’s how many students have made this mistake.  There are actually 4 of them left in a secluded meadow in Ireland; it is up to you to make sure they do not go extinct.”

Homework that we do in class:

Radical Files from megcraig.org(file here) I obviously typed this right after reading a tweet about allowing students to make choices in problems to do.  It actually worked out better than I had planned because they would say stuff like, “oh, wait, this has a zero exponent, that one’s going to be easy!” As in, they were actually looking at all the problems and evaluating how they would be solving them. (Although some of them just did the first 10).  I did the same thing throughout the chapter, but I just gave the instructions verbally.

Next up, let’s work with radicals!

Radical Files from megcraig.org

Radical Files from megcraig.org(file here)  I also changed this up this year.  Instead of spending one day where all the radicals were perfect, then another day when they weren’t, I started with perfect radicals but then gave them a tricky problem at the end of their practice row (#17-24).  Then we discussed how we would go about simplifying them. I think it worked out pretty well. This took us most of two days to finish front and back, then we did some practice:

Radical Files from megcraig.org

(file here) which pulled questions from this homework:  (I think I called it homework because a lot of students were absent for some reason? Then they felt like they should do it rather than, “Oh we just practiced in class, nothing I need to make up.”)

Radical Files from megcraig.org(file here) Now it’s time for some binomials, again, I mixed everything together (and this was before I read Make it Stick about varied practice!):

Radical Files from megcraig.org(file here) And homework:

Radical Files from megcraig.org(file here)  And a review:

Radical Files from megcraig.org(file here)

Ok, we’re almost there, guys!  We need to talk about rational exponents:

Radical files from megcraig.org

(file here) and homework:

Radical files from megcraig.org(file here)  And then solving!

Radical files from megcraig.org Radical files from megcraig.orgDay 2:

Radical files from megcraig.org Radical files from megcraig.orgFinally it’s time for the last quiz of the chapter!  Review:

Radical files from megcraig.org

(Due to weird scheduling issues this year, we started the next chapter before we quizzed.)

(File here)

Of course there’s a powerpoint!  It’s more of an overview (i.e. not the same probs as study guide).

Radical files from megcraig.org(file here)

So, holy cow, I have a lot of stuff for radicals. Kudos for you to reading til the very end!

Lines in Algebra II: SRSLY, You Should Know This By Now.

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(The continuation of my posting all of my resources for Algebra II.  See more files and FAQs here.)

So the students have seen lines in middle school, Algebra I, and Geometry, so this should be a nice easy review, right?  “Hey guys, let’s graph a line! And then let’s start in point-slope form and write an equation of a line!”

Lines 0

Until they find a cure for Math Amnesia, I guess we’ll start from scratch!  Starting with functions:

Lines from megcraig.orgFile here. (sorry for the some wonkiness in the scans… I use a roller scanner. If my phone had any sort of storage/wifi/3Gsignal in school, I’d use CamScanner instead, but alas..) Next year I’ll be sure to use the vending machine analogy that Justin (@JustinAion) shared on twitter:

Lines from megcraig.orgComplete File here.

And some homework:

Lines from megcraig.orgFile here.

What is it about domain and range that students have such a hard time with it? As you can see, I try both the “flatten the graph to the x or y axis” method and the “box in your graph” method.

Lines from megcraig.orgFile here.  (Legal, but it’s sized to shrink to 2/letter sized).

Somewhere about this time, I like to do graphing stories:

Lines from megcraig.org File here (not my original idea, I just made it into a worksheet instead of card sort).  Also, yeah, #6 always gets me. I want to say it’s the curved line, but only by process of elimination. I’ll have to do some filming of a bathtub one day.

However, I think I’d like to replace it with this graph matching instead:

Lines from megcraig.orgLines from megcraig.orgFile here. Again, not mine and I haven’t tried it yet, but it seems a little bit higher level.

Ok, now it’s time to get started with those lines!

Lines from megcraig.orgFile here. I used HOYVUX this year, but not sure if it’s my favorite. I usually go with “x(or y) = #” means we cross the x(y)-axis at that number. Also, note the hearts around #5) y = x. I tell them it is my favorite graph of all time, the graph all others graphs originate from and aspire to.  And because it’s my favorite, it will be on every single test until every single student gets it right, which usually means it is on at least 4 tests.

This year, instead of my normal writing equation notes, I did this Translating Lines discovery instead (yes, it’s very similar to the Precal one I shared because it’s awesome)

Lines from megcraig.org

File here. Next year, I’m going to add some more practice like the first 11, but wait on parallel/perpendicular/two points until the next day to reinforce the “new” point-slope form (they all learned it as y – y1 = m(x – x1) instead of y = m(x – h) + k) and also work a little on getting it into slope-intercept and standard (since that is what a lot of standardized tests use).

However, even without that, most did well on the in-class practice:

Lines from megcraig.orgFile here.

Or maybe you’d like a scavenger hunt with graphing, functions, and equations in slope-intercept and standard form?

Lines from megcraig.org

This is just the first 2 pages; it goes all the way to X and takes most of a period to finish.

File here.

Or maybe you’d like to stop here and give a test?  Well, here’s a study guide

Lines from megcraig.orgFile here.

Now let’s actually use these lines!  Next year, I think I’d like to start with Mathalicious Domino Effect, or at least make that the first type of problem on the notes.  Actually, I need to change a lot of the problems on here. I teach in a suburb, so my students have no idea about the ride fare of a taxi. Also, don’t set yourself up to talk about “expanding rods” in high school.  And look how quaint #4 is–a toll phone call!

Lines from megcraig.orgFile  here.

If you look below, I did take someone’s (??) suggestion about the new way of finding slope with a table and labeling the slope.  Also note the mad-libs portion of the worksheet describing what the slope and y-intercept tell us, an idea I got from Mimi (@untilnextstop) (side note: I miss regular posts from Mimi! If you haven’t read her entire blog, you are missing out on some AWESOME activities and teaching ideas. I’d say I get at least 1/3 of my ideas from her. Also, she lives the most adventurous life!)

Lines from megcraig.orgThen we did some linear regressions on the calculator (of course you could also use desmos), again practicing some “Math-Libs”  on what part of the equation tells us.  Note: next year, I need to add a negative correlation example.

Lines from megcraig.org Lines from megcraig.orgFile here.  I must say I like the two part version of #5, where we find more data = more accurate (or at least a better picture).  Which always reminds me of this xkcd comic:

This year I did Mathalicious Reel Deal (members only), which talked about movie length over the years.  It didn’t go as well as I had hoped; there was a lot of handholding throughout and little “oh, I get it now!” moments.  Maybe because it was the first time I’d done a Mathalicious lesson?

Then it was time for some absolute value, a discovery lesson that actually went well!

Lines from megcraig.org Lines from megcraig.orgI typed up the first part of the first sheet, file here.  It looks like the rest came from the Louisiana Comprehensive  Curriculum, the pdf file is here. The homework file is here.

Alternatively, if you’d just like some notes:

Lines from megcraig.orgFile here. Also, check out that nice vertical stretch work on #4.  I’d almost like them to do the chart and change the y-values instead of thinking of it as slope, since that won’t work for any other function.  But the discovery activity was also really nice…hmmm….decisions, decisions.

Well, at least I know how I like to teach graphing inequalities:

Lines from megcraig.org

Hey, look, it’s my favorite graph again!

File here. (The shading on the last row usually prints nicely from the printer–I think it was a copy of a copy that I was using, so you couldn’t see it very well.) Also, no, we don’t have time for test points, we just go above and below.  Hint: make them put their pencil on the line and then move it above (or below).  That seems to help for when they secretly want to go left/right.

And finally it’s a study guide!

Lines from megcraig.org Lines from megcraig.orgFile here (print it out on legal, then copy two-sided and cut in half).

My thing

Ok, this is going to seem like a weird thing, but have y’all tried the Command adhesive shower products? They are seriously awesome. I hate the suction cup caddies that either (a) slowly slide down the wall or (b) quickly crash to floor (usually in the middle of night).  We’ve had these in our shower for almost two years now and they haven’t slipped a bit!  So treat your shower to a makeover this summer and install some of these. You can thank me later when you’re not woken up in terror at the sound of a burglar shower caddy falling.

All aboard! Destination Function Junction!

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choo-choo-choose-you_14205450311(This post is part of my attempt to get all of my resources online for y’all. See more files and FAQs here.)  Are you ready to get this Precal train rolling???

I used to start Precal with the “Chapter P” in my book: basically, hey, remember this from Algebra II? No, ok, let’s teach it again. Because there’s no better way to start off a course than with absolute value inequalities, amirite?

Right….so I changed it up this year because (1) I had only PreAP Precal and I do expect them to be a little more prepared that regular and (2) the first full week of school we had a week of grade level meetings and picture days, which meant I saw some periods all five days and some three and some kids in and out throughout. So we did a review of equations instead. I just gave them the sheet and said “have at it! See what you remember and what your group can figure out, then we’ll follow up on the rest!”

Functions from megcraig.org

A former coteacher made this, which is why it’s written with the new equation editor (HATE) and also why it says “Pre-Cal” instead of “Precal”.

File here (4 to a page!).  I think I remember it taking about 3-4 days for them to finish.

Then we started with the “Chapter 1” business: lots of definitions and descriptions and probably stuff they should know from Algebra II but don’t, e.g. functions.

Functions from megcraig.orgFile here. I really need to start using NAGS (Numerically, Algebraically, Graphically, Sentence) more throughout the year.

The homework for the chapter:

Functions from megcraig.orgFile here. (I print it 2 to a page)

Then everything you could possible want to know from a graph:

Functions from megcraig.orgFile here.

Oh, wait, actually that wasn’t everything you could want to know about a graph!  How about relative extrema and even/oddness?

Functions from megcraig.orgFile here.  I use a powerpoint to introduce even and odd graphs:

Functions from megcraig.orgFile here.  The animation on this is actually pretty neat. Once you download it, be sure to watch the actual slideshow (not just scroll through the slides) so you can see it.

Then we spend some time playing “Math Pictionary,” where we break out the whiteboards and make graphs that meet different conditions:

Functions from megcraig.orgFile here. If time allows, I also show a funky graph and have groups come up with as many descriptors as possible.  Then we go around and each group shares one, no repeats. The last group to have one to share wins!

By this time, everyone should have had a chance to get a graphing calculator, so we start using it:

Functions from megcraig.orgFile here.  Next time I may put some graphs on there that require changing the window to see everything.

I know I said I wasn’t going to reteach Algebra II (or, ahem, Algebra I) topics, but being able to write the equation of a line is just too important a topic not to spend a day on. Also, using this new way of graphing-by-translating and writing point-slope form is a nice (re)intro to (h, k).

Functions from megcraig.orgFile here.

Now I’m going to stop here because I think you may have just breezed by this without trying it, thinking “oh, just another graphing/writing equations worksheet” but it’s not! I promise!  Try this first problem:

1) Graph y = 2x. (it’s ok if you do it in your mind, but feel free to get paper. I’ll wait.)

2) Translate to the right 4 units by moving each point 4 to the right.

3) What is your new y-intercept?  Write your equation in slope-intercept form.

4) Now factor out the GCF.

WHAT?!?!?!?!?!?!?!?  Yeah, that just happened.  The mystery of why h is negative in y = a(x – h) + k is solved with a simple four-step line problem. (Ok, maybe not “solved” because that will come a little bit later, but for now it’s pretty cool, eh?)

Ok, enough amazement, back to work with piecewise functions (we haven’t started doing them in Algebra II yet, so this is the first time they’ve seen them):

Functions from megcraig.orgFile here. Regular Precal file (more graph practice, no writing equations) here.

Yeah, check out those first graphs…the gray graphs are already on there for them so we can focus on the restricting the domain of each one before we pull it all together and graph from scratch (which I do by graphing all the functions with dotted lines, then filling in the parts that I need, so it’s very similar to the first examples).

Then it’s time for Average Rate of Change, which didn’t go so well this year (I was also out for a meeting on the second day so that didn’t help).  And it started so well when I let them loose on this:

Functions from megcraig.org

I print 2 to a page so this was all on the front.  And they were rocking and rolling. Then we flipped to the back:

Functions from megcraig.org

(file here) I HAVE NO IDEA HOW TO DO ANY OF THIS BECAUSE IT’S USING DIFFERENT WORDS THAN THE FRONT!!!!!  WHAT IS THIS MADNESS?????? And now you want us to do these practice problems?

Functions from megcraig.org

(file here)  We spent 3 days on this (again one of them I was out), and I can’t help but thinking if we started with this like I usually do:

Functions from megcraig.org(file here) and then spent two days applying it, they’d have a better feel for what AROC is than they did. Or maybe some hybrid of the front of the first worksheet, this, then the back?

As a side note, using The Biggest Loser as an example of AROC is great. It always hurt my head if the graph went up and down, but the AROC was zero. “It can’t be ‘no change’!  We were totally changing the whole time!!” But if you think of it as a contestant gaining weight and then losing the same amount of weight in a week, at the weigh-in she’ll have 0 change.  (This is also a good time to do a PSA to your students about not being obsessed with your weight every day.)

Finally (finally!) it’s review:

Functions from megcraig.orgPowerpoint file here

and the study guide:

Functions from megcraig.orgFile here.

Whew! I think I’m going to get off the train at Function Junction and take a break, but stay tuned for the next installment of “Wow, Meg Wasn’t Lying When She Said She Killed a Lot of Trees.”

#MTBoS Sunday Summary

My week:

Algebra II w/ Trig

We took our first test, which was a GREAT segue[1] into growth mindset and overcoming setbacks.  Although they didn’t knock it out of the park, I had fewer really low grades than previous years, so that’s a good sign.  This is usually a tough test and I think I need to either (a) break it up into 2 quizzes or (b) actually follow the notes that I leave myself every year that say “DO MORE DAYS OF ____”

Then into functions with this beautiful Note-Taker-Maker[2] that I stole bits and pieces of from everyone, but I think mostly from mathequalslove.

Capture1.doc file here

Friday we did the ever-popular graph stories, but I made it into a worksheet because I do not have time for scissoring.  They worked in groups and I paired groups up as they finished to discuss answers.  Then when everyone was done I brought out the talking dog.

No, it didn’t actually talk. I’ve been having an issue with students not listening to other students talking, so I thought I’d take a cue from GWWG and Mr Healy and start the rule that you can’t talk unless you have the talking dog.

So the dog was passed around as we explained and discussed our answers to each graph.  And what do you know?  It worked!!  Not 100% of the time, but they were listening a lot more than before. Some of the kids had great explanations, too, including “we first thought it was this graph, but then when we met with another group, we realized….”  Win!

PreAP Precal

Holy moly, definitely some ups and downs this week with discovery learning.  All of the kids are working super hard at trying to figure out what I throw at them, but they are lacking in (a) math skills and (b) seeing math connections.

We spent about 1.25 days on translating linear equations to discover our “new” point-slope form: y = m(x – h) + k.  Worksheet here.  It went really well and was a good lead-in to piecewise (“isn’t it so much easier to write the equation of this guy using point-slope since we don’t know the y-intercept?”)

We also spent 2 days on average rate of change, worksheet here. (I’m pretty sure I stole this from someone, too–I really need to work on documenting my sources!)  The first side of it went great, but I need to reword it so they understand we’re doing the exact same thing on the back!  Just calling it a different name!  Really!  That’s it!  Then I wonder to myself, would they have made better connections within the same time frame if I had lectured for 20 minutes on AROC, definition, formulas, etc, then let them work with applications of it for 1.5 days?

So even though it may get me kicked out of the MTBoS, I think I’m going to stick with introducing a topic as a class first, then letting them loose on going deeper, rather than letting them loose to discover the topic but running out of time to go deeper.  Plus, I think based on this: zone(poster available here) I’ve been in the panic zone as a teacher way too much this year and need to scale it back to be in the learning zone.  I need to keep reminding myself I don’t have to try everything that I learned at TMC during the first month of school!

[1] I’m not going to lie to you; I just found out last year this is word that people are using when they say “segway” as in, “Using (person, birthday) as a function example allowed me to segway into mentioning I don’t like Starbucks.”

[2] Notetakermakers (or “NTMs”) are what I call my graphic organizers.  Yes, I make one for each section.  Yes, I do use a lot of paper.[3]

[3] Dude, I don’t know why the superscript is showing up as just tinier script.  I’m using “sup” and everything.  Here’s a subscript example: [1] which looks to me exactly like the superscript.  Sometimes I hate everything about wordpress except for the fact that I get to use “megcraig.org.”