This lesson expands on the three techniques to computing with integers by covering multiplication and division. Students will: 
- multiply and divide integers. 
- investigate story problems that involve negative numbers. 

- How can mathematics help to quantify, compare, depict, and model numbers? 
- How can mathematics help us communicate more effectively? 
- How are relationships represented mathematically?
- How are expressions, equations, and inequalities used to quantify, solve, model, and/or analyze mathematical problems?
- What makes a tool and/or strategy suitable for a certain task?

- Integer: A real number that does not include a fractional part. 
- Negative Number: A number with a value less than zero. 
- Opposites: Two numbers whose sum is 0. (For example, 3 and −3 are opposites because 3 + −3 = 0.) Also knows as additive inverses. 
- Positive Number: A number with a value greater than zero.

- black and red chips 
- copies of the Multiplying Positive Numbers by Negative Numbers sheet (M-6-1-3_Multiplying Positive Numbers by Negative Numbers and KEY) 
- copies of Multiplying Negative Numbers by Negative Numbers sheet (M-6-1-3_Multiplying Negative Numbers by Negative Numbers and KEY) 
- copies of Multiplying Negatives Problem Set (M-6-1-3_Multiplying Negatives Problem Set and KEY) 
- copies of the Dividing Negative Numbers sheet (M-6-1-3_Dividing Negative Numbers and KEY)

- Monitor student performance on the negative number worksheets used in the lesson to assess skill level. 
- Informally assess students by monitoring their interactions and performance in group activities and lessons. 

Scaffolding, Active Engagement, Modeling and Explicit Instruction 
W: Students will study rules for multiplying and dividing negative numbers. 
H: Allow students to discuss and answer a word problem involving a negative number. Distribute the Multiplying Negative Numbers worksheet, and have students complete the first page. Have students look for the pattern that develops in their answers. 
E: Engage students by exploring the pattern that developed in their answers. Continue to the second page of the worksheet, which contains multiplying a negative number with a negative number, and take note of the pattern. Discuss multiplying more than two numbers and the results of an even or odd number of the integers being negative. 
R: Provide an example of division with one of the numbers being negative. Use the colored chips as needed to visualize the problem and to determine that the answer will be negative. 
E: List division problems for students to solve. Discuss the results when one or both numbers in the problem are negative. Distribute the Dividing Negative Numbers worksheet for students to complete. 
T: Use the Extension section to personalize the lesson to the students' specific requirements. The Routine section includes strategies for reviewing the lesson topics throughout the school year. The Small Group section contains suggestions for students who could benefit from more examples or practice. The Expansion section contains suggestions for making the lesson challenging for students who are willing to go above and beyond the requirements of the standard. 
O: This lesson aims to help students understand the rules for multiplication and division of negative numbers and apply the rules for successful multiplication and division. 

The activities in this lesson are intended to provide students with real-world examples of multiplication and division of negative numbers, as well as to demonstrate how working 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 clarify the concept of a negative number. 

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

"You're 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 do you owe your parents? How would you express this mathematically as a problem?" Allow students to discuss these questions among themselves, and encourage them to use a negative number anywhere in the setup. (You owe 20 dollars, or −20; (-5) • 4 = −20.) 

Activity 1: Multiplying with Negative Numbers

Distribute the Multiplying Positive and Negative Numbers sheet (M-6-1-3_Multiplying Positive Numbers by Negative Numbers and KEY) and have students complete the table. Assist students as needed in filling out the answers, and encourage them to see the pattern that develops. Continue the lesson when students have completed the page.

"While working with the tables, you may have observed that you were following a pattern. For problems like 3 • 1 and 3 • 0, note that when you reduce the second number in the multiplication problem by one , you reduce the result by three. To continue the pattern, it makes sense to answer 3 • (−1) as −3." (Reduce 0 by 1 and the answer by 3.) "This is the setup you use to multiply negative values. If you need to multiply a negative number by a positive number, you multiply them as usual, but the result is negative.

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

Once students have finished the activity or enough time has passed, explain, "As you worked through these tables, you may have noticed a pattern developing. What happened when you multiplied a negative number by a negative number?" (The result was a positive number.) "When you multiply a negative number by a negative number, the result is positive.

"Let's see what happens when there are more than two numbers to multiply. If that happens, solve the problem two numbers at a time, following the rules we just discussed." Work through the following example with the students:

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). Allow the class to finish the worksheet, guiding students as needed. If you run out of class time, assign this page as homework for the following class session. After students have completed the worksheet, ask them, "Do you notice any patterns when multiplying a series of positive and negative numbers?" (Yes, if there are an even number of negative values, the product will be positive. If the number of negative values is odd, the product will be negative.)

Activity 2: Dividing by Negative Numbers

Describe the following situation. "You owe a total of eight dollars to four friends. You owe the same amount to each friend. How much do you owe each friend? How would you express this as a math problem?" (2 dollars for each friend, -\(8 \over 4\)  = -2)

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

After completing the problem, tell 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." Create a pile of eight-red chip and divide it into four groups of two red chips to demonstrate this concept. "Another way to think about it is as follows: If you divide a negative number by a positive number, you divide normally, but the result is negative." 

Allow the class to work on the following problems for a few minutes, then have a few volunteers give their answers. If students require more practice, you can use the black and red chips to help them understand the concepts.

-\(12 \over 2\)    (-6)

-\(25 \over 5\)     (-5)

-\(40 \over 10\)   (-4)

-\(7 \over 7\)       (-1)

After the class has finished these tasks, say, "You may need to divide a positive number by a negative number instead. Based on what you've seen thus far, how do you think we should solve the problem -\(12 \over 3\)?" (Divide the problem as usual, but because one of the numbers is negative, the solution is negative.)

"Whenever you divide two integers and one of them is negative, the result is negative. Based on this, what is the solution to \(25 \over -5\)?" (-5)

"What about \(36 \over -2\)?" (-18)

"You may also have to divide a negative number by a negative number. Based on what we've done so far with multiplication, what do you think the answer to the problem \(-25 \over -5\)?" (5)

Allow the class to discuss this problem for a minute. Say, "To answer this problem, you must know what times −5 is −25. What happens if you multiply a positive number by a negative number?" (The answer is negative number.) "If you're working on a division problem and divide a negative number by a negative number, the result is positive. It is important to remember that the rules for negative numbers apply to both multiplying and dividing with negative numbers."

Divide the class into two groups and distribute each student a copy of the worksheet Dividing Negative Numbers (M-6-1-3_Dividing Negative Numbers and KEY). Assist students as needed while they work through the problems. After 15-20 minutes, have some class members volunteer to give their answers. Clarify any concepts if needed.

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 answer a question about computing a problem with negative numbers. The representative must give the correct answer; if s/he does, the team will receive three points. Then you will give the following question to the other team and alternate until all of the questions have been answered.

Each question will be presented as a story problem, with the team representative having 60 seconds to solve the problem. The representative can pass the question to the next team, but will lose one point (-1). The other team must then try to solve it. An incorrect answer costs a team three (−3) points. 

Play the game until all of the questions have been answered or the time limit has gone. The team that scores the most points wins this game. 

Questions to ask about the game:
1. "You have five dollars, and your friend returns 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 December day, the high temperature was 10 degrees and the low was -25 degrees. What was the difference between the high and low temperatures?" (35)
4. "A submarine dives below the surface at a 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 drops by five degrees per hour. What is the temperature outside 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 were completed?" (1) 
7. "You owe $15 to 3 distinct friends, and you owe the same amount to each of them. How much money do you owe each friend?" (−5) 
8. "A frog moves 3 feet every 5 minutes, but after 15 minutes, a strong wind takes it up and moves it backwards 30 feet. How far has the frog come from its starting point after 20 minutes?" (−18) 
9. "Over a four-week period, a business had the following gains or losses: 100, -250, 200, and -50. What is the business's gain (or loss) after four weeks?" (0) 
10. "You owe your bank ten dollars but receive two dollars per day from your parents. How much money would you have after eight days if you repay the bank what you owe it?" (6)

Extension:

Routine: As real-world situations involving negative numbers happen during the school year, have students discuss and solve the problems in class. This will help students remember the concepts of baseline and negative number computation.

Small Groups: Students who would benefit from more learning opportunities may be assigned additional practice problems or permitted to use black and red chips while working through problem sets. For extra problem sets, see the website listed below. Select any of the items in the Integers section. http://www.adaptedmind.com/Sixth-Grade-Math-Worksheets-And-Exercises.html?tagId= 
The following website can be used for online practice games. The fruit shoot game can 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 write their own math stories that include positive and negative numbers for multiplication and division. Then, students can switch with a partner and solve each other's problems. Students receive immediate feedback as they check over each other's work.
Students can also take their partner's math stories and rewrite them using the relevant operation. For instance, one student wrote the following problem:

A submarine dives below the surface at a rate of five feet per second. How many feet below the surface is the submarine after 70 seconds? 
The student's partner may solve the original problem and find the answer, which is -350 feet. Then rewrite the problem as a related division problem. 

A submarine was -350 feet below the surface (-350). If the submarine traveled at a rate of five feet per second, how long did it take to reach that depth below the surface? (70 seconds)