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Notes Packet for Positive/Negative Number and Adding Integers Negative numbers Positive numbers Number line counts up from zero going right. Number line counts down from zero going __left___. Numbers to the right of 0 are _positive__; Numbers to the left of 0 on the number line are _negative__, zero is neither _positive__or __negative_____. Mark the following numbers on the number line above: 7, -2.5, ½ , -9.5 Two numbers that are the same distance from zero but in opposite directions are called opposites. Example: -6 and +6 are opposites, + 3.5 and -3.5 are opposites. Name a pair of opposites: _______________ Opposite numbers have the same _value_____, just opposite _signs____. The sum of any number and its opposite is _zero_____. Example: 6 + (-6) = 0 Work: -8 + 8 = ______ 4 + (-4) = ______ -0.3 + 0.3 = ______ The _sign____ of a number refers to whether it is positive _ or _negative______ (“+” or “-“). The sign____ of 32 is “+” or positive______. The sign or -5 is “-“ or _negative______. What is the sign of?: -5 ____ 7 _____ 3 ____ -4.2 ____ |-10| = 10 |8| = 8 Absolute Value_ is the distance a number is from zero. Absolute value is always a _positive_________ number. Specify absolute value by putting number between 2 _vertical________ bars. Absolute value of x is shown as |x|. Think of it as “How much is it worth?” (The $20 dollars you owe someone is worth the same amount as the $20 in your piggy bank.) or “What is the distance?” (If you are two floors below ground, you are the same distance from ground level as if you are two floors above ground.) What is the absolute value of the following: |-7| = _____ |9| = _____ |-6.5| = _____ |4.2| = ______ If someone is standing at -9 on the number line and they walk to 6 on the number line, how far did they travel? |-9| + |6| = 9 + 6 =15 If someone is standing at -4 on the number line and they walk to 9 on the number line, how far did they travel? |-4| + |9| = _______ INTEGERS Integers are whole numbers (positive, negative, zero). Fractions, mixed numbers and decimals are not integers. Note to the wise: BE ABLE TO DEFINE AN INTEGER! The numbers on a telephone keypad are integers. The number of kids in each class in the school is an integer. Circle the integers: 9 -7 6.3 -4.5 3½ 0 274 -1 75.2 Comparing Integers Less Greater Integers that are farther to the right on the number line are greater than those to their left. A positive integer is always greater than a negative integer (1 > -100) The farther to the left a negative integer is from zero, the smaller its value (-1 > -100). Fill in the blank with > or < to make the statement true: 9 ___ 11 5 ___ 4 -3 ___ 7 -5 ___ -9 20 ___ -25 3 ___ -7 -4 ___ -3 1 ___ -100 Adding Integers You can show the sum of two integers by using arrows on a number line. When we add a positive number, we move right on the number line. When we add a negative number we move left on the number line. Careful: This is different than adding absolute value. Here the sign gives us direction, left or right. Adding two positive integers: Find the sum of (+5) and (+3). Starting at zero, first move 5 units to the right, then 3 units to the right. +5 +3 Therefore (+5) + (+3) = (+8). (Remember, positive integers are most often written without positive signs: 5 + 3 = 8. The sum of two positive integers is positive. Students, please use a straight edge to draw your arrows. Show on the number line below the sum of (+2) and (+4), then write out the number sentence: _________ + ________ = ________ Adding two negative integers: Find the sum of (-5) and (-3). Starting at 0, first move 5 units to the left, then 3 units to the left. -3 -5 Therefore, (-5) + (-3) = -8. You can add two negative integers, but because you are moving in the opposite direction, the sum is negative. The sum of two negative integers is negative. Show on the number line the sum of (-4) and (-2), then write out the number sentence. ________ + _______ = _______ Adding a positive integer and a negative integer: Add (+7) and (-4). Step 1) move 7 units to the right. Step 2) move 4 units to the left. Step 2) -4 Step 1) +7 Therefore (+7) + (-4) = ____. Add (-10) + (+8): Step 2) +8 Step 1) -10 Therefore, (-10) + (+8) = _____ Add (+5) and (-7): -7 +5 Therefore, (+5) + (-7) = ____ Add (-3) + (+9): +9 -3 Therefore (-3) + (+9) = ______ Add on the following number line (-4) + (5), then write the number sentence: _______ + _______ = ________ Remember we said before that a number and its opposite add up to zero. When you add a positive integer with a negative integer, which ever number is smaller in absolute value cancels out that same number from the number of greater absolute value. That is, when we add (-4) + (5), the 4 negative units cancel out 4 of the positive units, and we are left with (-1). Please pull out your set of Algebra Tiles. Each yellow square is marked “+1” and represents a positive 1 integer. Each red square is marked “-1” and represents a negative 1 integer. If we add (-4) + (5), we lay out 4 red tiles to which we add 5 yellow tiles. Match up each red on top of a yellow. Each red/yellow pair adds up to zero and can be removed. We are left with one yellow tile. (-4) + (5) = (1). Use tiles to work out the following problems: (-8) + (6) = ____ (7) + (-3) = ______ (10) + (-6) = ______ (-2) + (5) = ____ (2) + (6) = ______ (1) + (-3) = ______ (-3) + (-4) = ______ (9) + (-7) = _______ What is the rule for adding a positive integer with a negative integer? ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ______________________________________________________________________________ ____________________________ Stations Packet Station 1: Computers set to this website: http://nlvm.usu.edu/en/nav/frames_asid_161_g_2_t_1.html. Works like Algebra Tiles. Follow “Instructions in upper right hand corner.” Work 4 problems. Write them here. _______ + ________ = ________ _______ + ________ = ________ _______ + ________ = ________ _______ + ________ = ________ Station 2: Money Station: Banker steps you through income and payment steps. What is your financial situation along the way? Write out the math sentences and solve. You start with $50 savings. Good for you. ___________ You dogsit for a week and get paid $100. Cool! Feeling rich! ___________ Your balance: Your balance: Got to have that $200 itouch. Mom charges it. You pay her what you have now and owe her the rest. Your balance: ___________ You broke your sister’s itouch. Mom says you must pay for it. Your balance: ___________ You babysit every Friday and Saturday night for a month and make $300. Your balance after paying mom what you owe her: ______ Station 3:Thermometer: Large wooden, standing thermometer with Start Temperature Clip and End Temperature Clip. Be careful to use the right temperature scale! 1) One cold morning in DC, the temperature is -6°C, but rises by 18°C to get to the afternoon high. What is the afternoon high temperature? Mark start and end temperatures on the thermometer and write out the number sentence: (-6) + (18) = _____ 2) One afternoon in NYC, it is a lovely 68° F when a cold front moves through and drops the temperature 30°F. What does the temperature get down to? Mark start and end temperatures on the thermometer and write out the number sentence: (68) + ____ = _____ 3) One day in January in the Mojave Desert, the temperature reaches 18°C during the day but drops to -6°C that night. How much of a drop is that? Mark start and end temperatures on the thermometer and write out the number sentence: ______ + ______ = ______ 4) The low temperature in Fairbanks, AL in January 2010 was -41° F recorded on January 12th. The high temperature for that month was 58° F higher than the low and was recorded on January 7th. What was the high temperature for that month in Fairbanks? Mark start and end temperatures on the thermometer and write out the number sentence: ______ + ______ = ______ Station 4) Elevation: Large standing poster with picture of a mountain next to the ocean (Sierra Nevada Mountains next to the Mediterranean Ocean in Spain—from high elevations, look across the Mediterranean to Africa). One the side there is a ruler marking the elevation (11,000 ft. to the top of Mt. Mulhacen, sea –level and the depth of the Mediterranean (average 5000 ft). Use Start Clip and End Clip to mark the problems. Write out the math sentences and solve. How much of an elevation change is there from the bottom of the Mediterranean to the top of Mulhacen? (-5000’) + _____ = (11,000’) How much of a drop is there between the mid-way station on Mulhacen (marked on poster) and the submarine in the ocean (also marked on poster)? _____ + (-____) = _____ A drop of 7000’ from the summit of Mulhacen gets you to what station? _________ _______ + (-7000) = _______ How far is the seagull above the octopus on the poster? ______ + _____ = ______ Station 5: Laptop playing scene from movie Stand and Deliver where new teacher, Jamie Escalante, teaches bright, but previously under-taught student how to add positive and negative numbers by envisioning digging holes in the sand. “Fill up the holes,” he tells him. Students watch scene then use Duplo board filled completely with one layer of duplos to represent untouched sand (zero). Student takes away duplos (buckets of sand) to represent negative numbers, and then fills them in by adding duplos (buckets of sand) representing positive numbers. Start with 3 Duplos removed from the board. Add 5 Duplos first filling in the holes. (-3) + 5 = ______ Now add (-8) Duplos. _____ + (-8) = _____ Add 10 Duplos : ______ + ______ = ______ Homework: Addition of Integers (Positive and Negative) Define an Integer:_______________________________________________________ _____________________________________________________________ _____________________________________________________________ _________________ Fill in the blanks with < or > to make the number sentence true: 1) -100 ____ 1 2) 70 ____ -75 3) -3 ____ 2 4) 5 ____ -4 Complete the number sentence to make it true. 5) (-7) + (7) = ______ 6) (-3) + (4) = ______ 7) (9) + (-15) = ______ 8) (6) + (12) = ______ 9) (-7) + (-4) = ______ 10) (-20) + (11) = ______ Complete the following word problems. You will be given one point for a drawing showing the problem, one point for a correct math sentence, and one point for the correct answer. 11) Mt. Everest, the highest elevation in Asia, is 29,028 feet above sea level. The Dead Sea, the lowest elevation, is 1,312 feet below sea level. What vertical distance would be traveled going between these two elevations? 12) In Buffalo, New York, the temperature was -14°F in the morning. If the temperature dropped 7°F, what is the temperature now? 13) A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? 14) If I am $570 in debt, but I get paid $1000 by my employer and pay off my debt, how much money will I have left? (Picture can just be a number line).