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Unit 1:Integers and divisibility In this lesson you will learn about: Integers and hierarchy about how to operate with them Powers, roots and their properties Absolute value Vocabulary you need in this lesson: You are going to work with some vocabulary related to this lesson and the problems you will work with. The same words appear in the crosswords, in the word scramble and in the word search. Some theory Integers are the set of whole numbers and their opposites. The number line can be used to represent the set of integers. Look carefully at the number line below and the definitions that follow: Definitions The number line goes on forever in both directions. This is indicated by the arrows. Whole numbers greater than zero are called positive integers. These numbers are to the right of zero on the number line. Whole numbers less than zero are called negative integers. These numbers are to the left of zero on the number line. The integer zero is neutral. It is neither positive nor negative. The sign of an integer is either positive (+) or negative (-), except zero, which has no sign. Two integers are opposites if they are each the same distance away from zero, but on opposite sides of the number line. One will have a positive sign, the other a negative sign. In the number line above, +3 and -3 are labelled as opposites. Rules of operations Addition 1) To add two numbers with the same sign, add their absolute value and write the common sign. 2) To add two numbers with different signs, subtract their absolute values and use the sign of the number with the larger absolute value. Subtraction 1) Change the subtraction symbol to addition. 2) Write the additive inverse of the number being subtracted. 3) Use the rules for addition. Multiplication/Division 1) If both numbers have the same sign, the product (quotient) is positive. 2) If the numbers have different signs, the product (quotient) is negative. Powers and their properties A power whose base is an integer and whose exponent is a natural number is the product of a number by itself as many times as the exponent indicates. Properties a 0 a 1 a a a a a 1 a b a b m n n a n n m n a b a m a n a n ´ b n n m n n a m m Square root If b is a non-negative integer: b/ameans: a 2 b a 2 b n Now it´s time to practice 1. Exercises 1. Put these temperatures in order on the thermometer: 16°C, 37°C, -33°C, -6°C 25°C, 2°C, 25°C, 1°C, 14°C 2. Tell me an integer to describe each situation: 5 degrees below zero a loss of 7 pounds a gain of 10 yards 3 feet below sea level e. 2 degrees above zero a. b. c. d. 3. Tell me the opposite of each integer: 3, -5, -151, 1420, -1 4. Write 4 consecutive, even integers beginning with each of the following integers: -26, 20 5. What number are you on if you: a. Start at 2 and take away 4 b. Start at -7 and add on 3 c. Start at 5, take away 8 and add on 1 d. Start at -3m take away 3 and add on 4 6. .- Write the correct symbol ( < or > ) between each pair of numbers: a) 3 5 b) 4 1 c) 6 10 d) 7 1 e) 1 6 f) 7 15 7. Work out: a. 1 ( 7 2 10 ) (3 8) b. 13 8 ( 6 3) 4 3 ( 7 ) c. 2 7 5 2 3 (10 4 ) d . 1 3 4 15 12 4 2 7 5 e. 2 7 f. 3 2 4 3 27 5 36 - 2. Problems Intergalactic Temperature Quest 1. Your Space Commander sends you on a quest around the universe to collect temperatures for research purposes. You have the use of a spaceship: …and a thermometer: First find the difference between the hottest and coldest temperatures you collected for each planet. Copy the table into your book. The first one is done for you. Planet Hottest recorded temperature Coldest recorded temperature Difference Kark 10°C -5°C 15°C Hikl 126°C 37°C Topa 54°C 12°C Uni-7 -13°C -179°C Jolarg 54°C -68°C Vortan 104°C -97°C Intop -95°C -214°C Yurg -27°C -163°C 2. The record high temperature in Nowhere, USA is 98 degrees and the record low is -23 degrees. What is the temperature variation between the record high and the record low? 3. Jenny has $2. She earns $5, spends $10, earns $4, then spends $3. How many dollars does she have or owe? 4. The highest elevation in North America is Mt. McKinley, which is 20,320 feet above sea level. The lowest elevation is Death Valley, which is 282 feet below sea level. What is the distance from the top of Mt. McKinley to the bottom of Death Valley? 5. The temperature in Anchorage, Alaska was 8°F in the morning and dropped to -5°F in the evening. What is the difference between these temperatures? 6. A submarine was situated 800 feet below sea level. If it ascends 250 feet, what is its new position? Use integer numbers to solve this problem. 7. The Roman Civilization began in 509 B.C. and ended in 476 A.D. How long did the Roman Civilization last? 8. The Punic Wars began in 264 B.C. and ended in 146 B.C. How long did the Punic Wars last? 9. A proton has a charge of positive one. An electron has a charge of negative one. Find the total charge of an ion with 14 protons and 19 electrons. Some interesting web sites! In the next page you will find a presentation of how to represent integers on the number line: http://www.nebo.edu/misc/learning_resources/ppt/6-12/ integers_numberline.ppt#256,1,Integers and the Number Line To take a self-test and check if you have learnt about integers and absolute values correctly, go to: http://www.glencoe.com/sec/math/studytools/cgi-bin/msgQuiz.php4? isbn=0-07-829633-1&chapter=3&lesson=1&headerFile=4&state=na To read a bit more and do some more practice, take a look at this page: http://www.glencoe.com/sec/math/msmath/mac04/course2/ study_guide/pdfs/mac2_pssg03.pdf To play a game with integers, click here: http://www.glencoe.com/sec/math/msmath/mac04/course2/ group_activities/lesson.php/na/3l To take a quiz about integers, where you can choose the length and the difficulty, go to: http://www.thatquiz.org/ If you want to read about integers across history and how they appear and are used by mathematicians, click here: http://www.bookrags.com/research/integers-wom/