Subtracting Negative Integers Notes: Comparison of CST questions to the skill of subtracting negative integers. 5 th Grade/65 NS2.1 Add, subtract, multiply and divide with decimals; add with negative integers; subtract positive integers from negative integers; and verify the reasonableness of the results. 6 th Grade/65 NS2.3 Solve addition, subtraction, multiplication and division problems, including those that arise in concrete situations, that use positive and negative integers and combinations of these operations. 7 th Grade/65 Total # of Total # of Concept Questions Benchmark Questions 22 0 (1 Subtraction w/- answer) 17 0 (4 Adding /- Integers) 14 0 (3 Absolute Value) NS1.2 Add, subtract, multiply and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers. Although the skill itself is not emphasized in the test, experience tells us that students still struggle when working with the concept of negative and positive numbers and operations. This is the order in which to introduce integers: 1. Using a number line to add integers (developing the language used for working with integers) 2. Tile spacers (concrete) Page 1 of 26 MDC@ACOE 10/17/10
3. Introduce the rules of adding integers (by now students have figured it out, anyway) 4. Introduce subtracting integers: number line, tile spacers, decomposing, rules Once students have developed the number sense necessary to succeed in adding and subtracting integers, they can transfer that skill into adding and subtracting positive and negative decimals and fractions and combining like terms. Regarding this lesson: This lesson assumes students have had little exposure to different strategies, and a general weakness in working with integers, so there is a lot of scaffolding embedded. Please be sure to begin the lesson where your students needs are. Page 2 of 26 MDC@ACOE 10/17/10
Subtracting Negative Integers Warm-Up Quadrant II: 3 rd Grade #46, NS2.8 Tony had $20. He paid $8 for a ticket to a baseball game. At the game, he bought a hotdog for $3. What amount of money did Tony have then? Quadrant I: 5 th Grade #47, AF1.2 If n = 31, what is the value of 6 n? Show two ways to solve. Quadrant III: 6 th Grade #27, NS2.3 Solve: One morning, the temperature was 5 below zero. By noon, the temperature rose 20 Fahrenheit (F) and then dropped 8 F by evening. What was the evening temperature? Show at least two ways to solve. Quadrant IV: 7 th Grade #59, AF4.1 Solve: What is the value of x if - 3x 2 = - 7? What are some errors students might make? What are some errors students might make? Today s Objective: Develop a variety of strategies to solve addition and subtraction problems, while providing opportunities to develop number sense regarding adding and subtracting integers. Page 3 of 26 MDC@ACOE 10/17/10
Subtracting Negative Integers Warm-Up Solutions Quadrant II: 3 rd Grade #46, NS2.8 Tony had $20. He paid $8 for a ticket to a baseball game. At the game, he bought a hotdog for $3. What amount of money did Tony have then? Quadrant I: 5 th Grade #47, AF1.2 If n = 31, what is the value of 6 n? 6 31 6 31 = 6 6 25 = 6 (-31) = 0 25 (- 3) (- 8) = -25 = $9 9 12 20 = $9 _ = 24 negatives = -24 Quadrant III: 6 th Grade #27, NS2.3 Solve: One morning, the temperature was 5 below zero. By noon, the temperature rose 20 Fahrenheit (F) and then dropped 8 F by evening. What was the evening temperature? - 5 20 (- 8) = 20 (- 5) (- 8) = 20 (- 13) = 7 13 (- 13) = 7 0 = 7 Possible errors: adding positive integers Quadrant IV: 7 th Grade #59, AF4.1 Solve: What is the value of x if - 3x 2 = - 7? - 3x 2 = - 7-3x 2 = - 7 2-2 - 3x = - 7-2 - 3x = - 9-3 x = - 3 3 x = 3 Possible errors: adding 2 to both sides, 7 2 Page 4 of 26 MDC@ACOE 10/17/10
Standard: 7NS1.2: Add, subtract, multiply and divide rational numbers (integers, fractions and terminating decimals) and take positive rational numbers to whole number powers. Objective: Students will be able to add and subtract integers. Define Integers: Any member of the set I= So, integers are the positive and negative whole numbers, including zero. APK (Activate Prior Knowledge): Where do we see integers in the real world? See bullet note for possible student responses be sure to include ideas that students may have missed. [weight gain/loss, football yards gained/lost, stock market gains/losses, bank deposits and withdrawals, temperatures, elevation above and below sea level, elevators, escalators, parking garages ] Show a number line and ask: Where are the bigger numbers on the number line? [right, right of zero] The smaller numbers? [left, left of zero] Where are the positive numbers on the number line? [to the right of zero] The negative numbers? [to the left of zero] Do positive numbers have a greater value than negative numbers? [yes] How do you know? [they are to the right of negative numbers, any number to the right of another number is greater, when you move to the right on a number line the numbers increase in value] Page 5 of 26 MDC@ACOE 10/17/10
Which number is greater: 3 or -2? [3] How do you know? [3 is to the right of -2] Which number is greater: -7 or -2? [-2] How do you know? [-2 is to the right of -7] Students should be able to articulate and justify their responses: [the number that is the furthest to the right on a number line has a greater value, if both numbers are negative, the number closest to zero has the greatest value] Check for Understanding: List several pairs of numbers and have students state either the bigger or the smaller of the two. List three numbers and have students order the numbers from greatest to least, least to greatest. Number Line Concept Development: If I were to add two integers, 3 and 5, using a number line, what would be a good strategy for me to do this? [start at 3 and move 5 to the right, end up at 8; 3 5 = 8] So, if I am adding positive integers, I move to the right. What if I am adding negative integers? Which direction would I move? [to the left] So, let s add a positive 3 with a negative 5: 3 (-5). What number do I start with? [3] (- 5) What am I adding to that? [-5] Is this a positive or negative? [negative] Page 6 of 26 MDC@ACOE 10/17/10
If I were adding a positive 5, I would move to the right, but I am not. I m adding a negative 5; which direction do I move? [left] Where do I end up on the number line? [negative 2] Let s try again: 7 (-3). Where do I start on the number line? [7] (- 3) What am I adding to that? [-3] Is that a positive or negative? [negative] If I were adding a positive 3, I would move to the right, but I am not. I m adding a negative 3; which direction do I move on the number line? [left] Where do I end up on the number line? [4] So, 7 (-3) = 4. Again: -4 (-5). Where do I start on the number line? [-4] (- 5) What am I adding to that? [-5] Is that a positive or a negative? [negative] Page 7 of 26 MDC@ACOE 10/17/10
If I were adding a positive 5, I would move to the right, but I am not. I am adding the opposite of 5, or a negative 5, so which direction do I move on the number line? [left] Where do I end up? [-9] So, what is our equation? [-4 (-5) = -9] Let s try another: -6 5. Where do I start on the number line? [-6] 5 What am I adding to that? [5] Is that a positive or a negative number? [positive] Which direction do I move on the number line? [right] Where do I end up on the number line? [-1] What is our equation? [-6 5 = -1] Let s try a word problem together: One morning, the temperature was 5 below zero. By noon, the temperature rose 20 Fahrenheit (F) and then dropped 8 F by evening. What was the evening temperature? What integer represents five degrees below zero? [-5] What does it mean when a temperature rose twenty degrees? What math operation would that be? [adding 20, positive 20] Page 8 of 26 MDC@ACOE 10/17/10
Finally, what math operation represents dropping eight degrees? [negative 8, subtracting 8] So, what is the equation for this word problem? [-5 20 (-8) =] 20 15-5 20 = 15 10 15 (- 8) = 7 5 0-5 Where do I start on the number line? [-5] What are we adding to that? [20] Is that a positive or negative number? [positive] Since this number line is vertical, which direction do we move on it? [up] Where are we on the number line? [15] What are we adding to that? [-8] Which direction do we move? [down] Where are we on the number line? [7] Page 9 of 26 MDC@ACOE 10/17/10
What is our equation? [-5 20 (-8) = 7] You Tries: a) 7 (-9) = -2 (- 9) b) -15 22 = -37 (Holt 7, L 1-5) (- 22) *Note: At this point, some students may see that they will keep the sign of the number that has the greatest absolute value. Take a moment to frontload students with the concept of zero pairs. Zero Pair Concept Development: Imagine I am standing at zero on a number line. If I take one step to the right, where would I be on the number line? [positive 1] Now I will take one step to the left. Where am I on the number line? [zero] Now imagine you are on the ground floor in an elevator and you take it one floor up (1). You stay on the elevator and ride it one floor down (-1). Where do you end up? [ground floor, back where you started, zero] Page 10 of 26 MDC@ACOE 10/17/10
What if you are at the top of the staircase, and you take 7 steps down (-7); you forgot something upstairs and took 7 steps up (7). Where did you end up? [back where you started, zero] These examples illustrate zero pairs. A working definition for the concept of zero pairs is: any number or variable and its opposite equals zero. The value of a Positive 1 and Negative 1 create a zero; therefore we have a zero pair. The value of a Positive 27 and Negative 27 create a zero; zero pair. The value of a Positive x and a Negative x create a zero: zero pair. Ask for students to give examples of zero pairs. Tile Spacers Concept Development: Let s work with integers in a different way. If I am adding 3 5, how many positives do I need to represent 3? [3] Draw three positives. How many positives do I need to represent 5? [5] Draw five positives next to the three positives. How many positives do I have altogether? [8] What is our equation? [3 5 = 8] Page 11 of 26 MDC@ACOE 10/17/10
If I am adding -3 5, how many negatives do I need to represent -3? [3] Draw three negative symbols. - - - How many positives do I need to represent 5? [5] Draw five positive symbols next to the three negative symbols. - - - I now have 3 negatives and 5 positives. Remember our examples with going up one floor and then down one floor what number do we have when there is a number and its opposite, like a positive 7 and a negative 7? [zero] If I matched one negative with one positive, what concept would that represent? [zero, zero pair] How many zero pairs do we have? [3] Take them away. What is left? [2 positives] - - - Page 12 of 26 MDC@ACOE 10/17/10
We keep the sign of what we have the most of; since we have more positives than negatives, our answer is positive 2. What is our equation? [-3 5 = 2] Let s try another: 3 (-5). How many positives do I have? [3] Draw three positives. How many negatives? [5] Draw five negatives next to the three positives. - - - - - Are there any zero pairs? [3] Take them away. What is left? [2 negatives] - - - - - Since we keep the sign of what we have the most of, what is our equation? [3 (-5) = -2] Page 13 of 26 MDC@ACOE 10/17/10
You Tries: a) -4 (-3) - - - - - - -4 (-3) = -7 We ve added (combined) the numbers and kept the sign of what we have the most of. - b) 18 (-7) - - - 18 (-7) = 11 (Holt 7, L 1-4) We found zero pairs (combined the numbers) and kept the sign of what we have the most of. - - - - Page 14 of 26 MDC@ACOE 10/17/10
Concept Closure: Give students the following two stems and have them work with an elbow partner to see if they can articulate the rules for adding integers: If the signs are the same If the signs are different Give students time to process, and then debrief. Ask groups to share their ideas. [If the signs are the same, add the numbers and keep the sign. ] [If the signs are different, subtract the numbers and keep the sign of what you have the most of. ] Using this language is key. Subtracting Integers Have students compare the following two problems: 3 (-5) and 3 5 Use the number line and tile spacers to model. Traditional: Signs are different; subtract and keep the sign of what you have the most of. 3 (-5) = -2 (- 5) - - - - - = -2 Page 15 of 26 MDC@ACOE 10/17/10
3 5 = -2 (- 5) The rule for subtracting integers is to re-write the subtraction problem into an addition problem, and then add the opposite of the integer. So, 3 5 becomes three plus the opposite of five: = 3 (-5) which are the same two problems we ve compared. We will need to re-write all of our subtraction problems so that they are addition ones. You Try Rewriting: What is the opposite of -8? [8] -5 (-8) = -5 8 What is the opposite of 35? [-35] 32 35 = 32 (-35) What is the opposite of -25? [25] 75 (-25) = 75 25 But, why is subtracting a negative the same as adding a positive? Check student responses. If I have gone into debt in my checking account because I am $25 overdrawn, do I have a positive balance or a negative balance? [negative, -25 dollars] Page 16 of 26 MDC@ACOE 10/17/10
If someone was going to take my debt away as a gift to me, what would they have to do? [they would take away negative $25] What would the balance be in my account? [zero dollars] Even though I don t have any money, is that better than having -$25 in my account? [yep!] What would that look like as a math problem? Write the following: -25 (-25) = 0 If someone was willing to help me out like this, what would they have to do in order to bring my checking account balance to zero if I have - $25? They would have to give me $25! Positive 25. So, knowing the integer rule for subtraction is to re-write a subtraction problem into an addition one and then adding the opposite integer, we would need to think about my banking situation like this : Write the following: -25 (-25) = 0-25 (25) = 0 Does anyone see an example of a zero pair? [yes!] Where is it? [-25 and 25, negative 25 and positive 25] If I were using a number line for this situation, Write the following: -25 (-25) = 0 Page 17 of 26 MDC@ACOE 10/17/10
Where would I start on the number line? [-25] If I was going to add negative 25, which direction would I go? [left] But I m not, I m doing the opposite of adding negative 25, I m going to subtract negative 25, so which direction do I go? [right] Where do I end up on the number line? [0] - (- 25) When adding a negative number, move to the left. Since we are subtracting a negative number, we move to the right. Let s try one more guided practice, Write the following: 14 8 = We need to change our subtraction to addition and add the opposite integer. Write the following, and ask for students response on rewriting: 14 8 = 14 (-8) [14 plus -8] Page 18 of 26 MDC@ACOE 10/17/10
Now, where do we start on the number line? [14] We are adding the opposite of 8; is this a positive or negative 8? [negative] Which direction do we move on the number line? [left] Where do we end up? [6] (- 8) You Tries -17 (-10) = -17 10 10 = -7 Traditional: Signs are different; subtract and keep the sign of what you have the most of. - - - - - - - - - - - - - - - - - = -7 Page 19 of 26 MDC@ACOE 10/17/10
You Tries (continued ) -14 (-14) = -14 14 (- 14) = 0 Traditional: Signs are different; subtract and keep the sign of what you have the most of. - - - - - - - - - - - - - - = 0 Decomposing Concept Development: Let s try one more strategy with some problems we ve already worked through: decomposing. 3 5 = 3 (-5) = -2 Traditional: Signs are different; subtract and keep the sign of what you have the most of. 3 5 = 3 (-5) = 3 (-3) (-2) = 0 (-2) = -2 Given Rewrite as addition problem Decompose -5 to create a zero pair Identity Property of Addition Answer *Note: The use of the third column frontloads students for success in geometry, where they will be required to justify their work with twocolumn proofs. In this context, having students write the justifications is an option, but explicit discussion of the math reasoning and properties is highly recommended. Page 20 of 26 MDC@ACOE 10/17/10
-17 (-10) = -17 10 = -7 Traditional: Signs are different; subtract and keep the sign of what you have the most of. -17 (-10) = -17 10 = -7 (-10) 10 = 0 10 = -7 Given Rewrite as addition Decompose to create a zero pair Identity Property of Addition Answer 14 8 = 14 (-8) = 6 Traditional: Signs are different; subtract and keep the sign of what you have the most of. Same problem 14 8 = 14 (-8) = 6 Traditional: Signs are different; subtract and keep the sign of what you have the most of. 14 8 = 14 (-8) = 6 8 (-8) = 6 0 = 6 14 8 = 14 (-8) = 8 6 (-8) = 6 8 (-8) = 6 0 = 6 Given Rewrite as addition Decompose to create a zero pair Identity Property of Addition Answer Given Rewrite as addition Decompose to create a zero pair Commutative Property of Addition Identity Property of Addition Answer Page 21 of 26 MDC@ACOE 10/17/10
You Tries (Holt 7, L 1-5) -18 (-25) = -18 25 = 7 Traditional: Signs are different; subtract and keep the sign of what you have the most of. -88 (-10) = -88 10 = -78 Traditional: Signs are different; subtract and keep the sign of what you have the most of. -15 x; x = -10-15 (-10) = -15 10 = -5 Traditional: Signs are different; subtract and keep the sign of what you have the most of. -18 (-25) = -18 25 = -18 18 7 = 0 7 = 7-88 (-10) = -88 10 = -78 (-10) 10 = -78 0 = -78-15 (-10) = -15 10 = -5 (-10) 10 = -5 0 = -5 Given Rewrite as addition Decompose to create a zero pair Identity Property of Addition Answer Given Rewrite as addition Decompose to create a zero pair Identity Property of Addition Answer Given Rewrite as addition Decompose to create a zero pair Identity Property of Addition Answer Page 22 of 26 MDC@ACOE 10/17/10
Now let s practice it all! (Holt 7, L 1-5) Traditional Number Line Tile Spacers Decompose -15 (-10) -15 (-10) -15 (-10) -15 (-10) = -15 10 = -15 10 = -15 10 = -15 10 = -5 10 - - - - - = -5 (-10) 10 Traditional: Signs are different; subtract and keep the sign of what you have the most of. - - - - - - - - - - = -5 0 = -5 = -5-5 (-15) -5 (-15) -5 (-15) -5 (-15) = -5 15 = -5 15 = -5 15 = -5 15 = 10 15 - - - - - = -5 5 10 Traditional: Signs are different; subtract and keep the sign of what you have the most of. = 0 10 = 10 = 10 Page 23 of 26 MDC@ACOE 10/17/10
-15 (-12) -15 (-12) -15 (-12) -15 (-12) = -15 12 = -15 12 = -15 12 = -15 12 = -3 12 = -3 (-12) 12 Traditional: Signs are different; subtract and keep the sign of what you have the most of. - 3 - - - - - - - - - = -3 0 = -3 - - - - - - = -3 Page 24 of 26 MDC@ACOE 10/17/10
-18 (-25) -18 (-25) -18 (-25) -18 (-25) = -18 25 = -18 25 = -18 25 = -18 25 = 7 25 = -18 18 7 Traditional: Signs are different; subtract and keep the sign of what you have the most of. 7 - - - - - - - - - = 0 7 = 7 - - - - - - - - - = 7 Page 25 of 26 MDC@ACOE 10/17/10
15 (-20) 15 (-20) 15 (-20) 15 (-20) = 15 20 = 15 20 = 15 20 = 15 20 = 35 20 = 10 5 20 Traditional: Signs are the same; add and keep the sign. 35 = 10 20 5 = 30 5 = 35 = 35 Page 26 of 26 MDC@ACOE 10/17/10