Teaching Integer Operations Activities and Strategies

I love teaching integers! I think it's probably my favorite concept to teach. I've already blogged 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 help 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 two-sided chip 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. I had to fall back on the two-sided chips.

I knew from previous experience with my OCR classes that my LLD class would have a hard time remembering which color was positive and which was 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 can drag out as many as needed, just like the app. We discuss what a “zero pair” is and first relate it to money:

“If I had $1, but owed my friend $1, how much money would I have left? Right, $0!”

I repeat 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 equal...?” 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 students write it on their individual whiteboards. Then I give them chips and ask them to set up or model the problem before attempting to remove any zero pairs.

I'm a real stickler for having them set up their chips neatly and lining up the chips under the correct number on their whiteboards. Once I know they can model the problems correctly, I have them start the “game.”

“Can you make a move and take away a zero pair?”

Once they do that, I ask, “Can you still make a move, or is your turn over?” They like thinking of this as a game. Once they cannot make any more moves, they count the remaining chips to find the final answer.

Addition with a Number Line

Then we move to the number-line method. I explain that most students end up liking this method better. It's one they can use anywhere because, as long as they have scrap paper, they can draw their own number line.

I even show them that if there is a coordinate plane somewhere on a test, they 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 use a template with “start” and “move” boxes and a number line.

I give students communicators with the same number line and plenty of empty space at the top for them to write the problem. I have them draw a dot at the start number, then use the move number to travel the correct number of spaces in the correct direction.

I remind them that negative means move left and positive means move right, just as negative numbers are located to the left and positive numbers are located to the right on a number line. Most students pick up on this strategy very quickly.

Subtraction

Once we finish addition, I tell students that I do not like subtraction, so we're going to change subtraction problems into addition problems. That allows us 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 students work with subtraction for a day, I give them a mix of addition and subtraction problems. They must pay attention to the operation and determine whether they need to change subtraction into addition before using one of the strategies.

This is where I see the most difficulty. Students may forget to change subtraction to addition, change the operation but forget to change the sign of the second number, or attempt to use “add a line, change the sign” on an addition problem.

I make sure to give them plenty of practice so they become comfortable switching between addition and subtraction. For my LLD students, I kept the operations separate on the integer-operations quiz and clearly indicated which problems were addition, which were subtraction, and when they should use “add a line, change the sign.”

One way I provide practice with integer addition and subtraction is through a friendly game of bingo. Students love the friendly competition and don't mind doing the work when it's fun.

I use BingoBaker.com to create my online bingo games. I also blogged about Bingo a few years ago.

Multiplication and Division

On to the easy part! I explain the rules for multiplication and division and tell students that they can use a basic four-function calculator for the fact part, but they must use the sign rules to determine whether the answer is positive or negative.

I emphasize that for addition and subtraction, they should use the number line rather than the calculator. For multiplication and division, they cannot use the number line and should either solve the fact mentally or use the calculator.

I still saw a few students try to use the number line for multiplication and division or the calculator for addition and subtraction, but not too many!

Integer Operations Centers

On Wednesday, we completed centers for review. This is my favorite day! The centers included:

1. Integer Operations Cubes
2. Integer Operations Notes and Puzzle
3. Math Practice iPad App

Students had about 10 minutes at each center. Partners worked together on the cubes and puzzle, while students worked independently on the app.

Integer Operations Quiz

While students worked on the study guide Thursday and took the quiz Friday, I left the integer-operation rules on the board for them to reference.

Now that the unit is over, I will allow my students to use scientific calculators so they can use the negative button to complete calculations. Integers are such a big part of the eighth-grade curriculum, so I want students to understand that working with negative numbers is not the same as working only with positive numbers.

At the same time, I'm realistic. Even if students correctly learn the procedures for solving equations, they may still get an incorrect answer if they make a mistake with the integer operations. Calculator use is also included in each of their IEPs.

Looking for Integer Operations Activities?

These hands-on resources give students extra practice with adding, subtracting, multiplying, and dividing integers.

View Integer Operations Cubes View Integer Notes and Puzzle

"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.