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UNIT 2: INTEGER OPERATIONS I Can … Order integers on a number line Add integers and solving problems involving integers. Subtract integers and solve problems involving subtraction Multiply and divide positive and negative numbers Solve addition and subtraction equations Solve multiplication and division equations Solve inequalities with integers Solve problems written in exponential form Apply the properties of exponents to solve problems Solve problems involving scientific notation 1 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples Vocabulary *Positive & Negative Numbers in real life Integer--- Example *Write an integer to represent 10 below zero A gain of $2.00 per share Positive Number -Example Graph the #’s on a number line(2, -3, 4, 0 4, 3, -2) --------------------------------------- Negative Number -Example Compare the numbers using an inequality sign (< or >) -5______ 0 6______-2 -4_______-3 Absolute Value -Example Order from least to greatest 5, 3, 0, 6, -2, -4, 1, -1 Evaluate the expressions on the board NAME THREE METHODS TO ADD INTEGERS: 1. 2. 3. 2 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ Notes and Examples Adding Integers with a number line DRAW A NUMBER LINE TO ADD (-6) + 2 Vocabulary Opposites – Example (-3) + 7 (-3) + (-2) Adding Integers with ALIKE signs I need to _____________ the integers like Normal and ____________ the sign. Examples Additive Inverse Property – Example ADDING THREE OR MORE INTEGERS Group the ___________ together and Group the ___________ together Examples: Adding Integers with UNLIKE signs I need to ______________ the integers And take the sign of the _____________ Looking numbers. Examples 3 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples Vocabulary Subtracting Integers Modeling Opposite – Try these examples: 6–7 -17 – 18 21 - ( -24) -36 – 47 4 - l-6l -13 - (-25) What do you notice about 3 – 5 and 3 + (-5) -25 – 15 Subtraction is the same as _____________ The opposite. Word Problems If the overnight temperature was - 14 degrees but rose to 8 degrees during the day, what was the difference in the daytime and night time temperatures? Rules—learn this chant Same Change Opposite Examples 9 - 12 7 - (-15) -6 - 8 2–6 3–8 -30 - (-20) The boiling point of water is 212 degrees while -460 degrees is its absolute lowest temperature. Find the difference between the boiling point and the absolute lowest temp. 4 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples Vocabulary The WEMS football team did not have a very good first half against Moss. On the first 4 plays of the game, WEMS lost 5 yards a play. How could we write an expression to represent five yards lost for 4 plays? Dividing integers – the rules are the same + * + = __________ + * - = __________ Look at the pattern on the board What rule do we discover? - * + = __________ A POSITIVE integer multiplied by a POSITIVE integer results in a _____________ product. - * - = __________ Examples 2 A POSITIVE integer multiplied by a NEGATIVE integer results in a ______________ product. -4 (-2) 5/(-1) A NEGATIVE integer multiplied by a NEGATIVE integer results in a ________________ product. Multiplying and Dividing Integers Examples from class -4(-3) 2(-9) 6(4) -5(-8) -20(2) -3(2) (-3)² -36 9 8/(-2) 10 (-5) -15/(-3) -100/(-50) ALGEBRAIC EXPRESSSIONS! Use all of our rules or strategies to evaluate these algebraic expressions if a = -3 and b = -5 -2a - b a + 2b 5a - b (-5)² What about 3 or more integers? 4 • (-3) • (-2) -2 • (-3) • 6 -2(3)(-9) -1(-9)(-4) (-2)³ 5 UNIT 2: INTEGER OPERATIONS INTEGERS IN THE REAL WORLD!!! Vocabulary At the WEMS football game versus Moss, the 5 plays resulted in the following yards, a loss of 5, a gain of 10, a loss of 15, a loss of 5, and a gain of 5 yards. What is the mean yards of the five plays? Mean– Example Chris Allen reported that the temperature is dropping at a rate of 5° per hour. At this rate, how long will it take the temperture to drop by -25°? At a golf match, the scores for the top five players were 6 below par, 4 below par, par, one above par, and 5 above par. What was the mean score for the five players 6 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples Vocabulary You checked your bank statement today and it had a negative balance of $5.00. Yesterday, you had taken out $20.00. What type of equation would represent the amount that your account began with? Write and solve your own equations If you decrease a number by 20, the result is -14 the sum of n and 25 equals -18 Refresher -- how do we solve these equations? if you decrease a number by 8 the result is -14 Isolate the _________________ by “____________” the operations. Addition Equations with Integers w + (-4) = -5 s + (-3) = -8 the sum of a number and 40 is 9 Solving division and multiplication equations with integers 8g = -32 -35 = -7r 6a = -42 -2 + n = -6 Subtraction equations with integers x - 8 = -3 y-2=5 b - 4 = -10 n - 5 = -6 n = -12 2 x =4 -5 t = -8 -6 7 = p - 12 -8 = d - 11 A scuba diver can dive at a rate of -10 feet per minute. If the diver is at a depth of -60 feet, how many minutes has he been diving? Write and solve an equation. r - 20 = -4 7 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples Vocabulary To keep ice cream from melting, the temperature of the freezer must be no more than 32 degrees. If the COEFFICIENTS are NEGATIVE!!!!!! How do we write that as an inequality? WARNING: Suppose the current temperature of the freezer is -10 degrees. How many degrees can the temperature rise to keep the ice cream from melting? Write an inequality to represent this situation. Solving addition and subtraction inequalities When multiply or divide BY a NEGATIVE number, you must FLIP the sign -4x < 16 -2r -10 r < -3 -3 w>5 -4 -5m -25 a -2 -10 Solve just like equations y+7 1 h - 2 < -1 x + 2 < -4 p - 4 -2 r + 3 -5 j - 5 -2 Solving inequalities by Multiplying and Dividing with Integers 7h < 63 9r -45 2m -4 x >6 2 r < -4 3 w -3 5 Using inequalities A local weather forecast stated that it would be 12°F tonight and at least 10° colder the next night. Write an inequality to show how cold it will be? The temperature at ten oclock was -5 degrees. The temperature at 2 o'clock was at least 20 degrees below the temperature at ten. Write an inequality to represent the possible temperature at two. Notice that the _______________ ARE ALL ______________! 8 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Notes and Examples What are exponents? What do they mean? Vocabulary Base -- Rewrite using exponents Example 3(3)(3)(3)(3) 4x4x4 10 x 10 8x8x8x8 Rewrite without exponents in as a product N5 63 34 26 Exponent -- Example Power – Example Rewrite using exponents a•a•a•a a•a•b•b•b a•b•b•b•c•c More Examples Complete this pattern 10³ = 10³ 10² = 101 8² 100 10-1 10-2 Any exponent of 0 has a product of _______ Negative exponents can be written as one divided by the number to the nth power 2° 54 2²•5² 3-3 5° 6° 3-1 4-2 9 UNIT 2: INTEGER OPERATIONS Section Title: _________________________________________________ I CAN… Simplify these fractions Notes and Examples Can you rewrite these expression using exponents? 4(4)(4)(4)(4) 4(4)(4) 7(7)(7)(7) 2(2)(2)(2)(2)(2)(2) 2(2) 4(4) 3(3)(3)(3)(3)(3) Rule: To _____________ powers with the same___________, keep the _______ and ______________________. 2(2)(2) If 4(4)(4) = 4³, then what do you think about 4 * 4² 6 If 3(3)(3)(3)(3)(3) = 3 , then what do you think about 3² * 3 So here is our rule when multiplying exponents To ______________ powers with the same ________, keep the __________ and ________________________. example: 5² * 5³ = 5 16 * 16³ = 16 Your turn 6 6² 12³ 12² 8 8³ There are about 10 molecules in a cubic meter of air at sea level, but one 10 molecules at the altitude of 33 km. How many more molecules are there at sea level than at 33 km? a² * a² = a 3² * 4² = ? Your turn 8 * 8² 2² * 2 10 * 10³ A light-year is the distance that light travels in one year. A light year is 10 centimeteres. To convert this number to kilometers, you must divide it by 10 . How many kilometers is a light year? r³ * r 10 The weight of the planet Mars is 8 tons. The weight of Venus is 8 tons. How many more tons is Mars, than Venus? A rectangle has a side of 15² inches and another side of 15³ inches. What is the area of the rectangle written with one base? More Complex 8 x 10 4 x 10 3x5 2x5 The population of the United States is 3 x 10 and the population of the world is 7 x 10 . How many times larger is the population of the world than the population of the United States? 11 UNIT 2: INTEGER OPERATIONS Section Title:____________________________________________________________ I CAN… Notes Vocabulary Why do we use scientific notation? Scientific Notation— Writing Numbers in STANDARD FORM (________________________________) 5.34 x 10³ the exponent is 3, so you move the decimal 3 places in a positive direction WORD PROBLEMS The distance from Earth to the new nonplanet Pluto is 4,280,000,000 miles. Write this number in scientific notation Try these 7.42 x 10² . A googol is a number written as a 1 followed by 100 zeros. Write a googol n scientific notation. 6.1 x 105 3.7456 x 104 The smallest unit of time is the yoctosecond, which equals 0.000000000000000000000001 second. Write this number in scientific notation. 6.2897 x 106 What if the power is a negative number? 2.3 x 10-2 AN OXYGEN ATOM HAS A MASS OF 2.66 X 10-23 GRAM. WRITE THIS NUMBER IN STANDARD FORM. 3.68 x 10-4 6 x 10-7 11.2 x 10-3 Writing numbers in scientific notation Step one -- move the decimal to a point that makes the number greater than or equal to ____________, but less than ________ Step two -- count how many places you moved the decimal 37.5 0.00876 3, 725, 000 0.114 4,999,999,999 0.000316 14,140,000 12 JUMPSTARTS UNIT 2 13 14