• The Adding and Subtracting Negative Numbers worksheet (M-6-1-2_Adding and Subtracting Negative Numbers and KEY.docx) will help students practice the concept. Observation during this activity will allow you to clarify any misconceptions.
• Informal assessment by observation of student interactions during activities, class discussion, and lesson will aid in determining which students might benefit from additional exercises.

This lesson will show students additional ways to represent negative numbers and how to add and subtract using negative numbers. The use of real-life concepts to represent negative and positive numbers gives students some background on why learning about negative numbers might be necessary and useful; it also provides students needing opportunities for additional learning with the number line visual. Using colored chips gives students a tangible way to represent negative numbers, and it helps to develop the idea that negative numbers may be subtracted. At first subtracting negative numbers using chips may seem contrived, but once students are able to follow the model through, the chips help to demonstrate the idea of pairing up positives and negatives, which is a concept at the heart of addition and subtraction with integers. Once students are comfortable using the black and red chips to solve problems, they are then presented with problems to solve without using the chips, which forces students to apply the concepts they have learned.

Example 1

Begin the lesson with the following scenario. “You earn 20 dollars for raking leaves, but you owe your best friend six dollars and your sister five dollars. You also have two dollars hidden in your shoe. How much money will you have once you repay your friend and sister?” Allow students a few minutes to discuss this problem amongst themselves; then explain to the class that this can be solved as a math problem in the following way:

“If you take the 20 dollars from raking leaves and add the two dollars from your shoe, 22 dollars is the total amount of money you have at the moment because 20 + 2 = 22.”

“If you subtract six dollars to pay back your best friend, you now have $16 because
22 − 6 = 16.”

“If you subtract the five dollars you owe your sister, you have a total of 11 dollars left because 16 − 5 = 11.”

Ask, “Is there another way we could have solved this problem? The way we worked it out just now was 20 + 2 − 6 − 5, but could we have done it another way?” (Students are likely to suggest combining numbers in a different order.)

“One important thing to realize about subtraction is that subtracting is really the same thing as ‘adding the opposite.’ For instance, +2 − +6 is the same thing as +2 + −6. In fact, we can rewrite any subtraction problem as an addition problem by changing the subtraction sign to an addition sign and switching the sign of the second number. That means this problem can also be thought of as adding a series of positive and negative numbers. We could have represented the problem as 20 + 2 + (−6) + (−5) and solved it that way. Debts we owe other people are often represented as negative numbers, and we can add them together to find the amount of money we have left after all debts are repaid. In this lesson, we will explore this concept and how this kind of addition works.”

Activity 1: Adding Positive and Negative Numbers, Dollars Simulation

For this activity, you will use the Mock Dollar Template to make sets of mock dollars and IOUs (M-6-1-2_Mock Dollar Template.docx). Make sure each student has five mock dollars and five mock IOUs.

Explain to the class that each IOU means “I owe you” and cancels out one mock dollar. If they have three mock dollars and three IOUs, they really have no money because once they pay off the IOUs, no money is left. Say, “We can represent +3 with three dollars.” Hold up three mock dollars. “However, it is also true that we can represent +3 with five dollars and two IOUs.” Lay out five mock dollars and two IOUs. “The two IOUs for one dollar cancel out two of my mock dollars. Similarly, if I have three IOUs and one dollar, I have −2 dollars and I end up owing two dollars. We can say the net value is −2 dollars. Net value is another way of saying what the solution is after taking into account the positives and the negatives.”

Model this understanding of adding positive and negative numbers using a number line. Draw a number line on the board and show students how the number line can be used to solve this problem as well. Students can be given a number line (M-6-1-1_Number Line Template.docx) to follow along at their desks. Five mock dollars and two IOUs can be shown by starting out at positive 5 and then moving to the left 2 spaces on the number line to represent the two IOUs, which means you take away two (negative). Students will see that you end up at positive 3.

Another example is three IOUs and one mock dollar. On the number line start at negative three because IOUs are negative. Then move one space to the right since a mock dollar represents a positive number. Students will see that you end up at negative 2.

After the examples, have students represent the following values with their mock dollars and IOUs:

• −4
• 2
• 1
• −1
• 0 (Students must have some mock dollars and IOUs in their pile.)

Answers to each will vary. As students get the idea, try to encourage them to come up with multiple answers to represent each number.

Activity 2: Moving Along the Number Line

To reinforce the concept of net value, use problems similar to the examples below. Tape a number line on the floor to help students visualize what each problem represents.

• “A football team gained 6 yards and then lost 10 yards when the quarterback was tackled. What is the net yardage?” (6 + (−10) = −4. The net yardage is −4 yards, meaning that the team is farther back from their goal than when they started.)

If a number line is taped on the floor, have a student start at 0. Then have the student move 6 spaces on the number line in a positive direction since the football team gained 6 yards. Then have the student move 10 spaces in the negative direction (to the left) because of the loss of yards. The student should end up at −4.

• “You earned $15 for cutting the lawn. You spent $9 on a CD. Your sister paid you back $5 that she owed you. What is the net amount?” (15 + (−9) + 5 = 11. The net amount is 11 dollars. You have 11 dollars more than when you started.)

Have a student start at 0 on the floor number line. Then have the student move 15 spaces on the number line in a positive direction to show that the person earned $15. From that spot have the student move 9 spaces in the negative direction because spending money can be represented by going in a negative direction on the number line. At this point the student should be at 6. From that spot, have the student move 5 spaces in the positive direction since the person’s sister paid back money. The student should end up at 11, showing a net gain of 11 dollars.

• “A kite ascended 300 feet into the sky. Then the lack of wind caused the kite to descend 75 feet. The wind picked up and the kite ascended another 25 feet. What is the net height of the kite?” (300 + (−75) + 25 = 250 feet. The net height of the kite is 250 feet.)

Since a number line this length cannot be taped on the floor, discuss how it can be shown. “The kite starts on the ground which would be represented by 0 on the number line. So let’s say I am standing at 0. Then I would move 300 feet in the positive direction since the kite ascended 300 feet. Ascended means to rise or go up.” (Move a distance away from 0 to the right.) “I am now at 300 feet. Then the kite descended, or went down, 75 feet. I will move to the left 75 feet. I move to the left because descended is like going in a negative direction. If I start at 300, then go down 75 feet” (move to the left), “I am now at 225 feet. Then the kite ascended, or went up, 25 feet. So from 225 feet, I go to the right 25 feet because ascend means to go up. So 225 + 25 = 250 feet.”

In small groups have students create problems similar to the ones shown above. Then have them “act out” the problem by moving along a number line or with drawings. While students are working, monitor interaction and ask students questions similar to those listed below.

• “How do you know where to start on the number line?”
• “How do you know whether to go to the left or to the right on the number line?”
• “Give an example of something that can ascend. Give an example of something that can descend.” (airplane, cost, kite, temperature)
• “What does the word net mean when dealing with positive and negative numbers?”
• “What is a real-world situation that shows you can go in a positive and negative direction?” (temperature, yardage in football, stock market, bank account, above/below sea level)
• “How can you tell if your answer is reasonable?”
• “What does your answer mean?”
• “Is there a spot where you got stuck when trying to ‘act out’ the problem?”

Demonstrate the following:

“Black chips represent positive amounts and red chips represent negative amounts. Remember that negative is the opposite of positive. If we have positive one and negative one, what do they add up to? Zero. They are opposites.” Hold up a black chip and a red chip. “This black chip is positive one and this red chip is negative one. When I add them together, what do I get? Zero.” Hold up two black chips and one red chip. “What do the two black chips represent? What does the red chip represent? When I add them together, what do I get? The one red chip with one black chip equals zero so we can take those away. What remains? One black chip, or positive 1. So 2 + −1 = 1.” Demonstrate the following three examples on the board or on a projection device:

• 7 − 2                Show 7 black chips; then cross off or cover 2 of them. Five chips remain.
• 7 + (−2)           Show 7 black chips; then add 2 red chips. Explain that pairing black with red results in zero and zeros can be removed. Cross off or cover 2 red chips and 2 black chips. Five chips remain.
• 7 − (−2)           Show 7 black chips; then explain that we can’t cross off 2 red chips because we don’t have 2 red chips. Remind students that pairs of chips are equivalent to zero; they can be referred to as “zero pairs.” Since we need 2 red chips, bring in 2 red and 2 black chips. Now you can cross off the 2 red chips and see that 9 black chips remain.

Leave these three examples posted in the room so that students can refer to them when they get confused. Solve more examples if necessary.

Ask students to work out the following subtraction problems with their piles of black and red chips:

• 4 − 2                (2)
• 4 − (−2)           (6)
• 5 − (−3)           (8)
• −4 − (−2)         (−2)
• −6 − 3              (−9)
• −2 − (−5)         (3)
• 7 − 9                (−2)
• 2 − (−7)           (9)
• 5 − (−15)         (20)      [Note: This example will probably be unsolvable for most of the class due to not having the necessary black chips. Use this problem to transition to Activity 3.]

Activity 3: Adding and Subtracting Negative Numbers (Using Chips Only to Check Answers)

For this activity, leave students in their pairs. Explain, “We do not always have black and red chips to help us visualize how to add and subtract negative numbers and positive numbers. Therefore, we need other approaches to add negative numbers.”

Write 3 + 3 on the board and ask, “How do we usually solve this problem?” (Students will likely suggest using mental math to add the two numbers together to get 6). “We solve this one by adding the 3 and 3 together to get 6. We can extend this to adding negative numbers together, as well.”

Write −2 + (−2) on the board. Ask, “How could I solve this problem?” Give students time to think. “We solve the problem with the black and red chips by starting with two red chips and adding two more red chips, and we end up with four red chips, or −4.”

Write on the board:

• −2 + (−2) = −4
• 3 + 3 = ___                  (6)
• −5 + (−1) = ___           (−6)
• 3 + 5 = ___                  (8)

As a class, briefly solve the remaining three equations. Ask, “Do you notice a pattern when we add two numbers with the same sign?” (Yes, when adding two numbers with the same sign, the answer keeps the sign of the two numbers being added.)

Show the class the following problem on the board to illustrate the idea, using the red chips to illustrate the idea as you work the problem: −4 + (−5) = −9. Once the problem is worked and the class has had time to ask questions, clear the board and go to the next part.

Write 9 + (−2) on the board. Ask, “Does anyone see a way we can solve this problem without using black and red chips?” (Rewrite the problem as 9 − 2 and solve it that way.) Once someone gives the correct answer or enough time has passed, explain, “Adding a negative number to a positive can be solved using subtraction. When we solved 9 + (−2), we got 7 for our answer, which is the same answer we would get if we solved 9 − 2” (have the pairs verify by using the black and red chips). “This is true whenever you are adding a negative number to a positive number. You can rewrite it as a subtraction problem and solve it that way. The absolute value rule states that when adding numbers with different signs, you find the difference of the absolute values of the numbers. The sum has the sign of the number with the greater absolute value.”

Demonstrate this by solving the following problems on the board. Where possible, students can verify that the steps work using their piles of black and red chips:

• Problem 1: 7 + (−4); rewrite as 7 − 4; solve to get 3. Another way to show this is using the absolute value rule: |7| − |(−4)| = 7 − 4 = 3. The answer is positive 3 since the number with the greater absolute value is 7 and 7 is a positive number.
• Problem 2: 20 + (−5); rewrite as 20 − 5; solve to get 15 (Note: The groups will probably not have enough chips to solve this problem.) Another way to show this is using the absolute value rule: |20| − |(−5)| = 20 − 5 = 15. The answer is positive 15 since the number with the greater absolute value is 20, a positive number.
• Problem 3: (−2) + (3); switch the order to 3 + (−2) because the commutative property of addition states that addition problems can be solved in any order; rewrite as 3 − 2; solve to get 1.
• Problem 4: 6 + (−10); rewrite as 6 − 10. “This time we have a problem, as 6 − 10 does not look solvable. However, using our black and red chips we would get −4 as an answer for the problem 6 + (−10). Can anyone see how we can solve this problem without using the chips?” Allow students time to answer; then proceed.

“We can solve this problem without using the chips. Our answer will be negative because we only have 6 objects, and we can’t take 10 objects away. However, we can figure out how many objects we are short by subtracting the 10 from the 6 like this: 10 − 6 = 4, which tells us we are four objects short of being able to compute 6 − 10. Therefore, we represent the answer as −4 to show we are four objects short. If we end up with a subtraction problem where we want to subtract more than we have (in this case taking away 10 from 6), we can find the answer by subtracting the smaller number from the bigger one (in this case, 6 from 10). Then we make our final answer negative to represent that we were taking away a larger amount from a smaller amount.

“Another way to show this is using the absolute value rule: |6| − |(−10)| = 6 − 10 = (−4). The answer is −4 because the number with the greater absolute value is −10, and the difference between the two numbers 6 and 10 is 4.”

Proceed to demonstrate this concept by solving these problems on the board:

• Problem 5: 6 + (−9); rewrite as 6 − 9. Say, “Since we cannot take nine away from six, we can solve the problem by computing 9 − 6 and making the answer negative to represent the fact we were short by that amount. The answer is −3. Can someone explain how the absolute value rule can help us solve this problem in another way?”
• Problem 6: 11 + (−16); rewrite as 11 − 16; solve to get −5. “Can someone explain how the absolute value rule can help us solve this problem in another way?

“Let’s think about subtracting negative numbers again. For the subtraction problem  5 − (−2), does anyone see another way we can find the answer?” (Change it to 5 + 2; work the additive inverse.) “When we are subtracting a negative number from another number, we can rewrite the problem as an addition problem, such as 5 − (−2) = 5 + 2 = 7. Another way to think of it is that subtracting a negative number yields the same answer as adding by that number.”

Demonstrate to the class how to work the following problems:

• Problem 7: 6 − (−3); rewrite as 6 + 3; solve to get 9.
• Problem 8: (−3) − (−4); rewrite as (−3) + 4; solve to get 1.

Integer Coverall Activity

This activity can be used to reinforce adding and subtracting positive and negative numbers. Give each student an Integer Coverall Board (M-6-1-2_Integer Coverall Board.docx). Then display the Integer Coverall Expression Bank (M-6-1-2_Integer Coverall.docx) or give each student a copy. Have students fill in nine expressions from the Integer Coverall Expression Bank on their Integer Coverall Board in a random order. Once all students have completed their Integer Coverall Boards, they can begin play.

After cutting apart the Integer Coverall Game Chips (see Integer Coverall Game Chips on page 2 of M-6-1-2_Integer Coverall.docx), randomly pull a game chip and call out the value. Students see if they have the expression on their Integer Coverall Board that represents the value. If they do, the students can cover the expression using a game marker. The first student to cover his/her entire card is the winner. Check student answers before declaring the winner. Alternate options instead of doing a coverall include four corners, diagonal, L-shape, and T-shape.

Keeping students in their pairs, distribute the Adding and Subtracting Negative Numbers worksheet to each student (M-6-1-2_Adding and Subtracting Negative Numbers and KEY.docx), and have students complete the problems. If there is not enough time to finish the worksheet, assign it as homework and have students discuss their answers in the next class.

Extension:

• Routine: The understanding and use of negative numbers is an important element in this lesson and should be emphasized throughout. It is important that students realize they can add negative numbers by rewriting the problem as a subtraction problem, and they can subtract negative numbers by rewriting the problem as an addition problem. The black and red chips are used as an initial aid for the class to understand the concept, but they are then removed so students can learn how to solve the problems without them. Still, if you find some students need opportunity for additional learning, assign them additional problems with the black and red chips until they grasp the necessary concepts.

• Small Groups—Adding Positive and Negative Numbers with Black and Red Chips: In this activity, positive and negative integers are represented by black and red chips; black chips representing positive numbers and red chips representing negative numbers. This is a way to represent addition and subtraction of negative numbers visually. Model the following representations of integers using black and red chips:
  • 2 (show the class two black chips in a stack or draw a circle and write B two times to represent two black chips).
  • −2 (show the class two red chips in a stack or draw a circle and write R two times to represent two red chips).

Remind students how the black chips are similar to the mock money used earlier in the lesson; the red chips are similar to the IOUs used earlier in the lesson. Next show students nine black chips and two red chips in the same pile. Explain to students this represents 7. Ask students to explain how this pile of chips can be written as a math problem using positive and negative numbers: 9 − 2 = 7; 9 + (−2) = 7.

Model using another example: 5 + (−2) = ___. Explain how to represent adding positive and negative numbers as combining piles of black and red chips. Take five black and two red chips. Since a black chip and a red chip cancel each other, remove a black chip and a red chip from the pile at the same time. This leaves five black chips and two red chips. Then remove one more black and one more red chip, which leaves four black and one red. Then remove one more black chip and one more red chip. This leaves three black chips. Represent this with the equation: 5 + (−2) = 3. Have students practice this concept on the Adding Positive and Negative Numbers worksheet (M-6-1-2_Adding Positive and Negative Numbers Using Chips.docx).

• Expansion: Give students who demonstrate proficiency the opportunity to attempt some problems adding four or more integers. You can also present them with problems involving fractions and decimals.