#SwDMathChat Integer Activities and Strategies

I love teaching integers! I think it's probably my favorite concept to teach. I've already blogged a few weeks ago about the Wii activity I did with my LLD class to introduce real life integer situations, so I thought I'd write this post about the rest of the activities and strategies I use to teach integer operations.  Since this is my LLD class, I really take my time to ensure my students understand each operation before moving on to the next. I try to give them as many tools and strategies as possible to make them feel successful.

Addition with chips

I spend a few days on integer addition.  One integer addition strategy that many teachers use with students is the two-sided chips manipulatives.  When I started teaching, this was my go-to activity for integer operations.  Then I discovered the Interactive Integers iPad app which I've blogged about previously.  Unfortunately our tech people were re-imaging all of the iPads so they were not available to use, so I had to fall back on the 2-sided chips.  I knew from previous experience with my OCR classes that my LLD class would have a hard time remembering which color is positive and which is negative so I wrote + and - on each side with a Sharpie.

I begin by demonstrating on the SmartBoard using an infinite cloner for the chips so I could drag out as many as needed, just like the app.  We discuss what a "zero pair" is and first related it to money - "If I had $1 (+1), but owed my friend $1 (-1), how much money would I have left?  Right, $0!"  I repeated this a few times with different amounts.   Then I go around the room asking each student what their favorite number is. For example, if they tell me 7, I say "so positive 7 and negative 7 equals...?" and wait for them to answer 0!  I repeat this with each student.

Now it's their turn - I write an integer addition problem on the SmartBoard and have them also write it on their individual whiteboards.  Then I give them chips and ask them to just set up or model the problem before attempting to take away "zero pairs".  I'm a real stickler for having them set up their chips neatly and lining up the chips under the correct number that they've written on their whiteboards. Once I know they can model the problems correctly, then I have them start the "game".  "Can you make a move (take away a zero pair)?"  Once they do that I ask again, "can you still make a move or is your turn over?"  They like thinking of this as a game. Once you cannot make any more moves, count up your chips for your final answer.

Addition with a number line

Then we move to the number line method.  I explain to them that most students end up liking this method better.  It's one that you can use anywhere, even on the PARCC test because as long as you have scrap paper you can always draw your own number line.  I even show them that if there's any coordinate plane somewhere on the test, you can use the x-axis as a number line.  All of my students' desks have these stick-on number lines from Nasco:



On the SmartBoard, I have this template set up with the "start" and "move" boxes and the number line:

I give them communicators with the same number line and plenty of empty space at the top for them to write in the problem.  (Note to self: as I'm writing this, I realized that I should create a new template for them with the "start" and "move" at the top instead of just empty space.)  I have them draw a dot at the "start" number, then use the "move" number to move that amount of times in the correct direction (I remind them that - is left and + is right, just like the all the negative numbers on the number line are to the left and all the positive numbers on the number line are to the right).  Most students pick up on this strategy very quickly.

Subtraction

Once we finish addition, I move on to tell them that I do not like subtraction, so we're going to change any subtraction problems to addition problems in order to use either of the two addition strategies we just learned.  I use the slogan "add a line, change the sign" to teach them how to do this.  After they have worked with subtraction for a day, the following day I give them a mix of addition and subtraction problems so they have to be aware of the operation and whether they have to change subtraction to addition before using the strategies.  This is were I see the most difficulty.  Students either forget to change the subtraction to addition and just solve it as if it were addition, or they change the subtraction to addition but forget to change the sign of the second number, or they attempt to "add a line, change the sign" in an addition problem.  I make sure to give them a lot of practice time with this to get comfortable with switching between addition and subtraction.  For my LLD students, when I gave them a quiz on integer operations, I made sure to keep the operations separate and specifically indicate whether they were addition or subtraction problems and which ones they should "add a line, change the sign".

One way I provide practice for integer addition and subtraction is to play a friendly game of bingo!  Students love the friendly competition and don't mind doing the work if it's fun.  I use BingoBaker.com to create my online bingo games.  I blogged about Bingo! a few years ago - check it out!

Multiplication and Division

On to the easy part!  I explain the really easy-peasy rules for multiplication and division and tell them that they can only use the calculator (basic 4-function calculators, no negative button!) for the fact part but must use the rules to determine the sign of their answer.  I emphasize that for addition and subtraction, they should use the number line (none of them this year liked the chip method better) and not the calculator, but for multiplication and division, they can't use the number line and should either do the fact in their heads or use the calculator.  Still I saw some students try to use the number line for multiplication and division, or try to use the calculator for addition and subtraction, but not too many!

Centers 

On Wednesday, we did centers for a review.  This is my favorite day!!  The centers are:

1.   integer operations cubes
2.   integer operations puzzle
3.   Math Practice iPad app

They have about 10 minutes at each center.  Partners worked together on the cubes and puzzle, and independently on the app.

       

Quiz

When they were working on the study guide Thursday and taking the quiz on Friday, I left the rules on the board for them to reference.


Now that this unit is over, I will allow my students to use the scientific calculators so they can use the negative button to do the calculations.  Integers are such as big part of the 8th grade curriculum so I want them to be aware that they are not the same as positive numbers, but then again, I'm realistic and know that if they must continue to do these operations without the aid of a calculator, even if they correctly learn the procedures for solving equations for example, they will get an incorrect answer if the integer operations are done incorrectly.  (Not to mention that use of a calculator is in every one of their IEPs.)

#IMMOOC "Pre-Reflecting"??

One of the new practices I have this school year comes from the ideas I got from my summer reading.  Before I begin a new topic this year, I am "pre-reflecting" (not sure if that's a real word or if I just made it up).  I am more cognizant of the prerequisite skills my students need coming into each topic and thinking about how I can incorporate more real world applications to help my students better relate to and understand how it applies to their lives.  Sometimes, with math topics as abstract as they can be at times, this is difficult.

George Couros quotes Chicago-area teacher Josh Stumpenhorst in his book The Innovator's Mindset: “Innovative teaching is constant evolution to make things better for student learning.”  George rephrases this as “What is best for this learner?”  The students in my resource classes come to me with all ability levels, so this is a question I am constantly asking myself about each and every one of my students every day.

Most "reflections" are done post-lesson, looking back at what went wrong or right.  I'm calling this "pre-reflection" because I'm looking ahead at what I can do to make the lessons I'm teaching this week better for my students before I even teach them.  I'm rethinking activities I have used in the past and modifying them to fit my new teaching/learning model.  In the past I have used a lot of technology, but after being a part of the amazing #IMMOOC group the past few weeks, I realized innovation is not all about the tech that you use.

I’m defining innovation as a way of thinking that creates something new and better. Innovation can come from either "invention" (something totally new) or "iteration" (a change of something that already exists), but if it does not meet the idea of "new and better," it is not innovative.   - George Couros

So innovative doesn't necessarily mean I have to throw tech at it.  It just has to be a change that makes it better than it was before.  My goal is to get my students from point A to point B and sometimes I don't need tech to do that.  I've incorporated more hands on materials this year to help my learners.  As Katie Martin said during last night's #IMMOOC live session, "You can't measure innovation but you can measure student outcomes".  I've narrowed down and fine-tuned my list of go-to tech apps this year and I'm only using the ones that really help my students understand the concepts better and improve their learning outcomes.

Innovation is something new and better.  This blog started out as me writing about all of our 'tech adventures in a middle school math class', but this year is a turning point for this blog, hopefully new and better.  I was even thinking of changing the name of it but couldn't come up with a good title.  George says "change is an opportunity to do something amazing".  I hope I'm making my students math experience this year at least somewhat amazing and memorable.

My new approach to "Do Nows"

"Do Nows", "bell ringers", "warm ups"... whatever you call them, I've been wanted to change my daily routine for a few years now, but up until this year my Do Nows always consisted of a quick review of what we learned the day before. While there's nothing wrong with this approach, and I occasionally still revert to this, what I've been wanting to do is to use this time to focus on improving my students number sense and mental math abilities. It's something I've always wanted to work on with them but could never find the time while trying to "cover" (my math supervisor hates that term!) the required curriculum. This summer after reading a lot about building number sense and daily routines, I decided the perfect place to build this practice into our daily routine was to steal the time from the Do Now time.

My three favorite resources for these new routines are Jessica Shumway's Number Sense Routines, Sherry Parrish's Number Talks, and a PDF I discovered online from San Diego City Schools entitled "Middle Level Mathematics Routine Bank".

The second week of school I started out with the "Number of the Day" routine from the "Middle Level Mathematics Routine Bank". I gave my students a random number and they had to write down 3 different ways to express that number. At first, they looked at me like I had two heads -- they had no idea what I was asking them to do. So I gave them a few examples. Unfortunately, most of them wrote down the super easy "+1" or "x 1" expressions, so I had to include in the directions that they could not use 1 as one of their numbers. The next day I gave them the stipulation that each of the 3 ways had to use a different operation. Gradually throughout that week I expanded the task so they had to use multiple operations in their expressions. I was very proud of one student who thought out of the box and was using exponents and parentheses!

I started out the third week of school with what I thought was a fairly easy routine - using mental math to add 10 to a random number, such as 37 or 148.  While most of my students found this easy, there were a few students for which this was quite a challenge.  I gave those students a hundreds chart and showed them that adding 10 just means moving down one row on the chart.  After a few days, they began to understand this concept.  So after the two-problem Do Now of adding 10 to a number, we did a "Count Around the Room" activity (Shumway calls it "Count Around the Circle" in her book).  They actually loved this and never wanted to stop!  I ended up going around the room twice each day.  Each day that week and the next I upped the ante and increased the number they were counting by to 20, 30, 40, and 50.  Some days we would count forwards, and some days we would count backwards.  The trickiest turns were the ones that required them to jump across a century (197 to 207 for example), but they improved on this as time went on.

The next week I changed it up and we multiplied a random number by 10, 20, 30, 40, and 50.  We discussed strategies for doing these calculations mentally.  I had explained to them that I wanted them to improve their number sense and be able to take apart numbers and put them back together again, so this was the perfect activity to demonstrate how that works.  We had "Number Talks" about all the different strategies they had used.  This brought us into the following week's Do Nows in which I taught them how to use the area model for multiplication so they were breaking numbers into their hundreds, tens and ones, multiplying using our mental math strategies from the previous weeks, and putting the numbers back together again.  That week I also showed them how to use partial products method instead of the traditional multiplication algorithm.  At one point we had 5 or 6 ways recorded on the board for solving one problem -- being the math geek that I am I was super excited!

What frustrates me is that after a few weeks of Do Nows consisting of "Counting Around the Room" by 30s or multiplying a number by 20 successfully, when my students are faced with the same type of problem in the content we are currently learning, they automatically reach for the calculator for these problems.  They are super dependent on their calculators, so trying to break them of this habit is going to be difficult.  I try to anticipate when they are going to reach for the calculator and head them off at the pass!  Hopefully after another month or so more of doing these types of Do Nows, they will gain more confidence and realize that they don't need the calculator for these mental math problems.