Fraction Remix...

I had about 20 minutes after a test today, and RIGHT before the test, a high school student in precalculus was confused about how to divide 2 fractions, and she couldn't remember the method: "do you multiply the top and bottom?" were words that came out of her mouth.

So. I thought I'd give the following short lesson on why the "flip and multiply" that they seem to spout works.

First I asked them the following questions:

Simplify:   (7/3) / (4/5)

Then after they did that and checked, I asked them to discuss WHY it worked, and "my teacher said so" or some such thing wasn't an option.

Then we walked through the following 4 scenarios:

 So to recap, it basically boils down to seeing how many groups of the denominator fit into 1 unit (that's where the "flip" comes from), and then how many 1's fit into the numerator, and then you combine the 2 numbers and MATH.
 

Boring Ceiling. Fixed That For You.

Something needed to change. I could no longer stand the travesty that was my ceiling. How could we spend day in and day out under such a boring mess? How was learning taking place? How did we all not just fall asleep under the endless white scandal that was looming over our heads?

After searching the internet for ideas and finding an art teacher that had her kids paint black and white designs ON the tiles and realizing I didn't want to go THAT far, I came up with:

 I am happy with the results. I still need more, as not all kids chose the opportunity. And as a side note ... for all the questions I get about bonus points, you'd think I'd be drowning under a pile of drawings, but, nope. Nada. I am not one for bonus points, and this was a rare instance, but, hah, go figure.

 

Intermediate Value Theorem

I have never liked my unit as a whole for the Intermediate Value Theorem in Calculus. My explanations, I like, my homework and the time the kids spend thinking about it and then quickly forgetting it, I don't like.

Attempt this year:

 I like this first page because the kids have to stop and process the whole IVTh and think about the parts and then analyze the parts and see what goes wrong. I heard some good discussion.

I like this second page because for 5 and 6 they then have to start the thinking process graphically on how to apply the IVTh. Then problems 7-9 are what I just used to do by themselves. Hopefully, this year after they've processed things, then 7-9 should go better. We will see.

In other news, apparently I make kids feel dumb - based on a conversation I had with a parent. Oy! They interpret how I talk to them as just expecting them to know things already. Of course it's an obvious logical conclusion on my part that "one kid" implies THE WHOLE CLASS. Go me! Because of this, I made the following handout, and we had a discussion, and I asked them to tape this in their notebooks (or staple it to their hearts) to remember:

Trig Identity Match Up Activity

I'm on my quest to try different things for the Trigonometric Identities unit. They explored last class, and today I wanted to get them started on simplifying various trig expressions using identities. Here's what I did. I made this set of 2 sheets for every student, put it on colored paper, and then made them cut them up (my first class) when they started.

They're mixed up on the sheets, and there are FOUR total problems that start with the "squiggly" boxes, but I didn't tell them that first. I said they should look for 2 cards that pair up in such a way that they are ONE step away from each other either by an algebra law or by an identity. I did one example with them and took questions. Then they were on their way.

Since they sit at tables, and I didn't want to do different colors of sheets of paper (though I guess I could have), I had them draw dots in different colored pens (one per set) at the bottom right corner of their cards to avoid mix up.

Once they got as many pairs as they could, and I mentioned that depending on a variety of things they may not get pairs. Then I said they could try to find a "triple" pair for their remaining lone cards. After some time for this, then I asked if they can "pair up" the pairs to get a set of four .... Ultimately, I then said, or they noticed the "squiggle" cards, and we got to the point that there are FOUR separate problems, that each start with a squiggle card, so see if you can set up 4 different strings of "one steppers".

 

During all this they had their "notes" on pink paper that they were going to eventually tape in their notebooks.

This all took about an hour or so to give them think time, and not everyone was successful with all of them, but we came together as a class and talked it out and got the 4 sets of problems, and celebrated any successes they had. Some problems had 7 steps, some 6, some 4. Then students taped the notes in their books, and we copied down 2 of the problems into their notes with "reasons". Hopefully, this will aid in their homework tonight where they have to do a string of one steps for each problem.

Trig Identities

We have started our unit on Trigonometric Identities in precalculus. Every year is going to be THE YEAR where it all comes together nicely, and the sun comes out and the birdies chirp and the kids embrace their Identities. Yes. It will happen. Chirp. Chirp.

I took some extra time at the start of the unit this year for them to explore instead of me walking them through things and them nodding their heads pretending to understand.

This took most of a block period (2 pages), and it was a nice refresher of a few things: 

* proof rules from geometry

* fraction rules for adding/dividing

* exponent rules

Hopefully, the extra time we spent will be worth it in the end as we skip and hop through solving/simplifying various equations/expressions involving trig.

Learning is Hard

Students are learning to solder in Digital Electronics. I let them tinker and figure a lot out for themselves. I do give them tips, and we had a safety discussion, and we saw a soldering video and we talked some more. But basically, once they have the kits, they have to jump right in to something they are not comfortable with. 

There are casualties. 

Hopefully, this is not a symbol on how they feel inside.
 

Basic Derivative Rules

Last year our school got "schooled" on Kagan strategies, and one I liked was "Quiz Quiz Trade". I used that today for about 10 minutes for my calculus students to help each other learn some basic derivative strategies for monomials.

They always struggle if there are radicals or x's in the denominator or fractional exponents. I have tried in the past to get them to rewrite the f(x) first as ax^b. But after they nod their heads "yea yea", they go back to messing up.  

Quiz Quiz Trade to the rescue. They get practice, and they have to explain, and they have another student helping them, and they want to understand. I think it helped this year, since I had less questions when they started their homework.

Here is my template for cards that looked like this:

Quiz Quiz Trade Rules:

You have to model this and keep an eye on them and move them along.

1. You raise your hand high when you need a new partner and you search for 

    another hand and you pair off.

2. You greet each other with eye contact (social skills)

3. One person holds their card, question side facing partner.

4. The quiz-ee tries to answer it.

5. If correct, you praise the quiz-ee (social skill).

6. If wrong, you give hints on how to get to answer .... until answer correct, 

    then praise.

7. The quiz-ee is now the quiz-or and does 3-6.

8. When both done, TRADE. and go to 1.

Back to School Night

It's that time again: Back to School Night. Our school is following the "no schedule" rule, so that after a brief overview in the "cafetorium", the parents can wander the halls and pop in on various teachers at their convenience as opposed to rushing from class to class and being talked to for about 5 minutes before they zip off to another class.

Because it's not always possible to talk with every parent for as long as they would like, I have made up a half-sheet hand out that sums up why the courses their kids take from me may be useful or interesting. I didn't list everything, obviously, but just enough for "dinner party" conversation.

I'll also have some projects laying around and our notebook of things learned so far (or maybe last year's notebooks).

Stolen Ideas....

I found a "study sheet" of one of my Calculus students practicing for their trig quizzes, and I totally stole her idea for today's trig lesson.

(excuse the sideways issue!) I feel it will be a great visual resource when they do these special radians in their heads.

Templates HERE.

Unit Circle and Radians

Breaking News: Kids struggle with fractions!
Alert: Kids can't describe how 3π/4 relates to π.
News at 11: Radians? What new torture is THIS?!

I tried something new today in precalculus to help cement the "special angles" around the unit circle. Based on the spontaneous comment of one of my students ("this REALLY helped."), I am hoping for greater and sooner fraction success this year.

First I made templates and printed them on various colors of papers (one color for each special "radian"):

Each student trimmed the circles and pasted the yellow "unit circle" in their notebooks along with a blue slot to store wedges:

 

We then had a discussion about π/6 and I had them place various radians in standard position and we discussed reference triangles and values around the sides:

I made sure to do some negative angles:

We finally discussed the π/6 angles and worked through a problem:

 Now I sit back and wait to see if THIS year they are fraction geniuses!

Function Notation Error

No matter how clearly I think I explain that f(x + Δx) ≠ f(x) + f(Δx), I still have the "believers".

Yesterday, I drew a graph to explain it to a student, and she had the big, "OH!" moment, so hopefully, THIS will work. Here's a short video of what I did:

Yearly Goal

Aaaaaannnd we're back into the swing of things at school: frenzied hamster on the wheel looking over her shoulder while madly scrambling faster, so that the big boulder does not plow into her because she did not complete a task.

Fun things for the 1 week of school so far:

* Love my students.  Because I haven't taught all the geometry kids now for a couple of years, I have tons of new student (to me) in precalculus.

* My advisory students are just funny and sweet.

* I've purged a bunch of things I've basically never used but felt I needed or "would use someday" from my classroom, including the teacher desk, and now it feels more roomy.

* I keep thinking of this quote, and it gets me through just about everything.

* I think my operating theme for THIS year is: take care of each other above all else.

It's more important to have kids feel heard and valued and loved than to "get through my to-do list". People before paper! Be present! Enjoy the little whipper-snappers.

School School School

Back to school on Monday for the start of PD week. Our fun admin has decided to "flip our lessons" for the week, so that we have more time to chat and plan together. That should be new and intriguing. 

And we have 2.5 new math teachers to show all the ropes to. Yay for new blood and new ideas. Boo for sleep deprivation. Yay for getting back to more people to talk to than just my mirror's image, so I can alleviate the loneliness of summer isolation. Boo for sleep deprivation. Yay for actually moving around to justify eating calories. Yay for goofy kids and spontaneous laughter.

This year (don't have the official word yet, though, you know, since it's more than a handful of days before school starts), I will be teaching Precalculus, AP Calculus AB, Digital Electronics, and Computer Science 1. A nice mix to keep me from getting bored. 

I'm working on doing some form of SBG in DE and CS1. I had grandiose plans of mapping it all out during the summer. Yea, about that ..... apparently, it will be business as usual, and I'll think about to and from work and in between and map it out as the year goes on. I don't know why I'm so continually deluded into thinking that I'm a different person than I really am.

Other than that, hopefully it's another epic year in the making. Good luck and good math to everyone else who is starting or has already started school. Go Us!
 

In Class 1st Day Activity

Okay, after 17 years of teaching, maybe I WON'T leave everything until the last minute, because I think I've just finished all my 1st day things. Hah! Who am I fooling. This is a fluke. Soon it will be back to business as usual. This year is different, though, because I think I'm going to fly out for the weekend before school starts to see my turning-80-years-old dad, so I won't have those 2 days to get ready.

I did just remake my 1st day in-class activity for precalculus. It really has nothing to do with precal, but that's the class I'll use it in. I like my calculus activity from last year and will do it again, and for digital electronics, I'm going to have them do a DE thing, and for CS, they will be coding independently from day 1.

I was inspired by this activity from KFouss. Woot! Internet Searches! 

Here is my version:

These will be done by the kids in class while I'm snapping pictures and calling roll and passing out a seating chart and passing out more information.
 

Who Vacuumed Up All the Summer?

Sheesh! I just realized that I only have 2 unstructured-ish weeks left of summer. Must. Soak. Up. Fun.

I did enjoy doing a crafty thing by knitting and felting a "bowl" for a friend (just look away from the glaring error that SOMEONE was too "zen" to fix):

And I have revised my first-day homework assignment:

And I'm trying to sit less by sprinkling exercise in throughout the day via a checklist:

 Other than that ....

Studying Tips for Students

I recently attended a CS workshop for teachers, and if you live in TX, I highly recommend it for next summer. We learned a variety of things, and though some topics were not immediately adaptable to taking back to the classroom, they still were fascinating things you could mention to your students or adapt in some way to your population. Example, did you know that your brain doesn't have to process how to walk (for the most part), the mechanics of the motion are stored in your spinal cord.

One of the talks was on effective teaching, and the presenter mentioned VARK, which I had never heard of. This stands for visual/aural/read-write/kinesthetic, and, of course, refers to learning styles. What I found useful in the talk was actual suggestions on how students could study for and take exams more effectively than they may be doing currently. The hand out was not in a form I would hand out, so I made my own version on half pages for the students to tape in their composition notebooks on the first day of class for future reference. I culled from the VARK website and others that offered tangible ideas.

Online they have a questionnaire for the kids to take to determine which type they are (yay, first night's homework!). I like that you can choose more than one answer for each question and that you could be a "multi-modal" learner. There are different versions of the questionnaire for adults and such (do a search on VARK).

I want the students to tape in ALL four styles into their notebooks because they may find helpful ideas from a mix of strategies.

Geometry Idea

Next year will be the 2nd year in a row I haven't taught geometry, but I had an inkling of an unformed idea for a possible exploratory activity. Whoa! Doesn't that sound like you just want to jump on it and use it the FIRST day of school (far far away from now).

I was in an Airport Shuttle and staring mindlessly out the window and saw a building wall made up of tessellated hexagons. Then I remembered how learned about angle sums of polygons in class by separating various polygons into triangles from one vertex and noticing a pattern [(#vertices - 2)*180]. 

Let's use the pentagon for an example of my future I-can't-use-it idea. Kids most likely don't know that there are 3*180 or 540 total degrees in all the interior angles of a pentagon. I was thinking of (for the pentagon group) having the kids use 10 pieces of spaghetti taped on 5 different scraps of paper to form 5 different angles. Then they would measure the angles and get a total sum of the 5 angles. Then they would try to manipulate the 5 taped down "2 sides and a vertex" scraps of paper to try to form a pentagon. 

I'm thinking there would be 4 or so people in a group, and they would each have a different sum they would most likely come up with. Most likely a pentagon won't be able to be formed unless they happen upon a 540 degree sum.

Then maybe they could try again. Then maybe they could see what works and get that sum (having a human error discussion at some point for measuring mistakes). I'm also thinking there would be different groups with 4 sides and 6 sides (3, too??).  In this way, they can see the tactical things that go wrong with angles that don't "fit". Then there would be a follow up summary and pattern.

Next post: things I could teach in 6th grade math ..... KIDDING

Playing with Math

I just finished a nonfictional Romance Book called "Playing with Math", edited by Sue VanHattum. It's filled with passion (for math), joy (of games and puzzles), good-looking heroes and heroines (math makes you sexy), conflict and tension (hello math haters and standardized tests), and a warm and fuzzy resolution (math conquers all). Okay, it's not really a romance novel, but the passion about math is threaded throughout the whole book. There's basically something for everyone, whether you are a math teacher of any sort, a parent, or a math enthusiast.

I teach in a public school to goofy high school students, so I can only give my perspective as a non-mother, non-math-circle-participant, non-unschooler, non-homeschooler. But just like in math, where you see patterns in one problem and can then extrapolate to solve OTHER problems, in this book, I could read about the variety of settings, and always find something that I can either take back to my classroom or take back to my mind to work on. 

Every other chapter is a math problem of some sort. These range in levels of difficulty and variety of math concepts. Many times I found myself putting the book down and pondering the cool problems and figuring them out. There are also a ton of resources listed, from books, to online options, to people you could e-mail and start a discussion with. This wealth of goodies in itself is a reason to read "Playing with Math".

Before I read this book, I only had my biased, public-school-teacher opinion of homeschooling and had never heard of unschooling. Through the various sections and vignettes shown, my opinions were swayed with an understanding of how such situations could be in the best interest of different students, and how kids could thrive. And even if I won't rush out and un-enroll my non-existent offspring and start unschooling their imaginary selves, I did find myself taking notes on each chapter to remember great quotes and philosophies and teaching techniques that I'd like to remember and adapt to my classroom.

I also enjoyed to peek into things I didn't know existed. There is a section on taking a math circle to a prison, and the nonthreatening and successful way the teachers engaged with the men. There is a section about the nonlinearity of learning math, written in a nonlinear and creative fashion, and it's filled with words I want to mull over and process. And these are just a few examples.

Whether you want to have a new source of good math puzzles, or find out how to start a math circle, or re-engage your love of math, or find more things/thoughts/techniques to bring back to your class this coming fall, or simply to stay awhile in the company of others that love math as much as you do, I think you can't go wrong with this book.

Binary Addition

I would like to move towards SBG for various basic topics in my Digital Electronics class next year. To that end, I will spend this summer creating and/or finding online practice tools for my students to hone their skills. 

One of the topics they learn is Binary Addition. After searching for "just the thing" online and coming up short, I created a Scratch program for them:
 

I think I'll leave it as such, but if you have any suggestions for improvement that can easily be done by a summer teacher, thanks for sharing.

Also, I had originally created this program with Java with the intention of making an applet to put on our school website. This turned out to be a pain full of "security issues" ... "bad!" .... "don't even think you can open or share this applet". Don't know if I'm doing something wrong, or if that's the way of applets these days, and I have to pay money or some such to be able to have that message not come up. Does anyone know?

Viva La "Inflated Sense of Self"

And by the way, Viva La End of the School Year.

This week our school got together as a team and discussed children and things that may benefit them next year and what resources they may need that we should provide. Threaded through the conversation a handful of times was that so-and-so had an "inflated sense of self" with the meaning that the kid may be in for some wake up call in terms of college applications and acceptances. I nodded my head in agreement because I knew the kid(s), but then later I started to wonder if we have it all wrong.

I think back to my childhood, and alternately absent and alcoholic and human and goofy as my parents were, they ALWAYS made us feel like we could do anything and why shouldn't we and of course it's possible. Maybe we were living in a (champagne) bubble and other adults were going around tsking and shaking their heads and wondering when our big fall would come. I know my 6th grade teacher despised me and thought I scraped the bottom of the barrel, and I know a few HS teachers and college professors that would scrunch their faces if they knew what I accomplished, but being "bubble girl" (and I now realize "inflated sense of self girl") I still plowed on. 

Maybe that's one of the important things we can provide for the kids we teach, "of course you can do it" ... "why not you" ... "don't be silly, it's totally within your reach". This, obviously, has to be sprinkled with lessons on hard work and persistence and grit and the power of failing and getting back up and such. But how sad the alternative would be, a kid looking at you and your shaking head and attitude of "bless your heart" and taking that to heart and giving up on whatever before they even put out their whole effort into the endeavor.

There are students I have who I don't know if they'll "do it", but who am I to decide. And mostly, I hope what comes out of my mouth and heart is, "DO IT" "you can do it" "obviously it's within your reach". I just do this without thinking all year long, but I'm reminded of the power of this when I get a smattering of end-of-the-year cards that state, "thank you for believing in my and not giving up on me". I am also reminded of this when a student, just by happenstance, gets put into a position to shine, and they soar way beyond what we've seen of them before. These may not necessarily be the kids that think they are all that, but whatever "that" is, hopefully I don't, as a teacher, contribute to snuffing it out before it has a chance.