Learning to add, subtract, multiply, and divide positive and negative numbers is critical in moving forward in mathematics. How can you evaluate expressions and solve algebraic equations without mastering this critical concept? After several years of teaching 7th grade, this is, in my opinion, the most important unit to master. After a short review of fraction and decimal operations, factors and multiples, I always start my school year with integers. I spend a good ten weeks each year making sure integer operations are solid.
What lessons do you need, and in what order?
My first lessons include the vocabulary of rational numbers. We complete frayer models for natural numbers, whole numbers, integers, and rational numbers. These vocabulary graphic organizers are then cut and glued into the interactive notebooks. Students then use another graphic organizer to sort numbers into the correct categories. FREE Rational Numbers Sort I also begin with a (clothesline) number line I string across one wall of my classroom. Students are given index cards with numbers they must clip on to the number line in the correct place, and spaced properly. Including fractions and decimals, both positive and negative, give students good practice ordering numbers.
Real-life Examples
A great hook for introducing integers is to give students some examples of integers in the real world. I let students come up with their own examples and then add some of my own. Temperature, elevation, sea level, bank balances, buying and selling, etc. are all examples of how students will use integers in their lives. We then add another frayer model for Absolute Value to add to their interactive notebooks. (This will prepare them to add integers with different signs, where they will take the sign of the number with the greatest absolute value.)
Integer Chips
Students love learning to add and subtract integers with integer chips. If your students are in person, you can pass out the red and yellow chips, and let the students practice adding positive and negative numbers and taking out zero pairs. I also add my digital version as homework. In 2020, I had to create a resource that allowed students to move these colored chips on their computer screen. Now that my students are back in person, I find that it’s a great supplement to the actual hands-on lesson. Adding Integers with Integer Chips
Adding Integers on a Number Line
The next lesson that is important to teach is adding integers on a number line. I model for the students how to start at zero, then move left on the number line for negative numbers, and right on the number line for positive numbers. And then I supplement this lesson again with a digial version. The digital version is fun for students. They move the red and yellow number lines on their screen, and learn to shorten or lengthen the arrows to model the addition. This leads next, to temperature on the number line. Showing addition of integers on a vertical number line is important for students to visualize the number line in a different orientation. (As they move to graphing on the coordinate plane, students need to be comfortable with both vertical and horizontal number lines. Adding Integers on a Number Line
Real-life examples of a change of zero
Introducing the vocabulary of opposites, and additive inverse, leads students to the next lesson I cover in my integer unit. Students will practice identifying real world examples of a change or a sum of zero. I give students examples, and they must decide whether the situation models a change of zero. “If it snows 6 inches on Wednesday, and on Thursday it melts 4 inches, is this a change of zero?” (No) “If a tree branch grows 3 inches, and then you trim 3 inches from the branch, is this a change of zero?” (Yes). There are many examples to model this, such as hair growing and hair trimming, adding 2 cups of dog food to a bowl, and a dog devouring it, rain and evaporation, etc.
The Rules for Adding Integers
I like to lead my students to creating the rules for adding integers. This starts with number sentences. Students must come up with number sentences in with 2 positive addends and a positive sum, 2 negative addends and a negative sum, one positive and one negative addend with a positive sum, and one positive and one negative addend with a negative sum. I also throw in two impossible number sentences: 2 positive addends and a negative sum, and 2 negative addends with a positive sum. Let students think for themselves and come to the realization that it is not possible. This will lead them to the rules: If the signs are positive, add and keep the sign. If the signs are different, subtract and take the sign of the number with the greater absolute value. They add these rules to their interactive notebook.
Fractions and Decimals with Negatives
I find this is the best to time to introduce negative and positive fractions and decimals. My digital lesson gives a pair of fractions, or a pair of decimals to add, and have students determine the sign of their answer. This can be extended to find the actual answer. However, I think it’s important to first make sure students are able to look at this addition and determine immediately the sign of their sum. You can then add positive and negative fractions and decimals for students to place on the clothesline number line. Fractions and Decimals with Negatives
Real-life examples of Adding Integers
To complete the addition of integers, I give real life examples and have students find the solution. For example, “If the temperature at 9:00 a.m. is -5 degrees, and it rises 10 degrees by 11 a.m., what is the temperature now?” “If a swimmer dives down 10 feet, and then swims upward 2 feet, how many feet below the surface are they now?”
Subtracting Integers: Student Number Lines
I introduce students to subtracting integers, by having student volunteers, model this on a large classroom number line. This can be placed along a wall, or using tape on the floor. (The wall works best for all students to be able to see the number line.) Starting at zero, the student will then walk to the first number. (Other students can give the instructions to the volunteer in order to model this subtraction.) If they are subtracting, they then turn and face left. If they are subtracting a positive number, the student will walk forward the same number as the second term. If they are subtracting a negative number, they will walk backward the same number as the 2nd term. The students will be able to see that subtracting a negative number means they move right on the number line. I then give them number lines and have them work with a partner to show integer subtraction on the number line.
Subtracting Integers with Integer Chips
How do you show subtraction of integers with integer chips? When subtracting integers you now subtract red chips when you subtract negative numbers, and subtract yellow chips when subtracting positive numbers. In this case, you may need to add zero pairs in order to have chips to subtract. For example, -8 – 5. To show -8, you show 8 red chips. But you can’t subtract a positive 5 because you don’t have 5 yellow chips to subtract. So you add 5 zero pairs. Add 5 red and 5 yellow chips. Now you can take away 5 yellow chips, and you are left with 13 red chips. -5 – 8 = -13. Give students several examples to practice. I also have my digital version to add as homework. (Another way to explain subtracting a negative is to”take away a negative”. When we take away a negative, we are left with a positive. Math Antics has a great video explaining this idea.) Subtracting Integers with Integer Chips
Change a Subtraction Problem to an Addition Problem
Once students understand subtracting integers conceptually, you can give them some tools to use to make this subtraction easier. Have students change their subtraction problem to an addition problem by adding a plus sign after the first integer. For example, 9 – 15 is the same as 9 + -15. -6 – 4 is the same as -6 + -4. Since students have already learned adding integers, this will give them a tool to use to make solving subtraction problems easier. Give several problems for students to convert to addition, and then give them problems to convert and find the solution. For the integer problems in which students are subtracting a negative, have them circle the two negative symbols that are next to each other, and change to a plus sign. For example, 8 – (-9) = 8 + 9. Also give students several problems of this type to solve. Subtracting Integers by Changing to an Addition Problem
Multiplying Integers
I begin showing multiplication of integers by using arrays.
3 x -5 (3 groups of -5). (-5 + -5 + -5 = -15).
-3 x 5 (Using the commutative property, we can switch the integers to 5 x -3.) (5 groups of -3) (-3 + -3 + -3 + -3 + -3 = -15)
-3 x -5 (The negative symbol can mean “the opposite of”. So -3 x -5 would be the opposite of 3 x -5 or positive 15)
You can model these with integer chips. After doing several of these arrays or with integer chips, have students come up with the rules with what the observe. If the signs are the same, the product will be positive. If the signs are different, the product will be negative. Multiplying Integers with Integer Chips
I then give students an Integer Multiplication Chart with 4 quadrants. Students fill in all of the squares and then make observations about each quadrant. Quadrant I will be positive x positive, so all products will be positive. Quadrant II will be negative x positive, and all products will be negative. Quadrant III will be negative x negative, and so all products will be positive. And Quadrant IV will be positive x negative, so all products will be negative. This visual gives students a more clear understanding of how this integer multiplication works. We also add foldables to the interactive notebooks. INTEGER FOLDABLES
Once students know the two rules for multiplying integers, I tell students the rules are exactly the same for dividing integers and then give them several problems to solve of both multiplication and division.
Multiplying a Series of Integers
When teaching a complete unit on Integers, it’s important to also show students how to determine the sign when multiplying a string of integers. They count the number of negative terms. If there is an even number of negative terms, the product will be positive. If there is an odd number of negative terms, the product will be negative. Give students several practive problems where they are multiplying a series of numbers. First have them just determine whether the product is positive or negative, and then give a string of integers for students to find the product. Multiplying a Series of Integers
You can then extend multiplying integers to multiplying fractions and decimals with negatives. There are many videos you can use to supplement your unit on Integers. Math Antics and Khan Academy are great for more visual lessons. I also make my own instructional videos with Screencastify so students can watch my lessons later to review.
Dividing Integers
The rules for dividing integers are the same as with multiplying integers. To give students a more conceptual understanding of dividing integers, teach with integer chips using the game of golf. Follow a professional golfer over a 3 or 4 day tournament. The best golf scores are in the negative numbers! Find the final score of the tournament by averaging each daily golf score. Add the total score for each day of the tournament, and divide by the number of days. Integer division! Dividing Integers with Integer Chips
Integer Slide Game
In order to review all year, I created an Integer Slide game which I assign all through the year. Different versions with different problems are themed for the school holidays: Halloween, Thanksgiving, Christmas, New Year’s, Valentine’s Day, St. Patrick’s Day, Easter, as well as Monster, Basketball, Football, and Pirate-Themed games. Played in “Slideshow” mode, these digital slides give students instant feedback for integer addition, subtraction, multiplication, and division. Pirate Version – Integer Slide Game
Complete your Integer Unit with a Final Project (PBL)! Integer Project: Hike the Appalachian Trail
If you’re searching for more lessons, puzzles, foldables, and games to supplement your lessons, click on the link to view the previews: