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Learning Target: I can… Solve problems involving integers and absolute value Definition Integers: the set of positive whole numbers, their opposites, and zero (…-3, -2, -1, 0, 1, 2, 3…) Real life example: If it is 10 degrees below zero, it is -10° 20 Feet below sea level = -20 Representing Integers Gain of 5 yards Loss of 6 yards A profit of $18 A decrease of 14 cents Use your number line to help you! -8 -7 -6 -5 -4 -3 Negative Integers -2 -1 0 1 2 3 4 5 6 7 Positive Integers 8 Examples 1. Put the following integers in order from least to greatest: -4, 8, -2, -6, 3 -6, -4, -2, 3, 8 2. Put the following integers in order from greatest to least: -2, 0, 7, -1, -5 -5, -2, -1, 0, 7 Comparing Integers Fill in the blank with greater than (>) or less than (<). 1. -10 4 2. 3 -9 3. -15 -20 4. -1 0 5. -2 -4 OAA Practice The table below shows the low temperatures of four cities on one winter night. Low Temperatures of Four Cities One Night City Boston Lowell Springfield Worcester Temperature 3°F 0°F -8°F -5°F Which city had the lowest temperature that night? A. Boston B. Lowell C. Springfield D. Worcester Review - Practice 1. Put the following numbers in order: -2, 3, 0, 6, -5 2. Fill in the correct sign for the following: (>, <, or =) 3 -4 -1 1 0 -3 Brainpop Review – Least to Greatest -8 -7 -6 -5 -4 -3 -2 -1 0 1 2 Negative Integers 1) -6, 4, -12, 0, 5 2) -3, 1, 9, -2, -8 3 4 5 6 7 Positive Integers 8 Absolute Value: The distance of a number to zero Example: │6│ = 6 │-6│ = 6 Absolute value is always positive! You try it! Find the absolute values! │-9│ = │10│= Quick Check Compare the numbers │-9│ a. b. c. d. ≥ < > = │5│ Compare the numbers │18│ a. b. c. d. ≥ < > = │-19│ Absolute Values: Evaluating Expressions 1) |18| ÷ 2 × |–3| = ______ 2) |–3| + 10 = ______ 3) (12 + |–8|) ÷ |–5 | = ______ 4) |–16| – |–16| = ______ Ordering with Absolute Value When you order integers, make sure you put the number back in its ORIGINAL form when put into order. 1) -2, │-3│, 0, │-4│, -5 2) │-7│, -3, │-3│, -9, -1 Adding Exit Question Fill in the blanks with >, <, or = 1. 6 │-6│ 2. │-5│ 4 3. │3│ 2 4. │-3│ │2 + 1│ Learning Target: I can… Solve problems involving integers and absolute value Homework? Summer Assignment? Parent Signature? Team Points… Dividers: 1. Notes 2. HW/CW 3. Weekly Reviews 4. Quizzes 5. Tests Absolute Value Review Definition? The distance of a number to zero │2│ = │-2 │ = -│-2 │ = -│2 │ = Clicker Practice 1) │13│+ │-5│ = 2) -│-5│ = 3) │2│ + │-2│ = 4) -│10│ + │10│ = Additive Inverse What you add to a number to get zero. Examples: -5 + ____ = 0 13 + ____ = 0 12 + (-12) = ____ 1) -7 + _____ = 0 2) 15 + ______ = 0 3) _____ + (-5) = 0 4) 13 + (-13) = _____ Real Life Examples Opposites combine to make zero Example 1: A deposit of $50 combined with a ____________ of $_____ Example 2: A gain of 5 yards after a ________ of ______ yards. Clicker Word Problems 1) Which statement best describes a situation in which opposite quantities combine to make zero? A. Shawna made 8 cups of soup and divided the soup into 8 containers. B. Melanie deposited $10 in her savings account and then withdrew $10 from the account. C. Peter scored 2 goals in the first period of a hockey game and 2 goals in the second period. D. Marcos missed 4 questions on a test in which each question was worth 4 points. 2) Which statement best describes a situation in which opposite quantities combine to make zero? A. Angie had 100 sheets of paper and divided them evenly into 5 folders. B. Mandy deposited $200 in her savings account and then added $200 her following pay period. C. Paul and Josie are on a team together. Paul scored 5 points on his turn and Josie lost 5 points on her turn. D. Mario bought 2 sodas, which cost $2 each. 3) Steve was golfing at Quail Hollow Resort. After the first nine holes, Steve was 3 under par. After the back nine, he had a score that was even to par. How many over par was he on the back nine? A. -3 B. 3 C. 0 D. 6 Monica budgeted $235 for “back to school” stuff. She went school shopping with her mom and spent $130 on clothes and $105 on supplies. Is this an example of the additive inverse? A. Yes B. No http://nlvm.usu.edu/ https://www.youtube.com/watch?v=31Q2_ETqxzc Learning Target: I can… Add integers using counters and number lines Any Number Plus Its Opposite Equals? ZERO! This is called the ADDITIVE INVERSE! Real Life Additive Inverses For example, a hydrogen atom has 0 charge because its two constituents are oppositely charged. (a single positively charged proton and a single negatively charged electron) Using A Number Line Step 1: Plot your __________________________ number. Step 2: If adding a positive number, move _________________. If adding a negative number, move __________________. Step 3: Record your answer. 1) 2 + (-4) = 2) -1 + (-2) = 3) -7 + 5 = 4) -3 + 4 + (-9) = Using Counters Step 1: Assign one color to be ______________________ and one color to be ____________________. Step 2: Lay out counters to represent each of the numbers in the expression Step 3: Cancel out any ________________________ __________________ (1 positive and together). 1 negative 1. -8 + 4 = 2. 5 + (-12) = 3. -1 + (-9) = 4) -5 + (-4) + 9 = 5. 8 + (-2) + (-3) 6. -4 + (-2) + (-1) Part B. Complete with your partner Exit Question Draw out the question and find the sum below: -5 + 3 = Learning Target: I can… Solve word problems involving integers REVIEW Which number line models the problem -3 + 2? │3 + -4│ When you can’t use the number line Step 1: Ask yourself if the signs are same or different Step 2: If they are the same: ADD, if they are different: SUBTRACT Step 3: Decide whether your answer will be negative or positive Example 1) -99 + 18 = 2) -10 + (-48) = Quick Check! 3) -45 + 8 = 4) -13 + -43 = 5) 87 + -11 = Multiple Numbers 1) -5 + 17 + (-20) + 36 + (-13) You try! 2) -97 + (-65) + 15 + (-4) 3) 67 + (-54) + 90 + (-35) + 47 Word Problems (Key Words) Positive Negative Adding Integers Word Problems Key Words Below sea level Below 0 Rise Drop Spend Earn Adding Integer Word Problems 1. During the football game, Justin caught three passes. One was for a touchdown and went 52 yards. The other was for a first down and was for 17 yards. The other was on a screen pass that did not work so well and ended up a gain of -10 yards. What was the total yardage gained by Justin on the pass plays? 2. It was a very freaky weather day. The temperature started out at -9°C in the morning and rose 15° by noon. It stayed at that temperature for six hours and then dropped 8°C. What was the temperature at 6 p.m.? Integers Word Problems 3. You owe $225 on your credit card. You make a $55 payment and then purchase $87 worth of clothes at Macy’s. What is the integer that represents the balance owed on the credit card? 4. A monkey sits on a limb that is 15 ft below sea level. He swings up 10 ft, climbs up 6 ft more then jumps down 13 ft. How far below the ground is the monkey now? 5. Mary has $267 in her checking account. She writes checks for $33, $65, and $112. What is the balance in her account now? 6. A submarine dove 836 ft. It rose at a rate of 22 ft per minute. What was the depth of the submarine after 4 minutes? Exit Find the sum without using a calculator, number line, or tiles. 1) -13 + (-12) = 2) 10 + (-15) = 3) 29 + (-9) = Learning Target: I can… Subtract integers without a calculator Review – Adding Integers The temperature is -4°F in Alaska on a day in January. If it is 8°F warmer here, what is the temperature here? a. 12°F b. 4°F c. 5°F d. -12°F The Golden Gales football team loses 2 yards on the first down. On the second down, they gain 5. What is their net yardage gained? a. 7 yards b. 3 yards c. -7 yards d. -3 yards A number plus its opposite Review The answer to -10 + 5 will be: a. Negative b. Positive c. Zero The answer to -11 + (-6) will be: a. Negative b. Positive c. Zero And again! Adding and SUBTRACTING Integers Brainpop Subtracting Integers To subtract integers, say “cha-ching!” Step 1: Change subtraction to addition Step 2: Change the second number to its opposite Step 3: Add Example 1 4 – (-7) = 4 + (+7) = = 11 Example 2: -7 – 2 = Cha-ching! -7 + -2 = = -9 Example 3 6 – 10 = Example 4 4) -7 – 4 Examples 5) 3 – (-9) 6) 2 – 10 7) -4 – (-6) 8) -9 – 1 Word Problem 1) On back 2) On back The highest point in the United States is Mount McKinley at 20,320 feet above sea level. The lowest elevation in the country is Badwater Basin at 280 feet below sea level. What is the difference between these two elevations? 3) On Back 4) On Back Discuss - Extended Response Example Alberto subtracts different negative integers from 13. Show three different examples of what Alberto does. Explain what happens when you subtract different negative integers from 13. EXIT 1) 16 – (-1) = 2) -5 – 1 = 3) -3 – (-2) = Exit What is 14 – (-8)? Learning Target: I can… Solve word problems involving the adding and subtracting of integers Review Remember cha-ching! Review! Kelly has $40 in her account. If she withdraws $30, then deposits $50, how much does she have in her account now? And again! Ohio State gains 7 yards in the first down. If they lose 10 yards on the second down, and gain 6 yards on the third, what is their net yardage for this series of plays? In January, the temperature hit a low of 2°F. The next day, it got 3 degrees colder! What expression could be used to show the temperature on the second day? a. -2 – (-3) = 1° b. -2 + (-3) = -5° c. -2 + 3 = 1° d. -2 – 3 = -1° Word Problems Stacey is a cliff diver. She starts 90 feet above the water, dives and falls to 25 feet below the water. Write an expression to find the total distance Stacey traveled. Exit Mount Graf is located 1,547 feet above sea level. Lake Graf is 120 feet below sea level. What is the difference in altitude between the two locations? Learning Target: I can… Multiply integers without a calculator Review Adding and Subtracting Integers To add ask yourself: Signs same or different? Should I add or subtract? Will the answer be negative or positive? To subtract: Cha-Ching! Change the subtraction to addition and the second number to its opposite Examples -85 – (-13) = -14 + (-5) = -8 – 13 = -23 + 5 = 13 – (-14) = Multiplying Integers Using Counters Multiplication is really repeated addition. 2 · 3 means 2 sets of 3 2 · -3 means 2 sets of -3 -2 · -3 means the opposite of 2 sets of -3 You Try! 1) 3 · (-1) 2) 5 · (-2) 3) -3 · (-4) 4) -2 · (4) You Try! 1) -3 · (-2) 2) -3 · (2) 3) -4 · (-4) 4) (-3)² Multiplying Integers Rules 1. Positive x positive = positive example: 2 x 3 = 6 2. Positive x negative = negative example: 2 x -3 = -6 3. Negative x positive = negative example: -2 x 3 = -6 4. Negative x negative = positive example: -2 x -3 = 6 Why? 1. 2 x 3 means two groups of 3, which is 6 2. 2 x -3 means two groups of negative three, which is -6 3. -2 x 3 means the opposite of two groups of 3 or -6 4. -2 x -3 means the opposite of 2 groups of -3, which is 6 You try! No calculators! -4 ∙ 2 = -5 ∙ -3 = 2 ∙ -3 = You try! More Than 2 Numbers -7 · (-5) · (-4) = _________ -1(9)(-5) = _________ -3(5 · 2) · -2 = ______________ More than 2 Numbers 1. When multiplying an EVEN number of negatives, and zero is not present, the product will be ___________________________________ 2. When multiplying an ODD number of negatives, and zero is not present, the product will be ___________________________________ Practice Again (-1)(-7) ∙ -4 = -3 ∙ 4 ∙ -2 = (7)(-2)(2) = EXIT -3 ∙ 4 ∙ -2 = Exponents (-2)² -2² (-2)³ (-2)⁴ Exit 1) -3 ∙ 1 ∙ -7 = 2) (-3)² 3) -3² Learning Target: I can… Divide integers without a calculator Review - Multiplication (-5)² -5² (-5)³ Counters - Discuss When we divide a positive and a positive the quotient is positive. When we divide a positive and a negative (in either order), the quotient is negative. When we divide a negative and a negative, the quotient is positive. Examples -4/2 = __________ -15 ÷ -3 = _____________ -27/9 = ____________ You try! No calculators! 50/(-10) = (-144) ÷ (-12) = (-33) ÷ (-11) = Order of Operations P E MD AS 1) -45 ÷ (-5) + 7 2) -3 – 12 ÷ (-3) 3) 2 (5) (20) 2 (3) 4) 3(4) (20) 5 (3) Word Problem The table below shows the number of points scored in the first ten minutes of Jeopardy. Find the mean number of points scored. Person James Jenny Howard Arie Points -100 300 -500 200 The West Side Dolphins football team gained 8 yards, then gained 4 yards, then lost 4 yards, then gained 15 yards, then gained 3 yards, then gained zero yards, then lost 2 yards, then gained 8 yards. How many yards did the Dolphins gain or lose on average per play? EXIT EXIT A negative number multiplied by a negative number will result in a product that is __________________. Real-Life Integer Problems Learning Target: I can… Solve real life integer problems Addition/Subtraction A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? The lowest point of the St. Lawrence River is 294 feet below sea level. The top of Mt. Jacques Cartier is 1,277 feet above sea level. How many feet higher is the top of Mt. Jacques Cartier than the lowest point of St. Lawrence River? Word Problems Ohio State gains 7 yards in the first down. If they lose 10 yards on the second down, and gain 6 yards on the third, what is their net yardage for this series of plays? Multiplication/Division A failing company lost $200 for 5 months straight. What is their net profit? Expression (-200)(5) a. -$200 b. -$1000 c. $1000 d. -$205 Ten people went to dinner. As a group, they spent $120. If they are going to split it equally, how much will each person owe? A little bit tougher… Jack just got his license. In his bank account, he currently has $2000. He paid $500 as a down payment on a car. He also plans on spending $50 a month in gas. How much will he have left in his bank account after 1 year? Justin gets $100 a week from his job. He puts half in his savings account and half in his checking account. If he spends $20 a month all from his checking account, how much will he have left in each account after 1 year? Exit Which expression could be used to show that Jan spent $20 every week for 3 weeks and then earned $50 at the end of the month? a. 20 x 3 + 50 b. (-20 x 3) + 50 c. 20 + 3 – 50 d. (20 x 3) - 50 Stations! Learning Target: I can… Review for my integers quiz Exit James subtracts different negative integers from 10. Show an example of what James does. Is the answer always positive, always negative, or always 0? Exit On four consecutive days Arnold has the following transactions: Monday, he deposited $20.00, Tuesday he withdrew $5.00, Wednesday he deposited $10.00 and Thursday he withdrew $25.00. Write a number phrase for this situation and solve. Review (answer numerically) (6)(-7) = (-60) ÷ (-12) = Learning Target: I can… Do well on my integers quiz BONUS MATH 1. Addition -9 + 1 -9 + -1 2. Subtraction -9 – 1 -9 – (-1) 9 – (-1) 3. Multiplication -3 · 2 -3 · -2 4. Division -45 ÷ (-5) -45 ÷ (5) -(-3) │-9│ = -│-9│= Exit Simplify the following expression: -│-9 + 3 – 1│=