• Monitor student performance on the negative number worksheets used during the lesson to evaluate skill level.
• Assess students informally by observing interactions and performance during group activities and lessons.

The activities in this lesson are designed to show students real-life examples of multiplication and division of negative numbers, and to show how operating with negative numbers is a logical extension of the rules of mathematics. Students should understand how to add and subtract when negative numbers are included; the use of colored chips should help solidify the concept of a negative number.

Students will learn how to multiply and divide negative numbers by understanding that the rules are a logical extension of the rules of arithmetic they already know.

“You are at the fair and need money to go on the rides, buy food, and play carnival games. You borrow five dollars from your parents four different times. How much money do you owe your parents? How would you represent this mathematically as a problem?” Let students discuss these questions among themselves; prompt them to use a negative number somewhere in the setup. (You owe 20 dollars; or −20; (−5) • 4 = −20.)

Activity 1: Multiplying with Negative Numbers

Distribute the Multiplying Positive Numbers by Negative Numbers sheet (M-6-1-3_Multiplying Positive Numbers by Negative Numbers and KEY.docx) and have students fill in the table. Assist students as needed with filling in the answers, and challenge them to pay attention to the pattern that develops. Once students have completed the page, continue with the lesson.

“As you worked with the tables, you might have noticed that you were continuing a pattern. For problems like 3 • 1 and 3 • 0, notice that as you reduce the second number in the multiplication problem by one, you reduce the answer by three. Therefore, it makes sense that the answer to 3 • (−1) would be −3 to keep the pattern continuing.” (Reduce 0 by 1; reduce the answer by 3.) “This is the setup you use when multiplying negative numbers. If you need to multiply a negative number by a positive number, you multiply the numbers as you normally would, but the answer is negative.

“Now let’s explore what happens if you need to multiply a negative number by a negative number.” Distribute the Multiplying Negative Numbers by Negative Numbers sheet (M-6-1-3_Multiplying Negative Numbers by Negative Numbers and KEY.docx). “Complete the table by filling in the missing products.” Give the class some time to complete the sheet and assist students as needed.

Once students have completed the task or enough time has elapsed, explain, “As you completed these tables you may have noticed a pattern developing. What was occurring when you multiplied a negative number by a negative number?” (The answer was a positive number.) “Whenever you multiply a negative number by a negative number, you multiply the numbers as you normally would, but the answer is positive.

“Now let’s take a look at what happens when you have more than two numbers to multiply. If that occurs, complete the problem two numbers at a time, applying the rules we just talked about.” Work out the following example with the class:

3 • (−9) • (−2) • 4 • (−1) =

(−27) • (−2) • 4 • (−1) =

54 • 4 • (−1) =

216 • (−1) =

−216

Distribute Multiplying Negatives Problem Set (M-6-1-3_Multiplying Negatives Problem Set and KEY.docx). Have the class complete the worksheet, assisting students as needed. If you run out of class time, assign this page as homework for the next class session. After students have had a chance to complete the worksheet, ask “Do you notice a pattern when multiplying a series of positive and negative numbers?” (Yes, if there is an even number of negative values, the product will be positive. If there are an odd number of negative values, the product will be negative.)

Activity 2: Dividing by Negative Numbers

State the following scenario. “You owe a total of eight dollars to four friends. You owe the same amount to each friend. How much do you owe to each friend? How could you represent this as a mathematics problem?” (2 dollars to each friend,)

Ask, “Why should you use −8 in this problem and not +8?” (Answers will vary, but you are looking for something along the lines of the money owed is negative, so negative numbers are appropriate here.)

After completing the problem, explain to the class that it was an example of dividing a negative number by a positive number. Say, “When you divide a negative number by a positive number, you are splitting up the negative number into as many groups as you are dividing by and seeing how many will end up in each group.” Demonstrate this concept by making a pile of eight red chips and then separating them into four groups of two red chips. “Another way to think of this is as follows: If you are dividing a negative number by a positive number, you divide as you normally would, but the answer is negative.”

Have the class work on the following problems for a few minutes, and have a few volunteers share their answers. If students need additional practice, you can use the black and red chips to help them understand the concepts.

•     (−6)
•    (−5)
•    (−4)
•      (−1)

After the class has completed these problems, say, “Sometimes you will need to divide a positive number by a negative number instead. Based on what you have seen so far, how do you think we would solve the problem ?” (Divide the problem as usual, but since one of the numbers in the division problem is negative, the answer is negative.)

“Whenever you are dividing two numbers, if one of them is negative, then the answer is negative. Based on this, what is the answer to ?” (−5)

“What about ?” (−18)

“You may also have to divide a negative number by a negative number. Based on what we have done so far with multiplication, what do you think the answer is to the problem ?” (5)

Let the class discuss this problem for a minute. Say, “To solve this problem, you need to know what times −5 is −25. What happens when you multiply a positive number by a negative number?” (The answer is a negative number.) “If you are working a division problem, and you are dividing a negative number by a negative number, the answer is positive. It is important to remember that whether you are multiplying or dividing with negative numbers, the rules for negative numbers are the same for both operations.”

Separate the class into groups of two, and give each student a copy of the worksheet Dividing Negative Numbers (M-6-1-3_Dividing Negative Numbers and KEY.docx). Assist students as necessary as they work through the problems. After 15–20 minutes, have some volunteers from the class share their answers. Clarify any concepts as necessary.

Activity 3: Game Show

For this activity, divide the class into two teams. Explain to the class that they are going to play a game in which each team will have one student come forward and receive a question that involves computing some problem with negative numbers. The representative will need to give the correct answer; if s/he gives the correct answer, the team will get three points. Then you will give the next question to the other team and alternate until all the questions have been answered.

Each question will be stated as a story problem, and the team representative will have 60 seconds to attempt to figure out the problem. The representative can pass the question to the next team but will lose one point (−1). The other team must then attempt to solve it. An incorrect answer gives a team negative three (−3) points.

Play the game until all the questions are answered or time has expired. The team with the most points is the winner of this game.

Questions to ask for the game:

1. “You have five dollars and your friend repays you five dollars. How much money do you have?” (10)
2. “You borrow five dollars from four friends; how much money do you owe your friends?” (−20 or you owe $20)
3. “On a cold day in December the high temperature was 10 degrees and the low temperature was −25 degrees. What was the difference between the high and low temperatures?” (35)
4. “A submarine dives below the surface at the rate of five feet per second. How many feet below the surface is the submarine after 70 seconds?” (−350)
5. “The temperature is 10 degrees at 2:00 p.m. and decreases by five degrees every hour. What is the outside temperature at 5:00 p.m.?” (−5)
6. “A football team runs three plays. Players rush for a loss of five yards, and then pass for a gain of nine yards. Then they are sacked for a loss of three yards. How far did the football team move after the three plays are over?” (1)
7. “You owe 15 dollars to 3 different friends, and you owe each friend the same amount of money. How much do you owe each friend?” (−5)
8. “A frog moves 3 feet every 5 minutes, but then after 15 minutes a strong wind picks up the frog and moves it backwards by 30 feet. How far has the frog moved from its starting point after 20 minutes?” (−18)
9. “A business made the following gains or losses over a four-week period: 100, −250, 200, −50. What is the businesses gain (or loss) after the four-week period?” (0)
10. “You owe your bank ten dollars, but you are receiving two dollars a day from your parents. How much money will you have after eight days if you repay the bank what you owe it?” (6)

Extension:

• Routine: As real-world situations involving negative numbers arise throughout the school year, have students discuss and solve the problems as a class. This will keep the concepts of baseline and computation with negative numbers fresh in students’ minds.
• Small Groups: Students who might benefit from additional learning opportunities may be assigned additional practice problems or be allowed to use black and red chips while working through problem sets. The following Web site may be used for additional problem sets. Select any of the items from the Integers section. http://www.adaptedmind.com/Sixth-Grade-Math-Worksheets-And-Exercises.html?tagId=

For practice in the form of an online game, the following Web site may be used. The fruit shoot game may be played on a timed basis or an untimed basis. http://www.sheppardsoftware.com/mathgames/integers/FS_Integer_multiplication.htm

• Expansion: Have students who demonstrate proficiency create their own math stories that require the use of multiplication and division of positive and negative numbers. Then students can switch with a partner and solve each other’s problems. Immediate feedback is provided as students check over each other’s work.

Students also can take their partner’s math stories and rewrite them using the related operation. For example, one student wrote the following problem:

• A submarine dives below the surface at the rate of five feet per second. How many feet below the surface is the submarine after 70 seconds?

The student’s partner can solve the original problem and determine that the answer is −350 feet. Then the problem can be rewritten as a related division problem.

• A submarine was −350 feet below the surface (−350). If the submarine was traveling at a rate of five feet per second, how many seconds did it take the submarine to get that far below the surface? (70 seconds)