Tuesday, September 19, 2017

Talk Like a Pirate (and Practice Order of Operations!)

    Today was the best day.  One of those days that your lesson goes exactly like you want it to, the kids are amazed at what you're doing, and it all just falls into place.

     The first part of the day that was so awesome was related to the fact that is was International Talk Like a Pirate Day.  I've been looking forward to Talk Like a Pirate Day for awhile for two reasons.  Reason #1...my son has an awesome pirate hat that I looked forward to wearing to school.  I also had an old Pi Day shirt (Pi-Rate, When Good Numbers Go Bad) to wear...so perfect!

Reason #2....I had this idea this summer of making a pirate name generator.  I figure I could make up a problem (I used an order of operations problem) and the kids could roll dice, and plug the numbers into the problem.  The answer to the problem then generated the kids' pirate names.  So for example, in the first problem, the kids rolled 4 numbers and plugged them into this expression (___ + ___)^2 + ___*___.   So let's say you rolled 4, 2, 3 and 5 then your answer would be 51.  Then I had a table that told them different names for different number ranges.  So 51 meant the first part of your name was "Thieving".

Click on the picture if you're interested in purchasing this pirate name generator.

My pirate name generator had two parts.  My absolute favorite nickname of the day was Salty Fishlips!  Some of the other awesome nicknames:  Parrot Plankwalker, One-Eyed Devil, Jolly Dog, Bearded Cutlass, Gold-Toothed Buaccaneer....it was a blast!  And the kids got a little bit of order of operations practice in.  As I walked around, I really enjoyed hearing students explain to their classmates how to do the problem as they tried to get their pirate name.  

The only disappointment was that I really wanted a name that involved Scurvy Legs or Plankwalker, and the dice never let that happen for me!

Now, the other really awesome part of today's lesson was the part where I showed the kids how to use a spreadsheet....and they got it, and they were as amazed as I thought they should be at the power of spreadsheets.  But I'll leave that for another post!



Friday, July 7, 2017

Area Model in the Middle School Classroom

In my last post, I talked about using the multiplication chart as a tool in the middle school classroom. I really love this idea of building on elementary tools and techniques in our middle school classrooms. Making these connections to prior knowledge is important for students, and it makes our lives easier. So, today I want to talk about another elementary tool that can be useful in the middle school classroom: the area model. When students are first learning multiplication and area,  the area model are foundational for building understanding. Here are a few ways that I like to use the area model to help teach middle school concepts:

 1. Distributive Property--We all know that this is an important concept moving forward, but it can sometimes be tricky for students to wrap their minds around. I use lots of different strategies to help kids understand the distributive property, but the area model is definitely one of them.
The representation below can be seen as two rectangles, a 5 x 8 with an area of 40 and a 5 x 12 with an area of 60.  Or you can see this as one rectangle, a 5 x 20 with an area of 100.  This is a concept that is understandable for students, and it is a good way to reinforce our abstract ways of showing this concept.
          
area-model
area-model
Learning abstract representations of math can be one of the major challenges as students transition from elementary to secondary math, so connections like these can be helpful.

2.  Factoring--This is the natural extension of using area model to teach distributive property.  By simply leaving out the shared side length, we encourage students to factor, and help them see the connection between factoring and the distributive property.

It's good to start with an example that only shares one common factor, like this one.
area-model
         Students can see that the side length has to be the same number.  Next, we want them to make the connection between the same side length and a common factor of 35 and 56.  Student thinking might be like this: 
 What 5 times what equals 35?  8 times what equals 56?  
Finally, we want to students to make connections between the picture and the to the abstract work: 
35 + 56 = 7(5 + 8).  

Now, you can move to examples that have more than one common factor that could be factored out. 
area-model

40 + 60 = 5(8 + 12).  
Connecting the drawing back to the work is important....where do you find the 40, the 5, the 8, and so on in the picture?   

3.  Battling Common Misconceptions--If you give your students the problems (8)(4.5), would you be surprised to have some students give the answer of 32.5?  Me neither!   But the area model can again help us out.  
If students have been using area model to show the distributive property, this representation should be familiar.  This shows that the area is 36 and gives a visual illustration of why we can't multiply 8 x 4 to get 32 and simply add 0.5.

4.  Reinforce proportional thinking--If I had to pick one topic that was the most important thing we do in middle school, it would be proportional reasoning.  Every chance I get, every way I get, I want to reinforce proportional reasoning with my students.  I want to give them different ways to see it.  So what about this?
Since the side of 3 is the same for both rectangles, if you double the 4 to get 8, it also doubles the area from 12 to 24.

5.  Move towards algebraic thinking--Ultimately, our middle school students need to be ready for the demands of algebraic thinking.  The area model can also give us another way to get students using variables in middle school.  Consider the progression of the examples shown below.



If students are consistently using the area model as a representation in our middle school classrooms, hopefully the jump to the last two representations will be easier.  

      So we've looked at multiplication charts and area model...what other elementary models and tools can continue to be helpful in middle school?





Thursday, June 22, 2017

Multiplication Charts in the Middle School Classroom

     Every year when I put up my posters, I put a multiplication chart near the front of my room.  Until a few years ago, it kind of bugged me.  You know how it goes...kids coming in to middle school should know their multiplication facts, why should I need this, etc....   But then I finally realized I needed to start looking at the good old multiplication chart not for what it might have been in elementary school (although, yes, some kids still use it like that), but for what I could use it to show in middle school.  Because now what I see when I look at that multiplication chart in the front of my room is patterns, patterns, patterns!  That's what math is all about.  Here are some of my favorite ways to use a multipication chart in my middle school classroom.

1.  Equivalent fractions--The multiplication table is filled with row upon row of equivalent fractions.

Here you can see a simple multiplication table that I created on Google Sheets.  The thing I love about my Google Sheets multiplication table is that I can customize it in whatever way is useful.  So if I want to talk about equivalent fractions for 3/8, I can highlight those rows.  But then I can easily change to something else.    What a great way for my students that may struggle with equivalent fractions to have a quick reference to find them, but also a visual way to see that the reason by 12/32 is equivalent to 3/8 is because both 3 and 8 were multiplied by 4!  These realizations that may at times seem to be no big deal for teachers can absolutely blow the minds of our students.

2.  Equivalent ratios/ratio table--Just as we can use the multiplication table for equivalent fractions, it can also be used for scaling ratios up and down to find equivalent ratios.  So now our students are trying to answer some proportional reasoning question:  "It takes James 4 minutes to solve 7 problems.  How many problems can James solve in 12 minutes?"  Again, this idea of using the multiplication chart flexibly, even attaching a label or meaning to some quantity, can be a stretch for kids at first.
This would also be a great way to teach students to think about if the answer to a proportion question like this is even reasonable.  For example, what if the question had been, "It takes James 4 minutes to solve 7 problems.  How many problem can James solve in 11 minutes?"  Now the answer is not actually on the multiplication table...but a sense of what is reasonable sure is.  If 8 minutes is 14 problems and 12 minutes if 21 problems, then the answer must be between 14 and 21 (but closer to 21!).  That sort of amazing proportional reasoning can be supported by a great visual tool like the multiplication chart.

3.  Proportional relationships versus non-proportional (but linear) relationships--Let's say you were focusing on the proportional relationship of 6x = y.  What about showing the multiplication chart as a place to see this?
The multiplication chart can show the table of values for a proportional relationship by simply looking at the column with the correct constant of proportionality.   This gives another way to think of a proportional relationship....it is a relationship that if you had an infinitely large multiplication table, it would have a row on there.  You could also build the connection between 6x = y and 6x + 1 = y by having students add one to all of the values in the 6 column.

       Ok, and honestly, some students will use the multiplication chart because they don't know their multiplication facts.  As much as I wish this weren't true, it just it.  So rather than fighting against it, I've decided to help all of my students see that the multiplication table can be a great tool to help us learn about a lot of middle school concepts far beyond simply multiplying.  After writing all of this, I think this year, we just may create a digital multiplication table in the first few weeks of school to establish right away what a great tool it can be.

How do you think multiplication tables can be helpful in middle school?  What other "elementary" tools do you rely on to make your classroom a better place?

Wednesday, June 14, 2017

Beginning of Class Routine Revamp

        At NCTM, I got several ideas that I wanted to incorporate into my beginning of class routine, and I've been finding others as well.  Here is my beginning of class routine for next year.



Wonder Monday:  This idea is the culmination of a lot of reading and listening that I have been doing.  Jo Boaler's Mathematical Mindset, as well as her growth mindset course have really opened up my eyes to the need for math to be an open and creative field.  I've also been reading "Becoming the Math Teacher You Wish You'd Had", which talks about the importance of a "notice" and "wonder"...what do kids notice about a problem?  What do they wonder about?
           So this is my thought for how to get kids thinking creatively, as well as how math is woven in to so much that we do.  My plan here is to find a crazy or interesting picture each week, and just letting the kids start to wonder about it.  I think it will get their creative juices flowing, and hopefully start to see math as an open subject, with a place for interesting questions.  I think this will be a fun way to start each week!


Two Way Tuesday:  This one came directly from a wonderful session I went to at NCTM.  The idea of the two-way puzzle is that you add going horizontally and vertically.  I think the puzzle aspect of this will keep kids engaged, and I can see it being useful for all kinds of review content....fractions, decimals, whole number, integers, and combining like terms are the first few that come to mind.

In this example, the missing box in the top row would be 22, since -8 + 22 = 14.  The bottom left square would be -5, since -8 + 3 would be -5.  From there, you can fill in the rest of the squares.
What's the Question Wednesday:  I got this idea from another blog I was reading.  Basically, you give the answer, and the kids brainstorm what the question might have been.   Again, I think this could encourage creativity and help kids see that there are all kinds of ways to get to any given answer.


Number Talk Thursday:  This is something else that I've been reading about, and something that I heard about at NCTM.  The idea is basically that you give kids a problem to solve mentally, and then you let kids share their strategies for how they solved the problem.  I tried this out a couple of times toward the end of last year, and I was amazed at what a great use of class time it was.  The kids were highly engaged, and had tons of great strategies.  It also allowed for great discussion as we compared strategies.


Quick Draw Friday:  This is also something that I got at NCTM.  The idea behind it is that you give kids a short look at a geometric drawing, and they try to reproduce it.  Then you give them one more look, and a chance to revise.  Then let kids share their vision for how they saw the picture, and how they re-drew it.  I think this one can really lead to some great vocabulary, and my artistic kids will love it!  The idea comes from this e-book.


       So these are the ideas that I plan to use next year. If you would like a copy of the Google Slides shown above for this beginning of class routine, click here.

One other idea that I would also love to incorporate (but ran out of days!) would be to have a day each week dedicated to looking at a graph and focusing on what story it tells.  I think this is really important as we live in a world surrounded by data, with graphs everywhere trying to convince us of one point or another.  I may try to work this in somehow to my routine, but I can't decide what to give up!  Why is there always more to do than there is time?????

What routines do you use at the beginning of class that you love?


Sunday, June 11, 2017

A Good Math Class Discussion: Part 2

    In my last post, I talked about my presentation norms that I use in my class.  Today, I'm going to address another important part of a class discussion:  listening.  For most kids, listening is a passive activity.  It is our job to teach them to be active listeners.  These are the strategies I use to teach my students to be active listeners in class.

1.  Listen carefully.  The first one is pretty obvious and speaks for itself.  If you're not paying attention, it's hard to hear what someone else has to say!

2.  Write down questions, comments or notes.  I think we all fall into the trap of thinking that we will remember what we want to say, what question we wanted to ask, etc.. when it is our turn to contribute.  The reality is that if we jot down notes to ourselves, we are far more likely to remember things.  Making sure that students always start with a piece of paper in front of them, even if it's just a scrap of paper or a post-it, is very important in making sure that students are active listeners.


3.  Be ready to summarize what the speaker said...    This requires a focused kind of listening.  This requires that students be more ACTIVE in their listening. As students try to do this, I think it also requires them to really think about whether or not they understand the explanation that is being give.  This leads to the second half of this expectation.

4. ......or ask the speaker a question.  It was really important to me that my classroom listening norms leave room for students to NOT understand.  I always want to send the message that it is OK to struggle and not understand, as long as you're still trying and working.  At the same time, I want students to know that not understanding doesn't mean that you don't participate.  This expectations gives students a way to stay active and involved even when they don't understand.
 
5.  Think about how your strategy compares.   I want a classroom that is open to many strategies.  By comparing strategies, students can see more clearly how strategies compare.  The more students get used to comparing strategies, the more likely they can start to pick the best strategy for the given problem.

A Good Math Class Discussion: Part 1

      Good discussion is so important in class, and it supports the standards for mathematical practice.  Yet, we all know that good discussions don't just happen by accident.  Over the years, I have learned that I need to spend time teaching my class how to have a good discussion so they can really get the most out of it.  In this post, I'm going to focus on the presentation norms that I use in my classroom.

1.  Speak loudly enough for everyone to hear.   This one is pretty obvious, and yet we all have students that seem to speak at a whisper.

2.  Speak at a reasonable pace.  Again, seems obvious, but I know that students really seem to struggle with this for a variety of reasons.  For one thing, when kids get excited, they often rush when they are talking!  Unfortunately, that can really get in the way of other people getting understanding what you're so excited to share with them.

math-practice-smp6-critique-reasoning

3.  Pause after each step and make eye contact.  This one goes hand in hand with speaking at a reasonable pace.  I can't tell you how many times I have had students completely lose everyone in the room (even me!) trying to explain their method.  I find that there are two common reasons why kids get lost during another student presentation.  One reason is that presenters give all of their steps at one time, and this puts everyone's brain on overload if they're still trying to process the second step, and the presenter is talking about the fifth step!  The other common reason that happens is that a student doesn't understand something early on, so they either stop understanding or stop listening.
         For these reasons, I teach kids that they need to pause after each step and make eye contact.  This way, the listeners have a chance to process what you're saying as you pause.  Hopefully, when you make eye contact it will be obvious if the people that you're talking to are lost!
       I also find that it is very important to tell my class that this helps everyone....including me.  I like having my students see that I also have to ask people to slow down, repeat a step, or answer a question to clarify their method.  I think it is so important to normalize the process of understanding, and that needing someone else to slow down does not make you "dumb".

4.  Ask for questions from the class.   This one closely follows the last one.  If you are pausing after each step, it is a natural time to let people ask questions.  Hopefully when you continue, there is a better chance for your audience to understand what you're saying now.  Also, if you have more chances for questions, there is a better chance more people will understand by the end.

5.  Show visuals.  This can help for different kinds of learners.  It is also helpful to have it as a reference throughout the presentation.

At the beginning of the year, we spend time talking about and practicing these norms.  In my next post, I'll look at the other side of the discussion:  listening norms.



Tuesday, April 25, 2017

My NCTM Experience Part 3: Number Talks

  When I saw sessions on number talks in the program,  I knew that I wanted to go to one of them.  We are planning a statewide book study for that will launch at the KATM conference next year, and the 4-10 book topic is on number talks.  I've looked over a copy of the book that we're planning to use, so I know the basic idea of a number talk, but really wanted more information about putting it into action.

       The idea of a number talk is fairly simple: you give students a problem, and give them time to work the problem mentally....no pencil, no paper, no calculator.  Then have a discussion about different ways that students solved the problem.


        I was eager to try this idea in my classroom, but somewhat reluctant to give up the time (isn't it always about time!).  After attending a session on number talks in middle school, I was convinced that I wanted to make this part of my classroom.  It seemed like a fairly easy idea to implement and one that could really be the center of lots of good discussion.

       The session that I went to for math talks was a good introduction.  We watched some video clips of the instructor doing number talks in a classroom and analyzed them.  One of the most helpful things that we did was practice recording the thinking of our partners.  Some of the ideas were easy to record, but others were a bit challenging.  It was definitely helpful to spend some time thinking ahead about some of the best ways to record strategies to help students understand abstract representations.

         So this week, I actually tried out a number talk for my warm up the last two days, and it was awesome!  I will definitely be incorporating number talks into my warm ups a couple of days a week from now on.  The conversations we had around different ideas was phenomenal.   My first piece of excitement came from the wide variety of hands that I had in the air of students eager to share their strategies....and some of them were kids that definitely do NOT make a habit of raising their hand.   I have one kid that has been completely disengaged since spring break....like this kid's goal for state assessment was "To try and stay awake".....and he has had his hand in the air the last two days, sharing his ideas.  Is that not amazing???!!  :)

       The other thing that was so exciting was the huge variety of strategies.  The first problem I picked was 18 x 5, which I think was a suggestion I got from the session.  It was a great problem and it led to lots of different strategies.  Our discussion has included some of the following strategies:

  • 10*5 + 8*5 = 50  + 40 = 90
  • 20*5 - 2*5 = 100 - 10 = 90
  • (2*9)(5) = (2)(9*5) = 2(45) = 90
  • (9*2)(5) = (9)(2*5) = 9(10) = 90
  • 18 + 18 + 18 + 18 + 18 = 90
  • 18 + 18 = 36, 36 + 36 = 72, 72 + 18 = 90
  • counting up by multiples of 5
  • counting up by multiples of 5, starting at 60 since they knew that 5 x 12 -= 60
I was very pleased with this many strategies coming to the surface on our very first attempt!  And this one number talk brough up important ideas and vocabulary such as distributive property, associative property and commutative property.  

        So on day 2, I chose the problem 15 x 8.  I intentionally chose a problem that had an even number and a multiple of 5, hoping to encourage rearrangement of factors  to get to a multiple of 10.  Again, I had tons of hands in the air, and a wide variety of strategies.  As with the first problem, I had a variety of strategies used.  The most common ones were probably these:
  • 10*8 + 5*8 = 80 + 40 = 120
  • 15 * 2 = 30, 30 x 2 = 60, 60 x 2 = 120
  • 15 + 15 = 30, and there are four groups of 2 15s, so you would have 30 x 4 = 120
My favorite one, however, was the very last one of the day.  It came from a student that had already shared one strategy, and as he looked as the wall, he said, "Or you could use a clock.  The 15 is like 15 minutes, and there is 4 of those in an hour.  So it would take 2 hours to have 8 sets of 15 minutes, and I know that 2 hours is 120 minutes."  I mean seriously.....could I have asked for anything more!  What awesome, creative reasoning!

          So, after 2 short days, I am quickly a believer in number talks in the middle school classroom.  I can definitely see a ton of advantages to making these a part of my classroom from day 1 next year.



Monday, April 24, 2017

My NCTM Experience Part 2: Jo Boaler

       The very first day of the NCTM conference, I was so excited about all of the sessions that I forgot to leave myself time to eat lunch.  That's not even true--I knew that I hadn't left a lunch break but I just couldn't help myself!  Jo Boaler was speaking at 12:30 and I was NOT going to miss that.  It as well worth it (and I did manage to find time to eat a quick sandwich after Jo spoke).

     Earlier this year, I read Mathematical Mindsets and it was a truly amazing read.  Listening to her talk was equally amazing.   Jo talked about many of the points from the book but I also had a few different take-aways.  Here are some of the most important points I took away from this hour.

1.  "If you're not struggling, you're not learning."--I talk a lot about making mistakes with my kids at school, and I think I have done a decent job of helping them realize that mistakes are a good way to learn.  But this phrasing helps me realize I need to take that message a step further...I need to normalize the struggle, and not just the mistakes (or right answers).




2.  "Math is not about speed, it is about depth and multiplicity of ideas."--Again, this is not a new message for me, but hearing it at this session just helped reinforce how important this is.  According to Jo, much of math anxiety onset begins  with timed tests. Interestingly, she said that math anxiety most affects the high achievers.  This matches with my beliefs....I've always been one to give kids as much time as needed.  Looking at this made me realize that although I have never really associated  speed with being good at math, this is not something that I talk a lot to the kids about.  I need to do a better job of verbalizing this message to my kids....math should be about deep thinking and understanding over speed.


3.  Teach kids to be skeptics--I love this idea.  What a great way to encourage great discussion and listening skills.  Jo gave three levels of being a critic.....convince yourself, convince a friend, and convince a skeptic.  I'm trying to figure out exactly how to incorporate this into my classroom norms for next year, but I definitely love this idea.

4.  Math freedom--This was one of my biggest take-aways.  Jo showed several clips of kids from her summer math camp, and so many of them talked about freedom being the reason that they liked the camp when they didn't like math in school.  Jo expanded on this idea into two types of math freedom:  organizational freedom and math freedom.



  • Organizational freedom included several things such as how you handle talking, sharing, recording, spending your time and movement in your room.  I'll be honest....this one gives me pause as a classroom teacher.  I understand that kids like freedom in moving around and how they spend their time....but I also know that in my classroom, structure and procedures have always been a bedrock that help my room run effectively.  I don't want to discount this idea, but I do think it is easier to do some of these things in a summer camp setting versus a regular classroom setting.  This is one I will take some time to reflect on this summer and think of ways that I can use this.
  • Math freedom included things like interpretation of problems, how kids see problems, learning new ideas, how we think about mistakes, and ideas about inquiry and creation.  I really loved this idea of math freedom....that kids begin to see math as a subject that is not just a set of rules, but there is freedom about where to start and how to proceed.  I want kids to see that math can be creative in how you think about a problem and that it needs to make sense.
I was so inspired by all of these ideas that since I've gotten back, I've taken Jo's free online course for students call "How to Learn Math for Students".  It had such great messages for students that I'm trying to figure out how to incorporate this awesome material into my classroom next year.  I also enrolled in "How To Learn Math for Teachers and Parents".  This one was not free, but I'm so excited to see what else I can learn.  I've just started the course, and I look forward to all that I will learn.

I'm already starting to have some ideas about how I want to change up some things in my classroom next year.  One definite thing is that I will be starting next year with some form of the free online course.  The other big thing I have been considering is changing up how I do homework.  I really want to make it more self-directed...I think I'll blog more about this idea later as it is still forming in my head.  I just know that I'm wanting to move towards something that is differentiated and puts it to students to examine where there are at and push themselves.

Sunday, April 16, 2017

My NCTM Experience Part 1: Dan Meyer

I feel so lucky to have been sent to the NCTM Annual Conference in San Antonio last week by the KATM Board.  I am going to do a series of blog posts about my favorite sessions and biggest conference take-aways.  I'm starting with one of the last sessions that I went to.  

 Last week I sent an email to the generic Desmos email.  Imagine my excitement when I not only got the answer I wanted, but the email came from Dan Meyer!  Yes, I in my own geeky kind of way was soooo excited.  Fast forward to the NCTM conference when I was talking to another math teacher who starts to tell me about someone (can't remember who!?!) and said, "She's my math crush.  Who's yours?"  And while I may not have thought of it in those exact words, I had to admit that it was Dan Meyer.  Now fast forward to the 8 am session on Saturday morning of the NCTM conference....what a way to start my day!

Dan with Kira and me.  He has perfect long arms to take a selfie!
The title of the session was "Math is Power, not Punishment".  The big idea of the session was based around the idea that we need to create intellectual need in our students for what we are teaching them.  As Dan said, "Math is the aspirin, but what is the headache?"  He had some really great, quick activities to illustrate this point.  The most powerful one involved the coordinate plane.  Dan started with a slide of a bunch of dots on the screen and told us each to choose one of them.
         


Then the screen changed....all those dots were there, but there was a bunch of others.  Dan got a couple of volunteers from the audience.  Volunteer #1 had the job of trying to describe which dot was hers to volunteer #2...and let's just say that was a tough job!
Then it was Volunteer #2's turn, and here is what happened.  I thought this was a great illustration of the idea of creating intellectual need.



The examples that Dan used in the presentation were very meaningful for me as a middle school teacher....the need for the coordinate grid, and another activity that looked at the value of combining like terms before solving equations.

This idea is not just powerful for secondary teachers however.  As I left this session with my K-2 Math Enrichment colleague, it got us talking about a lesson she had been telling me about earlier in the day.  She had done a lesson using non-standard measurement units, such as hands, feet and so on. As she and I talked, we realized that this lesson on non-standard measurement units would be a great way to create intellectual need in her students.  See what headaches can be cured by using standard measurement units.

As teachers, we want to help our kids find the easiest way to do things.  But perhaps we are taking some of the value of process away by not letting them experience some of the headaches first.
This is truly a powerful idea....that if kids see the value of what a method saves them it will be more meaningful to them.  Think of all those things you teach.....why did mathematicians invent those things?  What headaches did they help cure?

Sunday, March 12, 2017

4 Wins Teaching Integers This Year

     I just finished teaching my integers unit this year, and overall I'm  really pleased with how things went.  The kids overall did pretty well, and some of my kids that struggle sometimes really hit this one out of the park!

     I've been trying to reflect on what I did this year that set my students up for success, and here are some of the things that I think helped.
integers-number-lines-formative-assessment



1.  Number lines, number lines, number lines
      We used them a lot!  One of the very first lessons in our Accentuate the Negative unit is a unit based on the number line.  It focuses quite a bit on looking at opposites on the number lines, and comparing which of two numbers are farther from zero.
     This year, I decided it was a great day to get kids moving.  So instead of doing the lesson out of the book, we did the lesson on a human number line.  I set our pieces of construction paper numbered from -5 to 5, and gave each of the kids an index card with a number on it.  Some of the numbers were integers, some fractions, and some decimals.  Then I called up 3-4 kids at a time to find their "spot" on the number line.  Once the kids were on the number line, I started asking the same types of questions that were in the lesson in the book: Who is farther from zero?  How do we know that Robbie and Ashley are opposites?  Where would Zoie's opposite stand?  Whose number is largest and how do you know?  How far apart are NiJa and David?
integer-number-line-#modelmathematics

     I was so excited at using this as an introductory lesson.  I was amazed at how much more engaged the kids were just because I got them out of their seats....they were totally into this lesson.  Also, it really got them thinking.  On day 2 of the integers unit, I was able to start asking really high-level questions because the kids could see it.  They were totally making sense of how far apart -2 and 3.9 are on the number line, and it was exciting to see them making sense of subtraction so early in the unit.  I think this also set the stage in student's minds for the number line being a helpful tool that we would rely on during this unit.

2.  New way to introduce the chip model
     I've always introduced both the chip model and the number line.  I really like the chip model and think it is helpful to make sense of things like why taking negatives away makes your answer bigger.  But my kids often struggle with the chip model, especially when they are having to take away more than what they have (problems like -3 - -6).  This year I changed the way I introduced the chip model.  Our team at school uses tickets, so I talked about a new "ticket" system.....where students can get positive tickets, which can be used kind of like money.  But now there are also negative tickets that would cause students to owe chores.  Then I posed situations to push their understanding. I really tried to get students to see connections....if a student does something good, then a teacher could give them a good ticket or take away a bad ticket.  If a student does something they shouldn't, the teacher could give a bad ticket, or take away a good ticket.  Students were able to see the connections between adding a negative and subtracting a positive, as they both had the same overall effect.
      I also had a card matching activity that I think really helped the kids make connections between addition and subtraction.  In the activity, the students had a copy of a number line with a problem on it.  They had to match with with an addition problem (example, -4 + -4), a subtraction problem (example, -4 - 4), the answer (-8), and a statement like "starts at -4, decreased 4 in value".  I think this really helped kids to see that -4 - 4 and -4 + -4 are really both the same problem, since they are both starting at the same place and moving in the same direction.
number-line-integer-#modelwithmathematics


3.  Lots of short, frequent assessments
      I gave quizzes almost every day....they were very short but it really helped me to keep track of what my kids were learning and what they were still struggling with.  I used quia or Google forms for most of the short quizzes, to make the grading lots easier.
     I also did a couple of days with addition of integers based on this model, and the kids did a great job with it.  Right after we talked about adding on the number line, I had a series of short assignments and assessments for kids to work on.  This was a great way to differentiate work and push kids to do as much as they could.  I had a series of Practice Problems with Exit Slips.  I had the kids work on the practice problems and I had the answers posted.  When they finished the practice problems, they checked the answers.  Then they got the exit slip which I checked myself to see if they were really understanding it.  These progressed in difficulty:  small integers, larger integers, fractions with common denominators, simple decimals, harder decimals, fractions with different denominators.
     I was SHOCKED at how much the kids enjoyed this.  One of my least motivated students was on fire during this activity....he was one of two students that finished all of the exit slips and I saw him push himself far beyond what he usually does.  They really wanted to work through the levels and worked really hard.  I was able to keep track of who I needed to work with in a small group because each exit slip was focused on a specific skill and I knew what they needed.  This simple activity was a total win!  I'm thinking I will upgrade this activity when I get a chance....make it more like a video game where kids can "LEVEL UP" for each activity, maybe build an avatar or something.  But even in its simplest form, it was really helpful.

4.  Visualizing the number line
       This super simple strategy was really effective. I need to do a LOT more of this in the future.  When we got to the point of using integers that wouldn't "fit" on the numbers I had available, I started asking the kids to visualize the number line, and the moves they needed to make.  When the kids took the time to actually do it, it really helped....even my kids that were struggling.  Hopefully once the kids realize this number line is always with them, they will rely on it even more!

These are all strategies I definitely hope to continue next year.  As I have moved into equations with negatives now, I'm still finding that these strategies are paying off.  As we talk about how to solve equations, I frequently find myself saying things like "Ok, when we subtract 5, is it rising or falling in value?"  or "Picture this on the number line....what direction would we go?".  It's been nice because it has made equations feel more connected to our work with integers.


Sunday, February 5, 2017

Confessions of a Mom with a Disorganized Kid

       This post is from the perspective of a mom, and what I've learned from that as a teacher.  My son is in 7th grade this year, and as a 7th grade teacher it's been interesting for me to see things from the other side.  It has really made me stop and think about some of the assumptions I have made about some students and parents over the years.

      First, let me tell you a little about my son.  He is loving and devoted to family.  He still hugs me goodbye every morning and waves to me as I pull out of the driveway.  He loves to play video games and board games, and he loves watching superhero TV shows together as a family.  And he has ADHD.....like he gets distracted in the middle of getting dressed.  He also has slow processing speed.  I've always known this about him.  If I ask him a question that I know he knows the answer to, like "Do you want macaroni and cheese?", he still stops to think about it for like 10-15 seconds.  I'm just starting to realize that this slow processing is a real thing, and not just for my kid.  Oh, and he has anxiety too.  Like we're talking him throwing up every morning before he went to school for the first month or so of school.

     So with all of this information, I was terrified about sending him to middle school.  I was terrified that he would be late to every class.  I was terrified that every assignment would be late.  I was terrified that his locker and binder would be a war zone.  I was terrified that his teachers would think he was being defiant when they asked him a question and he seemed like he wouldn't answer.  I was terrified.

      Some of this has turned out ok, and some of it....not so much.  Let's start with the good news.  Luckily, he has not been tardy.  I'm pretty sure this is because he has an amazing and flexible group of teachers that have absolutely taken him under their wings and done what they could to help him be successful.  So, no tardies for the win!  Shockingly, his binder and locker don't seem to be huge disaster areas.  He loses a few papers, but certainly not all of them.  So that is good news too.  As mentioned, he has awesome teachers who have gotten to know him.  They appreciate how difficult it is for him to answer in class, and build his confidence when he answers.  Honestly, his teachers have been nothing but supportive and wonderful this year.

       But let's be honest....this year has been an adventure.  And not always a good one.  There's been lots of issues that have caused this year to be hard and I don't really know when I see some of them getting better.  It's given me a lot of time to reflect on my practices as a teacher.  These are my take-aways as a teacher.

#disorganized


1.)  The more communication from the teacher the better.  
I will admit that as a teacher I don't always love communicating with parents.  It's a hassle sometimes.  You often worry that your attempt at alerting a parent to a problem will somehow turn into a storm that you didn't mean to cause.  But those daily e-mails from the teachers are a life-saver for me.  I can predict how the homework conversation will go every day....I ask my son if he has homework.  He tells me no.  I tell him what the email said, and he tells me the teacher didn't mention that was homework OR that he doesn't know what that is.  I go through his binder and find something resembling what was in the email, and we try to get it done.  EVERY. DAY.  Thank goodness I have the email, or my son would NEVER get anything done.
So as a teacher, I've tried to do better, even if just to send a quick email about that kid I'm worried about.  Yes, I still worry that it will cause a storm, but I'm hoping if I tell why I'm concerned, with no trouble attached, the parent will take it the way I mean it.

2.)  Parents that email all the time are not trying to bug me, and they probably did try to get an answer or action from their child.
I can hear all of the things I've said over the years. "It's right there on the calendar." "Why can't you just ask your kid?  Why do you have to ask me?"  "OMG, 's mom just emailed for the fourth time this week!".    
And right now as a parent, I wish I could take all of those things back.  I pause before I send every single email.  I worry that the teachers think I'm an idiot, or that I should talk to my kid, or that my kid should be handling some of these things.....all things I have thought as a teacher.  But as a mom, I'll just tell you....I do talk to my kid.   And he doesn't have useful answers, so I try to figure it out by piecing together what's in his binder compared to the email.  But every now and then, I need to hear from the teacher.

3.)  As a teacher, I have no idea what has happened at home when that assignment doesn't get returned.
I think this has been the hardest lesson for me.  My son has struggled with late work, and I think he will continue to struggle because homework is slow for him.  Painfully slow.  So teacher, I know that you gave time to work in class, and I know that the assignment is a reasonable length.  But an assignment that takes most kids 10 minutes takes my son an hour. I would say we spend two hours a night on homework, almost every night.  Plus a couple of hours each day on the weekend.  In two hours, we're lucky to get two things done.  Once or twice a week, I get my son up half an hour early (5:00!) so that we can get a little extra homework done.  And sometimes that is still not enough time to get his assignments done.  Do his teachers know this?  Probably not, although they have been great about letting me shorten some of his assignments for him.
As a teacher, I'm trying to do better to not just assume that kids didn't get their homework done because they didn't try.  Maybe that's true sometimes (maybe even most of the time).  But sometimes, that kid might have tried their best and simply not been able to get it finished.  And it doesn't cost me anything to give kids the benefit of the doubt.  Plus, even when we do get something finished, it often takes two or three days before my son remembers to turn it in.  Every morning, I clip the completed assignments to the outside of my son's binder so he won't forget to turn it in.  And each afternoon, with a little bit of dread, I ask if he turned in the completed assignments.  It's pretty hit and miss....about half the time he remembers.  If not, I cross my fingers for the next day.

4.)  Just because a kid has missing work doesn't mean the parents aren't checking their grades.
At any given moment, I know almost of my son's grades.  And I know what assignments he is missing.  And he still has missing work.  I look at his grades every single day.  But when new homework gets assigned every day, and we try to get that done, we have little time left for missing work.  It doesn't mean I don't care.  We're doing the best we can.  We're working hard.  This is our best.

5.)  If teachers can coordinate days that they assign homework, it really is helpful.
I've already mentioned that I love the daily email from the teachers.  But I will also tell you that I approach it with a certain amount of dread.  When I open that email and see homework in four classes, my stomach ties in a knot.  Because I know that we will never be able to finish that much homework in one night, and I know that my kid was not able to finish any of the assignments labeled as "homework if not finished in class".  So then I start to try to figure out which assignments will get done, and which ones will have to wait for another day. It isn't that we want to get further behind, but after two hours my son starts to melt down, and I can't really blame him.
I know that as a teacher, there are times that a certain assignment has to come on a certain day.  But if there are ways to give families more time and flexibility, maybe I could help out.

So there it is.  That's our school year.  It's been a rough year, but maybe I'll be a little more understanding moving forward.


Friday, January 27, 2017

Kahoot or Quizlet Live or Quizizz

I'm a big fan of Kahoot.  I love the instant feedback that I get.  Plus, let's be honest...the kids love it!  Recently, I've found two other things that are along the same lines as Kahoot, but each with their own advantages and disadvantages.  Below, I'll outline what I liked and didn't like about three online class games:  Kahoot, Quizizz and Quizlet Live.

Kahoot
Like I said, I love Kahoot.  I think it's a great teaching tool with formative assessment built in to what you're doing.  When we play Kahoot in my classroom, I have 3 rules:  1.  I have to be able to identify you from your nickname.  2.  Move to a place where you can see.  3.  It needs to be quiet between questions so I can talk.
Kahoot-classroom-technology


Advantages
I think the best thing about Kahoot is that the whole class is answering every question at the same time, so you can teach in between each question as needed.  You can give kids real time feedback about their answers, and often specific feedback about what mistake they made based on the wrong answers they chose.  There is really nothing like this, especially if you'e got multiple questions of the same type in a row.  As a teacher, you find out right away what is going wrong, and get to see if your explanation leads to improved results.
Student engagement is another great advantage.  The kids love it and stay involved....can't beat that!

Disadvantages
There are a few things that I don't love about Kahoot.  One is the general noise level.  My class gets LOUD when we play Kahoot.  Granted, it's because the kids are having fun, but still....that can get exhausting.  But really the main disadvantage to Kahoot for me is time.  My kids are always in a rush on a Kahoot to get the answer quickly so they can get more points.   Kahoots are also difficult to do with problems that take much time to solve, as it's a game of speed.

Quizizz
I just found out about Quizizz recently, and I've only had the chance to use it once.  In some ways, it is very similar to Kahoot.  You design a series of multiple choice questions for students to work through.  However, unlike a Kahoot, where the whole class is playing each question at the same time, on a Quizizz students are self-paced.
Kahoot-classroom-technology



Advantages
One cool thing I found about Quizizz is there is really easy ways to pull questions from different existing quizzes and merge them into one.  I'm usually really picky about my questions, so I loved this feature!  Noise level was another advantage.  Kids are basically working on the questions by themselves, and as soon as they finish, the next question comes up.....so less down time for things to get loud.  My students also liked not feeling rushed, and being able to work at their own pace.  The kids also loved the fact that it shows you a meme after each question.
Quizizz-classroom-technology   Quizizz-classroom-technology


Disadvantages
The biggest disadvantage is missing out on the teaching time between questions, since not everyone is on the same  question at the same time.  My original Quizizz was 16 questions long, but that seemed way too long when I wasn't getting to give feedback between questions.  I think if I were to use this again, I would make several small Quizizz games (2-4 questions each, probably) so then I could teach in between games as we reviewed the questions.

Quizlet Live
Quizlet Live has students working collaboratively on a set of questions, trying to match the answers. The way I understand it, each kid on the team gets the same question, and they all have several options to pick from....but only one of them has the right answer.  So they have to work as a group, seeing between their whole team who has the correct answers.  If they miss a question, they have to go all the way back to 0.  It is a race to see which team finishes first.
Quizlet-classroom-technology  Quizlet-classroom-technology


Advantages
The biggest advantage is that my kids absolultely LOVE this....maybe even more than Kahoot.  I also love that it adds the collaboration piece.  It also really makes them think, not even knowing if any of the answers are correct.  I like hearing the conversations my students have as they tell each other how they know that none of their answers are correct.  Great conversations!
Quizlet-classroom-technology

Disadvantages
This is a little embarrassing to admit, but one of my colleagues actually came and closed my door because my class was loud playing this.  But I'm also kind of ok with that because they were learning and having so much fun too.   The main disadvantage that I can think of is that to do quizlet live, you start out with a set of digital flashcards, and then it mixes the answers up.  I suppose there might be some content that is hard to figure out how to make it work on a flashcard.
Quizlet-classroom-technology



Friday, January 13, 2017

Teaching Percent of a Number: Is that the answer? I don't know....what is the question!

    I have been teaching percent of a number for the last couple of days.  As always when I teach this topic, I think that actually teaching the kids to find percent of a number is less than half the battle.  I used to teach the kids to change percent to decimal and multiply.  In the last several years, however, I've really become a fan of either percent tables or proportions.  I think they set the kids up to understand better what their answer means, and how to be flexible in working from that answer.

      Some years I have some percent stations that I use to introduce the kids to the different methods, but this year we used a pretty simple graphic organizer.  After going through a few problems together, they seemed to really be getting the hang of it....the math of it, that is!
percent tables #percentofanumber #modelwithmathematics
This is an example of one of the percent tables that we use.


      However, I can't tell you how many times I've been asked in the last two days, "So is that my answer, or do I have to add (or subtract) to get the answer?".   We talk many times about how reading plays a role in getting these questions right.   When a kid asks me that, I usually ask a question in return:  "Does your number answer the question?" and I read it back to them.  The tough part is that when I talk with kids, that makes sense, but then applying it on their own......my kids need something more.

     So at the moment, I'm feeling reflective, trying to think of ways to help this make more sense to my kids.  Here are 3 things I think I'm going to try....
percent of a number #erroranalysis


     1.  Highlighting---I think for my more visual kids, it might be helpful to highlight the percent in the problem, and what the answer is asking for.  Then students could look to see if what the question asks "matches" what the question gave them....if not, they need to do something else to get their answer.
percent of a number #storyproblems


     2.  Silly answer skits---I tried this once last year (with a different topic), and it was fun.  We basically turned a story problem into a short skit, and then acted it out with their "wrong" answers.  The last time I did this, it was based on the problem below.  When trying to answer the question how many peppers can you buy for $9 (which is right in the table!), many of my students told me the cost of 9 peppers.
absurd-math-errors-#storyproblems

So I thought if I had the kids act this out, they might realize their mistakes.  Here is the script we used.
When we acted this skit out, I think many light bulbs went on for students....helping them realize that their answer had to make sense.  I think I could use the same strategy for these types of problems. For example:  "Tax is 6%, book costs $10.  What is the total cost?"  I could have kids act out trying to buy a $10 book with only $0.60!  

     3.  Sorting cards--I'm excited about this idea.  I think I'm going to make my kids a card sort.  So, for example, if they have a card that says 6% tax, book is $10 then they will have to find cards that say $0.60 tax and $10.60 total cost.   I guess I know what I'll be doing this weekend.....making a card sort!

UPDATE:  I finished the sorting cards....can't wait to use them!


Thursday, January 5, 2017

Have I Mentioned that I Love Desmos.....

I know I've said this before, but I really love Desmos.  Today was one more example of that.

We are working on proportional relationships right now.  We've been working on  proportions and unit rates, but now it's time to move on to the next level of understanding.  I'm trying to get students to see the connections between what we have learned about unit rates, and tables, graphs and equations.

The last couple of days I've been doing a series of partner activities about proportional relationships.  The first activity was a breeze for the kids....find the unit rate.  The first time around all of the kids had the same unit rate so they could check each other's work.  Then we did another activity where they kids had different unit rates and had to compare and choose the better deal.

Then I had the kids graph the unit rates.  I don't know why I'd never thought to do this before, but it was a great addition to this activity.  On one graph, I had the kids graph both of their points (such as 12 books for $48 and 6 books for $24).  Then they graph the unit rate (1 book for $4).  It was a good chance for kids to make the connection about all of the points in a proportional relationship falling on a straight line.

With the second activity, I decided to save some time and show some graphs on Desmos instead of having kids graph them.  It may have been a split second decision, but it worked out great! In one class, the Desmos graphs brought up a major misunderstanding that I wouldn't have realized otherwise....and Desmos made it so much easier to clear up.  We were looking at this graph below, and my students though that the green line was a lower unit rate because the line was shorter...that was all they could see.
Luckily, with Desmos it was so easy to take care of this misconception.  I quickly changed from the tables I was using to equations, as shown below.  
As students watched me type the equation and saw the lines become longer, it was like I could see the light bulbs come on, and within a couple of questions they could explain not only which unit rate was better but what they had misunderstood.  This was definitely a case of technology supporting understanding, in an immediate way that other tools couldn't have done...this is technology at its best.  Simple, but awesome!