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Chapter 2

Chapter 2. Integers. Chapter 2-1-A Explore Absolute Value. Look at the picture. A scuba diver is diving at -130 feet. At the same time, a hiker is on top of a cliff at 130 feet above the water. Compare and contrast -130 and 130 feet.

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Chapter 2

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  1. Chapter 2 Integers

  2. Chapter 2-1-AExplore Absolute Value • Look at the picture. • A scuba diver is diving at-130 feet. At the sametime, a hiker is on top ofa cliff at 130 feet abovethe water. • Compare and contrast-130 and 130 feet. • What is a real-world situation with two values that can be represented with -10 and 10. Compare and contrast them.

  3. Compare and contrast walking from the house to school and from the house to the park. • Look up the definition of absolute in the dictionary. Explain how this could be applied to math. • REVIEW: What is the opposite of a number?

  4. Chapter 2-1-BIntegers and Absolute Value • The bottom of a skateboard ramp is 8 feet below street level. A value of -8 represents 8 feet below street level. • What would a value of -10 represent? • The top of the ramp is 5 feet above street level. How can you represent 5 feet above street level? • Numbers like 5 and -8 are called integers. • An INTEGER is any number in the set{…, -4, -3, -2, -1, 0, 1, 2, 3, 4, …} where … means continue without end.

  5. 0 • Is zero a negative number or a positive number? • Determine an integer for each real-life situation: • An average temperature of 5 degrees below normal. • An average rainfall of 5 inches above normal. • A temperature of 6 degrees above normal. • A temperature of 2 degrees below normal. • A depth of 50 feet below sea level. • A height of 75 feet above sea level. • Integers can be graph on a number line. To GRAPHan integer on the number line, draw a dot ON the line at its location. • Draw a number line from -10 to 10 (don’t forget zero). • Graph the following set of numbers on your number line: • {6, -4, 1, -9, 5, -2}

  6. Numbers that are the same distance from zero on a number line have the same ABSOLUTE VALUE. • ABSOLUTE VALUE • The ABSOLUTE VALUE of a number is the distance between the number and zero on a number line. • We use lines to represent absolute value. • │-5│ =5 • │5│ = 5

  7. Take 5 minutes and answer #1-10 on page 79 in your text book. • Let’s practice! • Evaluate each expression: • │-3│ • 3 • │2│ • 2 • │-8│ • 8 • │0│ • 0 • │-1│ • 1 • │-137│ • 137 • │234│ • 234 • │-1003│ • 1003

  8. Chapter 2-1-CThe Coordinate Plane • A COORDINATE PLANE is formed when two number lines intersect. The number lines separate the coordinate plane into four quadrants. The y-axis is the vertical number line. The ORIGINis the point of intersection of the two number lines. Quadrants are four sections of the coordinate plane. The x-axis is the horizontal number line.

  9. An ORDERED PAIR is a pair of numbers such as (-5, 2) used to locate a point on the coordinate plane. • (-5, 2) • The x-coordinate corresponds to a number on the x-axis. • The y-coordinate corresponds to a number on the y-axis.

  10. Count your spaces right. (RIGHT means positive) Start here. • When locating an ordered pair, moving right or up on a coordinate plan is in the positive direction. Moving left or down is in the negative direction. • When naming a point, follow the following steps: • Start at the origin. • Count your spaces left or right. • Count your spaces up or down. Count your spaces down. (DOWN means negative) So our ordered pair is: (8, -4)

  11. Helpful Hint: “You have to go down the hallway before you can go up or down in the elevator”. • When graphing an ordered pair, you follow similar steps. ( x , y ) • Start at the origin, (0, 0). • Move left or right. • Positive numbers = right • Negative numbers = left • Move up or down. • Positive numbers = up • Negative numbers = down. • Graph these on coordinate plan: • A(4, 5) • B(-3, 2) • C(-1, 6) • D(5, -2)

  12. Maps can be divided into coordinate planes. These can be used to give directions or find locations. What is located at (6, 5)? • What ordered pairrepresents the ThinkTank? • What is located atthe origin? • Take 5 minutes andanswer questions#1-9 on page 83 in your text book.

  13. Chapter 2-2-AExplore: Add Integers • In football, forward progress is represented by a positive integer. Losing yardage is represented by a negative integer. • On the first play a team loses 5 yards and on the second play the team loses 2 yards. • What is the team’s total yardage on the two plays? • Use the counters at your desk to figure this problem out: • Place 5 negative counters on your desk. • Then place 2 negative counters on your desk. • How many counters do you have in all? • 7 negative counters = -7 yardage =loss of 7 yards

  14. Before you begin modeling addition with integers, it is important to remember these two points: • When a positive counter is paired with a negative counter, it results in a zero pair. In other words, they cancel each other out. • = 0 • You can add or remove zero pairs because adding or removing zero does not change the value of the counters on the mat. • With your partner and your counters, complete problems # 1-17 on page 87 in your text book. • Let’s review your answers!

  15. Chapter 2-2-BAdd Integers • Atoms are made of negative charges (electrons) and positive charges (protons). • The helium atom shown has a total of 2 electrons and 2protons. • Represent the electrons in a helium atom with an integer. • Represent the protons in a helium atom with an integer. • Each proton-electron pair has a value of zero. What is the total charge of the helium atom? • Combining protons and electrons are a lot like adding integers!

  16. ADDING INTEGERS WITH THE SAME SIGN(also known as integers with like signs) • To add integers with the same sign, add their absolute values and use the common sign. • 4 + 3 = 7 • -4 + -3 = -7 • 5 + 6 = 11 • -5 + -6 + -11 • Try some examples: • -14 + -16 = ? • 23 + 38 = ? • -11 + -9 = ? • 12 + 18 = ?

  17. The integers 3 and -3 are called OPPOSITESbecause they are the same distance from zero, but on opposite sides of zero. • Two integers that are opposites are also called ADDITIVE INVERSES. • ADDITIVE INVERSE PROPERTY • The sum of any number and its additive inverse (or opposite) is zero. • 3 + (-3) = 0 • (-5) + 5 = 0 • (-1001) + 1001= 0

  18. ADDING INTEGERS WITH DIFFERENT SIGNSalso known as integers with unlike signs • To add integers with different signs, subtract the smaller absolute value from the larger absolute value. Then use the sign of the larger value. • The answer will be positive if the positive integer’s absolute value is greater. • The answer will be negative if the negative integer’s absolute value is greater. • (-10) + 6 = -4 • 10 + (-6) = 4 • 8 + (-7) = 1 • (-8) + 7 = -1 • Let’s look at some examples….

  19. Take 5 minutes and answer problems #1-10 on page 91 in your text book. • Find the sum of the two integers: • 10 + (-12) • (-13) + 18 • (-14) + (-6) + 6 • 3 + (-15) • Look at the picture below. How can we figure out the height of point D?

  20. Chapter 2-2-CExplore: Subtracting Integers • At a local aquarium, a popular attraction features dolphins. The dolphins jump through hoops that are 5 feet above the water. To prepare for their jump, they start at 6 feet below the surface of the water. What is the difference between the two distances? • We can use counters to represent this situation. • 5 – (-6) = ? • Start with five positive counters. • We don’t have any negative counters to take away though? How can we do this?

  21. We can use zero pairs to create negative counters without affecting the total. • We start with our original five counters… • Then we add six zero pairs because we need to take six negative counters away. + • Now we are left with this:

  22. Now we can take six negative counters away. • Remember 5 – (-6) = ? • Take away the six negative counters and we are left with…. • So 5 – (-6) = 11 • With your partner, complete the activity in your book on page 94, #1 – 11. • Let’s discuss the results and talk about what we saw.

  23. Chapter 2-2-DSubtract Integers -7 +6 • One way to subtract integers is to use a number line. • 6 – 7 = ? • = -1 • 1 – 5 = ? • = -4 • -3 – 4 = ? • = -7 • 0 – 5 = ? • = -5 -5 +1 -4 -3 0 -5

  24. Another way to subtract integers is to turn your subtraction problem into an addition one! • SUBTRACT INTEGERS • To subtract an integer, add its opposite. • 4 – 9 • changes to 4 + (-9) = -5 • 7 – (-10) • changes to 7 + 10 = 17 • 8 – 19 • changes to 8 + (-19) = -11 • 6 – (-3) • changes to 6 + 3 = 9 • In other words… • Subtracting a positive = adding a negative • Subtracting a negative = adding a positive

  25. Take 5 minutes and answer problems #1-12 on page 97 in your text book. • Find the difference: • 2 – 15 • -13 – 8 • 12 – (-6) • -21 – (-8) • Real world examples: • The temperatures on the moon vary from -173°C to 127°C. Find the difference between the maximum and minimum temperatures. • How do we set this up? • Brenda had a balance of –$52 in her bank account. The bank charged her $10 for having a negative balance. What is her new balance? • How do we set this up?

  26. Chapter 2-3-AProblem Solving Investigation • With your group, answer questions #1-10 on page 100-101 in your text book. • Your group may be asked to present your work. • Be ready to discuss all of your answers.

  27. Chapter 2-3-BExplore: Multiply and Divide Integers • The number of students who bring their lunch to Meigs is decreasing at a rate of 5 students per month. What integer represents the change after three months? • If the number is decreasing by 5 students, then we will use -5 to represent that change. • If we want to look at the change over 5 months, we will say -5 × 3. Three groups of -5 So -5 × 3 = -15

  28. Let’s look at (-4) × (-6) • This means remove four sets of six negative counters. • It can also be interpreted as removing six sets of four negative counters. • In order to do this, we have to start with enough zero pairs. We will start with four sets of six zero pairs. 1 set 2 set 3 set 4 set

  29. Now we can remove four sets of six negative counters: • Leaving us with 24 positive counters. • So (-4)×(-6) = 24

  30. Let’s look at one more problem: (-12)÷ 3 = ? • For this one, we’ll start with 12 negative counters and divide them into even groups of three. • When we do this, we are left with groups of -4. • So -12 ÷ 3 = -4. • Now with your partner, complete the activity on page 103 in your text book. Do problems #1-20. • Be ready to discuss your answers with the class.

  31. Chapter 2-3-CMultiply Integers • Sara’s ice cream cart decreases the temperature of her ice cream by 2°every minute. • What is the change in temperatureafter 5 minutes has passed? • Remember that multiplication is thesame as repeated addition. • (-2) + (-2) + (-2) + (-2) + (-2) = -10 • (-2) × 5 = -10 • The temperature drops 10 degreesafter 5 minutes.

  32. MULTIPLY INTEGER WITH DIFFERENT SIGNS • The product of two integers with different signs is negative. • 6(-4) = -24 • (-7) × 5 = -35 • MULTIPLY INTEGERS WITH SAME SIGNS • The product of two integers with the same signs is positive. • (-6)(-4) = 24 • 7 × 5 = 35

  33. Take 5 minutes and answer #1-13 on page 106 in your text book. • Let’s try some practice: • -12(-4) • 48 • 9(-2) • -18 • 5(-4) • -20 • (-5)2 • 25 • -3(9) • -27 • -7(-5)(-3) • -105 • -6(-8) • 48 • (-8)2 • 64 • (-2)(-5)(6) • 60 • -7(4) • -28

  34. Chapter 2-3-DDivide Integers • Division of numbers is similar to multiplication. • DIVIDE INTEGERS WITH DIFFERENT SIGNS • The quotient of two integers with different signs is negative. • 33 ÷ (-11) = -3 • (-64) ÷ 8 = -8 • DIVIDE INTEGERS WITH SAME SIGNS • The quotient of two integers with the same sign is positive. • (-33) ÷ (-11) = 3 • 64 ÷ 8 = 8

  35. Find each quotient • 20 ÷ (-4) • (-24) ÷ (-4) • (-81) ÷ 9 • (-9) ÷ (-3) • (-45) ÷ 9 • (-28) ÷ (-7) • Real-world examples: • One year, the estimated Australian koala population was 1,000,000. After 10 years, there was about 100,000. Find the average change in koala population per year. • How do we set this up? • The average temperature in January for the North Pole, Alaska is -24°. Use the expression to find this temperature in degrees Fahrenheit. Round to the nearest degree. • How do we set this up?

  36. Take 5 minutes and answer #1-9 on page 111 in your text book.

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