Intro to Algebra Unit 1

Transcription

1 Mon Tue Wed Thu Fri Aug 26 Aug 27 Aug 28 Aug 29 Aug 30 Introductions, Expectations, Course Outline and Explanation of Carnegie Review of Summer Packet Review of Summer Packet Set up of Carnegie Usernames & Passwords followed by Review of Summer Packet Quiz on summer packet Sept 2 Sept 3 Sept 4 Sept 5 Sept 6 Topic: 2-1/2-2 Integers/Adding Integers Topic: 2-3 Subtracting Integers Topic: 2-2 & 2-3 Adding/Subtracting Integers Topic: 2-4 & 2-5 Multiplying/Dividing Integers Carnegie Day HW: HW: HW: HW: Sept 9 Sept 10 Sept 11 Sept 12 Sept 13 Topic: Finding Averages Topic: Review of Operations with Integers; special focus on Application Quiz on Operations with Multiplying/Dividing Integers Topic: 5-1 Fractions & Decimals Carnegie Day HW: HW: HW: Sept 16 Sept 17 Sept 18 Sept 19 Sept 20 Topic: 5-1 Fractions & Decimals Topic: 5-2 Rational Numbers Topic: 5-2 Rational Numbers Quiz on Fractions, Decimals, and Rational Numbers Carnegie Day HW: HW: HW: Sept 23 Sept 24 Sept 25 Sept 26 Sept 27 Topic: STUDY GUIDE Topic: STUDY GUIDE Test Unit 1 Multiple Choice Test Unit 1 Short Answers Carnegie Day Please remember the dates above are tentative and may be changed. Students please sure to write the daily assignments on the calendar above to help yourself stay organized since we no longer hand out assignment books. 1

2 VOCABULARY DEVELOPMEMT: Word Definition Picture/Example Real Life Connection Opposite Additive Inverse Sum Variable Algebraic Expression Difference Product Quotient Mean 2

3 Video Lessons/Help Available: (BE sure to get your parent/guardians permission prior to using these sites SKIP ALL ADS, DO NOT DOWNLOAD Amusing, Helpful and Sometimes Annoying Songs for: Adding Integers (Row Row Row Your Boat): Adding Integer Blues: Subtracting Integers: Multiplying/Dividing Integers: All Integer Rules: Number Systems: Number Systems: Different Website you found: Lessons by other teachers: Adding & Subtracting Adding & Subtracting: Real Life Examples Using Integers: Real Life Examples Using Integers: Rational Numbers: Rational Numbers: (Ignore the Imaginary Number Part for now) Different Website you found: Interactive Games/Activities: Integers: Integers: Real Number Systems: Fractions/Decimals: 3

4 2.1 Integers and the Number Line This is called a. All of the points represent. Examples of integers: Positive Numbers: Negative Numbers: Zero: Opposites: **Another name for opposite is Additive Inverse because opposites always add to zero **What is a real-world example of when negative numbers are used? Examples: Write an integer for each situation. Together: You Try: feet below sea level feet under the ground 2. A temperature increase of 12 degrees 5. A weight loss of 6 pounds 3. A loss of $ weeks after birth

5 Adding and Subtracting Integers Activator: In football, forward progress is represented by a positive integer. Being pushed back is represented by a negative integer. Suppose on the first play a team loses 5 yards and on the second play they lose 2 yards. Use this number line to help you answer the questions. a. What integer represents the total yardage on the two plays? b. Write an addition sentence that describes this situation. The addition sentence we wrote above is an example of adding two integers with the same sign. Notice that the sign of the sum is the same as the sign of the addends. Rule for Adding integers with the same sign: Examples: 1. Find -4 + (-5) 2. Find -2 + (-3) 3. Find Find You Try: 5. Find Find Find -5 + (-4) 8. Find (-3) + (-6) 5

6 We can use a number line to help us add integers with different signs. **Remember on a number line, positive numbers move to the right and negative numbers move to the left. 7 + (-4) Step 1: Start at Zero Step 2: Move 7 units to the right Step 3: From there, move 4 units to the left 2 + (-3) Step 1: Start at Zero Step 2: Move 2 units to the right Step 3: From there, move 3 units to the left Let s look at these two sums and compare them. 7 + (-4)= 2 + (-3)= What do you notice about the signs of the sums? Rule for adding integers with different signs: Examples: First, state whether the sum will be positive or negative, then find each sum (-4) (-11)

7 You Try: First, state whether the sum will be positive or negative, then find each sum (-15) (-9) Adding More than Two Integers Two different methods: Examples: (-3) + (-9) (-3) (-6) + 2 You Try: (-4) (-3) + (-9) (-10) Summarize: Which method do you prefer and why? 7

8 Practice Problems First, state whether the sum will be positive or negative, then find each sum (-4) (-5) (-5) (-2) Find each sum (-1) (-2) (-6) (-8) (-8) (-4) (-14) (-6) (-15) (-2) (-3) (-5) (-12) (-43) (-5) (-4) (-7) (-6) (-32) Find each sum (-9) (-13) (-10) (-12) (-9) (-35) + 4 8

9 Subtracting Integers Recall: What is an additive inverse? (Hint: think back to opposites) Subtracting integers is very similar to adding integers. To subtract an integer, you add its additive inverse. Keep-Change-Change Example: 6 8 Using the additive inverse, 6 8 is the same as Let s use the number line to help us. Step 1: Start at zero Step 2: Move 6 units to the right (positive) Step 3: Move 8 units to the left (negative) You Try: -3 5 Using the additive inverse, -3 5 is the same as Step 1: Start at zero Step 2: Move units to the ( ) Step 3: Move units to the ( ) Find each difference:

10 Now, let s try subtracting a negative number 7 (-3) Using the additive inverse, this is the same as 7 + (+3) Keep-Change-Change Therefore we can conclude that subtracting a negative number is the same thing as Use the number line to verify this rule with -2 (-4) Examples: Find each difference (-4) (-7) (-9) You Try: Find each difference (-15) (-10) (-3) Evaluate algebraic expressions using subtraction You can use the same rule for subtracting using algebraic expressions. Examples: 1. Evaluate x (-6) if x = Evaluate s t if s = -9 and t = Evaluate p + q r if p = -11, q = 6, and r = -12 You Try: 4. Evaluate m (-2) if m = 4 5. Evaluate x y if x = -14 and y = Evaluate a b + c if a = 15, b = 5, and c = -8 10

11 Practice Problems Find each difference (-8) (-6) (-3) (-7) (-10) (-12) (-7) (-50) (-4) Evaluate each expression if x = -3, y = 8, and z = y y z z x 25. z x y 28. z y 29. x + y z 30. z y + x 31. x y z 32. During January, the normal high temperature in Duluth, Minnesota is 16 F and the normal low temperature is -2 F. Find the difference between the temperatures. 33. The highest point in California is Mount Whitney, with an elevation of 14,494 feet. The lowest point is Death Valley, with an elevation of -282 feet. Find the difference in the elevations. 11

12 Multiplying and Dividing Integers RECALL: Multiplication is the same thing as repeated addition. So 3(-5) is the same thing as (-5) + (-5) + (-5) RECALL: The commutative property of multiplication tells us that 3(-5) is the same thing as -5(3) Rule: The product of two integers with different signs is NEGATIVE. Examples: Find each product. 1. 5(-6) 2. -4(16) You Try: Find each product. 3. 8(-12) 4. -9(11) We know that the product of two positive integers is positive (Ex. 2 x 4 = 8). What is the sign of the product of two negative numbers? Let s use the pattern below to find out (-4)(2) = -8 (-4)(1) = -4 (-4)(0) = 0 (-4)(-1) = 4 (-4)(-2) = 8 Rule: The product of two integers with the same sign is POSITIVE. Examples: 1. -6(-12) 2. -4(-6) 3. 7(9) You Try: 4. -2(-7) 5. -4(-4) 6. 8(4) 12

13 Multiply more than two integers Examples: 7. -4(-5)(-8) 8. -9(-5)(2) You Try: 9. -3(-4)(-5) (-6)(3) Simplify or Evaluate Algebraic Expressions using Multiplication Examples: 1. Simplify -4(-9y) 2. Evaluate 4ab if a = 3 and b = -5 You Try: 3. Simplify -2(3x) 4. Evaluate -3st if s = -9 and t = 3 Practice Problems Find each Product (32) (-11) 3. 4(-8) (-3) 4. 9 (-8) (-4)(6) (-3)(-2) 6. 14(-5) (-8)(-3) (-7)(-4)(-3)

14 Simplify each expression x y 17. 6(-8a) 18. 5(-11b) s(-8t) 20. 2ab(3)(-7) 21. 9(-2c)(3d) j(3)(5k) Evaluate each expression n if n = s if s = ab if a = 9 and b = xy if x = -8 and y = During a 10-hour period, the temperature in Browning, Montana changed at a rate of - 10 F per hour. If the starting temperature was 44 F, what was the ending temperature after the 10 hours? 14

15 Dividing Integers Activator: You can find the quotient of -8 (-4) using a number line. To find out how many groups of -4 there are in -8, show -8 on a number line and divide it into groups of -4. a. How many groups of -4 are there? b. What is the quotient of -8 (-4)? c. What multiplication sentence is also shown on the number line? d. Explain how multiplication and division is related. Since multiplication and division are basically the same problem, the same rules apply. Therefore, Dividing Integers with the SAME sign = quotient Dividing Integers with DIFFERENT signs = quotient RECALL: What are two different notations to show 18 divided by 6? Examples: State whether the quotient will be positive or negative, then find the quotient

16 You Try: State whether the quotient will be positive or negative, then find the quotient Evaluating Algebraic Expressions using Division Examples: 1. Evaluate 4 if Evaluate 4 if 6 8 You Try: 3. Evaluate if Evaluate 6 if 12 8 In your own words, explain the steps you take to evaluate expressions: Step 1: Step 2: Step 3: 16

17 Practice Problems Find each quotient What is -91 divided by -7? 14. Divide -76 by -4 Evaluate each expression. 15. if if if if

18 Adding Integer Rules Same Signs Different Add the 2 numbers Subtract the 2 numbers Copy Sign of Larger Number # Problem Reasoning: CIRCLE ONE WORD FROM EACH PAIR (circle 3 words) = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative = The signs are same/different; so, we need to add/subtract. Answer is positive/negative Subtracting Integer Rules Keep the 1 st Change the to + Change sign on 2 nd number THEN FOLLOW ADDING RULES # Problem Rewrite Reasoning: CIRCLE ONE WORD FROM EACH PAIR (circle 3 words) (-13) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (-57) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (97) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (23) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (-13) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (-18) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (-17) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (-60) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (35) The signs are same/different; so, we need to add/subtract. Answer is positive/negative (121) The signs are same/different; so, we need to add/subtract. Answer is positive/negative 18

19 Multiplying/Dividi ng Integer Rules Same Signs Different Signs POSITIVE Answer NEGATIVE Answer # Multiply Divide Reasoning = 20 4 = The signs are same/different. Answer is positive/negative = = The signs are same/different. Answer is positive/negative (- 2) = - 18 (- 2) = The signs are same/different. Answer is positive/negative (- 3) = 24 (- 3) = The signs are same/different. Answer is positive/negative = 8 1 = The signs are same/different. Answer is positive/negative (- 3) = - 9 (- 3) = The signs are same/different. Answer is positive/negative (- 5) = 15 (- 5) = The signs are same/different. Answer is positive/negative (- 10) = 100 (- 10) = The signs are same/different. Answer is positive/negative = = The signs are same/different. Answer is positive/negative (- 4) = - 8 (- 4) = The signs are same/different. Answer is positive/negative. Word Meaning: Place the following words in the addition or subtraction category. Reduce, Down, Decrease, Up, Withdraw, Deposit, Rise, Increase, Win, Lose, Profit, Loss, Gain, Fell, Climbed. 31. Write the expression for the temperature dropped 5 from 37 from this morning. 32. Diane deposited $100 into her savings which was at $

20 Integer Word Problems 1) The temperature at noon on a winter day was 8 o C. At midnight, the temperature had dropped 15 o. What was the temperature at midnight? 2) In 2001, Standard Register reported a net loss of $43.9 million. In 2002, the company had a net income of $33 million. How much more money did the company make in 2002 than in 2001? (Source: Dayton Daily News) 3) On a recent glider flight, Lucy took a tow to 3000 feet above ground level. She found a good thermal and climbed 1800 feet. Then she hit big sink and lost 2200 feet. What was her altitude at that point? 4) An architectural drawing of a building shows the elevation of the basement floor to be 12 feet. The elevation of the roof is 32 feet. What is the total distance from the roof to the basement floor? 5) One day in July, the temperature at ground level at the airport was 90 o. A pilot reported the temperature at 10,000 feet was 50 o. How much did the temperature drop per 1000 feet? (Hint: find the amount of temperature change and set up a proportion.) 6) The high temperatures for the first 5 days of January in Fargo, North Dakota are listed below. Find the average high temperature for these 5 days. Jan. 1-4 o Jan. 2-3 o Jan Jan. 4-6 o Jan. 5-2 o 7) A football team gained 6 yards on a first down, lost 15 yards on the second down, and gained 12 yards on the third down. How many yards do they need to gain on the fourth down to have a 10 yard gain from their starting position? 20

21 8) While returning to the gliderport, Lucy descended for three minutes at a rate of 400 feet per minute. How much altitude did she lose? If her starting altitude was 2800 feet, what was her final altitude? 9) Carla s credit card statement showed that she owed $350. She made a payment of $200, and then she charged $23 for gasoline. She returned a sweater she had charged earlier for a refund of $35. Then she charged $15 at the bookstore. What is her new balance? 10) In 2000, Bob s 401(k) fund lost $9000. In 2001, it lost another $10,000, and in 2002, it lost $17,000. In 2003, it gained $16,000, and in 2004 it gained $12,000. How much more does it have to gain to be worth what it was at the start of 2000? 11) Katherine is very interested in cryogenics (the science of very low temperatures). With the help of her science teacher she is doing an experiment on the effect of low temperatures on bacteria. She cools one sample of bacteria to a temperature of -51 C and another to -76 C. What was the temperature difference in the two experiments? 12. On Tuesday the mailman delivers 3 checks for $5 each and 2 bills for $2 each. If you had a starting balance of $25, what is the ending balance? 13. You owe $225. on your credit card. You make a $55. payment and then purchase $87 worth of clothes at Dillards. What is the integer that represents the balance owed on the credit card? 14. If it is -25F in Rantoul and it is 75F in Honolulu, what is the temperature difference between the two cities? 21

22 15. During the football game, Justin caught three passes. One was for a touchdown and went 52 yards. The other was for a first down and was for 17 yards. The other was on a screen pass that did not work so well and ended up a gain of -10 yards. What was the total yardage gained by Justin on the pass plays? 16. James plays in the backfield of the Big Town football team. Last week he ran four plays from the halfback position. He made "gains" measured in yards of 3, 4, 1, and 5. What were his average yards per gain? Round your answer to the nearest tenth of a yard. 17. In golf, the average score a good player should be able to achieve is called "par." Par for a whole course is calculated by adding up the par scores for each hole. Scores in golf are often expressed at some number either greater than or less than par. Ms. Floop is having a pretty good day at the Megalopolis City Golf Club. Her score so far after 15 holes is -3. If par for 15 holes is 63, what is her score? 18. It was a very freaky weather day. The temperature started out at 9 C in the morning and went to -13 C at noon. It stayed at that temperature for six hours and then rose 7 C. How far below the freezing point (0 C) was the temperature at 6 p.m.? 19. The mailman delivered a $22 check and 3 - $14 bills today. He also took back 1- $5 bill. What is the total in the mailbox? 20. A monkey sits on a limb that is 25 ft above the ground. He swings up 10 ft, climbs up 6 ft more then jumps down 13 ft. How far off the ground is the monkey now? 21. Mary has $267 in her checking account. She writes checks for $33, $65, and $112. What is the balance in her account now? 22. A submarine dove 836 ft. It rose at a rate of 22 ft per minute. What was the depth of the submarine after 12 minutes? 22

23 Finding the Mean (Average) for a Set of Data Mean The mean (or average) of a set of data values is the sum of all of the data values divided by the number of data values. That is: Example 1: The scores of seven students on a mathematics test with a maximum possible score of 20 are shown below: Find the mean of this set of data values. Solution: Sum of all data values Mean Number of data values So, the mean is 15. You Try: The scores of nine students on a french test that had a maximum possible score of 50 are shown below: Find the mean of this set of data values

24 Instructions: Find the mean of each set of data, rounding answers to the nearest tenth. (1 place after decimal) 1) 2, 6, 1, -4, 7, 4, 4, 3, -6, 6 2) -4, 0, 3, -9, 0, 4, 5, -3, 9, 9 3) 14, -13, 21, 18, -24 4) 3, 17, 6, 14, 10, 22, 2, 1, 12 5) 30, -24, -30, 42, 24,

25 Fraction Convert (Show Your Work) Decimal Terminating or Repeating?

26 Fraction Convert (Show Your Work) Decimal Terminating or Repeating?

27 Fraction Convert (Show Your Work) Decimal Terminating or Repeating?

28 When there s a line use nine! This trick for repeating decimals works each time. 1. Put the repeating pattern on top 2. If it repeats 1 digit use 9, repeats 2 digits use 99; repeat 3 digits use 999, etc. So. 72 = 28

29 When comparing and ordering rational numbers, you must either get a common denominator or change both fractions to decimals. Example: Compare 4 3 and 6 5 Method 1: Finding a common denominator Method 2: Changing to decimals 29

30 30

31 Comparing and Ordering Rational Numbers Name Fill in each blank with <, >, or = to make each sentence true. Write the decimal notation beneath each fraction to check your answer Write the fractions in order from least to greatest. Write the decimal notation beneath each fraction as you did in problems