Integer Operations in Eighth Grade

It can be so difficult to teach kids what they think they already know.  Every year, integer operations seems to be one of those things.  Many of my students come to eighth grade knowing, "Same signs add and keep, different signs subtract, take the sign of the bigger number, and then you'll be exact" to the tune of "Row, Row, Row Your Boat."  Unfortunately, that doesn't always translate to accuracy when working with integers.  They know that "two negatives make a positive," but overgeneralize that rule to include integer addition.  I've struggled with helping my students build a conceptual understanding of how to work with integers when all they want to do is try to remember the rules they were taught.

Zero Game
To see what my students remembered about adding integers, I started with some problems where my students had to determine what value must be added to make zero.  I wrote problems on the board like the following:

-4 + ____ = 0

7 + ____ = 0

12 + (-9) + ____ = 0

-14 + 8 + ____ = 0

15 + (-20) + ____ = 0

We then played the Zero Game from Denise at Let's Play Math via Julie at I Speak Math.  The game is played with a deck of cards where red cards represent negative numbers and black cards represent positive numbers.  Cards are dealt to each player and the object of the game is to get a sum as close to zero as possible.  Read either of the posts above for more complete directions.  My students then completed this check for homework.

Positive/Negative Chips for Adding/Subtracting
The next day we began modeling how to add integers using positive and negative chips.  We modeled problems adding all positive integers first and my students did a Think-Pair-Share about their observations when adding all positive numbers.  Next we modeled adding all negative integers.  After a few examples, we did another Think-Pair-Share about what they noticed when adding all negative numbers.

Finally, we modeled adding numbers with different signs.  We moved all our zero pairs to the Sea of Zeros on this Integer Work Mat from Sarah Carter at Math Equals Love and students made observations about what happens when you add positive and negative numbers together.

Next we modeled subtracting integers using counters, adding in zero pairs as needed.  I tasked my students with using pictures to model what we had done in class for homework.

Adding/Subtracting on a Number Line
I love using a number line to think about adding and subtracting integers.  Adding means you move toward that end of the number line and subtracting means you move away from that end of the number line.  I like that this gets students thinking about why adding the opposite works for subtraction.

We did a few examples together and then my students completed this packet on their own.  The problems are paired so that students see an addition expression and its equivalent subtraction expression, and how the same movement is shown on the number line.  I wrote this packet up a few years ago and I'm not completely satisfied with it; however I haven't yet figured out how to change it.  All suggestions welcome!

When we reviewed this packet as a class, we spent a fair amount of time rewriting more examples of subtracting integers as adding the opposite.

Multiplying Integers Desmos Investigation
Next we explored multiplying integers.  This was my first time using Desmos with my students and I blogged more about that experience here.  We did Andrew Stadel's Multiplying Integers investigation.  Students modeled multiplication on a number line, looking at groups of positive or negative numbers and the opposite of groups of positive or negative numbers.  I've always been a fan of saying, "the opposite of" instead of "negative" and that tied in well with this lesson.

Integer Operation Rules Foldable
After four days of  what I hoped was a more meaningful look at working with integers, we were finally ready to summarize the rules for integer operations.  We put this foldable in their notes and did a few order of operations examples.  This foldable is available to download below.

One Incorrect Order of Operations Practice
The next day my students worked on this "One Incorrect" worksheet from Greta at Count It All Joy.  What I liked most about this worksheet was that my students knew that if they didn't get an answer of -13, they had most likely made a mistake somewhere since only one of the eight problems had an answer other than -13.  This was our first #VNPS day, and I let my students choose to either pair up or work individually around the room.

One extension to this activity was shared with me on Twitter: have students write their own set of "One Incorrect" problems.  I'm excited to try that the next time I use this practice structure. 

Two Truths and a Lie
Sarah Carter shared this template for a Two Truths and a Lie activity.  For homework, I had my students write their own statements about integer operations.  Some students chose to write about the rules, while others wrote numerical examples of using integers.  On the day of the quiz, we spent about half of the class reviewing the student-generated Two Truths and a Lie homework.  This gave us a chance to clear up some misconceptions as some papers had two or even three lies.

Integer Flashcards
Finally, I gave my students about ten minutes to quiz themselves using these integer operations flashcards from Sarah Carter before the actual quiz.

View/Download: Integer Operations Files