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Transcript
2-1 Integers and the Number Line Whole numbers: {0, 1, 2, 3…} Integers: {…,-3, -2, -1, 0, 1, 2, 3…} Ex. 1: Name the coordinate of each point. (Provide number lines) A D C F B E <--+---●---+---●---+---●---+---●---+---+---+---●---+---+---+---●-> -7 -6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6 7 8 A -6 B 4 C -2 D -4 Name the set of numbers graphed. Ex. 2: <--+---●---+---+---+---●---+---●---+---+---●---+-> -5 -4 -3 -2 -1 0 1 2 3 4 5 -8 -7 -6 -5 -4 -3 -2 -1 0 F 0 {-4, 0 , 2 , 5} 6 Ex. 3: ◄--●---+---●---+---●---+---●---+---●---+---+---+-> -10 -9 E 8 {…,-6, -4, -2} 1 Graph each set of numbers on a number line. Ex. 4: {1, 3, 5, …} Ex. 5: {integers between –2 and 4} To simplify on a number line: Remind students to get this from the website! 1st Start at zero 2nd Look at the first number and go in the direction of the sign that number of spaces. Put a line where you stop and use an arrow to show the direction you moved. 3rd From that stopping point, move the direction of the sign of the next number that number of times. th 4 Keep repeating until all the numbers are done. 5th When you have finished, circle the ending number and write the answer. Using a number line, simplify. Ex. 6: 6 + (-1) 5 Ex. 7: (-2) + (-5) -7 Ex. 8: 6 + (-9) -3 Ex. 9: 4 + 2 6 Ex. 10: (-5) + 8 3 Activity: Overhead problems Using a number line simplify. (Provide number lines) 1) 3 + (-7) 2) (-1) + (-8) 3) 6 + (-1) 4) 2 + (-10) 5) 4+5 Homework: 2-1 p. 75 #16-38 Even (Give number line worksheet) Algebra One – CP 2 Chapter 2 Notes 2-3 Adding and Subtracting Integers (Power Point) The absolute value of a number is its distance from zero on the number line. Since distance is always positive, the absolute value of a number is always positive. | -5 | = 5 |5|=5 |½|= ½ | -½ | = ½ Rules for adding signed numbers When adding numbers with the same signs, add the absolute value of each number and take the common sign. When adding numbers with different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the larger absolute value. Additive Identity Property (Add. ID. Prop.) Adding zero to any number will not change the value of that number. a+0=a * The additive identity is zero. Additive Inverse Property (Add Inv. Prop.) Adding a number to its inverse will result in zero. a + (–a) = 0 * If two numbers are opposites, they are additive inverses of each other. Simplify. State thought process (Add/Subtr/Add ID Prop./Add Inv. Prop.) 1) (-4) + (-10) 2) 6 + (-7) 3) 17 + 0 -14 add -1 subtr 17 add. ID Prop. 4) 11x + (-2x) 9x subtr 5) (-3) + 3 0 add inv. Prop 6) (-15x) + 4x -11x subtr Subtraction Rule: To subtract, change to adding the opposite and then follow the rules for adding signed numbers. Simplify. Show you are using the subtr. rule and state thought process (Add/Subtr/Add ID Prop./Add Inv. Prop.) 7) 3–5 8) (-4) – 12 9) (-6) – (-14) -2 subtr 16 add 8 subtr 10) 10x – (-7x) 17x add 11) (-9) – (-9) 0 add inv. Prop 12) (-3x) – (-10x) 7x subtr Evaluate if w = 14, x = -8, and y = -10. State thought process (Add/Subtr.) 13) |x| 14) |3–w| 15) |w|+|y| 8 11 24 Activity: Integer Line Up Homework: p. 90 #26 – 54 Even Algebra One – CP 2 Chapter 2 Notes 2-4 Rational Numbers A Rational Number is a number that can be expressed in the form a b , where a and b are integers and b 0. An Irrational Number is a number that cannot be expressed in the form a/b, where a and b are integers and b 0. (Decimals that do not repeat or do not terminate.) The set of rational numbers and irrational numbers together form the set of Real Numbers. Recall from 2-1: Rational Integers: {…, -3, -2, -1, 0, 1, 2, 3,…} Whole Numbers: {0, 1, 2, 3,…} Natural Numbers: {1, 2, 3,…} Irrational Integers Whole Natural State the set(s) each number belongs to: Whole (W), Rational (Q), Integer (Z), Natural (N), or Irrational (I). 1) ¼ 2) 3) -8 4) 0 18 Q I Q, Z Q, Z, W 5) 6) 16 Q, Z, W, N 8 3 7) Q 19.5 Q 8) 13 Q, Z, W, N Comparing numbers on a number line: Numbers on the left are less than numbers on the right. Numbers on the right are greater than numbers on the left. Comparision Property: sentences is true. a<b For any two numbers a and b, exactly one of the following a=b a>b Comparision Property for Rational Numbers: b > 0 and d >0: 1) if a/b < c/d, then ad < bc, and For any rational numbers a/b and c/d, where 2) if ad < bc, then a/b < c/d Replace each ? with <, >, = to make the statement true. (You must show work!) 9) 21/30 ? 7/10 10) 6/11 ? 8/9 11) -7/5 ? -11/6 210 = 210 63 < 88 -42 > -55 Write the numbers in order from least to greatest. 12) 7/18, 1/6, 1/3 1/6, 1/3, 7/18 Find a number in between the given numbers. 13) 1/9 and 3/5 6/45(2/15) – 26/45 Homework: On Green Worksheet Algebra One – CP 2 Chapter 2 Notes 2-5 Adding and Subtracting Rational Numbers Recall: * Rules for adding signed numbers -When adding numbers with the same signs, add the absolute value of each number and take the common sign. -When adding numbers with different signs, subtract the smaller absolute value from the larger absolute value and take the sign of the larger absolute value. * To subtract, change to adding the opposite and follow the rules for adding signed numbers. * To add or subtract fractions, you must have a common denominator. * To add or subtract decimals, line up the decimals Simplify. State your thought process (Add/Subtr.) and show that you are using the subtraction rule. Answers should be reduced fractions or mixed numbers. If the problem had decimals, so can your answer. 11 5 1 5 1) -8.007 + (-5.755) 2) 3) 16 8 6 6 -13.762 -2/3 1 5/16 4) 6.32 – (-3.4) 9.72 5) 1 6 2 7 5/14 Evaluate each expression. State your thought process (Add/Subtr.) Answers should be reduced fractions or mixed numbers. If the problem had decimals, so can your answer. 8 11 x , if x = 6) p – 7.1, if p = 5.24 7) 14 7 -1.86 5/14 Activity: Go over 2(1-3) Quiz Homework: Worksheet Algebra One – CP 2 Chapter 2 Notes 2-6 Multiplying Rational Numbers Rules for Multiplying with signed numbers: When multiplying numbers with the same sign, the product is positive. (+)(+)=(+) (–)(–)=(+) When multiplying numbers with different signs, the product is negative. (+)(–)=(–) (–)(+)=(–) Recall the Multiplicative Properties: Multiplicative Identity Property [Mult. ID. Prop] The product of a number and one is the number. a1=a * The number that is the multiplicative identity is 1. Multiplicative Property of Zero [Mult. Prop. (0)] The product of a number and zero is zero. 1a=a or 0a=0 or 1 a =1 a Multiplicative Inverse Property [Mult. Inv. Prop.] The product of a number and its reciprocal is one. a0=0 or Multiplicative Property of –1 [Mult. Prop. (–1)] The product of a number and negative one is the opposite of the number. a 1 =1 a a (-1) = (-a) or (-1) a = (-a) Simplify. State T.P. ( same/diff/ Mult. ID Prop./ Mult. Prop.(0)/ Mult. Prop. (-1) 1) (-2)(-10) 2) (11)(-4)(2) 3) (5)(-1) 20 same -88 diff, diff -5 Mult.Prop. -1 4) (3x)(-12) -36x diff 5) (0.3)(5) 1.5 same 6) (-⅞)(8) -7 diff 7) 12 1 -4 -48 mult. ID prop., diff 8) 4⅔ ⅜ 1¾ 9) (12x)(-½y) -6xy diff 11) | ⅔w + x | 2 12) xy – w2 -147 Evaluate if w = -12, x = 6, y = -½ 10) 10y – w 7 Activity: I have Who has Homework: Worksheet Algebra One – CP 2 Chapter 2 Notes 2-7 Dividing Rational Numbers (On Notes for LCD) The rules for dividing signed numbers are the same as multiplying. Therefore: When dividing numbers with the same sign, the quotient is positive. (+) (+)=(+) (–) (–)=(+) When multiplying numbers with different signs, the product is negative. (+) (–)=(–) (–) (+)=(–) The division rule says: Division is multiplying by the reciprocal. (You only have to use the division rule if at least one of the numbers is a fraction or if the quotient will not be an integer.) Simplify. Use division rule if necessary. State TP. 1) (-32) (-8) 2) 12 (-4) 4 -3 4) 40 ⅝ 5) 64 7) 8) 6) 9) 3x + 4y Overhead Problems 1) (-72) 12 56 (-8) -7 3) (-⅔) (-48) 1/72 4) (6/25)/15 2/125 22x + 44 -11 4 30 -3/40 20 50 y 10 -2 + 5y -6 2) 5) Homework: 12 x 16 y 4 16 2 8 9 16/21 8 2 7 28 Activity: (-⅔) (-⅞) 3) -2x – 4 2-7 p. 115 #14-36 Even, 2-8 p. 123 #40-50 Even Algebra One – CP 2 Chapter 2 Notes