Students will discover more ways to express negative integers with red and black chips and play money. Students will also continue to investigate the concept of absolute value and learn how to calculate it for a given number. Students will:
- learn how to add positive and negative integers with three or more integers in a single problem.
- 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.
- copies of the Adding and Subtracting Negative Numbers worksheet (M-6-1-2_Adding and Subtracting Negative Numbers and KEY)
- copies of the Mock Dollar Template (M-6-1-2_Mock Dollar Template)
- ten black and ten red chips for each pair of students (can be omitted if you wish to provide students with paper and have them write R for red and B for black each time)
- copies of the Integer Coverall Board (M-6-1-2_Integer Coverall Board)
- cut apart Integer Coverall game chips (see Integer Coverall game chips on page 2 of M-6-1-2_Integer Coverall)
- game markers to use during the Integer Coverall activity (chips, cubes, or pieces of paper)
- masking tape
- Adding Positive and Negative Numbers worksheet (M-6-1-2_Adding Positive and Negative Numbers Using Chips)
- The Adding and Subtracting Negative Numbers Worksheet (M-6-1-2_Adding and Subtracting Negative Numbers and KEY) will help students practice the concept. Observation throughout this activity will help you clear up any misconceptions.
- Informal assessment, such as observing student interactions during activities, class discussions, and lessons, will help determine which students could benefit from additional exercises.
Scaffolding, Active Engagement, Modeling
W: Begin the lesson by explaining that students will learn a new way to express negative numbers and how to add and subtract them.
H: Explain how to add negative integers in a story problem requiring addition and subtraction.
E: Distribute materials representing dollars and IOUs. Explain how they can offset each other, and how the two can be accumulated or combined to create larger and smaller amounts of money or debt. Then, distribute colored chips that indicate similar positive or negative amounts. Represent the colored chip combinations by simply adding their values, rather than adding and subtracting.
R: As a review activity, have students use the chips to answer addition problems with negative numbers.
E: Continue to use chips to illustrate addition problems and write number sentences on the board. Move away from using chips and instead answer problems with number sentences only. Have students complete the Adding and Subtracting Negative Numbers worksheet to assess their understanding.
T: Use the Extension section's suggestions to personalize the lesson to the students' requirements. The Routine section includes suggestions that can be used with all students throughout the school year to reinforce the concepts covered in this lesson. The Small Group section can be used by students who might benefit from additional learning opportunities. The Expansion section is meant to guide further training for students who are ready for a challenge that goes beyond the requirements of the standard.
O: This lesson demonstrates real-life examples of how to represent and combine negative and positive numbers in addition or subtraction problems. It is structured in a way that builds on prior learning.
This lesson will teach students alternative ways to express negative numbers, as well as how to add and subtract with them. The use of real-life concepts to illustrate negative and positive numbers offers students with context for why learning about negative numbers may be important and useful; it also provides students with chances for further learning through the number line visual. Using colored chips provides students with a concrete way to represent negative numbers and helps them develop the concept that negative numbers can be subtracted. At first, subtracting negative numbers using chips might seem contrived, but after students understand the model, the chips help to teach the concept of pairing up positives and negatives, which is important to addition and subtraction with integers. Once students feel comfortable using the black and red chips to solve problems they are given tasks to solve without the chips, forcing them to use what they have learned.
Example 1
Start the lesson with the following scenario. "You earn 20 dollars by raking leaves, but you owe your best friend 6 dollars and your sister 5 dollars. You also have two dollars hidden inside your shoe. How much money will you have after repaying your friend and sister?" Allow them a few minutes to discuss this problem between themselves before explaining to the class that it 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 repay your best friend, you now have 16 dollars because 22 - 6 = 16."
"If you subtract the five dollars you owe your sister, you have a total of 11 dollars because 16 - 5 = 11."
Ask, "Is there another way we could have solved this problem? We calculated it as 20 + 2 − 6 − 5, but could we have done it another way?" (Students are likely to recommend mixing numbers in a different order.)
"One important thing to remember about subtraction is that it is really the same as 'adding the opposite.' For example, +2 − +6 is the same as +2 + -6. In fact, we can convert any subtraction problem into an addition problem by changing the subtraction sign to an addition sign and switching the sign of the second integer. That is, this problem can also be thought as the addition of a series of positive and negative numbers. We might have solved the problem using the equation 20 + 2 + (−6) + (−5). Debts we owe other people are often represented as negative numbers, which we can add together to calculate the amount of money we have left once all debts are paid. In this lesson, we'll look at how this type of addition works."
Activity 1: Adding Positive and Negative Numbers, Dollars Simulation
To complete this task, use the Mock Dollar Template to create sets of mock dollars and IOUs (M-6-1-2_Mock Dollar Template). Ensure that each student has five mock dollars and five mock IOUs.
Explain to the class that each IOU represents "I owe you" and cancels away one mock dollar. If they have three mock dollars and three IOUs, they truly do not have any money since if the IOUs are paid off, there is no money 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." Prepare 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 will have -2 dollars and owe two dollars. We can say the net value is -2 dollars. Net value is another way of expressing what the solution is after taking into account the positives and the negatives."
Use a number line to model this understanding of adding positive and negative numbers. Draw a number line on the board and demonstrate to students how it may be used to solve this problem. Students can be given a number line (M-6-1-1_Number Line Template) to work with at their desks. Five mock dollars and two IOUs can be represented by starting at positive 5 and moving to the left 2 spaces on the number line to represent the two IOUs, which implies you take away two (negative). Students will notice that you end up at a positive 3.
Another example is three IOUs and one mock dollar. IOUs are negative, so start at negative three on the number line. Move a 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, ask students to indicate 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. Encourage students to generate multiple answers to represent each number.
Activity 2: Moving Along the Number Line
To emphasize the concept of net value, use problems similar to the ones shown below. Tape a number line to the floor to help students understand what each problem represents.
"A football team gained 6 yards before losing 10 yards when the quarterback was tackled. "What's the net yardage?" (6 + (−10) = −4. The net yardage is -4 yards, indicating that the team is further away from their goal than when they started.)
If a number line is taped on the floor, have a student start at 0. The student should then move 6 spaces on the number line in a positive direction because the football team gained 6 yards. The student should then 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 paid $9 for a CD. Your sister paid you back the $5 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 begin 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 there, have the student go 9 spaces in the negative direction, as spending money can be represented by going in the negative direction on the number line. At this point the student should be at 6. From that spot, instruct the student move 5 spaces in the positive direction because the person's sister paid back the money. The student should end up at 11, with a net gain of 11 dollars.
"A kite flew 300 feet into the sky. The kite then descended 75 feet due to the lack of wind. The wind came up, and the kite ascended another 25 feet. "What is the net height of the kite?" (300 + (−75) + 25 = 250 feet. The kite's net height is 250 feet.)
Because a number line this length cannot be taped on the floor, consider how it can be shown. "The kite starts on the ground, which is symbolized by 0 on the number line. So let's pretend I'm standing at 0. Since the kite ascended 300 feet, I would move 300 feet in the positive direction. Ascended implies to rise or go up." (Move a distance away from 0 to the right.) "I'm now at 300 feet. The kite then descended, or went down, 75 feet. I'll move 75 feet to the left. I move to the left because descending is equivalent to moving in the negative direction. If I start at 300, and then descend 75 feet" (move to the left), "I am now at 225 feet." The kite then ascended, or went up, 25 feet. So, from 225 feet, I move 25 feet to the right since ascend implies to go up. So, 225 + 25 = 250 feet."
In small groups, have students create problems similar to those described above. Then ask them to "act out" the problem by moving along a number line or using drawings. While students are working, monitor their interactions and ask them questions like the ones 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?"
"Please provide an example of anything that can ascend. Provide 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 or 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, while red chips represent negative amounts. Remember that negative means opposite of positive. What does the sum of a positive and negative one equal? Zero. They are opposites." Hold up two chips, one black and one red. "This black chip is positive, whereas this red chip is negative. When I add them together, what do I get? Zero." Hold up two black chips and one red chip. "What do those two black chips represent? What does the red chip represent? What happens when I combine them together? We can remove the one red chip and one black chip because their sum is zero. What remains? One black chip, or positive one. So 2 + −1 = 1." Show the following three instances on a board or 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 and then add 2 red chips. Explain that combining black with red results in zero, which may be removed. Cross off or cover 2 red chips and 2 black chips. Five chips remain.
7 − (−2) Show 7 black chips and explain why we can't cross off 2 red chips because we don't have them.
Remind students that pairs of chips are equal to zero and might be referred to as "zero pairs." Bring 2 red chips and 2 black chips. You can now cross off the 2 red chips, leaving only 9 black chips.
Keep these three examples posted in the classroom so that students can refer to them if they become confused. Solve further examples as necessary.
Ask students to solve the following subtraction problems using 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: Due to a lack of black chips, the majority of the class will most likely be unable to solve this problem. Use this problem to transition to Activity 3.]
Activity 3: Adding and Subtracting Negative Numbers (Using Chips Only to Check Answers)
For this activity, keep students work in pairs. Explain, "We don't always have black and red chips to help us understand how to add and subtract negative and positive numbers. As a result, we need alternative methods for adding negative numbers."
Write 3 + 3 on the board and ask, "How do we usually solve this problem?" (Students will most likely suggest using mental math to add the two numbers together to get 6). "To solve this one, we add 3 and 3 to get 6. We can use this to adding negative numbers as well."
Write −2 + (−2) on the board. Ask, "How could I solve this problem?" Allow 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)
Solve the last three equations together as a class. 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 result keeps the sign of the two numbers being added.)
To demonstrate the principle, show the class the following problem on the board, using the red chips to illustrate it as you work the problem: −4 + (−5) = −9. Once the problem has been solved 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 that way.) When someone provides 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, which is the same answer we'd get if we solved 9 - 2" (have the pairs verify by using the black and red chips). "This happens whenever you add a negative number to a positive number. You can rewrite it as a subtraction problem and solve it that way. According to the absolute value rule, when adding numbers with different signs, the difference in absolute values is calculated. The sum has the sign of the number with the greater absolute value."
Show this by completing the following problems on the board. Students can use their piles of black and red chips to verify that the methods work:
Problem 1: 7 + (−4); rewrite as 7 - 4; solve to get 3. Another approach to demonstrate this is to use the absolute value rule: |7| − |(−4)| = 7 - 4 = 3. The answer is positive 3 because the number with the greater absolute value is 7 which is a positive number.
Problem 2: 20 + (−5); rewrite as 20 - 5; solve to get 15. (Note: The groups will most likely not have enough chips to solve this problem.) Another approach to demonstrate this is to use the absolute value rule: |20| - |(−5)| = 20 - 5 = 15. The answer is positive 15, because the number with the greater absolute value is 20, which is positive.
Problem 3: (−2) + (3), change the order to 3 + (−2) because the commutative property of addition states that addition problem can be solved in any order. Rewrite as 3 − 2 and solve to get 1.
Problem 4: 6 + (−10), rewritten as 6 - 10. "This time, we have a problem, as 6 − 10 does not appear solvable. However, with our black and red chips, we would receive −4 for the problem 6 + (−10). Can anyone see how we can solve this problem without using chips?" Allow time for students answer before proceeding.
"We can solve this problem without using chips. We will give a negative answer because we only have six objects and cannot take away 10. Subtracting 10 from 6 like this: 10 - 6 = 4, indicating that we are four objects short of being able to compute 6 - 10. Therefore, we represent the answer as -4 to demonstrate we are four objects short. If we have a subtraction problem that requires us to subtract more than we have (in this case, subtracting 10 from 6), we can solve it by subtracting the smaller number from the bigger one. Then we make our final answer negative, indicating that we were subtracting a larger amount from a smaller amount.
“Another way to demonstrate this is to use 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 6 and 10 is 4."
To show this concept, solve the following problems on the board:
Problem 5: 6 + (−9); rewritten as 6 - 9. Say, "Since we cannot subtract nine from six, we can solve the problem by computing 9 - 6 and making the answer negative to represent the fact that we were short by that amount. The answer is -3. Can you explain how the absolute value rule can help us solve this problem in a different way?"
Problem 6: 11 + (−16); rewrite as 11 - 16; solve to get −5. "Can you explain how the absolute value rule can help us solve this problem in a different way?"
"Let's consider subtracting negative numbers again. Is there any way to solve the subtraction problem 5 −(−2)?" (Change it to 5 + 2 and apply the additive inverse.) "When 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 removing a negative number produces the same result as adding by that number."
Demonstrate to students how to solve 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 practice adding and subtracting both positive and negative numbers. Provide each student with an Integer Coverall Board (M-6-1-2_Integer Coverall Board). Then show the Integer Coverall Expression Bank (M-6-1-2_Integer Coverall) or distribute a copy to each student. Students should fill up nine expressions from the Integer Coverall Expression Bank on their Integer Coverall Board in random order. Once all students have finished their Integer Coverall Boards, they can start playing.
After cutting apart the Integer Coverall Game Chips (see Integer Coverall Game Chips on page 2 of M-6-1-2_Integer Coverall), select a game chip at random and call out its value. Students check to verify if their Integer Coverall Board contains the expression that represents the value. If they do, students can cover the expression with a game marker. The first student to cover his/her full card is the winner. Before proclaiming the winner, double-check the students' answers. Instead of a coverall, you can do four corners, diagonal, L-shape, or T-shape.
Keeping students in pairs, give each student the Adding and Subtracting Negative Numbers worksheet (M-6-1-2_Adding and Subtracting Negative Numbers and KEY) and have them complete the problems. If there is insufficient time to complete the worksheet, give it as homework and have students discuss their answers in the following lesson.
Extension:
Routine: Understanding and applying negative numbers is an essential element of this lesson and should be reinforced throughout. It is critical that students understand that they can add negative numbers by rewriting the problem as a subtraction problem, and subtract negative numbers by rewriting the problem as an addition problem. The black and red chips are initially used to help the class understand the concept, but they are later removed so that students can learn how to solve problems without them. Still, if you see that any students require further learning, assign them additional challenges with black and red chips until they understand the necessary concepts.
Small Groups - Adding Positive and Negative Numbers with Black and Red Chips: In this activity, black chips and red chips are used to represent positive and negative integers, respectively. This is a visual representation of adding and subtracting negative numbers. Model the following representations of integers with black and red chips:
2 (show the students two black chips in a stack, or draw a circle and write B two times to represent two black chips).
−2 (show the students two red chips in a stack, or draw a circle and write R two times to represent two red chips).
Remind students that the black chips are similar to the mock money used earlier in the lesson, while the red chips are similar to the IOUs from earlier in the lesson. Next, display students nine black chips and two red chips in the same pile. Explain to them that this represents 7. Ask students to explain how this pile of chips might be expressed as a math problem with positive and negative numbers: 9 − 2 = 7; 9 + (−2) = 7.
Consider the equation 5 + (−2) = ___. Explain how to add positive and negative integers by combining piles of black and red chips. Take five black and two red chips. A black chip and a red chip cancel each other out, therefore remove them 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, leaving four black and one red. Then, remove one more black and one more red chip. This leaves three black chips. Represent this with the equation: 5 + (−2) = 3. Students should practice this concept with the Adding Positive and Negative Numbers worksheet (M-6-1-2_Adding Positive and Negative Numbers Using Chips).
Expansion: Allow students who demonstrate proficiency to attempt problems using four or more integers. You may also present them with problems using fractions and decimals.
