Crazy Huge Puzzle

My freshmen, as you may have read in a previous post, really like puzzles.  They say that puzzles are way better than worksheets.

So I decided to make a crazy huge puzzle.  (Note:  I noticed when I ran this sheet a second time that it is kind of small.  The first time it was a full page.  You may need to play around with it.)  Then I made a sheet to explain the rules and to record answers on.  There are four given angle measures, along with some 90 degree angle markers.

What I like about this activity is that they need to use what they know about special angle pairs, triangles, exterior angles, etc. in order to do this.  There are also multiple ways to get the measures of many of the angles, so they can check their work.  But the very best part about this activity is that there is not enough given information to complete the puzzle!

I gave each group 5 chips (each group got a different color).  These allow them to buy an angle (or a hint) from me.  They had to choose a banker who would come to my desk to make a purchase.  Here is the answer key.  If I have any wrong answers, please leave a comment!

The minimum I had to "buy" in order to complete this was 2 angles.  I tried it several different ways, and each time I got stuck and need to buy 2 angles, but not always the same angles!

Kiddos were totally engaged for a full 45 minutes.  They begged for more time to complete it tomorrow.

Some great things I overheard:

"No!  Don't buy anything yet!  We need to be sure there is nothing else we can do on our own!"

"Wait: yesterday we did that exterior angle thing.  I think we could do that.  Y'know, add up the 2 far away angles?"

"Are you SURE about that?  How come?  Where did you get that number?"

Things I would do differently next time:

• reduce the number of chips to 3
• make sure the groups are no more than 3 people.  (I had one group of 4, and it just seemed the 3 person groups had all members engaged, where the 4 person group had one that just sort of sat back....maybe just the dynamics?  He was totally able to do the work, just chose to do more watching than doing.  I think 2 groups of 2 would have worked better.)
• Allow a little more time.  We have 80 minute classes.  If I had given a whole hour, they could have finished it.

Addition Number Talks and a Missed Opportunity

This week was a short week, because Veteran's Day was celebrated on Wednesday.  I had to really push to get what needed to be 5 days' worth of work into 4.  AND it is my policy to never give homework on a holiday, so whatever practice I wanted them to do on Tuesday, had to be worked into the class time.

The result was only one number talk this week, and a missed opportunity to point out how frequently they used the commutative and associative properties.

Here is the board shot of the problem 37 + 49:

The first student used the strategy of starting from the left, which I misnamed as the break apart method.  This is definitely the strategy most students (who didn't use the traditional algorithm) used.

The second is the traditional algorithm.  I asked her how she kept track of what she had and she said she could just mentally do this by looking at the numbers on the board and picturing them one above the other.

The third used the take and give strategy.  He took some from the 37 and gave it to the 49.  What amazed me is that he took 7 to give to the 49, not just 1 to make it 50.  This led to a number talk inside a number talk when I asked him how did he add 7 to 49 so quickly!  He said he could just do it: "I'm weird like that!"  Others begged to share how they did it (the favorite way was to add 1, then add 6 more), then got back to one more strategy.

This is the round and adjust strategy.  As I started to record what this student was saying, she said it might be easier if I showed it on that "number line thing you do".  Ah yes. It is getting through!

It was the third strategy where I really missed a great opportunity to discuss how breaking it apart and giving a piece to the second addend is really just the commutative and associative properties in action!  I am very disappointed in having missed this chance since this is an Algebra 1 class and we have been talking about the properties.  Sigh.

I will have to do another with them and hope for the same chance.  Next time:  3 digits plus 3 digits in the hope of them discovering that other strategies can serve them better than the traditional algorithm, or at least that they have a better understanding of what "borrowing" means.

Number Talks Chapter 5 reflection

We did some multiplication number talks this week.  I suppose it should not come as a surprise that a few students did the algorithm in their heads.

The problem was 13 times 12.

"I put the 13 above the 12.  Then I multiplied 2 times the 3 and got 6.  Then I multiplied the 2 times the 1 and got 2.  I put it next to the 6.  (Note: I put the 2 to the right of the 6, since she said 'next to the 6') .  In FRONT of the 6, I mean.  Next I put a 0 under the six."

"Can you help us understand why you put the 0 here?"

"I put it there because you are supposed to.....um,....., as a place holder."

"Ok, thank you."

"After that I did 1 times 3 to get 3 and put it in front of the 0, and last I multiplied 1 times 1 to get 1 and put it in front of the 3.  When I was done with multiplying I added up and got 156."

First of all, I was impressed that anyone could keep all that in her head.

Second, I asked if anyone wanted to share another strategy.
Thankfully, the very next strategy perfectly mirrored what the algorithm girl was trying to do.

After the students were done sharing, we went back and noticed how the two strategies were related.  I wanted them to see that the 0 isn't just something that you put there as a place holder.  I wanted them to notice what they were really multiplying by.  I didn't do any of the talking.  Other kids pointed out how the one related to the other.

All in all, a good talk.

For the next one, I want to pick a problem that will work nicely with the halving and doubling strategy.

"Way Better Than Doing a Work Sheet!"

We have been studying special angle pairs in my freshmen Geometry class. Yesterday I told them I LOVE puzzles and I gave them this homework.

For each angle they had to give a specific reason for the angle measure ("vertical angles with <1", etc).

When they came in this morning, they were all abuzz!  "Did you get angle 12?  yeah, but I got stuck on angle 15...." and other such snippets of conversation were overheard.

Finally one kid turned to me and said:  "This took me a long time, especially coming up with a reason for each angle, but it was cool!  I can see why you like puzzles:  they are WAY better than doing a work sheet!"

Gotta love 'em!!

Subtraction Number Talks

This week we worked through Chapter 4 in our on-line book study of Making Number Talks Matter.  This chapter deals with various subtraction strategies.  I was a bit nervous to actually jump in to this for real, but it has been really fun and my sophomores (2 classes worth) seem to really enjoy it.

My classes did two subtraction number talks this week.  I invited my principal to come see me do the second one.  I thought I would be nervous, but my classes really get in to these (I had done several dot talks with them) that I was pretty sure THEY would be fine.  I am such a novice at this, I wanted her to come observe me to see if I was making sure not to call on too many boys, not trying to correct "weird thinking", etc.  I sent her a picture of my second planning guide.  Here are both the planning guides:

Here are some board shots for the first number talk.  I neglected to get shots of the second talk!

The one over on the right really through me for a loop.  I asked her to help me understand how she was doing this, and this was the best she could do.  I thanked her for sharing and then took a few more volunteers.  

When we were all done I pointed out to the young woman that it looked to me like the person who shared below her share, did pretty much the same thing, except they noted that the 5 and 2 really means 50 and 20.  Now I am asking readers: how would you deal with the 8-3 thing?  Do you just point out at the end that you can just subtract in the order you see?  That would 60 - 20 to get 40 and 3 - 8 to get -5....combine these and you get 35.

We were done, wrapping things up and one young man asked if he could PLEASE share his strategy?  How can you refuse?  And it was gorgeous.

"I took 20 from 60 to get 40, then took away the 8 to finish the 28 and got 32, but I still was 3 short from the 60, so I added it back in to get 35."






The principal was really impressed by the number of strategies and the "OHHHS!" when someone shared a new strategy.  She is hoping that after I have had some more practice, I will feel comfortable being video taped.  I am also open to have people come in a visit to see how this is happening.  It is pretty amazing!

I am SO grateful for this book study. It is forcing me to do the talks, which could easily have gone by the wayside since there is SO much else I need to get done.  These have been worth it, though!!

Spontaneous Number Talk: Multiplication

Today's warm up did not include a Number Talk.  At least, it wasn't SUPPOSED to.  Today's warm up was an error analysis of a fictional student's work on an order of operations problem.

If the problem was done incorrectly, it involved multiplying 12 by 4.  For a number of my freshmen this is NOT a number fact they own.  And it is something that they would most likely be wanting a calculator to do....or at least use pencil and paper.  

The problem asked them to not only find where the student went wrong, but to correct it as well. We had a multitude of answers (which freaked me out considering how many of these we have done, but it was an excellent learning opportunity).  After we talked about it for a while and decided which answer was the correct one, I asked if there were some people who wanted to share where they got derailed.  I talked about how mistakes are when our brains grow.  We get the chance to go "OH!!!  I see what I did, but this is what I was thinking..." and how we ALL learn from that.

There was one young woman who was secure enough to put it out there.  In the process of explaining what she did, she mentioned that she multiplied 12 by 4 and got 48.  She is not one with real strong number skills so I wondered how she had done that so quickly. After we were done talking about the errors and how we were thinking, I asked her to share how she multiplied those numbers.

It involved halving!!  I nearly fell over with excitement (but contained myself very well, I think).

Hers is strategy #1.  "When there are even numbers, I like to cut them in half until they get to where I can multiply in my head.  I cut 4 in half to get 2, and I know that 2 times 12 is 24.  I did that twice because I broke the 4 into 2 pieces.  Then I just added them together."

When I asked for other strategies, hands went up!  One young man gave us strategy #2:

"I just did 4 times 11 because that is so easy, and then added one more 4." 

"Why did you add one more 4?"

"To make 12 fours.  And so 4 times 11 is 44 and 4 more is 48."

Then another young lady shared strategy #3. 

 "I multiplied 4 by 1 and got 4, then 4 by 2 and got 8."  

When I wrote it the same way I wrote the first one, I said "and that gives us.....?"  She could "see" that the way I wrote it was going to make 12, so she asked if she could think a minute.  Then she and her partner came up with the strategy #4. "Oh, the 4 is in the tens place, so it must be 4 times 10 to make 40, then 4 times 2 to make 8 for 48"

Honest:  this REALLY happened today....and I didn't plan it.  I told them I loved them.

Number Talks: Chapter 2

I am participating in the online Number Talks Book Study.  It has been immensely helpful re-reading this book, and being able to read other teacher's experiences with trying number talks on a regular basis in their classrooms.

This week I did some of the dot cards in a class of sophomores.  I was a bit nervous about it, especially the whole "put your thumb on your chest" thing.  But it went great and it was fun to look around the room and see a lot of students with one, two, three fingers out as well, meaning they had more than one method of "counting" the dots that involved grouping them in some way.

We started with something very easy:  just some dice like set ups.  I asked them how they all knew how many dots there were so QUICKLY??  "I just KNOW," was the most common reply.  I explained that they were using a term called subitizing.  They have done this since they first played board games and knew how many places to jump when they rolled this:

We finished up with this "dot" card from the book.  

One girl who never says ANYthing in class practically leapt out of her seat when I asked for volunteers to share how they were able to figure out how many dots there were.

Yup, I'm a believer!

Introducing a New Language

The first few days of my Geometry are usually my LEAST favorite.  I understand the need to get some basic vocabulary work down, but I just hate it and have been working for years to try to make it go faster and stick harder.

I began with Dan Meyer's Pick a Point.  They loved it and had a good laugh.

My students are frequently Special Ed. students who have issues with notetaking and too much reading.  I have been using Interactive Notebooks with them, which has helped a lot.  This year I did the INB part a little differently.

We started out with a  vocabulary sheet template and this sheet of definitions.  They cut the strips of definitions and match with vocabulary word.  Same with template for angle words and their definitions.  For some of these I drew in pictures in the appropriate slots, and for some I put in the appropriate symbols.  I did not make the pictures / symbols match the definitions they were next to, so they had a lot of cutting and matching.  Then they had to try to fill in the blanks with their partners.

Then I wrote the Chinese character for "man" on the board:

"Any one here know Chinese?  Their language is made of characters, referred to as ideograms.  This means they help you see the "idea" of the word.  Generally they came from pictographs.  This is what the word for man used to look like: " 

"Can you see the man?  Look at these:  Let's see if we can figure out what they mean.  Old character is first (more of a picture), then the modern version:"

These mean tree.

Try these: 

These mean mountain!


Finally I told them we were learning a new language we are calling "Geometrese".  It uses "ideograms" as well.  I put this up on the board:

"What does this symbol mean?"  

"Parallel!"  

"Yup!  I wonder why?"

"Because it shows 2 lines that won't intersect!"  "Because it has a pair of l's !!"(that is L's not ones)

"All great thoughts, but I am wondering why not like this?"

"Because that is an equal sign and it would be too confusing!" 

Then we looked at the symbol for perpendicular and tried to decide why they used that.  What was the "idea" in the ideogram.

And finally we looked at the congruent symbol and noticed and wondered about the equal sign that is part of that symbol.  They even asked about the squiggle on top: didn't that mean something?  We got to talk about the different ways the tilde is used in Geometry as opposed to Spanish.

When we were done, one young man looked up and said: "That was the best introduction to a lesson I have ever heard."

God bless him.  As teachers, we get lots of negative feedback ("This is stoooopid!").  We even get lots of neutral feedback (they appear engaged, not drooling).  But we don't always get feedback that tells us we nailed it.

The First Day of School 2016

Yesterday was our first day with our students.  Yes, we must have been the last school in the United States to start up!  (Oh wait, I did see some school out west...Seattle?....that went on strike.  So maybe NOT the last!)

I high fived my juniors and it was lovely!  The only word I can think of to describe their reaction is "delighted".  They lit up!  They laughed!  They high fived each other after they entered the room! It was so fun to watch.  Thanks, Glenn Waddell !

The best word(s) to describe my freshmen when I high fived them?  "Deer in the headlights" comes to mind....They high fived, but were very timid.  Not like the slaps the juniors gave me.  I am interested how long it will take.  Most were still timid "fivers" today!  I wonder how long it will take for them to start giving me a good "slap" high five!?

This class contains some of the "underperforming" freshmen.  We did the Four 4's yesterday and I forgot the most basic element:  that you have to use FOUR 4's to make an expression that will equal 1, then FOUR 4's that make an expression that equals 2, etc.  I just let them go at it: "You can only use 4's and any operation to make expressions that will equal the numbers 1 -10."  They were making strings of two 4's and strings of eight 4's and everything in between.  Duh.....

Well, I gotta tell you, it was a LOT of fun.  I may have messed up, but I did more formative assessment than if I had actually PLANNED it!  I found some who didn't know number facts, some who couldn't follow instructions, some who didn't know order of operations.  I found some who had some really unique and interesting ways of thinking, and some who could communicate really well, as well as some who had fabulous ideas and no way to talk about them.  I found some who could only write down the number 4 and then stare at it.  I found some who needed a calculator just to get going.  Amazing stuff!  And this was DAY 1 !!!

We worked on just the number 1 for a few minutes. First they worked on their individual white boards and right away I could see who didn't understand the instructions.  I asked someone who had an answer to put theirs on the board and explain how it came out to the number 1.  

"Oh! I did it differently!"  

(Be still my heart) 

"Would you mind showing us what YOU did?"  Up to one of the white boards she went.

"Oh! I have another!" I heard from the back. 

"Come on down!" I encouraged.  

This young man did the order of operations incorrectly.  Someone asked didn't this have to happen before that?  

"Could you explain what you are asking a little more?"  

"I think he wanted us to add up all those 4's and then divide that answer by 4."  And then he turned to the kid in question and asked: "Is that right?"

(I melted.)

"Yeah, that's right."

"So how could you get us to think like you?  Is there some way to make us do what you did FIRST?"

"Oh, I think I need parentheses here."

I let them all keep working.  As they came up with new expressions for various values, they added it to the poster paper.  I encouraged multiple representations, and they were happy to go at it.

Finally one kid looked up and said: "Hey, we can just make building blocks and we could make any number we want!"

The others all looked at him like he had 5 heads.  

"Could you explain that?  Maybe show us what you are thinking about?"

He took an expression that equaled 2 and added it to an expression that came out to be 3:  "This whole thing will equal 5."   A few wide eyes.  A lot more that were deer in the headlights again.  

"See:  this much equals 2 (he writes 2 above the appropriate expression) and this is 3 (does same and puts + sign between them)."

"OOOOOOOOOOH!"  And then it was bedlam.  And it was wonderful.

Exponent Spoon Game

In the spirit of MTBoS, I wanted to share this game I made up a bunch of years back and forgot about because I stopped teaching the class I made it for.

We don't start classes til after Labor Day (I know, I know: we will be in school until June 30th if we have a winter like last year's!!).  I am in my room today and bumped into this game.  I think I will use it on Day 1 with my PreCal cherubs, just to make sure they are up to snuff on exponents of all kinds.

I made FOUR of these decks of cards (yep, 52 x 4) by hand because the printer I had available at the time could not print card stock.

I couldn't get all the cards in one shot, so I did 2 pix:

For those of you who may not have played Spoons recently:  You put one less spoon in the middle than you have players.  The point of the game is to collect 4 cards of the same value (four 2's or four 7's). The deck is shuffled and each player gets 4 cards (which they don't show to anyone).  The rest of the deck is placed face down.   

The dealer picks up the top card, decides to keep it or slide it to the right hand player.  If he decides to keep it, then he must discard something from his hand and slide it to right hand player.  A player can only ever have 4 cards in his hand.  Player to his right decides to take this card or pass it to HIS right hand neighbor. This whole process keeps going: dealer grabbing card, making decision, passing; next player looking at this card, deciding, passing on a card.  The last person in the group puts his discarded card into a pile. If no one has collected 4 of a kind before the last card is played, the discard pile is shuffled and play resumes.  

AS SOON as someone gets 4 of a kind they grab for a spoon.  At this point everyone else has to get one too.  The person who doesn't get a spoon is out of the game.  Players remove one spoon, shuffle all cards and get ready for round two.  Play continues until the last round of two players and one spoon which is the CHAMPIONSHIP round.

Feel free to make a print copy of these cards.  I am just too lazy since I already have these!  And you can double check my math....  :)

Finding Space for Whiteboards

I work in a small portable classroom.  It is one room wide, so has windows on two parallel sides, and a door that goes into the next classroom in the back of the room.

I have 20 desks in this small space, and since there were no built in shelves or cabinets, have used wall space for storage units.

Therefore, here is my first attempt at creating vertical non-permanent space.  Most pieces are velcroed to the surface.

(my asst. principal, when he saw the velcro, suggested I cover one of the shelves of my bookshelf.)

(this is my laptop cart)

(this is btwn the outside door and one of the two windows on this wall. Peek out the windows and you can just see the railing of the deck that leads to my room.)

(I had 2 smallish extra pieces which I pieced together on the side of a metal cabinet.)

(my metal cabinet)

I do have 2 smallish bulletin boards which I plan to have pieces of white board that will hook over them.  This way I can have both bulletin boards and white boards in the same space.

9/10/15  A few more spaces for whiteboards!

Here is a before picture (note Velcro around edges)

 And here is what it looks like once the board is attached!

These two are attached over my only two bulletin boards:

TMC 15: Reflections on Amazing Professional Development

Before I can even talk about all the things I learned, I have to mention that once again I was outside my comfort zone.  This was my second Twitter Math Camp, and I was heading across the country to be with some people that I had met once before for 3 days and had spent time "chatting" with in 140 characters or less throughout the year, but being the shy person that I am, even that was extending myself, and I didn't do it very often.

Last year I was especially grateful to @jaz_math who took me on as a roommate, and introduced me to a number of people.  This year, I decided that I should do likewise and met the wonderful @numerzgal.  We found each other on the roomie "blackboard", and it turned out that not only would we room together, but we were both flying out of Boston on the same flight!  I got to introduce her to a bunch of people I had met before, and then we met lots and lots of others.

For some of you who have had the TMC experience previously, perhaps you would want to think about "mentoring" a newcomer.  I forwarded emails, replied to texts, and answered lots of questions for my roomie prior to heading out to California.  In return, she reminded me not to leave my backpack behind as we prepared to board the plane, and she ALWAYS had our room key with her!  I called her "Mom"!  :)

And now on to the reason for this post!

MY TMC15 TAKE-AWAYS:

• Greet the students at the door.  Every day.  With high fives. Or fist bumps. Or elbow taps. Or whatever.  But do it.  This will be a challenge for me since I teach in a portable classroom and the door the kids come through opens straight to the outdoors.  This means rain or shine or cold or hot, I need to be by that door!  Thanks, Glenn Waddell!

• Find what you love.  Do more of that.  I love teaching.  I love finding out new ways to get the message of the coolness of math to more students. This is my passion.  Thanks to Chris Danielson, I will do more of this. I will choose NOT to get caught up in other stuff that I DON'T love.  I have 7 years left in the classroom and I am going to do what I love.  And then I will do more of that.

• Number talks. For years I have done what I call "Noggin Math" - getting kiddos to do math in their "noggins":  1/4 times 20 (scary!) becomes 1/4 OF 20 (NOT scary!).  20% of $30 is "so simple"  because they can take 10%....twice!  (Or 15% because it is 10% plus half of the 10%.)  Before heading out to TMC15, I had read Making Number Talks Matter, a philosophy that a class can spend 15 minutes a day, or every other day, or.... and improve their ability to do quite a bit of pretty challenging math in their heads!   Thanks to Chris Harris, I believe I can do these with my students!!  This is way beyond my "Noggin Math", and I think it will be a great complement to it.

• Ask GOOD Questions. This is an area that I have been working on ever since reading Powerful Problem Solving, and thanks to a session with Rachel Kernodle and followed up by a session with Robert Kaplinsky, I have some MAJOR thinking to do!  I am sitting here right now trying to think of a way to ask this question: "Do you know how hard it is to ask a question that doesn't have a yes or no (or one word) answer?" in a way that would make you answer with a more thoughtful response than yes, no or maybe! So here is my attempt:  "I often find it challenging to come up with other ways to question students so they can't answer with just one word.  Could you share some of your techniques with me?" (Whew! That was tough!)

I learned so much more than just this, but these are the four things I think I can do and/or improve this year.  (Thanks for the challenge, @stoodle !)

Warm Ups This Year

I know this is absolutely nothing new, just something I am going to try with my Freshmen and Sophomores:  Warm ups that help them think, and talk about their thinking.

I am listing them here so all the links are in one place!

Day 1:  Which One Doesn't Belong?

Day 2:  Estimation 180

Day 3:  Number Talks  (has anyone made a collection of good problems for Number Talks?)

Day 4:  Visual Patterns

Day 5:  Story Problems from Math Forum

What I Hope to Get From Twitter Math Camp 2015 (and what I have already gotten!)

TMC15 is only 2 weeks away!

Yesterday I finally allowed myself to print out the sessions document.  Hubbie kept laughing at me because I would shout, "yes! Going to THAT one!"  and then "oh NO!  This one happens at the same time as THAT one!"

So I threw out a few tweets about some of the topics that I need to get information on and I was flabbergasted at how quickly people responded!

I was hoping to pick the brains of those presenting Desmos, maybe at a different session ( a Flex Session on Saturday afternoon?).  Jedediah (@mathbutler), Glen (@gwaddellnvhs) , Michael F (@mjfenton), and Bob (@bobloch), all jumped in with responses.  Beth Ferguson (@algebrasfriend) shared what she will be presenting to a group of teachers in her district.  All of them offered to do hallway conversations!

I am in the process of reading Making Number Talks Matter and saw that Chris will be doing a session on Number Talks in the elementary school.  Both Chris and Fawn have promised to yak with me about doing Number Talks with older kids.  This sparked quite a bit of interest among tweeps as many chimed in with "Me, too!"

So here are some other things I want to get out of these four days of amazing professional development:

• I hope to get my hands going in learning to teach the distributive property to students who struggle with this concept.  
• I also want to learn to plan units by piecing together some of the amazing resources found through the MTBoS, and/or just do a better job at planning a year, a unit, or a lesson.  
• I am hoping to get better at teaching Algebra 1, and I really want to pick the brains of my roomie, Jasmine, from TMC14 to see how to do a better job at linking Algebra and Geometry.
• I would like to help English Language Learners be more successful in their math classes.
• I want to ask better questions and learn to elicit better questions from my students.  Causing a fight in my classroom may be just the way to get started!
• Building number sense is a passion of mine, including fractions (!!), and I hope to see what others use as tools to help this happen.
• I have been doing Barbie Bungee for years, but Fawn and Matt have a new take on this!
• And I really, really hope someone can help me figure out how to incorporate Number Talks, Estimation 180, Visual Patterns, etc., and STILL have time to fit in "the other stuff".

How I will manage to get to all of these, without the use of time warp technology, is still unknown.

Can't wait!

Some Days I Am Bad at This Job

Yesterday I had only 3 days left with my sophomores before the state math test.  I had them today, then they go to shop for 5 days, then I see them for a single day and then, bam, they take this high stakes test, where THEY will be evaluated and I will be evaluated, too.

One young man (one of my sophomores) came back after school to work on solving systems of linear equations word problems.  Also happening in my room at that moment, was a test prep group.  They are a VERY needy, not always so motivated, group who show up faithfully because their teachers and parents make them.  It has been slightly torturous to get them to do anything, so we did trashketball.  It was slightly noisy, to say the least.

What on earth was I thinking:  I honestly was cycling through helping everyone, including this poor dear who had NO clue about solving systems of equations by elimination. This was the skill he was trying to practice.... with word problems. 

And I kept trying to get him to see how to eliminate one of the variables.

And he just had NO idea what I was trying to get him to do.

And I was frustrated but trying not to show it.

And he was feeling more stupid by the moment.

Finally I acknowledged that I thought maybe we needed a quieter time together to do this.  He agreed and we parted ways with a promise to get together on Wednesday afternoon.

I beat myself up all the way home.  "Honestly, Tina, you KNOW better.  You could have done this a hundred different ways. What were you thinking???"

Fast forward to this afternoon: a different young man arrived in study hall to do the SAME thing: work on solving systems word problems. ( NOTE: I am working through an on line course with the Math Forum called "Nurturing Powerful Problem Solvers".  Figured I had better start doing a better job!)

I grabbed a couple of very simple word problems off Kuta and asked him how he would figure out how many fancy shirts and how many plain shirts this young lady purchased.  He is adept at noticing and wondering, so it was not hard to get the conversation going.  

"How would you even get started on this crazy problem?" I asked.

"Guess and check I think."

"Excellent. Go at it."   So he guessed 4 fancy and 3 plain.  He explained that he did that because there were 7 shirts.  I asked him to figure out how much she spent.  He did the correct multiplication, and before even writing anything down, said it would be too much.  He changed that amount around: 3 fancy, 4 plain, and saw that he was moving in the right direction.  

At this point, I intervened.  "Let's just organize what you have just told me."  I made some labels on a table and had him fill in what he had just been doing with his calculator and his brain.

I wondered how he chose his values.  "I just started with 4 fancy, then knew I had to choose 3 plain to keep it at 7 shirts.  That came out to be too much money, so I figured I should choose fewer of the expensive ones.  When I went down one Fancy, I had to go up one on Plain.  That was still too much money, but I was going in right direction.  I went down one more Fancy, so had to be at 5 Plain.  That was just the right amount of money!"

He did several more problems:  

This problem was about how many chickens and pigs were in the barnyard.

This problem figured out how many vans and buses would be needed to transport 231 kiddos.

He seemed really comfortable with this method,  so I handed him the crazy quiz problem I had created for our class the week we had been working on this skill (he had been out with a concussion).  I was really worried because it is all made up words.  He read it carefully.  He read it again.

"You ok with the silly words?" I asked.

"Yep."   And he proceeded to produce this work of art:

Please, Dear God, let me have another chance with the FIRST young man who came for help. Dear Student:  I am sorry.  I was a horrible teacher yesterday.  I did you a total disservice, but I am prepared to try to do better when you come see me again.

"Three Strategies for Getting Students Engaged in Math"

Or something like that.  Dan Meyer's workshop had some or most of those words in the title.  I believe the words Common Core also appeared, but I don't care because the workshop itself was one of the most fascinating useful pieces of professional development I have done in a while (at least since last Thursday when the Math Forum crew were in Bridgewater MA!)

Anyone who has seen Dan work the room, knows what I am talking about here:  we are his colleagues, we are helping him out, we are creating a website, we are in on the joke about that "other group" a few years back that "got it wrong", and could we please share what that "other group" got wrong?

Oh. My. Word.  Six hours flew by and I have so much I want to play with, think about, create, delete, try again.  LOVE it.

The three take aways:

1. Create a fight:  cause controversy (oh yeah, hear that MP3?). If it is an opinion, no one can be wrong if answering "Which is best?"

2. Turn up the math dial: slowly!  Start with no words, no explanation (or minimal explanation, or broad question) and slowly add in some numbers, etc until they do some math or beg for a quicker way to do it.  Which leads to....

3. Create a headache:  "If the math you want them to do is the 'aspirin' (that which mathematicians use to make life easier), then what was the 'headache'?  What caused mathematicians to come up with this faster, quicker way of doing things?"

But most of all, Dan assured us that we are all good.  We work hard and not every day is going to be a stellar day and sometimes it takes 5 years to figure out how to create the headache!  But don't give up and don't beat yourselves up.  What a message of hope, inspiration, and validation.  Well done, Dan.  Thank you!

Noticing and Wondering about Systems Word Problems

The late winter was so broken up for us.  Snowstorm after snowstorm broke up the weeks: 2 days off here, 2 days off there.  Week after week this went on.  When we finally came out the other side, were hopelessly behind.  Then there were other things like state ELA tests and field trips that broke up the weeks.  So now we are further behind.

I have been trying to get my sophomores ready for the state math test. One of the things I still need to teach / review is solving systems of linear equations.  Solving by graphing was no problem.  They were comfortable graphing lines, and this made sense to them.  Any other method was like talking Klingon to these poor kids.

Finally I just went to some crazy word problems and let the kids solve them any way they could think of.  The problems were silly and funny and often involved "my friend, Max."  For years, "my friend, Max" was just a made up imaginary friend.  Maybe I was thinking about this cartoon character?

Then I met Max Ray and suddenly I had a real "my friend, Max" and HE came out to work with this particular group of sophomores early in the year.

Now every time they get stuck, I pull out "what do you think my friend, Max, would ask about this?" or "What did you do when my friend, Max, was here?"

One young lady asked if she could come after school on FRIDAY to get extra help on these systems problems.  No matter what I encouraged her to do, she was just guessing and getting lost:  we had tried drawing pictures, we had tried making tables or organized lists.  Nothing worked.

So when she came on Friday, I wasn't really sure what I could pull out of my bag of tricks.  However just the previous afternoon, the crew from Math Forum had presented at a seminar put on by the Mathematics and Computer Science Collaborative (MACS) over at a local college. (Thank you Steve, Suzanne, Annie, and Norma!)  And once again, Annie and Steve made me remember the value of Noticing and Wondering.

I pulled up some simple systems word problems from Kuta, and at first just let her read them out loud.  I wanted to see what she would do on her own.  She drew the boxes we had played with in class (for organizing some info) and then she just got stuck.  I said, "You know, there is a LOT of information in these problems.  How about we use some Noticing and Wondering here?  I am going to clean off my desk while you work on noticing some things about the first problem."  When I got back a few minutes later, she had done two problems worth!

She declared that was easy.  "Then let's sort the information.  What is the first way we could sort some of your noticings?"

"I noticed there were 2 types of shirts: Fancy and Plain.  I guess we could use those as the variables?"(she was building on discussions we had done in class).
"Sure, why not?"

"I noticed that there were 7 shirts in all."
"Cool.  What KIND of shirts?"
"Some were fancy and some were plain:  OH! Fancy shirts and Plain shirts = 7 shirts!"
"Those units of measure I keep nagging you guys about come in handy, eh?  So what else did you notice?"

"She spent $131 on shirts.  And fancy shirts cost more than plain shirts."

Here is where she got bogged down:  she said $28 + $17 = $131.  I asked her to check that on her calculator.  After some tapping she presented me with a frowny face.

"If you type in $28, how many shirts is that?"
"One fancy shirt."
"Yep, how many shirts did be buy for the $131?"
"Seven shirts."
"Hmmmm, could she have bought more than one fancy shirt? or maybe even several?"

She played around with the calculator for a while and then said in frustration ,
"But I don't KNOW how many fancy shirts she bought! And I noticed that she bought plain shirts, too!"
"Hmmmmm, do we have a way of saying 'I don't know how many?' "
"Oh! we could call it x !"

And then she was able to make more sense of all the algebra we had been doing the last couple of days / weeks.  She had been blindly "doing the math", with no understanding why she was doing it.

"We should have done Noticing and Wondering all along, " she informed (admonished?) me.  "It makes a lot more sense now!"

Here is what she presented me with:

One thing that came up during our chat was a question that really threw me for a loop.  As she worked on noticing and wondering she asked aloud:  "How many legs do chickens have?"  I have to tell you: we don't live in farm country out here, but we are not exactly city dwellers either.  It took a lot of restraint to not have surprise register on my face.  I answered her question and she was happy with that.  "And pigs have four, is that right?"

Later, I told her I was giving her one problem that would count as a "quiz" to see if she could really do this on her own.  Here it is. Look carefully at the Wondering column.  She had a question about goats :)

Which One Doesn't Belong and MP3

If you haven't been playing with Which One Doesn't Belong then get thyself over to Mary Bourassa's blog and read up on it and its origin.

The short version:  you are given 4 pictures, graphs, shapes, whatever, and you have to decide which one doesn't belong.  The coolest thing?  There is no right answer!  Or wait, yes there is: there are FOUR right answers!!

Try this:  Which of the 4 words you see below doesn't belong.  WHY? (oh yeah, that WHY is the very best part, the frosting on the cake so to speak!)

Last Tuesday on the Global Math Department, Mary and fellow collaborator Chris Hunter presented Which One Doesn't Belong? They encouraged us to participate with them as Chris was showing how he comes up with the four items that will make up the WODB.

My Geometry class has been doing investigations with special parallelograms.  I thought it would be fun to get them to a WODB on that topic.

Before I go on, let me tell you that my professional goal this year was to listen more and to get students to REALLY work on the eight Standards for Mathematical Practice.  In particular, I wanted them to be really comfortable with sharing their thinking and be able to critique their own reasoning as well as the reasoning of others (MP3). I want them to understand that critiquing includes telling what you thought was great and what you are confused about, not just disagreeing with what was said (though that, too, is an important skill.)

I introduced the students to the "game" of WODB by showing them the slide above, then.....

with MP3 in mind, I presented this slide:

I tweeted this slide out, too, and got some WONDERFUL comments about it and how to improve it.  I want to explain why there are no angle marks, no parallel markers nor tic marks:  I wanted to see if they "got" it.  Would anyone call me (or one of their classmates) on it?  One of my four class rules is Don't Make Assumptions.  

They had a blast: so engaged.  Kids who never say anything were practically jumping out of their seats (and I number the squares as though they were quadrants, btw, because I find that these cherubs don't know about Quadrant I, II, III, and IV!). 

"The one in Quadrant III has only 1 set of parallel lines.The others all have 2 sets."  

"Quadrant I is the only one with all equal sides."

"It appears that the one in Quadrant IV is the only one with diagonals that are not congruent."

We decided that I would have to go fix the "rectangle" in Quadrant II, so it would have something unique to it.  It originally had the dotted diagonals, same as the others. We just couldn't figure out how else to make it unique, though just as we about to give up, one student jumped up and said: "It's the only 'tall' one!  Taller than it is wide!!"  The others clapped.  One suggestion to "fix" the rectangle was to change how the diagonals were made, and another suggestion was to change its color.

This was all happening just before the bell and I praised the student who said "It appears....." because that was the closest to admitting there was no proof that any of these were any thing more than mere quadrilaterals and that almost ALL of us had broken Palmer's Rule #3.

After watching the conversation about this on Twitter (I was too busy at school today to be able to do more than glance!), I thought I would make a second slide with the marks on it:

When I do this with next year's class, I may just show the unmarked slide and then this second slide, and ask them which is better and why?

Which do YOU like better?  Is too much information too much??