This is the conclusion of the chapter on negative numbers for seventh grade. These chapters will eventually become a book on Traditional Math, to be published by John Catt Educational.
- Multiplication of Negative Integers
The multiplication of negative integers can be a confusing topic for students—particularly the rule that the product of two negative numbers equals a positive number. The main problem that students have is in seeing what multiplication by a negative number may mean.
I’ve found that providing examples of situations modeled by multiplication with negative numbers is effective in helping students understand the rules. I start with a review of negative and positive numbers in terms of changes in particular situations. That is, their daily experience with integers is how they describe things like temperature, electrical voltage, elevation above or below sea level, bank balances, and gains and losses.
Integers also can represent a change in the situation as they have seen with the “gains and losses” problems. That is, changes in money earned versus money lost, temperature increases and decreases.
To start, I ask students to describe various changes as positive or negative:
a) Ann gained 4 pounds in the last month. (4)
b) Jerome lost 14 pounds in a week. (-14)
c) Kathy lost $40 on a roller coaster ride. (-40)
d) Five minutes from now. (5)
e) Ten minutes earlier. (-10)
The next examples require a bit more thought; describe the change in terms of positive or negative:
f) The temperature changed from -3 degrees to 2 degrees.
g) The football team lost 5 yards on the first play and gained 10 yards on the second.
h) A bird was at 120 feet above the water to 30 feet above the water.
i) The water was turned on at 10:00 AM and turned off at 12 noon.
j) Janet finished the drive at 3:00 PM; she started at 1:00 PM.
Multiplication of two numbers in which one number is negative. Students know how to multiply positive numbers and know how to represent them as repeated addition. A problem like “Ted made $10 an hour for 3 hours; how much was his total pay?” is represented at 10 + 10 + 10 or 3 x 10.
After showing the above problem I ask students how they would write the following problem using repeated addition: “Sonia lost 3 pounds for 2 weeks in a row; how much did she lose after two weeks?” (If students need a hint, I will ask how they would represent a loss of 3 pounds.) Students will generally know the answer intuitively and upon hearing the answer of -6, I write on the board:
(-3) + (-3) = 2 x (-3) = -6.
Finding an example problem to represent (-2) x (3) is a bit more difficult since adding three -2 times does not make sense. Dolciani’s “Modern Algebra: Structure and Method” (1962) contains an example that I’ve used in seventh grade which provides meaning to the negative values.
The example is of a water tank with water flowing into it at the rate of three gallons per minute. I ask if the three gallons per minute is a positive or negative number. If it flows at three gallons for every minute, how much more water will there be in the tank after two minutes? Notice I am asking how “much more”, not what is the total amount of water. We are calculating the change in the amount of water in the tank and students are quick to give the correct answer of six. I now ask if the time, two minutes, is positive or negative. Both numbers are positive and we can represent the situation as 2 x 3 or 6 gallons more water than what was there before. This suggests the rule they all know:
2 x 3 = 6: Positive number x positive number gives positive number.
For the next scenario, I want to know how many gallons less was in the tank two minutes ago, if water is flowing into the tank at three gallons per minute. Intuitively, most students will know that there will be six gallons less, because the water in the tank is increasing for each minute. Therefore for each minute prior, there was three gallons less. In my experience students will shout out this answer. I ask how “2 minutes ago” is represented. I will hear someone, usually hesitant, saying “Negative two?” And that is correct. We know there will be six gallons less than there was, so we have:
(-2) x (3) = -6: Negative number x positive number gives negative number.
This example shows that the negative number can be the multiplicand (i.e., the number being multiplied) and the positive number the multiplier.
Now I tell the students to assume that the water is flowing out of the tank at the rate of three gallons per minute. I ask how we represent that, making sure they understand that it is -3, since it is representing a loss. The problem now becomes what is the change in gallons after two minutes. Students should recognize that the two minutes is a positive value. I ask how we represent this situation as a multiplication statement. Since this is similar to the opening problem about losing 3 pounds per week for two weeks, students should that the answer is once again 2 x (–3) = -6. This time the -6 represents a loss of 6 gallons. This problems suggests
2 x (-3) = -6: Positive number x negative number gives negative number
The final example is a negative number times a negative number. Before I present the example, I ask if anyone knows whether the product will be negative or positive. For those who say it’s positive I will ask why. Some explanations will be vague, but I’ve found that at least one person will extrapolate what we’ve done before. In such instance I then proceed with the example. If it doesn’t happen, it isn’t a problem. I just proceed and say something along the lines of “Let’s find out.”
In this case, the water is flowing out of the tank at the rate of three gallons per minute, represented as -3. I want to know if two minutes ago (-2) there was more or less water in the tank, and by how much. At this point, most will know the tank held six gallons more water. I now write:
(-2) x (-3) =6: negative number x negative number gives positive number.
At this point the rules are summarized:
- The product of two integers with different signs is negative.
- The product of two integers with the same sign is positive.
Admonition. After establishing the rules via the examples, I admonish students that the examples only suggest that these rules are true. There is a mathematical proof of these rules which I provide after they have learned about the distributive rule. For now, it’s all they need in order to get a sense of what’s happening when we perform these multiplications.
Nevertheless, there will undoubtedly be some students who will not understand how the examples work and why the rules are suggested by them. I tell these students that it will become clearer the more they work with such problems, but that for now should just follow the rules and “trust the math” that it is telling a true story.
Practice and extension. Students are now ready to work on guided practice problems, the first few of which summarize what we have just learned—problems such as (-5) x (-7), (-2) x (4), (-1)(5), and (-1)(5). After the last two problems I will give them the following: 5 + (-1)(5), and -5 + (-1)(-5). The first problem becomes 5 + (-5), which goes back to what they have just learned about adding negative integers. It is not unusual that they will look at the problem as if they have never seen it before. I ask them if they can remove the parentheses, and remind them that +(-) equals -. The problem then becomes 5 -5, or zero.
Similarly -5 + (-1)(-5) becomes -5 + 5 which is also zero. I will ask them to summarize what multiplying a number by -1 changes it into. There may be blank stares, and if so, I will remind them that 5 and -5 are called “opposites” which some have likely forgotten. It is not unusual for such lapses in memory, given the amount of new information that they are taking in. But it is important to link what they have learned in this lesson with what they know about addition and subtraction rules. To that end, other problems may include: 5 – (-2)(5); -4 + (3)(-2), and so forth.
So for 5 – (-2)(5), the number (-2) is multiplied by 5 which yields -10. The resulting problem is then 5 – (-10). From this point, a common mistake will be forgetting to include the negative sign next to 10, and writing 5 -10.
For these type of combination multiplication and addition/subtraction problems, it will be necessary to remind them of order of operations which they have had in sixth grade—multiplication operations are performed before addition or subtraction. Also, using parentheses to denote multiplication should be explained. This notation is easier to work with than the form 5 + (-1) x (5), since the parentheses make the order of operations more obvious. In addition, using parentheses for multiplication prepares students for algebraic notation. Later when they work with evaluating expressions like a + bc, by substituting numbers for a, b, and c, they will already have had experience with that form. Also, they will see that inan equation in the form 2x +4 =8, the 2x represents 2 multiplied by an unknown number.
Multiplying more than two numbers. Before I set them loose on homework, I put up a problem and ask how I would solve it: (-2)(3)(-5). Those who get it, I then ask to explain to the class. Such problems are broken down and solved by multiplying two numbers at a time; they above problem then becomes (-6)(-5), which is 30. A problem like (-5)(2)(-2)(4)(-1) becomes (-10)(-8)(-1), which is -80. After maybe two more, I will ask them if the number of negative numbers in the problem helps them determine whether the final product is negative or positive. I have them discuss that, leading them to see that an odd number of negatives will result in a negative product.
- Division of Negative Integers
It is definitely recommended that the warm-ups preceding today’s lesson include not only problems about multiplying negative integers, but also addition and subtraction. New information recently learned tends to eclipse older information. This can be seen by a common mistake that students will start making, and I’m sad to say will persist among some students well into the school year. That mistake is to conflate the rule stating that the product of two integers with the same sign is positive with the rule for addition of two integers with the same sign. Students will see a problem like -5 -5 and now in addition to making the mistake of saying it is zero, will then say it is positive 10.
Warm-ups should include straight multiplication problems like (-2)(5) and (-1)(-4)(3), but also a problem like 2 – (-2), 2 + (-5), and the combination multiplication and addition/subtraction problems that get at the problem described above. For example -5 – (-1)(-5), which becomes -5 – 5.
Going over the warm-ups will then serve as a review of what has been covered so far. At this stage I have found myself saying “Why did I ever think this was going to be easy?” Some classes will get it more easily than others, I’ve found, but there are always relapses, forgetfulness, and it will be necessary to repeat the rules.
Division as inverse of multiplication. Division of negative numbers is a straightforward application of the rules for multiplying negative numbers. The rules are the same:
- The quotient of two integers with like signs is positive.
- The quotient of two integers with unlike signs is negative.
I start the lesson by asking what 2 x (-3) is. Hearing -6, I then write 2 x (-3) = -6. Since division is the inverse of multiplication we can divide the produce, -6 by either 2 or -3 to get the other factor. That is -6 ÷ (-3) = 2, and -6 ÷ 2 = -3. In other words, when we divide -6 by 2 we are seeking a number which when multiplied by 2 yields -6. That number has to be negative. I will ask the class why and then wait while an uncomfortable silence pervades. I would like someone to say “To get -6, 2 has to be multiplied by -3 because a positive number multiplied by a negative number is negative.” I will settle for something reasonably close, however. If someone says that -3 x 2 equals -6 I’ll go with it and give further examples. I don’t want to spend a lot of time trying to get them to say the right thing and then have them forget what it is we’re learning.
Similarly if we divide 6 by -3, the number that is multiplied by-3 to yield 6 has to be negative. Why? Because the product of two negative numbers is positive.
I will then provide examples for the class to work on and include combination problems such as -10÷2 -3. Again, they will need to be reminded of order of operations; i.e., division operations occur before addition or subtraction.
The bottom line rules are exactly the same as multiplication: for two numbers which can be paraphrased into something succinct enough to fit on a bumper sticker: Like signs: positive. Unlike signs: negative.
Despite the fact that the rules are ultimately the same, students will still get confused. It is a matter of repetition and practice. Also, it is essential to include all types of problems that involve computations with negative numbers, not only in warm-up questions but in quizzes and tests given throughout the year.
I conclude this sub-chapter by providing examples of warm-up questions that I have given after covering all the topics discussed in this entire chapter.
- -3 x 3
- (-5 + (-20) ) ÷ (-2)
- -60 ÷ (-10)
- (5-10) x (4-8)
- (5-(-4) ) ÷ (-3) = ? + 6
Problem 5 is a more challenging problem and it is admittedly a “front-loading” of the type of problem they will be solving later when we cover expressions and equations. Typically this problem opens up discussion when we go over the problems despite the hints I gave students when they are working the problems.
Solving the left hand side results in 9 ÷ (-3) which equals -3. We now have to find what value the question mark represents.
I will sometimes give them a simpler problem that they’ve seen before such as 9 = 7 + ?. They will solve this quickly (usually—there are always exceptions!) and hearing the correct answer, I ask how they did it. The usual answer is “Two added to seven is nine.” Which is correct, though I’d much rather hear, “Nine minus seven equals two.” If I don’t hear that, I will say it; something like, “So if 9 = 7 + 2, we can also say 9 – 7 = 2 and 9 – 2 = 7.” This same pattern can then be applied to the problem at hand : -3 -6 = -9.