Call me slow. Go ahead, you wouldn't be the first.

Someone out there was calling for the worst topics we have to introduce. I would put rationalizing radicals right up there.  Then I had an epiphany. (A little late to the game in the scope of teaching, not proud, but happy that it came nonetheless)

Who cares about rationalizing radicals (and in particular, square roots)? I know from teaching both Precalculus and Geometry, why I want my students to be efficient at it, and I always struggled with; why not just use decimals?

Deep in a quandary about why my students were struggling with the Pythagorean Theorem and in particular, why is was hard for them to wrap their heads around why they needed to take the square root to find the length of the missing side, I realized I wanted my students to use simple radical form. I spent the first few days allowing the students to leave the radical unsimplified. Then I upped the anti and asked that they put all radicals in simplified radical form. (I tried starting with warm-ups of simplifying radicals, and found they were getting too hung up there when also trying to figure out the PT--yeah, some of the "getters" shake their heads and look with great sympathy (actually) at the kids who aren't there yet).

It is difficult to see the value of sqrt 192 When you rewrite it as 8 sqrt 3, and we know sqrt 3 is between 1 and 2, it gets easier to see its value when write an inequality and then build the value.

It is a start for them to place the value, and recall that the hypotenuse is the longest side.

Now we are preparing in Geometry for Special Right Triangles and Right Triangle Trig and it is time for the students to either learn or recall how to simplify radicals in the denominator. Again, why? Why can't the sqrt 2 stay in the freaking denominator.

Easy comparisons, that is why! What is
the value of

Is it more than one? Less than one?   When you start here
And move to here                    
The students give you a little, "oh." The strugglers get a number sense boost and the "Can we please just move ons" get to appreciate a little deeper why we torture need to rationalize a denominator.

7th Block...yeah you read that correctly...block...put 28 mostly 9th grade Algebra 1 students in a basement room after three 95 minute block classes, lunch, and no air conditioning and you get...my favorite class of the day!

We started with the premise that it sort of sucked to be us...and we rock! The basement is empty, everyone is gone. We watch the feet of athletes and the "lucky" students departing campus before we even start our warm- up ( our windows are below ground).  But guess what? We get to make noise! No one yells at us or sends us notes to be quiet! We hoop and holler about...MATH! We shout out answers! We eat lollipops and take teeny tiny breaks.

Today I was moving the students along this line of thinking:

So we did three old and one with a new twist together. They were getting it.

I stopped them and said, there is a story about why we study this kind of equation, and not mentioning it to you would be really mean on my part and would lead all of you into thinking Algebra was just smushing a bunch of numbers around (I said an exercise in futility, but I think I lost them).

So I told them the good ol' yarn about how it takes me 5 hours to paint the livingroom by myself because I am easily distracted and how it only takes my husband 3 hours because he is concrete sequential...Then we saw the POWER in doing a job together! The students were actually impressed...which led to a discussion about the alternative economy and what they could do to earn money (I said offer your neighbors to vacuum for $5 per room, or $10 to load the dishwasher and wipe down the sink after dinner). I also told them about the water treatment plant and how some liquid is pumped in and others are pumped out at different rates. (hence the possible minuses in the equations)

They WANTED to know how long it took Dom's uncle to replace the clutch on his '74 BMW by himself when it took Dom 8 hours by himself and took them 3.5 hours together. One of my squirrelly 9th graders (who will be repeating Algebra 1 next year) and who is beginning to think like a Sophomore said, "hey, I get this, this is kinda fun." 

May not be a 3 Act yet...but it was still a powerful afternoon.

My Student, A, is a nice kid. This doesn't say all there is to say about A, but it says a lot:
He mostly pays attention...he is a bit of a clueless wonder in Algebra, although his arithmetic skills are super strong. He is affable and wants to do better. He does an entire page of work...and it is all incorrect. My main theory about A. is that he comes from a family where he hasn't been shown how to study, that he is just under-exposed to "how to do school."

When Fawn Nguyen says Do These Two Things, you better listen, and here's why:

I was telling my students a story (1. Do talk to your students about yourself)
about how I spent my Saturday morning having breakfast with a bunch of fellow nerds at the Sonoma County Math Council's Breakfast. It was a rare lecture, usually we get to the business of getting dirty in some math activity. John Martin is an amazing lecturer, funny, light, thoughtful, inspiring...he must be from Midwest originally, he is just so nice. His talk was on the Golden Ratio, and he was giving a shout out to Fibonacci along the way. In addition to explaining a rather elaborate way to convert miles to kilometers using Fibonacci Numbers that only a mathematician can appreciate, he went on to show us that the 100th Fibonacci Number is in the quintillions! This is just astonishing to me! So I shared my astonishment with my students.

Alex A, thought it was astonshing too. Before you look at the picture below and really absorb the profroundity of what is happening here, re-examine the picture above:

Powerful image. We see so clearly what is right and now I have to ask myself what is wrong with A's picture and what can I do about it?

(PS, the next class is right before lunch and the kids are STARVING) A girl in front busts out a package of seaweed (we are from Sebastopol after all) and you know what...the ratio of length to width in those sheets is 1.55, intersting, eh?

Thanks John for the inspiration...and a little pat on my back for spending my glorious Saturday morning...in mathland.

Oh and anyone have a better title for this piece?

I am noticing all the very cool, thoughtful questions being thrown down on our blogs, so I thought I would add in and see what the next steps would be:


Okay, not sexy, but a little stretch for the test. Objective: Apply the Pythagorean Theorem.

Suggestions welcome!

Ha, ha, ha, ha, ha...

Well here is my idea for cramming in a topic we won't really get to cover until AFTER STAR testing...I am sooooooooooo ready for the Common Core Assessments, however they look. Real math, real thinking...

Evaluate:

for f(-2) and f(0) and f(4)

Evaluate:  

for f(-2) and f(0) and f(4)


What do you notice?


Now Simplify                                       
I am guessing I will get something looking like this:
(or whatever other permutations your imagination will drum up)



OH NO MR. BILL!!! What just happened?????

Hopefully now they will be a tiny bit motivated ( beg even?) for the math. 
 

This is the Friday Pic from the Smartboard. My kids worked hard this Friday and I let them play!

Cuppla things. On Tuesday evening, I came back from my last class in a 5 part series on Common Core Math Instruction, facilitated by Josh Deis and Krista McAtee (how cool is that, she is a former student, who is now paving the way...she teaches 4th grade). I came home so refreshed! Full of energy! Excitement for the next day!

We got around to talking about homework. Most of the teachers are elementary and middle school teachers in this cohort, AND, still, I find nearly all of our discussion totally relevant.

Josh showed us a typical textbook page: "Simplify," and then literally 44 questions about adding or subtracting fractions. Yip! Not! He explained a new way of thinking about these directions, as we are all money strapped and no one is getting new CCCS textbooks this year. So this is what I did with the icky picky hand-out from my department: (Stay with me, don't fall asleep yet!)

Side 1

 Side 2--Lest you think we forgot ANYTHING...and like the (!) will make doing all 28 any more fun.

Are you sleeping yet, please don't be, bare with me, because this is what was so freeing:

Shhhhh....I changed the directions.

1. Silently study all the problems but do not solve them: Put a CIRCLE around the number of the ones you will need to combine like terms for. Put a  TRIANGLE around the numbers for the ones you will need to rationalize the demoninator for. Put a HEART around the number of the ones that will be perfect square factors themselves. 

2. Now in pairs, come to consensus about problems you marked. 

3. Now get out your highlighters or crayons...make all the circle problems Green, make all the triangle problems yellow and make all the heart problems purple. Go home and do two of each kind. 

OH MY GOSH! WHAT GREAT CONVERSATIONS! MY STUDENTS WERE LOOKING FOR AND MAKING USE OF STRUCTURE, MY STUDENTS WERE CONSTRUCTING VIABLE ARGUMENTS AND CRITIQUING THE REASONING OF OTHERS! MY STUDENTS WERE LOOKING FOR AND EXPRESSING REGULARITY IN REPEATED REASONING!


I know just how geeky this sounds, but man, it was beautiful!

PS: I learned about "clock" partners. Anyone use this successful in Algebra 1 and beyond?

First week back from Spring Break is always daunting, don't you think? Kids are feeling the hints of summer freedom, they forgot that the "F word" is factoring, not the expletive, and the STAR is looming.

And it is also an interesting time for a maturity growth spurt for my Freshman Algebra students.

On Wednesday the third day back:

Student 1 to Student 2:       I am so going to tell Mrs. K. (their 8th grade Algebra Teacher) how much I hated taking all those notes but now I am SO glad I have them so I can figure out how to do this stuff.

Okay, I'll admit I was so butt hurt. Really. But then I got to thinking about it. Now they want to learn it. Why now, and why are they not using their interactive notebooks? Because maybe now they are motivated. Maybe now, they see big picture and they see the math in context, not just as a set of procedures. Does the SAT or a Physics textbook problem set say, "For this problem you will need to set the quadratic equal to zero and then you need to take out the GCF?" Of course I summarized for the students in a flipbook what it means to factor, but I don't give them a problem set with each step illuminated for them. I just keep repeating, "remember the flipbook we made about factoring? Read the first page!"


What do you think?

MS Funday Sunday is hosting a link to Math Stations:

This was the one of the most fun activities my colleague, Jessica, and I did with our Algebra 1A students.

We put Four 3 and and Two 4 digit pad locks at stations around the room. Then we made easy to horrendous Order of Operations problems that gave the correct combinations to unlock the locks. The students were so motivated! Talk about self -correcting! I guess we could have put candy inside a lock box or free homework passes or stickers...Eh...

One answer even had a fraction so that the answer 86 1/3 was the four digit combination, and of course, one decimal answer too.

What other kinds of problems can you think of to come up with combinations for the locks? You could have answers labeled 0-9 for any type of four problems per page and the answers that match those digits (say 3.375 seconds is an answer attached to digit 7). Hmmm...

Who can resist this picture?




Yeah, I drew it.


YOU: The kids thought that was pretty funny. I gave them one minute to write any questions they have about the math they would be using in their INB's. I took the quote from Phil Daro, "Writing means silence." I told them if they couldn't think of anything to write, draw the picture...theirs' were way better than mine. (Pictures later)

I have a monster fire truck...

Then I had the students to choose a partner and gave them 2 minutes to talk about the problem.

WE then had a group discussion. One young man walked up to the board and measured the width of the basketball to the width of the rim. "Is the hoop height regulation?" (Yay! someone asked!) yup, 10ft. "How far is he from the hoop?" "Is it allowed to have a fire truck on the court?" "What does he win?" (Easy, $1,000 per basket...of course they wanted to re-enact the whole thing...ya...no)

Without much prompting, someone asked, "how long does it take him to make a shot?" I revised this a bit, to "How long does it take the basketball to reach the hoop?"

I showed them this picture from Dan Meyers, To boost interest.


I told them the basketball was released from 18ft in the air. We Googled the average initial velocity of a basketball...it was 10ft/sec. I told them, since we weren't in motion, let's use 8ft/sec. (Makes numbers nicer for Zero Product Property.)

YOU/WE: Same thing again: one minute to write, two minutes to discuss, a few minutes to debrief. Many students had already written h = -16t^2 - vt +s. Sweet, right?

I pointed out the ideas I saw and heard that were productive and pointed out the ones that were on track, but needed some tweaking (desired height is not zero), and even some that were not at all relevant...um, like the circumference of the basketball...

I let them fuss and struggle and figure. It was so much fun! And it was too much fun to talk about why they got one negative root. We followed the arc of the ball behind the fire truck...there were a lot of "ahhhs."

At the end I showed this shot also from Dan,





Then the real discussions began, "That is totally Photoshopped...Dude...it was never going to make it."

It's going in!

I get it and can put it on the board.

I could have asked this:

There are 54 countries (most of the time) in Africa. There are 196 (mas or menos) countries in the World. What percent of the World's countries are African?

Instead I asked this "What percent of the world's countries do you think are in Africa?"
We recorded the guesses...

Then we did this:

How many countries are in Africa?

And then we did this:

How many countries are in the world?

And THEN we framed the question and wrote an equation and solved it. So much more fun!

We were also revisiting a poorly written test question (by me!) on purpose. Well it had a purpose that didn't work on the test, yet I was still able to use it when we discussed the "correct" answer. (I purposefully made the question so that there wasn't any one correct answer, however, I neglected to have the students explain their answer.)

Marianna said, "I wrote California if you are talking about size, and New Mexico if you are talking about percent of land," (Yah, Baby!) (I won't mention the the kid who said, "New Mexico is a state?")

Then from some lovely blogger who gave me the idea, I threw out there: let's add another column,

Forest Acres per person? Would that change our opinion?

I just love those teachable moments when everyone is arguing about...numbers...

What was your most recent teachable moment?

PS Sorry about the horrible image insert job, I am learning! 

I wasn't in that great of shape Friday morning. An unsettled night sleep made for a potentially groggy day. Ever have one of those? And yet I managed quite well. Amazing what a brainstorm at 2am will do for a gal! (I did get home that afternoon to discover that I had my cami on backwards--ie--the skinny bra part in the back was in the front...good thing it was cold enough to wear my droopy scarf all day.)

I knew I needed a really good Hook to (re) teach percent problems. I looked through this really old POW book and found this problem for my warm up: 



So I did my happy dance because I felt like I had purpose. I let the kids wrestle with it for 3 or 4 minutes. Then I stopped them and told them this TRUE story: my brother is a Pilot for X airlines. Back in the late 80's, early 90's, his airline started to get into financial trouble. I asked the students, "who works for an airlines?" We brainstormed and came up with a list (good kiddos!) I erased all the low skill jobs from baggage handlers to air hostesses and hosts (yup, sorry you all, low skill). That left mechanics and pilots. Next question, who can work anywhere else and get a high paying job? No-brainer for most (yeah, there are some kids who think non-military pilots are in high demand), MECHANICS! Cars, Planes, Trains, and Automobiles. So when the company asked folks to make concessions, it was up to the pilots. Airline X asks the pilots to take a two year temporary pay cut of 25% of their annual salary, where at the end of the two years, the company would return the 25%. (you math folks know where this going!)
Boy were the kids mad and shaking their heads. "That's not fair." Of course not, had the lawyers and union reps had paid attention during ALGEBRA, this never would have happened. Then I asked them to do this:

Which is a better deal? Taking 25% off a $100 purchase then paying 10% tax, or paying the whole $100 including tax and then getting the discount? Go! Same, same...now go back and have another look at your Warm-up. Is there anything you want to change, add? You have 120 silent seconds to move your pencils across paper (and/or use your handy-dandy calculator).

I got "blah!" I got "blah-blah!" What is the answer Ms. Z.? Who's right? "I have no clue, I haven't done the problem and probably won't get to it for awhile." Go home and ask your parents. Ask your neighbors. Survey your whole block! Now put all that fun away, you have a Chapter test to take!

The most important part for me, was not giving the answer right away. I wanted them to hit it several times and rationalize/reason for themselves.

It is curious to me, that the most likely-to-not-do-their-homework, most squirrelly, students were the ones with the most insight into percent problems...could "suss" them out in their heads and had a high degree of accuracy.

It was a long day of directed teaching...of course there was another way to approach the whole enchilada, and sometimes the subject is just so much fun to go exactly the way you want it to go.

Next time I would put one of Dan Meyer's 101qs and ask what questions the students have, and sometimes you just gotta give them their Chapter tests with enough time to finish.

Do you have a favorite way to teach percent problems?

I am thrilled. I am hopeful that in the next 4 years I see the real movement that I voted for to come about 4 years ago. Let's move forward powerfully and courageously.

I have tried many ways from discovery, to begging straight up: "these are the rules" to help my Algebra students work through properties of exponents. I had some success with a process I called, "1 by 1." We did 12 problems in shadowed boxes:

1. 
   

You can capture what is on your screen, like a screen shot, and save it as a Jpeg to use in...your blog posts!

I made this Tarisa Puzzle:

It takes a 18 x 24 piece of construction paper to complete: Here is the first part:
The next thing will be for me to write Tarsia Puzzle and this will be live for you to download. Anyone? Anyone?

Happy Martin Luther King, Jr. Day!