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.