Sunday, March 12, 2017

Books I've Read and Books to Read

I am devouring books this year.  I am on a good roll.  I don't want a dud.  I am loving talking about them too with everyone I meet.

Here is what I have read.  I recommend them all, even though I didn't care for one of the endings:
 


At #edcampBOS yesterday, one of my talks turned into a book club talk and I got so many more titles to read......Snow day on Tuesday???  = hit the library tomorrow!!!!

10 Books I Want To Read:

   




Saturday, March 11, 2017

#EdcampBOS recap

Thank you to the #EdcampBOS staff.  You did an amazing job and held it in a great facility at Microsoft and provided a delicious lunch.  Thank you. 

This was my third edcamp and my best experience yet.  Edcamp is a conference for teachers by teachers - all grades, all subjects and is free. 

It all starts with "The Board" and because we were at Microsoft the board was glass.  A grid of times and rooms is made and then we add the sticky notes - what we want to learn or what we want to share.  Here is the start.  Then, they type it into a google sheet and share it out.

 We would have five sessions, about an hour each with an hour for lunch.  For my first session, I went to one about Common Formative Assessments run by Rik Rowe @RoweRikW.  We had a great discussion about the benefits and challenges of formative assessments along with some websites to use.
   I put up a sticky for the 2nd and 4th session - - I didn't want to go too early or too late - jussssstttt right.  My 2nd session was on Desmos.  I love to share it.  I had about 8 people in the room.  It seemed the consensus was they have heard of Desmos but hadn't really played with it.  One just started using the activities and one has been using some of the activities.  I was like a kid in a candy shop wanting to teach and show them everything.  I started with just the calculator part and showed them the graph of a line, added sliders, added a constraint to the domain, showed the wrench to go to projector mode and how to change the window size.  We played a little.  I then switched them over the student.desmos.com and I was on teacher.desmos.com.  We started with polygraph lines, walked them through it, explaining the need for asking yes or no questions, showing them the teacher dashboard so I can see their use of vocabulary.  We moved on to marbleslides, my own teacher created factor sort, and Land the Plane.  It was so fun to share and walk around and see their abilities.  I met Sadie @relativelythink
    For my third session, Sadie and I moved on to one on what to do with snow days.  I just wanted to hear ideas.  It was a mix of challenges with putting into place a system that would have kids doing school work on snow days.  Some of these challenges are the inequity of teachers having to do work, but what do others like nurse, guidance, janitorial staff do?  They could do PD work?  Other challenge is the difference in levels of technological skills among staff.  If you have a staff member is low tech, what can they do?  Is a blizzard bag the answer when this might be a random filler of activities.  A better idea is to have it be current material for this day.  Get creative - have them take a picture and explain their snow day, write a blog post, write a tweet, do a small project.  It was food for thought.
    Off to lunch.  I was so happy to see #mathletes and #mtbos for a lunch room.  I grabbed a delicious lunch of pizza, salad, drink, popcorn, and a cookie and made my way to the room with about 6 of us math teachers.  Two had their knitting and I was bummed I didn't bring mine.  It was a knitting and math share session.  We shared our must follows on twitter, must read bloggers, and websites.  Thank you to our lunch bunch:


  The next session I put up on the board.  I thought I would share online games like Kahoot.it, Quizizz, and Quizlet Live and see what other ideas people had.  They added flippity.net which I use for random groups but have not looked at the other templates.  We played a bunch of movie related games to test them out.
    I'm not sure how, but we got off on a book tangent - have you read this - have you read this?  Since I love reading Young Adult, I didn't mind.  Everyone had something to contribute.  My list of books to read was drying up, although I have two on my desk waiting for me to start but now I have a new list to read.  We switched from game club to book club and I swallowed it up!  Thank you all!
   It was 2:30 and time for the last session. There was nothing jumping at me.  I felt like I had been at a Thanksgiving feast and was comfortably full.  I didn't want to go to a dud and wanted to leave on my high, so I used my feet (edcamp jargon) and left.  (It's also my husband's birthday, so I wanted to get home so we can go to dinner)

   Thanks #edcampBOS for a successful 2017 edcamp.  Nice work!

Wednesday, March 8, 2017

A Day in the Life of a Teacher #DITL: Post 9 An In-Between Wednesday

This is my 9th post on the 8th day of the month for the #DITL: Day in the Life of a Teacher blogging series. 

I titled it "An In-Between Wednesday" because it is between two important days of me applying for things.  Yesterday, I submitted my application for the second cohort of @Desmos fellowship and tomorrow I will have my class taped as part of my application for a PAEMST Teacher award.  More on this later.

Back to the beginning.  I woke up at 5:30 and I always start my day with a big glass of water.  It feels so good to rehydrate.  I listened to the DynamicCatholic's video for The Greatest Lent Ever to get me started.  I showered and dressed in red for International Women's Day.  Made my salad for lunch and headed out the door for work at 6:30.

6:40 am at work to do a few things before the day starts: make an answer key to my Foundations' class homework assignment, print one more make up test for a student who was absent, add colored paper clips to my grade book that I finalized last night.  Yes, it's March.  We started the new term at the start of February but I hate to write in the actual grade book until the kids are done finding their correct classes.  Start this blog post.  Think about what my day looks like:

First block is Per 6 Acc Alg 2 and I am teaching Law of Sines.  I taught it to my Period 3 class yesterday and it didn't go quite as planned so I have been thinking about what to do with this Period 6.  This was the day I was going to videotape and after Period 3 yesterday I am glad I didn't tape it.  Nothing went terribly wrong, it is just that Law of Sines - the ambiguous case is tough.  I woke up one morning with the bright idea of making a foam core poster with legos and pipe cleaners so they could see the opposite side actually hinge and why there could be two triangles to consider.

They already knew what Law of Sines was from Geometry last year and have been using it a bit without me reteaching it.  So, I thought we would be good to go.  I gave them some #VNPS problems.  The first was a regular solve the triangle, given a triangle picture.  The second problem, they were given just the values of a side, side, and angle.  The 3rd problem, they were given find the area of a triangle.  Before I knew it, most groups were onto #3, so I knew they missed the ambiguous case in problem 2.  So, I asked them to go back and solve the other triangle.  I didn't teach them how to do this because they had it last year and I wanted to see what they might remember.  I would say half of the class did.  I ran out of time to have the students finish the board problems and then do an exit ticket.  I asked them to come back to their seats, start the exit ticket with one ambiguous case on it, and then I wanted to collect them but they barely started.  So, now I am torn.  I don't think I will hand out the exit ticket.  I think I will save it to become an entrance ticket for tomorrow's class which will get videotape for the PAEMST award.  However, tomorrow's classes are shortened to 45 minutes from 60 minutes due to a college fair.  I think I will have them work on the problem while I am checking homework.  They need to know how to do the Law of Sines before we move onto Law of Cosines tomorrow but this way they will have had their homework as practice.  I am also going to pay closer attention to students starting #2 on the board and encourage them to think about it before erasing and moving on.

Period 7 and 2 will both be Accelerated Alg 1.  We are doing day 1 of a 2 day review for a quiz. We have been graphing quadratics and started solving them, so we need a couple of days to process this.  I have an A/B multiple choice review planned, 2 graphing webs, and then practice worksheets.  Also for period 2, we meet block period, so they have to bring in their catapults to test out.  This is just practice.  Students are bringing them in already this morning.  They will get some M&Ms and try them out, see if they shoot too high, too far, will something break, is something too loose?  Then take it home and remake it before our actual target shooting at the end of the month.  The kids really like this catapult project.

Period 3 is back to Accelerated Alg 2 and I am teaching the Law of Cosines lesson and going to try to make note of all my transition times so I can adjust to tomorrow's 45 minute taped class.  I purposely chose to tape period 6 so I could practice with Period 3 and adjust.

7:08 - last minute details before 7:20 start.  Let's see how the day goes.....

Fast forward to 9:47 am when I can sit again and I would say it went as planned.  I am glad I re-evaluated my Alg 2 class.  I taught the ambiguous case a little more slowly and as they worked on problems at the board, I was able to anticipate them missing the ambiguous case in #2, so I watched them as they worked and stopped them before they tried to move on, asking them to look at it more closely.  I suggested they draw it like me with the angle on the left and then the two sides on the sides, not the bottom, but some drew it on the bottom and then they couldn't figure out the height correctly to move forward.  In this class, I knew I wasn't going to get to the exit ticket so I let them work to the bell.  Good adjustments.

Per 7 - I was just going to read the homework answers and then do out 2 of the word problems.  However, I have a student who loves to lead the class whenever she can.  She asked to do a problem from the homework on completing the square.  Normally, for fear of losing class time, I have to say no, but I let her.  It was a word problem and she did it all exactly correct.  I was glad I gave her the opportunity to try it.

Now it is my prep.  I finish the answer key I started this morning.  I need to prep a lesson for Friday's Foundation class because I can really only do it one day at a time because I need to see how far we get.  I make a powerpoint and some copies.  I need to put last minutes fixes on my Law of Cosines videotaping lesson.  I did just clean my white boards to get them ready, too.

10:40 onto Period 2 Alg 1 plus catapult testing...

2:19 = exhausted and it is 54 degrees and I want to go home and get outside for a run but I need to process my Alg 2 class to get ready for tomorrow.

Period 2 went well.  As I was going over the homework in this class, I told them I let the girl in the other class go to the board to do this certain problem and she got it all right.  They know the girl because some of them were in class with her last term.  One girl said, "I know.  She told me."  So, again, glad I let her.  I think they learned a lot from the catapults as 2 fell apart, one went way too fast, and one went way too far.  They will take them home and make modifications to get ready for the real target day at the end of the month.

Period 3 - My Accelerated Alg 2 class - I was making notes all class long to see how long things would go in order to adjust to the 45 minute class tomorrow during the taping.  In 60 minutes I didn't even have enough time!  Yikes.  But, I did do a problem incorrectly, so I had to figure that out.  I will now do it correctly and save time there.  It took me 17 minutes to collect assignment sheets and check homework.  Good news is I don't have to collect the assignment sheets tomorrow.  I will check homework but I will do it more quickly.  I did have the class doing the old exit ticket as an entrance ticket in those 17 minutes too and I went over it.  It was one problem of the ambiguous SSA Law of Sines case.  I have decided to keep that until the next class.  It is okay that I spiral.  It took me 3 minutes to read the homework answers and there were no questions, so that is a good sign.  I handed out a discovery activity for the Law of Cosines and that took 10 minutes but definitely worth it to derive the Law of Cosines.  I will walk around and monitor their progress then discuss it.  We switched over to notes and it took 10 minutes to get to the first problem for them to try.  We wrote down the three types of Law of Cosines.  Talked about what it is, asked them to notice things about the rule, talked about when you can use it vs Law of Sines, discussed that you can find a missing angle too.  Then we tried one.  Some kids entered it in pieces on their calculator and undid the operations incorrectly so I tried to encourage them to do it in one entry with their calculator.  I gave them an example of SSS that I thought was going to have them fall into the ambiguous case but it didn't.  I was confused and trying to figure it out, so this took some time.  I was so happy I had this class to practice on them and they knew we were figuring it all out together.  That only left 10 minutes for the board problems and this is what I wanted to be the main part of my video, so I need to chop.  I think I will check the homework and save the Law of Sines exit ticket, do the discovery activity, chop my notes down a bit more, give them only one class practice problem together and then they are up and working.  We did struggle together but I want them to struggle in their groups of three and figure it out.  Wish I had one more class to practice with.  If it doesn't work, there is still time to pick another class to tape, but this one provides me with a lot to talk about, phew!

2:45 - leaving school, stop at the bank, still thinking about this lesson, think I am going to drop the one class problem I do, but then decide to keep it in there. 

3pm  I pull into my driveway and am still thinking about the lesson with my child #3 scares me through the window and knocks me out of this zone, which I needed.  The kids are heading golfing and I am going for a run which will have me back to thinking about this darn lesson.  I am way over thinking.  More importantly, whatever shall I wear???? 

3:08-3:45ish run about 3.8 miles - all the time thinking of this lesson.  I made one more decision.  Instead of walking around and then checking the homework then asking for questions, I will pass out the discovery activity and have the kids start WHILE I walk around and check the homework.  Anything to save a few minutes.  If I time my run correctly, I get back to the neighborhood at the same time my best friend across the street's little kids get off the bus and the kindergartner always greets me with a running hug, "Jennnn".  Again, another sign to break me out of this trace.  

Off to shower and into my pajamas, yep, by 4 pm.  My brain is done and my jammies are on.  Why not?  I don't have anything to go out to tonight. 

I pick out my clothes for tomorrow's taping.  I want to wear something that I am comfortable in, so I decide on a black short sleeved shirt with a gray sweater and I want black dressy pants with a design.  I have a few pairs and the ones that won by default are the only ones with a back pocket for the wireless microphone.  I never knew most of my dress pants don't have real back pockets.  Throw in a load of laundry, grab a piece of banana bread I made yesterday for a snack.  Check school email and find an invitation for our math team to go to States!  Wahoo!  I was hoping we would make it.  We are in the large school division on March 31st.  Now to figure out our top 8 kids to compete.  Next up, sit in the sun and read my new book "The Radius of Us" - A young adult book I am loving.

5:13 pm, finished my book, so good.  I hate when it ends.  I have been on such a great roll with books.  It's my 7th book this year, 2017.  I need to get to the school library tomorrow and get something new to read.  I bought a new journal at the beginning of the year, not sure what I would write in it, but I decided to make notes of the books a read - a book journal.  I'm recording the title, the author, the date I finish it and a brief synopsis with up to 5 stars.

5:20 pm Put a butternut squash in the oven to bake for a bit.  Making squash, rice, and steak for dinner.  It was 60 degrees today and listening to the news tell me snow and cold weather is moving in.  Glad I ran today.

5:24 pm Send an email to Youth Group reminding them of some payments due for Mystery Night Out and for Mission Trip.  Help my kid#3 fill out a sports application for college.  He is hoping to play golf at college.

We took a picture after school of those still around and wearing red:

  



6 pm dinner

More explanation on Desmos Fellowship and PAEMST.  Desmos had their 1st cohort of Desmos Fellows last year.  I saw it then and thought about it but did not apply.  When I heard people talking about it afterwards, I knew I wanted to do it.  When it was first advertised last week, I jumped on it.  I had to do a screen cast teaching Desmos how to use something on Desmos.  It was the first time I have created a screen cast and it took me many attempts but I was happy with the final product.  I had to answer other questions including why I want to do this.  And, I had to include some snapshots of something I created in Desmos, so I included my Factor Card Sort.  I will hear around May 15th.  All submitted and done.

The PAEMST is the Presidential Award for Excellence in Math and Science Teaching.  It is a national award.  A colleague nominated me.  I was totally surprised when I received the email at the beginning of February break.  I have seen the abbreviation on Twitter but didn't know much about it.  I read up about it and am honored just to be nominated.  It is a big process but one I am willing to try.  I listened to a webinar about tips on applying.  I researched and read some blogs of past winners.  They will award this to two teachers (Math and/or Science teachers) from each state.  It almost feels like applying to college (along with my kid #3).  I had to fill out information to see if I would qualify and I did.  I need three letters of recommendations and I have asked for them.  I have to type up a resume that can be up to three pages.  I have to video tape a class.  It has to be continuous, no editing.  It can be 45 minutes max.  Then, I have to write a 12 page paper about 5 different dimensions, some having to do with reflecting on the video.  I can include 10 pages of supplemental material.  So, I am videotaping my class tomorrow.  I had my friend tape one class yesterday more as practice for him, to test his camera and microphone and his placement in the room.  I emailed the parents and asked for the parents.  All the parents were fine with it and wished me luck.  If it doesn't go well for some reason, I can tape a different class, but I hope it goes okay so I can start the paper.  It is due on May 1st.  It has a nice, easy dashboard type website to submit everything, including a count down.  It is due in 53 days.  I won't hear about the winners for a while.  So, I will submit and then forget about it.

7:11 pm, getting ready to chill on the couch for the rest of the night

Answers to the DITL questions - abbreviated form
One of the questions is about the decisions teachers make - I made a million and one today, so that's enough for now.

Things I am looking forward to - this Friday is Soup and Stations hosted by our Youth Group at Church.  I look forward to trying all the yummy soups (snow coming on Friday) and having dinner with the youth.  Then, we go next door to the church and pray our way through the Stations.  It is a nice, quiet way to spend a Friday during Lent.  And, then Saturday, I am excited to go to #edcampBOS all day.  And, it is my husband's birthday, so I will come home and take him out to dinner.  Our youth group Mystery All Night is next Saturday, March 18th.  We have 31 kids and 6 adults traveling to two mystery locations and returning to the church at 1 am and then hanging out all night until 6 am.  I plan on sleeping all day on the 19th to recover.

A warm brownie for you if you made it all the way through this terribly long post.  Thanks for listening to me think aloud as I prepare for my lesson tomorrow.  See you next month for #DITL Post 10 on April 8th, a Saturday.  This is the day of our Youth Group big fundraiser dinner!


Tuesday, March 7, 2017

Two Truths and a Lie: Quadratic Style

I did this last year and forgot about it, but then got excited when I saw it in my plans for this year.  I really like it.  I like that it is a good opener.  It doesn't take the kids too long to make or to find the lies, but it lends itself to good discussion while they are trying to create the truths and the lies and good discussion as they walk around the room to look at the other teams' boards.

We started our unit on Quadratics two days ago.  We did Day 1: Teaching Graphing of Quads - Vertex Form and Transformations and Day 2: Graphing of Quads from Standard Form and all the properties.

Today was 2 Truths and a Lie.  Almost all of the students are aware of the regular game.  I have 6 groups of students and give them each a quadratic in vertex, standard, or even one was in factored form.  I ask them to do the problem and graph out on a big white board and figure out the properties, then come up with 2 truths and a lie about it.  Then they prop their big whiteboard on my class whiteboard and they pick up an individual whiteboard.  When all 6 boards were up, the students were asked to travel around to visit each one and determine the lie.  Then, I stood by each one and introduced the problem as Team One's problem.  I ask for someone not on team one to tell me what they thought was the lie.  I polled the class.  Then, I asked Team One if they were correct.  I thought Number 3 might have been the hardest because there were a lot of students clustered around it for the longest, but it turns our 4 was the one that stumped everyone.

I really like it.  And, then more fun to follow with Desmos polygraph with each student on their own computers followed by Desmos Marbleslides - 2 kids to one computer - more great conversations and use of vocabulary.

  

    

       


   
















Unit Circle #VNPS

Last week we learned about the unit circle in Accelerated Algebra 2.  They had it last year in Geometry and we are continuing it this year.  I explained where it came from by drawing out the special right triangle and then we created lovely paper plate unit circles, writing the degrees, radians, and points all around the edge. 
   When they returned yesterday, I wanted to see if they could create it on their own, so I grouped the kids into threes and had them all up at the whiteboards.  They were instructed to make the unit circle, including degrees, radians, and points and then to write 6 problems and solve them that are 6 trig ratios, a mix of degrees and radians. 
    It was interested to watch them go through the process. 
Did they draw the circle first or the coordinate plane?
How perfect did their circle have to be?
Some started using yard sticks to draw diameters through the circle to represent the different angles like I did in class.  But, some decided they only needed to put the points on the circle.
There were three people in the group. Some groups had one student do all the degrees, another do the radians, and then the points.  Some groups broke it up and each chose to work on a conic.  Some groups had one person write as they figured it out together. 
Then, they had to write their 6 trig ratios.  At first some thought I meant 6 trig ratios of 6 points or 6 trig ratios of each point.  Goodness, no, just 6 random trig ratios. 

I watched and admired their work and the thought process but I was starting to see some mistakes.  As the groups finished I extended my original idea and asked them to mingle around the room and see if they could find any mistakes.  A few were found.  The groups who drew them wanted to know right away, where is the mistake?  They didn't want to have any. 

It was a good 20 minute activity.

  

Monday, February 27, 2017

Siri: "Planes Overhead Now" #VNPS

I saw someone post on twitter that if you asked Siri "Planes Overhead Now", it would give you the following:


I was to meet period 6 before period 3 when we returned from vacation, so I decided to try this in class and see what happens.  It is accelerated Algebra 2.  They had right triangle trig in geometry last year and we are just starting trig now.  I wanted to see what they could make of it.  I didn't give them many instructions, just to draw it. 

This is what I got:
Group 1 - used variables and came up with an equation to create a table and then graph.  Very formal.

Group 2 -Nicely labeled x and y axes and planes.
 

Group 3 - Started out with just drawing the angle degrees, then guessed on where the dots for the airplanes would go.  I helped direct them and asked if they could figure out how far away from you they would be.  I did tell them they could use their calculators and at that point, they hadn't needed them, but now, they were "ooooh, that's why we would use our calculators."  They knew their final product wasn't quite drawn to scale but better than they started with.

Group 4 - Was similar to 3 in that they started with degrees, rays, and dots.  I encouraged them to add some numbers to their axes.  (I didn't tell them they had to put it on a coordinate plane, but they all did.)


Group 5 -I took this picture before they were done.  They didn't have numbers at first, they were starting to figure them out.
 

Group 6 - I am not sure why I let them off the hook so easily.


Group 7 - I like this.  They first drew a lot of smaller triangles and then put their information as points on the graphs without drawing all the rays (similar to number 1).

Group 8 - They color coded.  I love color coding.  Some groups didn't have any number labels on their axes or names on their airplanes, details, details.

Group 9 - erased.

I liked it.  It was a good conversation started.  Some people were using Law of Sines.  Okay, but they didn't need to because we had right triangles.  We discussed the distance away being the horizontal distance.  Some groups found their hypotenuse and used that though.

We live an hour West of Boston, so I was imagining all these planes departing Boston's Logan Airport.  I was wondering if any of the students might draw planes to the East and the West but they didn't.  I am going to do this again with Period 3 on Friday.  We will see their take on it.

Saturday, February 11, 2017

Persevering #Becoming Math

I am continuing to read Tracy Zager's book Becoming the Math Teacher You Wish You'd Had and am on "Chapter 9: Mathematicians Use Intuition", page 237.  It talks about doing Estimation 180 problems.  On page 238, it concludes, "Ultimately, what we really want them to do is to take this habit of mind and transfer it into their problem solving."

In my Foundations of Algebra I class yesterday, I was growing frustrated, losing my patience, feeling like I am not reaching them, they are not getting it.  Then, I read this and I regained hope and realized I am making a difference after I remembered our great discussions.

In class, I always start with the same opening structure, one Estimation 180 as a class, two Solve Me Puzzles as a class, and one Visual Patterns with all students at the boards in groups of 2 (there are 9 kids in class, one was absent).  Then, we get into our lesson.  Yesterday it was a paper card sort matching linear equations, graphs, words, and table of data.  But, let me back up.  The openers were great, the card sort, not so great.

We have done Estimation 180 since the beginning of the school year.  The kids question it - "Why do we have to do this?"  But, then sometimes, they really get into it.  And, whether they know it or not, they are getting really good at it and the discussions have been getting better and better. 

I invite one student per day to come up to my computer and run through the google doc with the conversation to poll the class for their range of too low and too high, then their guess.  The person at the computer gets final say as to the estimate entered.

Two days ago we did Estimation 180 Day 75 that asked "What is the weight capacity of the hospital elevator?'  There was some talk and some guesses.  We first had to decide what we were more familiar with - pounds or kilograms.  We decided on pounds.  We discussed how come it had to be so big.  We ended up guessing 3500 pounds and the answer was 5000 pounds.  The kids were shocked it was so high.

Yesterday, the Estimation 180 Day 76 question was "How many passengers can ride in the hospital elevator?"  with this picture given: 

Interesting conversation started right away.  I was trying to capture it all.  They knew the number 5000 pounds.  They said "Well, the average person might weigh between 140-180 pounds."  Another said, "It looks like 3 people can fit across the back."  I, myself, was thinking weight-wise, so I appreciated the visual idea of area.  Here we had some thinking about weight and some visualizing the space and area.  They decided that if you did 5000/100 pound person you would have 50 people, so 50 would be too many.  The students ended up guessing 24 because they thought it looked like 3 people could fit across and 8 people deep.  I loved it.  Great discussion, thought process, class coming to a good range and conclusion.  The answer is 33.  We were pretty close and they recognized that - "oh, that must mean there are 11 rows instead of just 8."

In Tracy's book, p. 239, Tracy is speaking with a teacher continues talking about the estimating process and says "we get to take that experience with us into the next problem."  The teacher says, "That makes perfect sense.  And that's why my students have so little intuition.  They've never been allowed to prove anything to themselves: they've just been told."

Next up were two Solve Me Puzzles.  If you haven't checked these out, they are great.  The kids can play on their computers but I take screen shots and put them into my class powerpoint. I put them on the quizzes and tests too.  I love the level of difficulty get increasingly more challenging and we are right at the point where the fractions are coming ("The fractions are coming! The fractions are coming!")  Solve Me naturally builds the need for fractions.  This was our picture yesterday with Puzzle #82: 
The kids automatically and quickly divide the top number in two and then in two again to determine the value of each branch.  Next they went to the far left with the branch of 8, divided it in half and looked at the three shapes.  Because the shapes are the same, I asked, "What same number added three times will give you 4?"  One student said to another student, "Can you do 4 divided by 3 on your calculator?"  I said, "Wait, what did you said?"  He repeated it. I said, "That is your answer."  At this point, the other student said "1.3" and this student then said, "yep, 1.3".  So, I wrote on the board, "So, 1.3 + 1.3 +1.3 = 4?"  I said, "you have to be careful because you are rounding that answer and rounded decimals are not exact.  Let's back up to what you entered 4 divided by 3.  How can you say that?"  Student asks, "4 thirds?"  I now wrote and asked, "So, what is 4/3 + 4/3 +4/3 or 12/3?"  Oh - four!  They say, "I still don't like fractions."  I say, "but they aren't that scary and they are more exact.  Just try them."

Here, I am not turning these into equations.  I know you can.  I feel that would turn the kids off to it.  Throw an x in there and they will shut down.  In Tracy's book on p. 239, she states, "It's true some special education students need more repetition, so what if we gave them extra opportunities to make sense?"  That is what I am trying here.  Take this picture and make sense of it, without throwing in the math and the variables.  And, they are doing awesome with it.

As you can see, our discussions have been long and these are just 2 out of the 3 openers but they are so valuable and worth it.  My third was this Visual Pattern #27. 
Again, I screen shot it and add it to my powerpoint.  They are given the first three steps and asked how many total squares are in Step 43.  It is always Step 43.  They know now that this is so they don't write all the way out to step 43.  They do enough to figure out the problem.  At this point, I have vertical whiteboards all around my room, so they get up to the board in groups.  Today I allowed them to pick their groups, but normally I would use Visual Random Grouping with flippity.net  I am kicking myself that I didn't take pictures of their work, but I remember their work well enough to replicate it in my own handwriting to share with you.  They have been working on these for about three month now.  Again, some love them, some hate them, a couple are really good at them, but we are all making progress.  Some draw lines to represent rows and columns, some color code, some use just numbers, some draw the actual shape (for example, a football helmet in a past problem).  They are allowed calculators, too.  Give it a try. 

Imagine this is in student handwriting.  It is student work.
At the front board, I had two students do this.  They are two who are really struggling with just getting started at any problem, but they did try and they did see something within the problem:

At the back board, I had three students come up with this.  They like to draw.  They always see it visually.  They love to color code.  They often will look for the number that it is increasing by.  If you look at the top, they are just doing the shaded blocks increasing because they already figured out the unshaded blocks.  There is always great discussion in these types of problems as to whether they should add the two numbers or multiply them.  Some are starting to see the difference. These students are.


This next work is from one student working by herself.  She used to draw pictures but once we starting looking around the room and comparing work, she realized she didn't have to really draw all the boxes.  She is one who is struggling with when to add or multiply.


And, this last student worked by himself.  He wanted it to be all him today.  He has gotten most of them recently, while working with a partner, but he wanted to show us, he could do it all by himself.  He is a man of few marks and uses lines for rows and columns.  He gets so excited for these problems because he is finding sucess and I see his confidence growing (All that makes it worth the time we are putting into it.)


When they are all done, I don't just share the answer to tell them if they are right or wrong and they know that by now, so they have stopped asking.  I ask them to look around the room and see what they notice.  I want them to learn there is often more ways to do a problem.  There are more ways to represent a problem.  I love the freedom they have on the whiteboards to spread out over space and the ability to erase gives them more confidence to get started. 

Then I ask the students to share their process.  The above was the ordered they shared in.  As the students noticed there were two of the same answer, so the two other groups figured out they must have done something wrong.  The first group realized they forgot to account for the gray blocks and the girl by herself recognized she didn't go up far enough.  They are self correcting.  They are self analyzing.  Again, I am not turning them into equations because that would stop them.  They are figuring it out with their own sense making. 

In Tracy's book on p. 240, it discusses this from Andrew Gael, "A lot of professionals are very good at scaffolding and creating step-by-step processes that lead to sucess for special education students.  Something I've been advocating for is that we don't scaffold out the mathematical thinking that goes into solving a math problem."  On page 241, Tracy continues going back to her master's program at U Washington and her professor Ilene Schwartz talking about the word "scaffold" - "imagine a building under construction with scaffolding all over it. The whole idea is, at some point, you take the scaffolding down, and there should be a building there that can stand on its own."

I could dive in and give my students a method, a process, an equation, but instead, I am just giving them an opportunity.  I am giving them an opportunity to make sense of it themselves because that is so much more powerful.  It is the experience they can take with them into the next problem (like from the beginning of this blog post.)  I really really hope they happen to have a lot of patterns on their state test next year, because I think they will rock them.

Thanks again Tracy for reminding me to stick with it.  It is paying off day by day.