Monthly Archives: July 2015

Managing Calibration Quizzes

One of my goals for this year is to use the idea from Make It Stick of giving many short quizzes (I don’t think I can handle daily, but maybe 2-3 times per week), but I don’t want it to turn into a paper/recording nightmare. So here’s my idea:

1) Make them easy to grade–just for wrong/right answers–and have students grade them.

2) Call them “calibrations” because we want students to use them to calibrate their knowledge–are they headed in the right direction, or do they need to refocus on a certain topic?

3) I want them to be able to drop their lowest scores, but I also want to put grades in for them every three weeks or so. So this is what I’m thinking:

calibration record(file here) I’m thinking making each calibration worth 4 points. 4 quick probs at 1 point each or 2 probs at 2 points each, depending on the topic. Students would grade their own and record their grade. Every three weeks, take the top 6 grades and record them and the sum. I collect them, record them, and return them (or go around the room and record while they are working on something else). Then after the next round, they take the top six of any unused grades and use those as their score.  Repeat again. At the final tally, they can also use any unused score to replace a previous lower one.  Also NO MAKE UPS. Because I HATE MAKE UPS WITH THE FORCE OF A THOUSAND SUNS.

Sure, they could probably figure out a way to cheat, but really, “they’re just cheating themselves.” Maybe make them do all of the quizzes on the same sheet of paper and have them turn it in with their calibration?

Hmmm, I’d also like to know how they did on these, though. Ooh, maybe I can bring in Plickers and have them hold up an “answer” corresponding to their points they made on the calibration?

4) I listened to the mathedout podcast featuring Jo Boaler and one thing that stood out was that you don’t have to give feedback to every student every time. So I’m thinking at least once a week one of the quizzes will be one longer problem. I collect 1 paper from a group member and write feedback. I return it to the group member and the rest of the group uses that feedback to discuss how they did on it. Or should I make that a group quiz if I’m going to do that?

So much to ponder!  Please leave a comment or tweet out any suggestions or pitfalls you can see in my plan.

Category: Uncategorized

Epiphany Part II: The Return of the Tranformation

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….

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.

#1TMCThing + I Can’t Count

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I decided that my #1TMCThing is going to be collaborating with Sheri Walker to make and USE some awesome Desmos stuff in Precal (and also Algebra II). But I have trouble counting, so it turns out I have more than one #1TMCThing. But hopefully some of them won’t be too difficult to implement!

  • Write daily learning targets that also include the Standard Math Practices (see Chris Shore’s examples on his 180 blog, starting at about day 160). Our school wants us to start using targets anyway, so this would be a good way for me to get excited about writing them. I’m meeting with our math coach Monday to help me get started. If you develop any (or want to discuss), we’re using #SMPTargets to connect.
  • Grow dendrites in class like Chris’s My Favorite.  Need to find brain poster and make dendrite stickers.
  • High-five at the door a la Glenn. I’m not that great at connecting with students early on in the school year, so I’m hoping this will jump start that. I really am a nice person, kids!
  • Use the hand signals from Chris Harris’s number talks (at the bottom of this PDF). And enforce not calling out. Although I am totally bad at that myself. I need to apologize to everyone in the blogging initiative seminar because I totally just blurted out stuff while others had a hand raised. Sorry, guys.
  • So simple…use Matt Vaudrey’s “End your discussions in 5…4…3…2…1” instead of “Ok, time’s up, guys!”
  • Using Brian’s quadrant plan for mixing up groups. I got lazier as the year went on and sometimes the kids got stuck in the same groups for a few weeks. Hopefully this would be easy to implement and keep up with-I could make a new quadrant plan every 9 nine weeks, and they could rotate through.
  • From Make It Stick #eduread, “Calibration Quizzes” 2x a week and spiral homework.
  • Finish posting all of my files, and continue to post new ones through the year! Thanks to all of you that let me know that you’ve been using some of my stuff or that my stuff helped make your stuff better.
  • Be proactive in welcoming newbies to MTBoS. Tweet out blog posts that I like so people know that someone is reading.  Keep an eye out for #nahf so I can send good vibes when needed.
  • Stay positive! Let others (students, teachers, and admin) see the nice and kind side of me that y’all saw at TMC.

I fear I may have too much on my plate, but I can’t decide what not to do. At least I’m super excited to go back to school!

Category: Reflections, TMC | Tags: ,

Important Things You Need To Know

[Updated to add #SMPTargets and t-shirt link]
[Updated to correct TMC16 date]
[Updated to include latest #eduread information]

There’s been a lot of talk about keeping the TMC spirit through the year, and to that effect there were a lot of links and hashtags thrown around. Here are some of them:

Hashtags to keep an eye out for:

#1TMCThing Choose one thing from TMC to focus on. Find other #1TMCThings that excite you. Check back in with those on October 26. This is not an exclusive thing…even if you weren’t able to be at TMC, you can still choose one thing you want to work on through the year. Some people are choosing big conceptual items, some are choosing smaller goals. Do what’s right for you. Mine is working with Sheri Walker to create and use some Desmos awesomeness.

#TMChange A lot of TMCers are starting at a new school next year. New things are hard. Hard things are less hard when you have supportive friends. Thus, #TMChange is for those that are changing and for those who have recently changed and want to give support to others. @sophgermain is also starting a slack (like a private twitter) to give y’all a safe space to talk. DM her your email if you want to join.

#nahf  A conversation on twitter last night led to a lot of openness about how there are so many people struggling behind the scenes. Mattie came up with “need a high five” (#nahf) as a “bat signal” to send out. Sometimes you just need some extra love with no explanation as to why you need it. (Even if you think it’s something about which #nobodycares)

#eduread ok, ok, so I’m doing a little non-TMC promotion. If you’d like to keep having wonderful, thoughtful, applicable discussions about math, please join us for #eduread. We are going to start reading What’s Math Got To Do With It and we should be starting next week. will be starting the first chapter discussion next Thursday, August 6th at 8E/7C. (We just finished Make It Stick which I would highly recommend, even if you just have time to read the last chapter!)

#SMPTargets If you attended Chris Shore’s presentation about using the Standard Math Practices daily, you will remember that he wrote learning targets that included the SMP focus for the day (for example, “I will persevere while solving right triangle trig problems.”) I am making writing those my 2nd #1TMCpage and Chris came up with this hashtag for others that would like to collaborate and/or support each other.

Important links!

TMC15 Wiki Check for presentation materials and info here!

bit.ly/MTBoSsearch the MTBoS search engine. Tweet @Jstevens009 if you’re blog isn’t listed UNLESS your blog is already listed on the TMC15 list. If so, it will be added in the near future.  If your blog is not on the TMC15 list, then go ahead and tweet him to add.

bit.ly/MTBoSbank A searchable database of activities sorted by grade level and topic. Share your activity by submitting it at bit.ly/MTBoSactivity

bit.ly/desmosbank A searchable database of Desmos activities.

bit.ly/TMC15archive If you wrote or will write about your TMC15 experience, add it to the archive (note: the actual archive will be up this Fridayish)

Blogger Initiation BrainstormingIf you want to help build the #MTBoS, add your thoughts to the document. A mentor idea was discussed, as well as helping people restart their blogs. The actual initiative is planned for late October/early November to feed off of presentations at NCTM Regionals.

Bonus t-shirts: At approximately 6:45, a few of us mentioned how we wanted “Find what you love. Do more of that.” on a t-shirt. At approximately 6:50, Mark (@hfxmark) had created it on spreadshirt. Here is the link!

I’m sure I missed some other important links, so please let me know and I will add them!

Oh, also, one other thing that may be important to know: TMC16 will be at Augsberg University in Minneapolis, MN, July 16-19 2016. See you there!

Category: TMC | Tags: , , ,

Make It Stick: Even Stickier with #eduread

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For the past few weeks, I’ve been joining the lovely gals of #eduread to discuss Make It Stick by Brown, Roediger, and McDaniel.

As for the book, the first 100 pages were really good, then it dragged quite a bit for the next 100. There were quite a few stories that were supposed to help elaborate their points, but really just seemed tangential. However, stick with it (ha!), because things get really good in the last 50 pages which has tips for students, teachers, and trainers as well as stories/examples that really let you see the tips in action.  I wish I could get a copy of this chapter into the hands of every student!

Here are some of the big ideas and how I want to implement them this year:

Calibrating-(we decided we liked this term better than “a-whole-lotta-quizzing”) Implement a lot of small, low-stakes quizzes so students can “calibrate” their learning-where they are and what they need to work on. Rereading does not actually help master material, but quizzing and flashcards (with correction) does. Although I don’t think I can handle daily quizzes, I’d like to try for maybe 3 times a week.

Spaced and Interleaved Practice- Although it doesn’t feel like it, trying to remember something a few hours/days/weeks strengthens your learning. Also, massed practice of one topic can lead to “illusion of mastery.” I’m guessing I’m not the first teacher who has had this issue–each day the kids are doing great on the topic, then you get to study guide day and all heck breaks loose because the kids are not used to the problems being all together. I’m going to try to do lagging homework (the homework for the night has topics from a couple of days previous) that also has a lot of spiral review in it.

Generation and Reflection- Generation is trying to find a solution before being shown and I’m assuming you know what reflection is. 🙂  I’d like to use these at the beginning/end of units: start each unit with some sample problems from the chapter and some leading questions: “what do you think the main idea of this unit is?” “what might we need to work these problems?” “how do you think these are different/same from previous chapters?” Then at the end of the unit (or maybe midway and again at the end), have them reflect on their original ideas and what they now know about the unit.

Mnemonics I also liked that they are a fan of mnemonics as a way to organize your learning, not in place of learning. There is one topic that I teach that some teachers use a mnemonic for that I wasn’t really fond of. Now I see that we can discuss the how and why of each part, then have the students use the mnemonic to remember all the parts–in essence, a to-do list of tasks, not the tasks themselves.

The really big takeaway from the book is that you must embrace the fact that “learning is deeper and more durable when it is effortful.” Many of these practices will seem hard to students (and me) and they will feel like they are not gaining anything, but the authors are pretty persuasive (with research to back it up, not just anecdotal accounts) that these setbacks are a sign of effort, not failure, and will make learning more meaningful and long-lasting.

Special thanks to @numerzgal, @algebrasfriend, @pamjwilson, @lmhenry, @fourkatie, @rachelrosales, @mary_dooms, and especially @druinok for such lovely conversation, debate, and motivation. What are we #edureading next?!?!?

 

Geometry Files: More Triangles? Are you serious?

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Yes, I’m totally serious.  (see more files and FAQs here)

But we already did triangles, you say. Sure, we looked at congruent triangles, but did we look at stuff that can happen inside triangles?!?!  There’s a lot of good discovery activities for these rules, also here are some rudimentary geogebra files I made:

triangle files from megcraig.org triangle files from megcraig.org triangle files from megcraig.org(files: medians, perpendicular bisectors, angle bisectors) then we can solidify everything:

Triangle Files from megcraig.orgTriangle Files from megcraig.org(file here) Oh, plus there’s this weird thing about medians:

Triangle Files from megcraig.orgI make mine draw out and label the median for each problem; the visual seemed to help a lot of them see the relationship between what was given and what we need.

Homework:

Triangle Files from megcraig.org(file here)

Then let’s put all this together!

Triangle Files from megcraig.org(file here) “Boy, Meg, sure wish there was a powerpoint to go along with this!”  Why, it’s your lucky day!

Triangle Files from megcraig.org(file here)

Then another powerpoint for exterior/interior angle and angle/side inequalities:

Triangle Files from megcraig.org(file here)

Then I guess I did more stuff with possible sides of a triangle?  Then some practice:Triangle Files from megcraig.org(file here) wiiiiiiiith powerpoint!

Triangle Files from megcraig.org(file here)

And more inequalities!!

Triangle Files from megcraig.org(file here)

ARE WE DONE WITH THESE TRIANGLES YET?!?!?!?!

Yes!  Now just a review!

Triangle Files from megcraig.org(file here) and powerpoint:

Triangle Files from megcraig.org(file here)

Finally!  There can’t be anything else we can do with triangles, right?!?!  (Shh…we won’t talk about trig coming up, ok?)

Category: Geometry | Tags: ,

2 Quick Ways to Help Kids Ask Questions

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If you haven’t been following #eduread on Make It Stick (by Brown, Roediger, & McDaniel) this summer, you are missing out on some good, thought-provoking conversation. (The next chat is Tuesday at 8 eastern/7 central on Chapter 6).  Last night we talked about avoiding “illusions of knowledge,” i.e:

This led into how do you get kids to get help/ask questions? I was somewhat successful with two methods this year and thought I’d share:

1) Question Pop Quiz

On the start of a practice or study guide day (or maybe just after you’ve learned something really meaty), alert the students that they will be having a open-note pop quiz. (This part is optional, but sometimes isn’t it fun to mess with their minds a bit?)  Hand out 1/4 sheets of blank paper (I always have a ton of one-sided scrap paper).   Then tell them for the next __ minutes (usually 2, but not more than 5), they need to write every question they may have about the section/unit/chapter. It can be general (how do you know when to do…) or specific (I need help on #4 from 5.3).  (Do I use too many parentheses?)  (Never!)  (Like I tell my kids, parentheses are protection, and they don’t work if you don’t use them.)

When time is up, I set the timer for 5-10 minutes to see if they can get the answers from their group members. If so, they can scratch them out, or if they have more, add them. Then I collect the “quizzes” and answer any unanswered questions as a class.

2) Personal Mr/Ms _______ Time

There is certainly something true about the magic of sitting down at eye-level with students. So during some study guide days, I bring my chair over to a group, set the timer for 5 minutes, and they get to ask any questions they have. If they don’t have any, I quickly scan their work to see if perhaps they do have questions but don’t know they do, but if not, then I let them “bank” their time for me to come back later, but I try to spend a whole 5 minutes with them at some point during the period. I tote around a mini-whiteboard in case I want to write something down that the whole group asks about, or I can help individuals one-on-one. If it’s a question like “AH WE HAVE NO IDEA WHAT WE’RE DOING” I may do a vocal poll of the rest of the groups and if a majority are in the same boat, go over it as a class so I’m not repeating myself 7 times (plus it doesn’t count against their time).  Pam even suggested getting a mechanic’s roller chair (aka “creeper chair”)…yes, they even have them with cupholders!

This was a good way for me to be more equitable about my time in class, but I still struggle with that when it’s a short group work time, or a day of whiteboard practice. I always end up inadvertently skipping a group, spending all the time with just a few vocal students, or hearing “I had my hand up before her!”. Anyone have suggestions on that situation? Maybe I can get some sort of ticket system like the deli has:

Category: Uncategorized | 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!

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!