*This is the first part of the unit on negative numbers for seventh grade. It includes a general introduction, gains and losses, and adding negative numbers. Tomorrow will be subtraction, and the next day multiplication and division.*

In my Math 7 classes, the beginning of the school year begins with the unit on integers, which includes operating with negative numbers. A new teacher in a new classroom, with new school supplies coupled with a topic students haven’t had before often has the same allure and excitement as that of a shiny new toy that holds great promise for many exciting and life-changing hours.

Students view me, their new teacher, just as they do their brand new school supplies including the graph paper notebook that I hand out on the first day. New notebooks (particularly graph paper notebooks) hold the promise of being filled with information that will make them smart. After the first few lessons, it has not been unusual for me to hear from some parents that their kids have remarked “This is the first time I’ve really understood math.”

As things become more complex, the feeling of newness and promise fades. Students go from saying “I finally understand it” to “I hate math” sometimes in the span of less than a week’s time. It is also not uncommon for students to understand and carry out a procedure perfectly the day it is introduced, only to have totally forgotten it the next day, with some students asking “When did we learn this?”

Nevertheless, there is good news. Based on what they have been doing in the lower grades with differences of quantities and computing changes in weight, amounts of money and so forth, students at this point know what losses and gains are and how to compute them. We build on this prior knowledge and intuition so that they are able to express and compute quantities in terms of negative numbers.

The first lessons build upon what they have learned previously. As the topic becomes more complex and students become confused, teachers can and should refer back to some of the introductory techniques as a way to underscore that what is being taught today is building on what they perfectly understood just a few days previous.

It is also important to continue refreshing the procedures for operating with negative numbers throughout the year, to ensure what has been mastered stays that way. Continued repetition and practice helps to lock in the procedure and ensure automaticity.

Some aspects of negative numbers will seem abstract to students to the extent that a procedure may not make logical sense or seem counter-intuitive. The mathematician John Von Neumann once said “In mathematics you don’t understand things. You just get used to them.” Like many things in mathematics, after experience and practice with a procedure and concept, what was once alien becomes familiar. The familiarity eventually allows students to see the concept as reasonable and accept it—and they may even wonder why they ever found it confusing. At that stage, it is not unusual for students to say “I understand it now.” This is particularly true with the topic of negative numbers.

**General Overview**

The general arc of progression on this topic is:

1. Introduction to negative numbers: Number line, order, and absolute value

2. Gains and losses

3. Adding negative integers

4. Subtracting integers

5. Multiplying and dividing negative integers

**1. Introduction to negative integers**

This unit focuses on the mathematical operations of addition, subtraction, multiplication and division with negative numbers. Some students have may have learned the operations with negative integers in sixth grade, depending on what textbook was used, and/or the inclinations and goals of their sixth grade teacher. For those students, this unit will be a review. Others may have had an introduction to what negative numbers are, but not operations with them.

I start this unit with a review of what all students may know about negative numbers. This review is generally not included in seventh grade textbooks and focuses on concepts with which they are familiar. It is an overview of what negative numbers represent, using examples such as temperature (degrees below zero), or depth (feet underground or under water). I generally take two days to do this review and overview.

On the first day, we cover the general concepts of gains and losses, and their representation. A gain of $10 can be represented as +$10 or $10. A loss is represented as -$10. A descent of feet can be represented by the number -50. A drop in temperature of 10 degrees: -10

Some students will observe that you cannot possess a negative amount of anything. If no one makes this observation, then I do. It is worth mentioning that negative numbers can be a comparison, or a relative amount. For students just learning about negative numbers, it is a new way of expressing comparisons and changes. For example, students will readily answer the problem “If it was 60 degrees yesterday and 40 degrees today, what is the change in temperature?”

I will give students this problem and upon hearing the correct answer of twenty degrees, I ask if it is twenty degrees more or less. Hearing “less”, I then point out that it is a loss of twenty degrees. Since they have learned that a loss can be expressed as a negative number, I then ask if I can say that the change is -20 degrees.

Negative numbers can also be used to indicate direction, or relative position. I will ask: “In a football game, if your team lost ten yards, how would you indicate that using a sign?” Of course I want them to say -10 and they usually will. The negative number tells us about the position of the ball relative to where it started.

**Number Line. **The discussion about negative numbers and direction directly relates to the number line, so I introduce it at this point. Students have seen number lines before, but now we look at it with respect to negative numbers.

There are two simple principles for number lines that I state:

1. Negative numbers are numbers that are less than, or to the left of zero; positive numbers are to the right of zero.

2. The bigger the number, the farther it is to the right. The smaller the number, the further it is to the left.

A number left of another number is less than that number: 5 < 7, – 7 < -5. A number right of another number is greater than that number. -2 > -8.

I might ask “If it is -20 degrees today and it was -30 yesterday, which day was colder?” It is obvious that -30 will be colder, and by plotting the points on a number line on the board, students easily see that -30 is to the left of -20.

I will show two numbers to the right of zero on the number line; say 5and 7. When we have two positive numbers, the number furthest from zero is the greatest number. I will then show -5 and -7. They will quickly identify -7 as the furthest from zero when asked. “Is this number the greatest number of the two?” They will see that the opposite is true. When comparing two negative numbers, the greater number is the one closest to zero.

**Absolute Value. **Students may have heard about absolute value in sixth grade, but now it is presented in more detail. I will have students draw a number line and plot two points on it; say -4 and 4. I may also other pairs, using different colors. I will ask if -4 and 4 are the same distance from zero, and similarly for the other pairs. These numbers are called opposite numbers. I will ask them then to give me some opposites: What is the opposite number of 100? of -50? of 25?

I will then pose a situation in which we say that the numbers represent a football team’s loss on a play, and positive numbers a gain on a play. The point zero represents the point at which the play originated.

A loss of 10 yards on a play is represented as -10. Looking now at the opposite pairs I have on the board, I will say that these represent losses and gains on a play. Although a loss of 4 yards can be represented as -4, the distance itself is 4 yards—the distance from the starting point, expressed as a positive number. To make it plainer, suppose someone is wearing a fit-bit and paces out the 4 yard loss. How many yards will show up on the fit-bit? Will it be -4? No, it will be a positive number.

Distance is always a positive number. We call the distance from zero on a number line (or, in general, the distance from the starting point) the “absolute value” of a number indicated by two vertical bars; e.g., So whether a number is to the left or right of zero, their distance from zero is always expressed as a positive amount. I have them do some examples at this point, mixing in yesterday’s discussion about how to determine whether a number is greater or lesser than another, with today’s discussion of absolute value:

Examples: Find the opposite number: -3, 5, -500,

Find the greater number: -3, |-6|; -2, -10; -2, |-2|; |-54|, |53|

This lesson provides a lead in to the next day’s discussion on net gains and losses.

**2. Gains and Losses**

This lesson introduces students to the concept of adding negative integers. The approach for this lesson comes from JUMP Math and is a very effective way of introducing students to the concept of adding negative integers through the concept of gains and losses—without students realizing that that’s what they’re doing. After the first few minutes of working with the problems, it is amazing to see them doing intuitively what they will be doing when learning the formal rules for adding negative integers.

When I first started using the technique from JUMP, I mistakenly thought “This is going to be easier than I thought.” As it turns out, while it did make things easier, it wasn’t a slam dunk. Students will still get confused, and will look at what they learned with this technique as something that happened in the distant past and no longer applies. Which is why this particular lesson must be continually brought into subsequent lessons as a reminder that what they did intuitively is what they will continue to be doing. It’s just that the formal rules appear as a different entity.

**Opposite Integers.** We start with a review of opposite integers. A loss of 5 pounds is represented as -5, and a gain as +5, or just 5. Does a person who loses 5 pounds and then gains 5 pounds end up weighing more, less or the same as their starting weight? Students will generally agree that they will end up with the starting weight.

Money is an easy and effective way to work with gains and losses. I start by having students say what the integer is that represents a gain or loss and then state what the opposite integer is. For example, the integer representing a gain of $6 is +6 (or simply 6), and its opposite is -6. I note that while the initial number has the dollar sign attached, when it is expressed as an integer it is without the unit. A loss of $7 is -7, and its opposite is +7, or 7. (Later in the lesson they will be instructed to write positive numbers without the plus sign.)

**Identifying overall gains or losses. **Writing +7 – 4 on the board, and explaining that the numbers represent money, I ask was more gained or lost? If there is a net gain, we call it a good day; a net loss is a bad day. They will see immediately that it was a good day. When asked how much was gained, students are quick to tell me $3.

Writing -4 + 4, I ask was anything gained or lost? Nothing was gained or lost, so zero represents “no change”.

We continue with examples. For each one I ask how they came up with the answer. For the problem -6 + 2, there is a loss of $4, (written as -4), which they will explain they derived by subtracting two from six to get a loss of 4.

I paraphrase what they’ve done:

*“You have more of a loss than a gain. That means you have more negative numbers than positives. So we write it as an everyday subtraction problem with the signs reversed: 6 – 2. We get positive 4, but since it is a loss, we write it as -4.” *

Nothing fancier than that for now. Students will operate intuitively with the exercises in this lesson.

I point out again that if the first number is positive, like +7-8, we don’t have to write the positive sign. If we write 7 – 8, it is assumed that the 7 is +7, and is positive. Additional examples help get them used to this, although it may take longer than you would like before there will still be no more blank stares when the “+” sign is omitted from the first number.

**Two gains or two losses. **I will write on the board -5 -3=__? and ask if it is a good day or a bad day. If the response is stunned silence, I will state the problem as “I lost $5 and then I lost $3. Good or bad day?” They will immediately see it is “bad”. I’ll then ask for the overall loss while keeping my fingers crossed that they give me the right answer. They usually do. It is an overall loss of $8, so -8 would be what is written in the blank.

After writing +2+2 =__ students see that the overall gain is $4, so 4 (or +4) is written in the blank. This will be revisited in the next day’s lesson and stated as a rule, that adding two positive numbers results in a positive number, and adding two negative numbers, results in a negative number.

A common error is to interpret two losses of the same number as zero; i.e., – 7 – 7 is mistakenly thought of as 7 – 7. This mistake will come up repeatedly, and the remedy that I have taken is to remind students of what it represents in terms of two losses: “I lost $7 and I lost $7 more; how much did I lose in all?”

**Adding more than two gains and losses.** Now we up the ante a bit, with a problem like +3-4-5.

This can be solved sequentially. That is, the first two numbers are evaluated: +3-4 which students will know is -1. Then we are left with -1-5 which students will know is -6. An easier way is to add the total gains, then add the total losses. In this way we get +3 -9. It is an overall loss of 6 or -6.

Other examples: 2-5-4+8-3. Adding the gains (and remembering that 2 is the same as +2), we get 2 +8; the losses are -5-4-3. Total gains equal 10, and total losses are 12, so we have 10-12, for a total loss of 2, or -2.

For homework I assign problems from JUMP Math (see Figure 1). Problems can also easily be constructed as a worksheet.

Figure 1: Gain/Loss problems from JUMP Math from AP workbook 7.1, Common Core edition, 2015; Toronto; (printed with permission)

**Adding Negative Integers on a Number Line**

The next day’s lesson now represents what we did with gains and losses on the number line. Also, no matter how much they were on track during the previous day’s lesson, that was yesterday, and today is entirely different.

Because this is a new representation, students may think that the number line method is to be used for some problems, and gains and losses for another.

It is therefore important to tell students that the number line method is a way to look at what was happening in the previous lesson when we worked with gains and losses. The warm up problems for the day should therefore focus on some of the gains and losses problems, as well as opposite integers:

*If a football team gains 8 yards on the first play, how would we write that? (8) If they lose 10 yards on the second play, do they have an overall gain or loss? (loss) By how much? (loss of 2 yards or -2)*

The above warm-up, written as a gain/loss problem would be 8-10. I like to use a warm-up problem to segue to the day’s lesson. Using the above problem, students now do the problem on a number line. Rather than writing it as 8-10, however, we write it differently: (+8) + (-10). The purpose is to emphasize that we are *adding a loss* of 10 yards.

The rules for showing this on the number line are kept simple: The first number, 8, is marked on the number line. To add a positive number draw an arrow to the right the specified number of units. To add a negative number we draw an arrow moving left the specified number of units.

For the above problem we would draw an arrow with a length of 10 units going left from the starting point of 8. It ends at -2 which is the answer, as shown in Figure 2:

Figure 2: Number line representation of (+8) + (-10)

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12. To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

++ = +, and +(-) = -; Examples: +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

**Summary of the Rules for Adding Negative Numbers. **

**The sum of two positive numbers is positive.**

Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

** 2.The sum of two negative numbers is negative.**

Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

**Adding integers with different signs: informal rule**

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

**Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive. **

Example: Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: **Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.**

This is illustrated with an example:

Example: -5 + 3. The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

**Common errors. **The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

**Overthinking and the Lead In to Subtraction. **I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day.

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.

Students then do various problems using the number line, including problems where opposites are added such as (+4) + (-4). I generally allow about 10 minutes for this guided practice. Included among the problems are the sum of two positive numbers and the sum of two negative numbers.

After it appears that students have the knack of doing problems on the number line (with full recognition that such appearance may be like the mirage of water on a highway that disappears as you approach it), I select a few number line problems that they have done and have them write the problems without the parentheses. For example, the problem (+7) + (-12), is the same as +7 – 12 (or writing the first number without the positive sign, 7-12). I repeat the explanation that this is the same as adding a gain of 7 and a loss of 12. To help them write these problems without the parentheses, the following mnemonic, which I write on the board, proves useful:

++ = +, and +(-) = -; Examples: +(+5) = 5; +(-5)=-5

Students now rewrite the selected problems without brackets, and solve the problem as they did yesterday with the “gain/loss” problems. Having them do this reinforces and builds upon their prior success in the previous lesson. Each problem must agree with what they obtained using the number line. If it doesn’t, we find out why. This part of the lesson leads to a formal summary of the rules for adding negative numbers and which I have made copies for gluing into their notebooks:

**Summary of the Rules for Adding Negative Numbers. **

**The sum of two positive numbers is positive.**

Example: I gained 3 lbs last week and 2 lbs this week. Total gain is 3 + 2 =5.

** 2.The sum of two negative numbers is negative.**

Example: I lost 3 lbs last week and 2 lbs this week. Total loss is -3-2 = -5

**Adding integers with different signs: informal rule**

Since students have been working with gains and losses to add negative numbers, the rule can be stated informally in terms of what they have been doing in this and the previous lesson:

**Determine whether the sum represents a gain or a loss. Find the difference between the numbers. If it’s a loss, then give the answer a negative sign. If it’s a gain, it will have no sign, since no sign means positive. **

Example: Our team gained 4 yards and then lost 6 yards. Are we ahead or behind and by how much? The sum is represented as 4 – 6. The amount of loss is greater than the amount gained, so there is an overall loss, calculated as 6 – 4. The loss of 2 is written as -2.

In addition to the informal summary above, I explain to students that on the board is the formal rule for what they have just learned: **Subtract the lesser absolute value from the greater absolute value. Then use the sign of the integer with the greater absolute value.**

This is illustrated with an example:

Example: -5 + 3. The absolute values of the two numbers are 5 and 3. The integer with the greater absolute value is -5; since the sign is negative, the answer is –(5-3)= -2

The formal rule will make more sense at a later time after they have sufficient experience with these type of problems. For now, I work with the informal.

**Common errors. **The error of thinking of -5 – 5 as zero will persist. Since they have been working with number lines, I show students who make this error what -5 -5 looks like on the number line, and then show +5) + (-5) and (-5) + (5)

**Overthinking and the Lead In to Subtraction. **I once had a student ask “If you can’t have a negative amount of anything then how can you add -10 to something?” I explained that you don’t physically have -10 of something; you are representing a loss of 10, just like we did in the gain and loss problems. “How can you add a loss?” This is an example of overthinking. For such questions, it’s good to remind them of the gain/loss problems they did on the previous day.

Von Neumann’s quote about understanding in math and getting used to things applies here. The silver lining is that such confusion can be exploited in the next lesson in which subtraction of an integer is defined as addition of the additive inverse of the integer.

I have observed a significant percentage of students struggle with processes such as subtracting a negative or the absolute value of the difference between a positive and negative number . However, when the number line is placed vertically, the students more easily grasped these ideas. It appeared that up = positive, down = negative was a “natural” visualization as opposed to left = negative, right = positive. We then proceeded to a horizontal number line, but occasionally had to revert back to a vertical one. I am curious to know if you have had similar experiences.

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I use the vertical for temperature, but mostly I use the horizontal number line. What you suggest makes sense, so I’d be curious to hear from others if they have the same observation as you. Thanks.

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