First Edition Extending the Number System
Understanding Integers Understanding integers on a number line.
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Section 1 Integers Key Terms integer - a whole number and its opposite negative number - a number to the left of zero on a number line, written with a (-) sign. opposite - a pair of numbers that are the same distance from zero; a pair of numbers when added together equal zero (also called additive inverses) positive number - a number to the right of zero on a number line. whole number - a number without fractions; integer. z Integers in the Real World Have you ever heard a weather report that talked about temperatures below zero, like the cold front will produce temperatures of 10 below zero. What does that mean? Look at the thermometer. Notice that there are is a 10 above zero and one below zero. The one below zero is written with a (-) sign to show that it is less than zero. Do you notice that both of the 10 s are the same distance from zero? They go in opposite directions, but are the same distance. Numbers with the same value that are on opposite sides of zero are called opposite numbers What is the opposite of 50 F? What is the opposite of -1 F? 130 120 120 110 110 100 100 90 90 80 80 70 70 60 60 50 50 40 40 30 30 20 20 10 10 0-10 -10-20 -20-30 -30-40 -40 0 3
Think about this: On the same winter morning, the temperature is -28 in Anchorage, Alaska and in 65 Miami, Florida. How many degrees warmer was it in Miami than in Anchorage on that morning? Click here LearnZillion Video Temperature Where else can you find use for integers? You use integers to determine how much money is in your pocket, account or wallet. You have $10.00 in your pocket.!!! $10.00 You owe your sister $5.00.!!!! -$5.00 How much money do you have to spend AND be able to pay your sister back. Money that you owe is a negative amount. Money you have is a positive amount. At a bank there are special words that describe your negative and positive money amount. A checkbook register is one way to keep track of your spendings and earnings. A place for the check number, date and description of a transaction are given to keep your organized. If you have a balance of $250.00 and need to pay your phone bill of $100.00, how much would you withdraw to pay the bill? How much would you have left? Click here LearnZillion Video Money INTERACTIVE 1.1 Check Register Number Date Transaction Withdrawal Deposit $ Withdrawal Deposit Checkbook A withdrawal is money you take out of your account, or spend. It is a negative amount. A deposit is money you put into your account, or save. It is a positive amount. 4
You also use integers in football. Have you ever heard of a 15 yard penalty? If so, what does that mean? It means you have to move back 15 yards in the opposite direction you want to go, or -15 yards from the line of scrimmage. Let s try writing a few integers and phrases that would represent them using the real world examples you have read about. Integer phrases practice: match the integer with the phrase Question 1 of 5 minus 10 degrees Fahrenheit a five yard penalty!!!!! -5 yds a gain of 10 yards!!!! 10 yds a loss of 12 yards!!!!! -12 yds A. -10 F B. +10 F a 10 degree dip in the temperature!! a 20 degree rise in the temperature!! -10 degrees 20 degrees C. -10 C D. 0 F a deposit of $300.00!!!! $300.00 a withdrawal of $75.00!!!! -$75.00 earning $15.00!!!!! $15.00 Check Answer you owe $3.00!!!!!! -$3.00 5
We can use integers when we talk about elevation. On Earth, sea level is often treated like 0 on a number line. Objects above that level are considered positive values. Objects below that level are considered negative values. Click Here Learnzillion Video Elevation Think about this: Denver, Colorado is called The Mile High City because its elevation is 5280 feet above sea level. Someone tells you that the elevation of Death Valley, California is -282 feet. a. Is Death Valley located above or below sea level? Explain. b. How many feet higher is Denver than Death Valley? c. What would your elevation be if you were standing near the ocean? 6
Integers in Math Now that you have an idea of where we use integers in the real world, we will consider how to use them in the math world. A number line is a great place to start - think of it as a sideways thermometer. In math, opposites are a pair of numbers that are the same distance from zero going in opposite directions. Much of the time in math you will refer to opposites as negative and positive numbers. However, sometimes the (-) negative sign is read as the opposite of the number it is naming. -3 would be read as the opposite of 3-16 would be read as the opposite of 16 Opposites: translate the opposites Question 1 of 3 the opposite of 12 What about 0 (zero)? Zero is neither negative or positive. Notice how the numbers match up. Move one space to the right of zero and your are at 1. Move the same one space in the opposite direction and you are at -1. 1 and -1 are opposites. 15 and -15 are opposites A. -12 B. 12 C. 21 D. 0 Check Answer 7
Independent Practice Directions: Write an integer to represent each situation. 1. A loss of 20 points 2. A gain of 14 points 3. A profit of $20.00 4. A loss of $18.00 5. An elevation of 500 ft. 6. 200 feet below sea level 7. 8 degrees below zero 8. 78 degrees 9. A decrease of $68.00 10. An increase of $55.00 Challenge problems Directions: Write the opposite of each integer described or written 11. A loss of 18 12. A gain of 22 13. -78 14. 999 15. -87 16. 30 feet below the surface of the ocean Extra practice http://www.onlinemathlearning.com/integer-number-line.html http://www.ixl.com/math/grade-6/number-lines-with-integers 8
Absolute Value and Ordering integers
Section 1 Absolute Value Absolute value is always a positive number. Absolute value is written as x The number is enclosed between two lines. Using this notation can be translated as the positive value of. Absolute value The absolute value of a number is its distance from zero. Example A -7 = 7 Solution: The absolute value of a number or an expression is ALWAYS positive or 0 The absolute values of opposite numbers are equal Example: -4 = 4 = 4 because both -4 and 4 are 4 units away from 0 Example B With absolute value the direction does not matter, only the distance. 10
**Important note!!! When a negative sign is written within the vertical lines, the answer will be positive -8 = 8 Fill in the rest of the chart, then answer the True False questions below. (Hint: think opposites)!! When a negative sign is written outside the vertical lines, the answer will be negative - 5 = -5 City State Elevation above sea level Elevation below sea level Example C - The opposite of the absolute value of 75, which is -75. The opposite of the absolute value of -25 is -25. Denver Colorado 5130? New Orleans Louisiana -8? Seattle Washington 0? Elevation True or False? Question 1 of 8 New Orleans is feet below sea level. Evaluate the following absolute values. a) c) b) d) A. True B. False Check you your Khan Academy App - Absolute Value Check Answer 11
Section 2 Ordering Integers The same applies when you have both +/- numbers in a group to be ordered. -12, 3, -4, 6 becomes -12, -4, 3, 6 in order from least to greatest Ordering integers You would have not problem if you were asked to put the numbers 13, 1, 15 and 8 in order from least to greatest. But, what if you needed to put -13, -1, -15 and -8 in order from least to greatest. What would you do? A number line would be a good place to start. Remember the farther to the left a number is, the smaller it is. The farther to the right a number is the larger it is. Draw the number line above on your own. Label the locations of -2 and -4. Remember the idea of opposites and that the farther to the left you go the smaller your value. Next, write the number is order from least to greatest. 0, 2, 4, -2, -4!!! 0 2 4 Comparing Integers From least to greatest: -15, -13, -8, -1-15 is farthest to the left so it is the smallest number In addition to ordering numbers in a set, you can use the number line to help you compare numbers using inequality signs (, ) How do we compare integers? -1 is the farthest to right so it is the greatest 12
rg/video/ordering-negativenumbers? There are a couple of important things to consider when comparing integers. 1. A positive number is ALWAYS greater than a negative number. The more positive a number, the greater it is. Let s look at an example. Example -6 2 Negative six is below zero. Two is above zero. Two is greater than negative six. -6 < 2 2. If two numbers are positive, the larger number is greater. Example 17 10 Seventeen is greater than 10. 17 > 10 3. If two numbers are negative, the number closer to zero is greater than the other. For two negative numbers, you have to think backwards. The larger the number the greater the loss is. The greater the loss, the smaller the number. Think about the number and its relationship to zero. Example -25-36 Negative 25 is closer to zero than -36. It is the greater number. -25 > -36 Check out your Khan Academy App - negative number basics Ordering integers practice: 1. 3, 0, -2, 6-1 2. -9, -3, 0, 2, 5, 6 3. 7, 4, 2, -19, 0, -12, 11 4. 4, -4, 5, 7, 0, 10, -7 Comparing integer practice: 1. -6 8 2. -99-9 13
3. 12 6 Think about this: You can use what you have just learned to organize data from the real world. REVIEW 2.1 Ordering and Comparing Integers Question 1 of 5 Which statement is true Mon Tues Wed Thurs Fri Sat Sun 5-1 -6-2 3 7 0 Look at the low temperature in Juneau, Alaska for one week. a. Arrange them in order from coldest to warmest temperature. b. On a winter day, the low temperature in Anchorage was 23 degrees below zero (in C ) and the low temperature in Minneapolis was 14 degrees below zero (in C ). Sophia wrote, A. -2-4 B. -4-2 C. -2-4 Minneapolis was colder because -14-23. Is Sophia correct? Explain your answer. c. The lowest temperature ever recorded on earth was -89 C in Antarctica. The average temperature on Mars is about -55 C. Which is warmer, the coldest temperature on earth or the average temperature on Mars? Write an inequality to support your answer. Check Answer Need more help? Click on this Learnzillion Video series (Click on any of the videos on the left side of the screen) 14
Section 3 Ordering Rational Numbers What is a rational number? A rational number is a number that can be written as a fraction; or p/q where q is not zero. Think to yourself: When moving to the RIGHT of zero, count UP. When moving to the LEFT of zero, count BACKWARD! -3 1/2 or -3.5! 1/2 or 0.5 Rational numbers include integers,simple fractions, mixed numbers, decimals and percents. Examples 1,2,3,4, 1/2, 1/3, 6/7, 1 3/5, 2 1/3 2.35, 5.5, 10.3, 60%, 125% Like whole numbers, fractions and decimals can have negative values. BUT, they are not considered integers. Remember integers are WHOLE numbers and their opposites. Ordering rational numbers is just like ordering integers. Tips for Ordering If you have fractions, percents and decimals in the same group of numbers to be orders, change them to all one type. Example: Change all numbers to decimals 3.5, 75%, -0.28, 1 1/3 3.5, 0.75, -0.28, 1.33 Now they can be put in order easily. 15
**Reminders Ascending order is another way to say from least to greatest (going up) Descending order is from greatest to least (going down) Ordering rational numbers Question 1 of 3 Which group of numbers is in ascending order? A. -1 1/2, -1/2, 0, 1, 4 B. 0, -1/2, 1, -1 1/2, 4 C. 4, 1, 0, -1/2, -1 1/2 Check Answer 16
Absolute value absolute value; noun 1 Mathematics the magnitude of a real number without regard to its sign. Also called modulus. Related Glossary Terms Drag related terms here Index Find Term
Integer ˈintijər noun 1 a whole number; a number that is not a fraction. 2 a thing complete in itself. Related Glossary Terms Drag related terms here Index Find Term
Negative number A negative number is a real number that is less than zero. Such numbers are often used to represent the amount of a loss or absence. For example, a debt that is owed may be thought of as a negative asset, or a decrease in some quantity may be thought of as a negative increase. Negative numbers are used to describe values on a scale that goes below zero, such as the Celsius and Fahrenheit scales for temperature. Related Glossary Terms Drag related terms here Index Find Term
Opposites a pair of numbers that are the same distance from zero; a pair of numbers when added together equal zero (also called additive inverses) Related Glossary Terms Drag related terms here Index Find Term
Positive number A positive number is a real number that is greater than zero. Such numbers are often used to represent the amount of a gain or increase. Related Glossary Terms Drag related terms here Index Find Term
Rational number In mathematics, a rational number is any number that can be expressed as the quotient or fraction p/q of two integers, with the denominator q not equal to zero. Since q may be equal to 1, every integer is a rational number. Related Glossary Terms Drag related terms here Index Find Term