Search Results for: transformations

Desmos Activity Builder for Function Transformations

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Find what you love: Function Transformations

Do more of it: Ok, how about working with Sheri Walker (@sheriwalker72) on an awesome transformation idea she had for Desmos’s new activity builder?

And here it is!

desmos activity builder

Using shifts, stretches, flips, and shrinks, get the red function to match the blue dotted one.

We even added two challenge rounds:

desmos activity builder 2

And we forced Eli to try it out before we left for lunch:

IMG_0419

He declared it awesome! (I have a feeling Eli declares everything “awesome,” though.)

Here’s what we LOVED about Activity Builder:

  • Easy to use–just like using the regular Desmos calculator
  • TEACHER DASHBOARD! Easily tell what all of your students doing. Easily share with the class (with projector) what everyone else is doing.
  • Easy to link up multiple graphs (before I had students turn off/on folders and Sheri ran into issues trying to link one Desmos graph to another in order because of how Desmos does its linking with edits)

Here are some things to be know when building an activity:

  • You don’t need to limit yourself to just graph screens! You can also do question screens and text screens.
  • You can also insert images into your graphs
  • Your default graphing window will remain the same when the students open it. However, projector mode is an option the students have to select themselves (if they desire to).
  • You cannot copy a previous step, for example, we used the same piecewise function and instructions for each step and had to reenter them. Shortcut tip: Open a regular desmos window and type your stuff in there. Then copy and paste each line into your steps.
  • Don’t hit cancel, even though it’s right next to done and even if you haven’t done any new editing!
  • You can visit the bit.ly/desmosbank to find some activities build by other teachers to use, but I don’t think you can (currently) edit them.
  • (Currently) students can see your work. We “hid” our transformation equation way down in a folder on line 49. It was also suggested we could take a picture of the graph and insert it as an image. (We were worried the students would accidentally move it, but now that I’m thinking about it, they would just have to make sure the axes lined up if they did happen to move it). Another suggestion was to use points to mark the new function (so it wouldn’t matter if the students saw the points because they wouldn’t be the “answer” like f(x + 2) would be). (This would be great for parent function transformations, but we wanted students to see the whole shape easily).
  • Student can (currently) accidentally (or on purpose) delete/edit input lines and there (currently) is not a refresh/restore button.
  • Based on tweets, the Desmos team is aware of these last three bullet points and are trying to work out the best way to address them.

So go ahead and give it a go!  Make an activity! Make a polygraph! I’m sure you’ll at least have fun making it, although maybe not as much fun if these rock stars aren’t there with you:

IMG_0428

IMG_0416

IMG_0418

 

Category: Uncategorized

Precal Files: Function Transformations, Compositions, and Inverses

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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.

Category: Alg II, Precal | Tags: , , , ,

Sunday Summary: I Love Transformations

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3-2-1 Sunday Summary:

3 Resolutions for 2015

1. Blog more. Remind self that I don’t have to type a novel every time, nor does anyone want to read a novel on a blog.  Keep up with short 3-2-1 summaries.

2. Exercise more.  I joined the #500in2015 challenge and did pretty good the first week.  To motivate myself, if I keep my goal of ten miles a week in January, then I get to buy the new Jessica Smith walking workout video set. (Right now I’m using this DVD from her. The Nike+ app seems to record the walk pretty accurately and I’m not going outside when it’s below freezing! 55 degrees!)  If you’re looking for a good indoor workout, check out her website–she has TONS of free full-length workouts posted, with special appearances by her dog, Peanut.

3. Leave school at school. This has been one of the more trying years I’ve had as a teacher and I’ve been bringing a lot of that home with me. I’m going to try to be better about shutting that part of my brain off.

2 good lessons this week!

1. One day this week, I ended up with four out of eighteen students in class. Instead of calling the day a loss, we got together in a group and worked through the notes together. It was so nice to talk with them one-on-one through the lesson and then we all worked together on the homework.  I need to remind myself to sit down with more groups as they are working, instead of just helicoptering around the whole room.  (Side note: I did use the exam study guide days to do this as well: each group got 5 minutes of Mrs Craig time to ask any questions; it worked really well!)

2. We started transformations in Precal this week. Coincidentally, Shelley Carranza (@stcarranza) asked if she could link to a previous post I had made about transformations and of course I said yes. (Here is her post.) As a bonus, she gave me a sneak peek of her desmos graphs which inspired me to change up my introduction graph:

transformation table

(Note: there is no table for the absolute value functions because my coteacher and I wanted them to thinking about those on their own for a bit).

Next year, I think I will use up some extra paper and recopy the table next to each graph.  Because being able to mark it up adds a wonderful visual to what happens when we affect the input, for example, f(x – 2).

transform x minus 2

THIS TOTALLY BLOWS MY MIND EVERY YEAR.  We are “reaching back” 2 to find the output value, which will “pull up” to where we are.  SO THAT’S WHY IT SHIFTS TO THE RIGHT WHEN SUBTRACTING.  We then talked about the “bonus” point of (8, -2) we could get from the original (6, -2).

Ok, are you ready for super mind-blowing?  Check out f(2x) (The green boxes are more “bonus” points.  A good question to determine these was, “where would this -8 output project to?”)

transform 2x
HOLY COW YOU CAN TOTALLY SEE THE GRAPH BEING PULLED IN!! We need to go out twice as far, then pull that answer back in to our x-value.

We spent two days on this, then did some more practice. Monday we’re doing a super-thoughtful-hope-they-all-ate-their-wheaties worksheet combining transformation, average rate of change, and area of the curve.  I will report back as to its success and/or not-there-yet-ness.

Here are the files: Table Worksheet  Table Desmos File

1 Thing I’m Looking Forward to This Week

More transformations!!!  Seriously, I love these.

Category: Alg II, Precal | Tags: , ,

The Things I Do For MTBoS (Posters & Function Transformations)

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Casey (@cmmteach) says, “MAKE ME A POSTER!”  and I say, “HOW SPACEY?”

Capture

.pdf  and .doc and yeah, because that would use a crapton of ink, bwpdf and bwdoc.

Then Julie (@jreulbach) says, “I NEED THAT IN A POSTER!” and I say, “I FEEL VAN GOGH IS SORT OF PICKY ABOUT USING HIS STUFF”

Capture2

.pdf and .doc

Then no one says “MAKE ME A POSTER OF THIS QUOTE, TOO!” but I think they just didn’t realize their classroom was incomplete without it so YOU’RE WELCOME, EVERYONE.

Capture4.pdf and .doc

Updated: I also made a color version of the above poster: .pdf 

Temporary Postponement of Success Color

Then Friday night on twitter we got into quite the discussion of (h, k) and function transformations, because that’s what cool kids do on Friday nights.  Julie was lamenting at kids not sure what to do first…reflect, dilate, shift?  So I mentioned what I call the S.S. method of graphing after one of the best math teachers with whom I’ve ever had the privilege to work.  I’m not going to lie to you; I didn’t trust it for a while. But then after a couple of years (!) she finally convinced me and it is beauty and efficiency all at once.  Then Julie says “GIVE ME AN EXAMPLE” and I say, “WELL HERE’S A WHOLE BLOG POST ABOUT IT ARE YOU HAPPY NOW?”

So you spend time talking about what the various parts do, maybe with some discovery desmos, or some crazy function match game with worksheet.  (hint: don’t use the t-tables! it just confuses the issue!), or the beautiful HOLY COW SO THAT’S WHY f(2x) COMPRESSES activity (which yes, here’s a quick showme video because Elissa said “CAN ANYONE HELP?” and I said “I WILL SHOWME YOU” even though other tweeps answered her much better than I did).


After all of that, it finally gets down to the nitty gritty of graphing a real function with all sorts of exciting things happening. And yes, yes, you could do all the shifting and dilating and reflecting, and substituting (M, N), but then I don’t know how that would be any quicker than just plugging points into the original equations after all that work (don’t get me started on the people that have them draw a graph for EACH TRANSFORMATION like we’ve got all the time in the world) and the Cal teacher would like them to be quick and efficient at these graphs.

But the good news is I can multiply and divide really well and I’m also really good at counting from the origin.  And that’s all that’s needed.

1) Mark (h, k) with a small x.  This is your new origin.

2) Take your basic t-chart (-2 to 2 normally does it for me, unless there’s domain or excitement issues).  Multiply/divide x’s/y’s as needed.  Feel free to multiply/divide by negatives in the SAME STEP if a flip is involved because we are that CRAZY GOOD AT MATHING.

3) Graph your t-chart from your new origin.  SHAZAAM.  Feel free to label points counting from the original origin if your teacher’s into that sort of stuff.

Here are some examples:

Capture5 Capture7The one catch–a negative INSIDE the function.  We just have to do some reorganizing:

Capture6

(and some of us need to learn the difference between flipping horizontally and vertically. I know which way to flip; I just always call it the wrong thing. I’ve started going with “flip across x-axis” or “y-axis”).

Anyway, maybe this is what most of you do anyway?  But I always have some teachers that are freaking AMAZED by it when I start graphing these babies at workshops so maybe you were, too. If you want more, here is the notetakermaker and the filled-in version.

Full disclosure: Of course y’all know Casey, Julie, and Elissa are three of the sweetest people on twitter and would never ask for things in the manner in which I implied. So I hope y’all consider this post just a feeble start to paying back all that MTBoS has done for me!

Category: Alg II, Classroom Decor, Precal | Tags: , , , , ,

Stars of the Week Vol 6 #SOTW

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Since my last SOTW post was over a month ago (!!!), perhaps I really should take Pat’s suggestion from my last post and rename it “Stars of Whenever I Want.”

If you’re a #SOTW, grab the badge using the code below if you’d like it for your blog!

Stars of The Week
<div align="center"><a href="http://www.megcraig.org/?p=1396" rel="nofollow" title="Stars of The Week"><img src="http://www.megcraig.org/wp-content/uploads/2016/08/Star-of-the-Week-Badge.jpg" alt="Stars of The Week" style="border: none;" /></a></div>

Remember you can always use the nomination form and I will put it in my next post!

If you teach Geometry, you definitely need to be reading  @lisabej_manitou‘s Crazy Math Teacher Lady blog. She has tons of great ideas, but also realizes that sometimes you just need some no-frills, get-the-job-done, quality materials (a girl after my own note-taker-maker heart). Check out her quadrilateral collection. (BTW, if you interested in more materials like these, be sure to check out the comments on @k8nowak ‘s post that Lisa linked.)

I know we’re not supposed to have favorites in the #MTBoS, but I do, and @TPalmer207 is one of them. Ever since our TMC14 dash-to-get-cupcakes-before-the-store-closed, her great outlook and humor have always been something I look forward to seeing on Twitter (and in person!). Not only that, but she’s definitely the type of teacher I wish I could be – in the moment, not afraid to try new things, and constantly creating great discussions in her classroom. Check out this gorgeous teacher move of slowly revealing all the information with the absolute twist at the end that no one saw coming (not even Tina!). I know what you’re thinking, sure, everyone gets lucky once in a while, but no, Tina makes her own luck, as you can see in this segment addition postulate lesson.  You know, she only has about 60 posts so if you’re looking for a way to up your teacher game, spend an afternoon reading the entire collection. You won’t be sorry.

Speaking of teacher moves, if you teach ELL students, @heather_kohn is your gal. She is a great advocate of doing what it takes to bring the ELL student up to the goal, instead of lowering the goal down to where they may be now. Her post on scaffolding open response questions will make you rethink how you can help your struggling students (who may or may not be ELL).

Hey, here’s a great teacher move that easy to implement into any lesson on Monday: Ask Me a Question from @dsladkey. It moves the “Do you have any questions” to “What questions do you have” change to a whole new level.

Another one that’s so easy to implement, but can change the whole classroom climate: @a_schindy‘s Nevermind Strategy.

Wait, you want more teacher moves? This next post comes with a warning: This article will lead to great feelings of inadequacy and make you think most of what you’ve created is crap. So if your current mood is: ugh, I suck as a teacher, you might want to skip the @Desmos Guide to Building Great Math Activities. But if you’re feeling pretty good and want to pick up some ideas that will help make your next lesson planning be a little more thoughtful, go read it.

If you did read it (or even if you didn’t) and you’re looking for good examples of rich problems, check out @algebrasfriend‘s Algebra II examples. I LOVE the parabola one!!

Speaking of parabolas, you know function transformation are near and dear to my heart. @jreulbach created a wonderful Desmos marbleslide for them, but the key move here is the creation of an additional real-life worksheet for students to reflect on their learning and can have it for reference. (Also check out her Function Notation QR Stations–great practice problems for something my students always struggled with)

Ok: One more teacher move: let’s make mistakes a starting point for rich discussions. @Dave_Sabol shows how he used the results from a Desmos Activity Builder to decode mistakes in Calculus. (Also I think Dave should be in the SOTW Hall of Fame for his How I Teach series.)

Man, I don’t know about you, but now I’m overwhelmed by all the new teacher moves I’m supposed to be doing. So let’s take a brain break, sponsored by @mathequalslove‘s brainteaser collection.

And then let’s eat our feelings by baking these Salted Caramel Pretzel Crunch Bars from Sally’s Baking Addiction.

Now onto some starred tweets!

For reals, take David’s advice:

Yes, Zippy won (Adorable) Scruffiness of the Week!

Joel’s smartass reply (does he have any other kind?) to the discussion of how to say “apothem”:

Speaking of smartass replies:

I cannot tell you how many times I’ve looked at this tweet and laughed out loud:

I told you Heather has the pro teacher moves:

And speaking of that pause feature:

And other features:

You know I love my shortcuts:

And this one was brand new to me and I’ve used it every day since then!!! I’m even using it RIGHT NOW.

It always warms my heart to know that people enjoy my stuff; especially when it’s someone who has so much great stuff herself!

And when someone uses my suggestion and then the result is heartwarming:

And then when someone uses an idea I had and their students do so much better with it:

Be sure to send some encouragement to Kristin next weekend!

Do I like CalcDave because of his gifs or in spite of them?

Insert picture of two beautiful math teachers here.

Now I know what the first 91 books I buy as a librarian will be.

Genius.

Genius Part II.

Genius Part III.

And I think I’ll let Casey wrap this (extremely long) edition up:

What? I’m not crying. YOU’RE crying.

Category: Stars of the Week, Uncategorized | Tags:

Life-Size Sine Curve

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Last weekend, there was some twitter chatter about making life-size graphs so students could explore points/transformations/what-have-you. That got my wheels turning, and after a quick trip to the dollar store and about an hour’s worth of work I ended up with:

trig graphSteps:

1) Buy four shower curtains at the dollar store. Bonus if they are prelined!

2) Tape together with packing tape. (hint: tape them down to the floor with washi tape to hold them down, then tape the seams on the front. When done with the whole thing, flip over and tape the back)

3) Use duct tape to mark the x-axis. Print labels (file here), cut out, measure your axis, and tape down with packing tape.

4) Apply colored masking tape for 1/2, root 2/2, root 2/3. (see next picture since the masking tape was at school)

Total cost depends on how much tape you have around the house. I used almost 3 full rolls of colored masking tape, but the good news is I bought it from naeir.org. Have you heard of this site? One of the teachers at school shared it with me. Basically companies donate overstock and you get to buy it for the cost of handling. You do have to spend at least $25 and shipping takes about 2-3 weeks, but holy cow, can you get a lot of stuff for $25!  My first shipment I got 8 rolls of patterned/colored masking tape, 2 packs of 12 small post-its pads, 8 post-it pop-up cubes, 12 correction tape thingies, a pack of sharpies, 3 sets of dividers, 2 packs of post-it labels, and I think some other things I’m forgetting. It’s crazy!

Anyway, in class, I handed out dry-erase pockets with a sheet that had an x value in it (0, pi/6, pi/4….2pi) (file here) and told the students to find sin x, 2 sin x, sin 2x, sin1/2x, and cos x. Holy moly. We could have easily spent the day doing that. No, if x = pi/3, sin2x does not equal 2pi/3. Once we got that sorted, we went out into the hall. I had all the students stand on their x-coordinate, then step to the y for the function I called out. It was very easy to find people who made wrong calculations! 🙂  Here’s what cosine looked like:

cosine curveAnd sin 1/2x (with an outlier!)

sin half x

By third time I ran through it, I had worked some of the kinks out:

  1. In my first class, I had more people than x-coordinates, so I gave coordinates that were more than 2pi. This did not go well. They were way far down and we couldn’t really see the pattern continuing. The next class I handed out 2-3 points per group and had them work together to find the values, then as we graphed we substituted people in who hadn’t graphed yet.  (The class shown had just one person extra.)
  2. I only did each graph once. We talked about max/mins, who didn’t move and why, how many cycles fit on the mat, etc. I think it would have been beneficial to do sin, then cos, then switch back and forth faster and faster. Then do the same for 2 sin x and sin x, sin 2x and sin 1/2x, etc. And also positive and negative. The last class we even tried sin x + 2 (“oh, that means we need to all step up 2!”)
  3. Have them write down noticing/wonderings as we are doing it, or a quick sketch of the graph (maybe have some axes printed on the back of their point card?) to help solidify the concepts.

The next day, when we went to graph, I asked them if it helped to visualize what we did yesterday. Only a few raised their hands. I told this to Mr Craig, wondering if I would do it again or if it would be more efficient to just jump into graphing then practice. He said, “Hey, you helped those 5 kids see it better! Plus sometimes it’s about the experience, not about being efficient.”  Sometimes that Mr Craig can be pretty smart. (Don’t tell him I said that, his head is big enough already.)

In other news, this happened on Twitter the other night:

8,006 tweetsDo you think I will hit 1,000 followers or 10,000 tweets first?

Category: Precal, trig | Tags: ,

Epiphany Part II: The Return of the Tranformation

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So the perfect storm happened at TMC. In case you missed it (because it was only in my head), here’s what happened:

1) I had an algebraic epiphany last month about using flowcharts to solve equations:

Flowchart math from megcraig.org At the very end, I had an inkling of an idea to tie it into function transformations:

flowchart math from megcraig.org

but I didn’t really know where to go from there or if it was a viable way to think about it.

2) I had a conversation with two people about how confusing it must be for kids to do transformations in Geometry where x + 2 means “move two to the right,” then move to algebraic transformations where seeing f(x +2) means “move two to the left.” We wondered how we could make the transition easier for students, but came up with nothing. (Side note: I can picture sitting at a table with a boy and a girl having this conversation, but I cannot picture the boy and the girl. I’m now thinking it was not at TMC, but maybe another workshop I was at this summer? But if you are reading this and it was you, let me know so it doesn’t keep me up nights anymore.) (Further side note: Um, somehow I never taught transformations in Geometry? Is that weird?)

3) I worked with Sheri Walker (@sheriwalker72) in the Going Deeper with Desmos morning session. We were tasked with making a new lesson using Desmos and she immediately turned to me and came up with an awesome idea because she knew that we both loved function transformations (who doesn’t, amirite?).  I also brought in a copy of my handouts from my session to share with her and then she just casually mentions how she approached function transformations and jots down something like this for the equation y = 5|2x – 6| + 7:

transformations 1(I know, it’s weird that her handwriting is the same as the Chowderhead font, but that’s just how cool she is.)

Ok, so at this point I begin freaking out because Sheri can obviously read minds. I mean, I just had this epiphany three weeks ago. So I was so amazed at her mind-reading ability that it was not until later that I realized the elegance and awesomeness of her next step.

Are you ready?

I really think you should be sitting down.

If you’re reading this on a tablet, make sure you have a soft spot for it to land when you drop it.

Maybe alert your in-case-of-emergency contact.

Here it is…

transformations 2Ok, she called it something other than “fancy form” (programming form? transformation form? input/output form?) but you get the idea. Or maybe you’re like me and you think it’s pretty but “getting the idea” will suddenly hit you three hours later. SHE JUST TIED ALGEBRAIC AND GEOMETRIC TRANSFORMATIONS TOGETHER.

giphy

Added bonus? No more worrying about whether the form is in f(bx – h) or f(b(x -h), because this takes care of that. Want to see it action? Let’s take that same equation, y = 5|2x – 6| + 7. Except this time, let’s just find (x, y) pairs from the original absolute value function and use the fancy form to transform them:

transformations 2transformations 3What? That just happened. Are you as amazed as I was?  Now pondering how to fit this in to my already over-long function transformations unit….

Category: Alg II, Precal | Tags: , , ,

An Algebraic Epiphany

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People, this post is why I love the #MTBoS.  You can’t read everything, learn everything, critically think about everything; but if you read blogs and tweets, then you can collect more of that knowledge than you would alone. So even though I am not participating in the #intenttalk book study/chat (Am I the only one who always thinks it’s Kimmie Schmidt on the cover?), I did see this tweet from Bridget:

I used that method a wee bit this year when I taught inverse functions and a few students really latched onto it. But now I’m thinking of starting this way on day one,  building on it, and tying it into Glenn’s three rules of mathematics. I sat down and played with it a bit for the last few days and all I can say is:

Are you ready for this?  Ok, let’s just dip our toes in:

Flowchart math from megcraig.orgThe main idea being that we think through the equation “forwards” and then work back to the solution using inverses. Another easy one:

Flowchart math from megcraig.orgI like (a) completing the circle of life by checking our answer and (b) each column showing equal values.

How about we try out the shallow end:

Flowchart math from megcraig.org Flowchart math from megcraig.orgYeah, I’m totally digging the two arrows for square root, too.

Flowchart math from megcraig.orgAre your ready to put your head underwater?  Ok, here it is….wait for it…

Flowchart math from megcraig.org

So one place where this method has problems is if there are variables on both sides. But I want to use this more as an introduction in each section, not a method for solving each individual equation. However, we can use the fact that each column is equal to set up the rest of the problem and finish with quadratic formula.

Now I thought for sure this could not work with quadratics. OR COULD IT?

flowchart math from megcraig.orgOk, so the weird thing here is that (a) my new erasable markers don’t like it when you rewrite over something you just erased and (b) we have 2 places that x is involved, so 2 starting points. But then I don’t know how they are going to add to equal 6. But (spoiler alert!) we do know what has to happen if we’re going to multiply to equal zero…

flowchart math from megcraig.orgHere the two back arrows from zero come from the fact we had two x inputs. Pretty powerful, eh?  Let’s try it on some other tricky problems, like rational exponents:

flowchart math from megcraig.orgOk, guys, we’re going to jump into the deep end now….ABSOLUTE VALUE!

flowchart math from megcraig.org

Update: I was so excited about “un-absolute valuing” that I forgot to “un-multiply”. -6 should turn into 3, which would then turn into -3 and 3; and finally -6 and 0 as the answers. Which I probably would have noticed if I followed my own recommendation to circle back through.

Holy cow I’m in LOVE LOVE LOVE with having to “unabsolute value” as a step, because of course to “unabsolute value” you go back to positive or negative.

But wait, what about….

flowchart math from megcraig.orgOk, ok, a little tricky, but not undo-able.

Now I did have trouble with this problem:

flowchart math from megcraig.orgI wasn’t sure if my beginning value should be x or 5. When I tried it with 5, I thought of it as “If I’m at 125, what root would I need to get to 5?  Oh, the third  root. That means the original operation in the top line needs to be the inverse of the third root, which is cubing, which means x = 3.”

But if I keep my beginning value as x, then it leads into a nice intro/need for logs:

flowchart math from megcraig.orgAnd then I went crazy with the log problems!  (Although not pictured is two logs equal to each other, e.g. log (x + 7) = log (2x – 4). I’ll leave it as an exercise for the reader; it really is quite pretty.)

flowchart math from megcraig.orgflowchart math from megcraig.org flowchart math from megcraig.org flowchart math from megcraig.orgThe last one being another case of, “Uh-oh, need to rewrite this as something isn’t so ambiguous.” Another case of that:

flowchart math from megcraig.orgOk, ok, I don’t know why I didn’t have two starting x’s and then divide them, but isn’t it just beautiful how it works out this way?  So I went some more down that path:

flowchart math from megcraig.orgThen I thought of other problems that cause students anguish, and immediately thought of the difference between 2sin(x) and sin(2x):

flowchart math from megcraig.org flowchart math from megcraig.orgAfter this, my brain was pretty much done for the day.  Or at least, I thought it was. Then I had a shower thought (where all problems are solved): hey, wonder if I could tie it to graphing transformations?

flowchart math from megcraig.orgGAH!!!!!  So you go through all the steps, then find your parent function, in this case absolute value. You have to use inverses to get to x (minus three, or in this case three to the left) and OH I SHOULD HAVE PUT = Y AT THE VERY END BECAUSE THEN YOU TRAVEL “FORWARD” (stretch 2, down 4) FROM THE PARENT FUNCTION TO GET TO Y.

Another one?  ANOTHER ONE!

flowchart math from megcraig.orgI don’t know why you would want it, but if you did want all of these examples in one pdf, here you go. Now there are some drawbacks as I’ve mentioned: things need to be simplified first, somethings get a little wonky, how will this work for trickier equations; but I think Kayne sums it up pretty nicely:

Would love any thoughts/opinions/comments/suggestions/epiphanies!

Category: Alg II, Precal, Reflections | Tags: , , , , , , ,

Alg II Files: Polynomials

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(see more files and FAQs here) This is one of my favorite chapters in Algebra II because it’s the first time we discover that:

We Can Do Hard Things Poster(poster file here)

I used to start with this page:

Polynomial files from megcraig.org(file here)

But then my co-teacher made this page that I liked and used this year:

Polynomial files from megcraig.org(file here) and I just did the bottom part of the first page in class on the board. And during the same class period, we jump into this:

Polynomial files from megcraig.org(this is the second page of the previous file). Then it’s time for some graphing!

If you want to do some discovery, I’ve had success with this page:

Polynomial files from megcraig.org(file here) but for the sake of time, I’ve been doing the same thing with this desmos file. Then we put all of our conjectures together and practice:

Polynomial files from megcraig.org(file here) after teaching this about a bazillion times, I now really like how it goes.  The only thing I may change next year (and maybe more so in precal) is talking about how x^3 has three roots at 0, with (x-2)(x-3)(x+1) we just translate those three roots to 2, 3, and -1 just like we translated (x +2)^2.  Is this even a thing or am I just seeing transformations everywhere? Also, yes, we do call cubic functions and triple roots “John Travoltas” (I stole it from someone on the #MTBoS) because:

Then we spend a day practicing:

Polynomial files from megcraig.org(file here).  I usually go around and stamp each row when they have completed it successfully, and then can only turn it in once it has all four stamps.

At this point I throw in solving sum/difference of cubics and quartic trinomials:

Polynomial files from megcraig.org(file here)  S.O.A.P is a handy mnemonic that I learned from my coteacher. It tells you the signs of the sum/difference formula: Same, Opposite, Always Positive.  It becomes a bit of a chant: “Cube root; cube root, square, multiply, square; same sign, opposite sign, always positive.”

Some homework:

Polynomial files from megcraig.org(file here) Because of scheduling, it was a good time to throw in complex numbers for a day or two:

Polynomial files from megcraig.org(file here) Ugh, now there’s something that can be taken out of Algebra II if you ask me (but no one ever asks).

At this point I usually take a break and quiz:

Polynomial files from megcraig.org(file here).  Yup, there’s a review powerpoint as well:

Polynomial files from megcraig.org(file here).

Then it’s time to really get our hands dirty with some division:

Polynomial files from megcraig.orgBut I really want to try the box method next year as promoted by @TypeAMathland (especially since I can probably get a tutoring session since Anna is going to be my #TMC15 roomie!).  But with just a bit of modification I can still use the same homework:

Polynomial files from megcraig.org(File here)

So the Algebra II book that we use likes to spend a section on “I give you a factor, you find all the rest” but that seemed like a waste of a day, instead I go with “I give you a factor, find all the zeros” as a lead-in for when “I give you no factor”:

Polynomial files from megcraig.orgI learned a while back that it’s handy to have them figure out how many answers there should be and write out that many blanks. Otherwise many would forget that the original given factor also told you about a zero.

Here’s another day that I’m not a fan of:

Polynomial files from megcraig.org(file here).  I finally took a stand and stopped teaching the “what are the possible number of real/imaginary roots this could have?” because WHY?  I almost want to take a stand on “hey, I’m only going to give you 2/3 of the answers and one of them happens to be imaginary so do you think you could figure out the third?” because WHY? but I’m pretty sure that is specifically in our course of study. At least it’s a nice breather after all the heavy lifting we’ve been doing.

Then finally the moment we’ve all been waiting for!  Let’s solve some polynomials!

Polynomial files from megcraig.org

After doing a couple without the calculator, we start using the graphing calculator to find the first zero (or the first two if it’s a quartic).

Then let’s wrap it up:

Polynomial files from megcraig.org(file here).  And of course a review powerpoint:

Polynomial files from megcraig.org(file here).

Are polynomials one of your favorite things?  Do your kids know who John Travolta is or do you have to do the dance for them? Wait, am I the only one doing the dance?

Category: Alg II | Tags: , , ,

Precalculus Files

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Here are all the posts that contain NoteTakerMakers for Precalculus. Although many are labeled for PreAP/Honors, I usually used similar ones in regular.

Frequently Asked Questions:

How do you use these?
I usually have a NoteTakerMaker for each section I cover. They are a blend of definitions, charts, example problems, and problems I want them to try individually or in groups.

Why are some parts of the worksheet in a really big font that doesn’t even fit in the textbox?
I use Calibri 97% of the time, but I also like Chocolate Covered Raindrops and Running for a Cause. If something looks REALLY BIG or doesn’t all fit in a box, it’s probably because it’s in that font.  Either download and install the fonts (you usually have to restart Word for it to recognize that it’s there now), or change the font and font size to something you like.

Why can’t I type and change this equation?
I use the old-school Equation Editor 3.0.  If you need to edit an equation, double click it to open the equation editor.  Learn how to better use Equation Editor here.

Did you make a mistake on this?
Although I’ve tried to fix all known typos, I’m sure there are still some around.  Part of the fun of the homework is to try and find the mistakes in the answer key. (But seriously, let me know if you find one–I really do try to fix them!)

Update: See this post for a day-by-day calendar of PreAP Precal, with links to all resources!

Functions

Quadratics and Polynomials 

Logarithms 

Trig 

Polar Graphs & Complex Numbers 

Vectors 

Conics 

Limits 

Derivatives 

Category:
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