Tag Archives: transformations

Alg II Files: Let’s All Translate Some Graphs!

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[More files and FAQs on my Algebra II files page!] One big change I made in Algebra II was making an entire graphing chapter. Usually, we would learn a function, solve it, then graph it, repeat. Now, using Jonathan’s model, they all got mushed together in one unit, which actually really helped them with (a) things that are similar with all the graphs (shifting, stretching) and (b) things that are different. It also solved the issue that I had before where if we were in the quadratics chapter, they would just write y = (x – 2) + 6 for the equation, leaving off the most important part! Now they realized why that was so important! I was reeeeeealy pleased with how well the students did on this unit. I needed to keep spiraling back to these through the rest of the year, though, because when you graph all at once it’s a long time before you graph again!

When we last left our intrepid reporter, she had just finished translations of linear functions, so now we’re ready to jump into absolute value. The first part that they did mostly on their own:

Unit 4 1

Second part where we made sure everyone was on the same page:Unit 4 2

(file and a practice WS file) I REALLY liked those questions on #1 that I stole from some worksheet; you’ll be seeing them for the rest of the chapter! Like on the quadratic NoteTakerMaker!

(file and homework) Again, we were just focusing on graphing by translating in this chapter. Let’s try translating some square root functions!

Unit 4 3

(file) And then it’s time for some John Travolta!

(file) This was the last function we were going to study, so we spent a day doing a Desmos “Match My Function” Activity Builder:

Unit 4 4

You can find it here. I think this was the first activity builder I made all by myself, so it’s not very elegant (it was before hidden folders so I had to monitor students not scrolling down to the answer!). I also had some students say it was too easy to just use sliders until it matched, so next time I would definitely add some Desmos-style questions like “Here is Addison’s (wrong) equation and graph. What would you tell her to correct so it matches?” and “How would you explain to your friend how to move a function left or right?”

Then a group speed dating day:

Unit 4 5

(file and yes the graph answers are included!) Some sort of dry-erase graph is a must for this activity so partners can see work! If you don’t have individual graphing whiteboards, take Tina’s (@TPalmer207) suggestion of buying a pack of job ticket holders and printing off graphs to put inside.

Then it was study guide day:Unit 4 6

(file and video key part 1 and part 2) As I said at the beginning, for the most part the score were GREAT on this test! Was it because we ended up going pretty slow through this unit? Or because they had graphed most of these before in Algebra I? Or because all the graphs were together? I don’t know the reason, but I will definitely put this portion of restructuring Algebra II into the “win” column!

Desmos Activity Builder Success

This summer I was so excited when Activity Builder was revealed at TMC. About the second week of school, I made a really quick one that involved the students graphing their names with lines. I reserved the mobile laptop cart that our school has just for Math and Science to use, wheeled it into my room, and was ready for the excitement to begin.

Oh, wait, except my classroom is in some sort of wifi blackhole, so it was taking students 10-15 minutes to be able to log in to the computer and open the activity, and even then only about 4 kids got to that step. It was the Friday before Labor Day; I didn’t have any backup plans, so I just threw in the towel and they had a chill day. I, on the other hand was not chill.  I was Over Technology. I was at Unstackable-Cups-Otter level of Not Chill.


It took me two months to venture into trying it again. This time I reserved the computer lab, where the computers are wired to the Internet. It went much better. Here are the two activity builders that I made:

Algebra II:

Writing Equations of Transformed Parent Functions

We are graphing 5 different functions (lines, absolute value, quadratics, cubics, and square roots) and the students were having a lot of trouble writing equations from given graphs. Enter in Match My Functions Activity Builder!

Match My Functions Activity

Issues: Students still do not know how to find the slope of line. This makes me sad. So they just tried decimals and then were upset because it seemed “like random decimals for the slope.” At least they were trying something, I guess? Also a couple students said it didn’t help them because they could just guess until they got it right. But all in all I would probably do it again; they actually did very well on their test writing the equations and recognizing the functions.  Also, a student who is in two math classes bragged about how fun/cool it was to his other math teacher. 🙂 Plus when it came to study guide day, at least a few students broke out the app on their phone to check their equations!

PreAP Precal

Introduction to Polar Graphs

I had actually done this activity last year as just a regular Desmos file paired with a worksheet that asked them questions about the graphs. It didn’t take too long to create an activity builder based off of it. We had graphed polar points and graphed a couple circles by plugging in values of theta, but they had not seen any other type of polar graph. I set them loose (in pairs) to work on Polar Graph Exploration.

polar exploration

I had them play around with sliders for each graph; then submit “I notice…” and “I wonder…” On the more complex ones, we then looked at each part individually (like the slide above that just focused on the n value). Trying to learn from the feedback from Algebra II, I made a few “quick check” text questions where they couldn’t just guess (although some just went back a screen and played around with the sliders until it matched)

polar exploration 2

The fact that, with no direct instruction, most of them realized there had to be a 3, 6, and sin involved in the graph above is pretty amazing. I really loved the notice/wonder part as well (I’ve actually never used it before in class #MTBoSDirtySecret).  I always had an issue with the petals on a sine rose alternating between positive and negative y-axis. One student noticed that if it is positive, more petals will be above the x-axis. So clever!

I did get a few complaints on the feedback, “why didn’t you teach us?” but I’ve come to expect that.  The other complaint was not being able to know if their work was correct on the quick check. Maybe I should add a screen after that with the graph that says, “check your work” or “how close were you?” (but then I also don’t want to discourage them if they were wrong–I only wanted to see how much they had learned from the activity).  The best part was the last screen where I told them to have fun!  They made so many cool graphs, and then I was able to show them off the next day in class.

Now I know Desmos gets a lot of love around here, but let’s not forget about Geogebra, which is powerful and useful in its own right. For example, this beautiful, awe-inspiring, oh-so-that’s-why file that compares the two versions of a trig equation (made by Mark Fowler):

geogebra activity

(Which I found the morning after we had done the Desmos activity, and now I’m debating if I want the Ss to play with this instead of the Desmos one! Or if Desmos could make this into a split screen activity with sliders that controlled both graphs?!?!) I used it as we talked about each graph, and was able to reference their responses from the day before, “Some of you were wondering what causes the inner loop…”

All in all, a fun week using technology in class. 🙂

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.


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.

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.

Sunday Summary: I Love Transformations

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.