Summer Math Packet for Entering Algebra 1 Honors Baker High School

Transcription

1 Summer Math Packet for Entering Algebra 1 Honors Baker High School *You should be fluent in operations with fractions involved (multiplying, dividing, adding, and subtracting). *You should know all of your positive/negative rules (including rules for multiplying/dividing, and adding/subtracting). *You should be comfortable not using a calculator on every problem you work. *You should be honorable enough to check your answers using solutions rather than just copying the answers down. Order of Operations: P/G E/R/AV M/D A/S Parenthesis/Grouping Exponents/Radicals/Absolute Value Multiplication/Division Addition/Subtraction Use Order of Operations to simplify each expression: 1) (6 + 2) 2 5(8) 4 2) [70 7(5)] ) (9 14) 4) 32(2) ) ( 5) 2 6) (5+11) 2(27) 4 7) (5 3) 2 5(5) 8) 1 5 ( 35) (54) 9) ( 6)( 1) 2 10) ( 3) ) 2 5 (40) (56) ) ( 5) (7 11)

2 To evaluate an expression, we plug in values for each variable then follow Order of Operations. The difference between an equation and an expression: An expression does not have an equal sign in the original problem. An equation does have an equal sign in the original problem. Evaluate each expression: 1) If a = -7 and c = 4 find 5ac - 7a 2) If x = -4 and y = -10 find -2x 2 + 6y 11 3) If m = 5, n = 6, and p = -9 find 9n + 4(m p) + 8p 4) If e = -1 and f = -5 5) If a = 4, c = -10, and e = -5 find f + 6e e 8f2 find 8a 2-7 c + e 6) If x =12, y = 11, and z = -3, x 6y z 7) If x = 8 and c = -4 find 2cx + 5c 2 + 9c 8) If m = -1 and y = -2 find 4m y + 7m 9) If w = 2, e = -5, and d = w + 35 e 9d 2 10) If a = 4, c = -10, and e = 8 find c 2 + 2a 3 e ) If x =4, y = -5, and z = -2, find 7 y z 4 + 3x ) If m = 1, n = 0, and p = -2 find (mp) 2 + 5n 11

3 Like terms are terms that have the same variables, with the same exponent on each variable. Ex: 4x and -7x Non-examples: 5x and 7x 2 8m 2 n 4 and 9m 2 n 4 4m 2 n 4 and 9m 2 n 3 To add or subtract like terms, we only add or subtract their coefficients. Combine like terms: 1) 8n + 9n 2) -10m 2 p 5m 2 p 3) 6x 2 9x + 5x 10x 2 4) -7ac 8a + 5c 10ac + 6a 5) 6mn 2 7m 2 n + mn 2 5m 2 n 6) 3xy 4y 2 + 8y 2 8xy 10xy 7) 4x + 8y 2y + 10x 5y 8) 20g 2 h 15gh 4g 2 h 4gh The distributive property allows us to multiply a term on the outside of a parenthesis by all of the terms on the inside. Ex: 10(5x + 4) -7(4x + 2) -(5x + 3) 10(5x) + 10(4) -7(4x) + (-7)(2) (-1)(5x) + (-1)(3) 1 7 (42x + 35) 1 7 (42x 1 ) (35 1 ) 50x x 14-5x 3 6x + 5 Use the distributive property then simplify any like terms. 1) 4(x + 2) 2) -6(7m 3) 3) (-9x + 2) 4) 8(x + 2) + 5(x 3) 5) 10(4m 2) 6(7m 4) 6) -3(5x 4) (x + 2) 7) -10(2x + 3) 5(4x 7) + 15x 8) 4(-2x 8) 7(3x 5) x 9) 1 (6x + 15) 1 (20x 24) ) 2 (30x + 25) + 3 (16x + 32) 11) 3 (24x + 48) + 6 (35x + 42) 12) 3 (50m + 20) 4(2x + 3) 2x

4 Words in Math: Addition Subtraction Multiplication Division Equal Sign sum plus total increased by more than added to increased by less than** subtracted from** decreased by minus difference reduced by **Means those are written backwards from how we read them in words. for per multiplied by times twice (Multiplied by 2) half (Multiplied by ½) product of (Like 3/4 of a number) ratio divided evenly divided equally divided by quotient of Equals Is equal to Same Is the same as Translate each verbal expression into an algebraic expression (using numbers and symbols instead of words). 1) Six less than a number squared 2) Four times a number, increased by seven 3) The sum of four times a number and the quotient of 8 and a different number 4) Twice a number, increased by the product of 5 and a number cubed 5) Two-thirds of a number 6) Half of c, decreased by 4 times a number 7) The difference of negative two times a number and seven times a different number 8) Three-fourths of a number of students 9) The total of seven shirts and three pairs of pants 10) Seven subtracted from three times a number Translate each word problem into an equation: (You MUST be able to create the equation, not just solve it.) 1) Harley had 63 songs on his phone before he downloaded some new ones. He now has 74 songs on his phone. How many songs did he download? 2) Katie spent $50 and now has $64. How much money did she originally have? 3) Kamry has $15 to spend on snacks at the movie theater, and the snacks cost $3 each. How many snacks can she purchase? 4) Logan had some money to spend on Christmas presents, and he bought a present for each of 7 friends. He spent $10 on each friend. How much money did he originally have? 5) Mrs. Rosie plants award-winning flowers. Last year, she measured a flower at 50 inches tall. By the time the competition came around, it was 64 inches tall. How many inches did the flower grow? 6) Harper is 12 years younger than her brother. Her brother is 27. How old is Harper? 7) Ashton had 4 dogs and then decided to foster some from the local shelter. She now has 11 dogs. How many did she choose to foster?

5 Two-step Equations: First: Isolate the variable term by performing the inverse operation on the constant term to move it to the other side. Second: Isolate the variable by performing the inverse operation to remove the remaining number. Solve each equation. Leave your answers as improper fractions. NO DECIMALS unless the original problem had decimals!!! 1) 5x + 3 = 13 2) 4x 3 = 12 3) 5x + 3 = 13 4) -9x 4 = -20 5) 7x 9 = -9 6) x 8 = -26 7) x = 16 8) a 5 3 = 8 9) 2x = ) x = 10 11) 0.75p = ) 0.2a = 9.8 Write a two-step equation for each word problem. 13) Owen opens a savings account with $60. Each week after, he deposits $20. In how many weeks will he have saved $500? 14) Kyle rents a car for one day and spend exactly $80. The charge is $20 plus $0.12 per mile. How many miles can he drive? 15) Katie wants to buy a bicycle that costs $129. This is $24 more than 3 times what she saved last month. How much did she save last month? 16) When 12 is subtracted from 3 times a number, the result is 24. Find the number. 17) If you multiply a number by 3 and then subtract 5, you will get 40. What is the number? 18) Joe went to the hobby shop and bought 2 model sports cars at $8.95 each and some paints. If he spent a total of $23.65, what was the cost of the paints? 19) Kendra is buying bottled water for a class trip. She has 16 bottles left over from the last trip. She buys bottles by the case to get a good price. Each case holds 24 bottles. How many cases will she have to buy if she wants to have a total of 160 bottles of water?

6 Two-step Inequalities: Solve and graph each two-step inequality (NOTICE the rule inside the repeated diamond shape above!!!): 1) 2x + 7 < 9 2) -3m 7 > 11 3) 4m 5 > 19 4) 10w 20 < 100 5) a 4 7 > 5 6) n ) a < 7 4 8) h > 6 5 9) -8g 10 > 14 10) 7 + 6e > 14 11) 15 2w > 27 12) x > 36

7 Multi-Step Equations: SPECIAL CASES Solve each multi-step equation. Leave your answers as improper fractions!! NO DECIMALS. 1) 5x 3 = 4x + 7 2) 6x 2 = 7x 2 3) 4x x = 2x + 4x + 8 4) -2x + 5(6x + 4) = 20 5) 5x + 15 = 5(x + 3) 6) 2x+5 3 = 6 7) 14x + 36 = ½ (7x + 21) 8) 7a 3 5 = 4 9) 2 5 x + 5 = 6 5 x 3 10) 5(2x + 6) = 4(-3x + 8) 11) 4(3x + 7) 20x = 5 + 7(2x 8) 12) -2(5x 6) + 12 = -5(4x + 1) 10

8 Multi-step Inequalities: Solve and graph each inequality. 1) 5(3x + 2) < 6x 10 2) 10m 6 > 2(5m 3) 3) 1 (15x + 21) > 2x ) 5x x 5) 2a 4 + 7a > 5a ) 4m + 18 > 2(2m + 9) 7) 4m ) -2(4k 8) 10 > 5k 50 9) 20e 5 < 8e 2 10) 2 x x ) 7 w + 9 > 5 w 1 12) 2 (30x + 45) 9x ) 5a + 3 7a < 10(a + 2) 14) 5y 2 + 4(2y 3) > 8 15) 10a < 5a + 2(3a 8) + 11a

9 Graphing Points: Name the ordered pair that corresponds to Graph and label each ordered pair. each labeled point. A. (-4, 8) B. (1, 3) C. (-5, 7) D. (-2, -8) E. (-9, -5) F. (6, -8) G. (3, -2) H. (-1, 5) I. (0, 4) J. (-7, 0)

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11 Slope is represented by the letter m. If we are given two ordered pairs, we can use the slope formula instead of graphing the points and counting rise/run. Find the slope between each set of ordered pairs.

12 Sometimes we get lucky, and the equation is already in slope-intercept form. Sometimes we have to change it to slope-intercept form to identify the slope. Identify the slope (m) and y-intercept (b) of each equation. Remember, you will need to change some to slopeintercept form first. Identify the slope and y-intercept in each word problem. Slope is the quantity that is changing constantly, and the y-intercept is always the starting amount. 7) Julia got $200 for her birthday, and she plans to start saving $25 each month. 8) Carson has a huge collection of 250 yo-yos. He plans to start giving away 10 each month to less-fortunate children. 9) Mary s puppy weighed 10 pounds, and it is expected to grow 0.5 pounds each week until it is full-grown. 10) Cammie is planning to buy 2 new shirts every month, and she currently owns ) Alexus has $10 but purchases a snack from school every day for $0.75.

13 Identify the slope and y-intercept of each line. Then write the slope-intercept form equation for each line.

14 Graph each line. Remember, if it is not in slope-intercept form, you must change it to slope-intercept form first.

15 Complete the table, and graph the points from the table to create the line.

16 Rate of change is a synonym for slope. Rate of change explains the meaning behind the rise over the run, using words and not just the numbers. The table shows Dayna s height at ages 9 and 12. 1) What is the change in her height from 9 years to 12 years? 2) How many years did it take for this change to take place? 3) Write a rate of change (fraction) for her height, comparing the change in her height to the change in her age. 4) Describe these numbers using words. This table shows the amount of money the Multicultural Club made washing cars for a fundraiser. 5) Write the rate of change for the fundraiser as a fraction. 6) Describe these numbers using words. 7) The table shows the number of miles a plane traveled while it was in flight. Use the information from the table to find the approximate rate of change in miles per minute. 8) Use the graph to find the rate of change 9) Use the graph to find the rate of change in miles in miles per hour when traveling on a highway. per hour when traveling in the city.

17 Write the slope and y-intercept for each graph. Then describe, in words, what is happening in the graph. 1) How much money was in the bank account when this person chose to start saving every week? 2) How many weeks passed when this person had $55 in the account? 3) Write an equation for this graph, in slope-intercept form. 4) If this person continued saving at this rate, how much money would they have at 15 weeks? 5) How many weeks would it take for this person to have $250? 6) How many minutes did it take for Drew s scuba tank to become empty? 7) How much air was in Drew s tank when it was halfway empty? 8) Did Drew s tank constantly lose the same amount of air? How do you know? 9) Describe the y-intercept (using words) for this scenario. 10) How many minutes did the candle burn before it ran out? 11) How tall was the candle when it was first lit? 12) How many minutes had the candle burned when it was 7 inches tall? 13) Why is this graph only in the first quadrant? 14) How many minutes did the candle have left when half of it had burned?

18 15) How many inches of snow was already on the ground when the storm began? 16) After 5 hours, how much snow had fallen? 17) When could these residents expect to have exactly 1 foot of snowfall? 18) How much snow fell every hour? 19) How much money did John choose to save every month? 20) How much money did he have in the account at 7 months? 21) How much money did he add to the account from the time he began saving every month until the end of 12 months? 22) Describe the y-intercept using words. 23) How much money did this person withdraw every month for this year? 24) If they kept withdrawing money at the same rate, which month (of the next year) will this person not have any money in the account?