Test 2 9 th Math Models

Transcription

1 Test 2 9 th Math Models

2 EQUATIONS VERSUS EXPRESSIONS Equations have an equal sign Equations can be solved for x, they have an answer Equa is the same in equation and equal EX: 2x + 1 = 5, 2(x + 1) = 6 Expressions do no have an equal sign Expressions can NOT be solved for x, only simplified by combining terms (x s with x s, numbers with numbers) EX: x + 2, 4x + 3y + 2z + 1

3 EQUATIONS VERSUS EXPRESSIONS 2(x + 2) 3(x + 1) First step: DISTRIBUTE 2(x + 2) 3(x + 1) 2x + 2 3(x + 1)

4 EQUATIONS VERSUS EXPRESSIONS 2x + 2 3(x + 1) We distributed the first set, now we have to distribute the second set of parentheses. ALWAYS check BEFORE a number to see what sign it is. Circle the number and what is before it to see what you are multiplying by. The number in FRONT now BEHIND 2x + 2 3(x + 1) This is a NEGATIVE 3

5 EQUATIONS VERSUS EXPRESSIONS 2x + 2 3(x + 1) 2x + 2 3(x + 1) 2x + 2 3x - 3

6 EQUATIONS VERSUS EXPRESSIONS 2x + 2 3x - 3 Apply the commutative property and simplify Combine like terms EX: puppies go with puppies, x prefers to be with x 2x + 2 3x 3 2x 3x = -x 1

7 EQUATIONS VERSUS EXPRESSIONS Final simplification 2x 3x x 1 There was no equal sign, X DOES NOT HAVE AN ANSWER This is an EXPRESSION, with NO equal sign Therefore your answer will NEVER be x =

8 EQUATION/EXPRESSION PRACTICE 4(2 + x) 2(1 + x) 2(2J+ J) 3(J + 1) 4(x + 1) -3(x + 1) = 4

9 FUN WITH WORD PROBLEMS At the local HUB s grocery store, the cost of a carton of tamales, c, depends on the number of tamales, t. The cost of the carton is $0.25 and each tamale is $0.50. a. What equation could be used to represent this situation???

10 FUN WITH WORD PROBLEMS Let s break this problem down: the cost of a carton of tamales, c, depends on the number of tamales, t. we know that c DEPENDS on the number of tamales Therefore, c would be our y value because it is DEPENDENT, and t would be our INDEPENDENT x-value We start every equation with. Y EQUALS y = x +.. Since c is our y variable here, we will start this equation with c =.

11 FUN WITH WORD PROBLEMS At the local HUB s grocery store, the cost of a carton of tamales, c, depends on the number of tamales, t. The cost of the carton is $0.25 and each tamale is $0.50. We know our equation starts with c =. We also know that we have to pay for the carton the tamales come in. So no matter how many cartons we buy, we always pay a rate of $0.25. This value does NOT change, and therefore will never be affect by our independent variable. Therefore it will NOT be connected to our t variable c =

12 FUN WITH WORD PROBLEMS At the local HUB s grocery store, the cost of a carton of tamales, c, depends on the number of tamales, t. The cost of the carton is $0.25 and each tamale is $0.50. Our equation so far is c =.25 + We know that t is INDEPENDENT and therefore CONTROLS the equations. We know the price goes up with every tamale. c = t

13 FUN WITH WORD PROBLEMS Our final equation is c = t Where c is the total COST and t is the NUMBER of tamales bought If a carton contained one dozen tamales, what would be the cost? We know the COST is represented by the variable c In this problem we are given the number of tamales, t Substitute 12 for t c = t c = (12) c = = $6.25

14 FUN WITH WORD PROBLEMS If the cost of a carton was $15.25, how many tamales were purchased? Go back through your variables. c is the total COST which is given here as $ We are asked to find the number of tamales t c = t Substitute for c = t Subtract.25 from both sides 15 =.50t 15/.50 =.50t/ = t The total number of tamales you can purchase for $15.25 is 30

15 MORE FUN WITH WORD PROBLEMS Surf City charges a base fee of $10 and $25 per hour to rent a surf board Just like the previous tamale problem, setup your variables. Again, the total cost to rent a surf board is represented by c and it is charged by the hour, h. The base fee of $10 does NOT change. It is NOT dependent on anything and it is NOT independently changing the situation.

16 MORE FUN WITH WORD PROBLEMS Surf City charges a base fee of $10 and $25 per hour to rent a surf board c = h If the surf board was rented for 5 hours, what is the total cost of the rental? c = h c = (5) c = = 135 If you paid $185 to rent a surf board- how many hours did you rent it? This time we are looking for the h value, as it cost us $ = h subtract 10 from both sides 175 = 25h divide each side by 25 to get h by itself 7 = h; You rented the surf board for 7 hours

17 DECONSTRUCTING WORD PROBLEMS DeeDee works at Toni & Guy as a stylist. She makes $30 per hour and works 40 hours a week. For every coloring session she does, DeeDee has to pay out to the company $5 for color (c). If her pay is a function of the number of hours she works, which of the following equations best describes DeeDee s paycheck? There s a lot going on here: HIGHLIGHT the sentence with the QUESTION MARK first

18 DECONSTRUCTING WORD PROBLEMS We want to find the best equation to fit this problem. What we know: She works 40 hours per week She makes $30 per hour She has to pay out $5 every time she does a service

19 DECONSTRUCTING WORD PROBLEMS If we were to setup this equations, we could set it up to reflect the number of hours she works and how much she has to take out. Let s create a situation for her paycheck, p p = ($30)(# of hours worked) - $5(hair treatments)

20 DECONSTRUCTING WORD PROBLEMS She makes $30 per hour and works 40 hours a week. For every coloring session she does, DeeDee has to pay out to the company $5 for color (c). If her pay is a function of the number of hours she works, which of the following equations best describes DeeDee s paycheck? Y IS A FUNCTION OF X ALWAYS Her pay (y) is a FUNCTION of hours (x) p = ($30)(# of hours worked) - $5(hair treatments) p = (30)(40) 5(c)

21 PUTTING WORDS INTO NUMBERS Special words to remember. Product = multiplication Quotient = division Sum, increased = addition Difference, decreased = subtraction A number = an unknown AN UNKNOWN NUMBER IN MATH IS A VARIABLE Example: x, h, c, etc

22 PUTTING WORDS INTO NUMBERS The product of five times a number increased by seven is seventeen. What is the number? We know product means multiplication An unknown number is a variable We ll use x here 5 times x increased by 7 is 17 5x + 7 = 17 Subtract 7 from both sides: 5x = 10 Divide each side by 5: x = 2

23 CONFUSING WORD PROBLEMS The larger of two numbers is 23. If three times the larger is 5 more than eight times the smaller, find the smaller number. Here we have TWO unknown numbers. Assign each one a variable. We have LARGER and SMALLER. Choose L and S.

24 CONFUSING WORD PROBLEMS The larger of two numbers is 23. If three times the larger is 5 more than eight times the smaller, find the smaller number. L = 23 We know that 3 times L + 5 = 8 times S 3L + 5 = 8S BUT we know L is = 23 Substitute 23 for L 3(23) + 5 = 8S = 8S

25 CONFUSING WORD PROBLEMS 3L + 5 = 8S BUT we know L is = 23 Substitute 23 for L 3(23) + 5 = 8S = 8S = 8S 74 = 8S Divide both sides by 8 74/8 = S

26 CONFUSING WORD PROBLEMS 2 If four times Catherine s age is subtracted from 5 times Jose s age, the difference is 32 years. Jose is sixteen. Find Catherine s age. FIRST assign variables C for Catherine and J for Jose 4 times C is subtracted from 5 times J is 32 We know that there is subtraction here They tell us the difference (answer when numbers are subtracted) is 32

27 CONFUSING WORD PROBLEMS 2 4 times C is subtracted from 5 times J is 32 4 times C is being TAKEN AWAY from 5 times J This means 5J is SUBTRACTING 4C NOT 4C subtracting 5J Subtracting from means taking something away from something else Our final setup is 5J 4C = 32

28 CONFUSING WORD PROBLEMS 2 Our final setup is 5J 4C = 32 They give us one final piece of information. That is, that Jose is 16 years old. That means J = 16 5J 4C = 32 5(16) 4C = C = 32; subtract 80 from both sides -4C = -48; divide -4 from both sides C = -12

29 LOTS OF VARIABLES TO PLAY WITH Joe, Carlos, Monses and Bryan all decide to go practice fútbol. Bryan practices 4 times as long as Monses. Carlos practiced half as long as Joe, and Joe played 30 minutes less than Bryan. The four guys played SOCCER for a total of 3 hours. How long did Carlos play fútbol? How long did Bryan play? Whoa- that s a lot going on. Let s see what we have and what we can put where.

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31 LOTS OF VARIABLES TO PLAY WITH Joe, Carlos, Monses and Bryan all decide to go practice fútbol. Bryan practices 4 times as long as Monses. Carlos practiced half as long as Joe, and Joe played 30 minutes less than Bryan. The four guys played SOCCER for a total of 3 hours. How long did Carlos play fútbol? How long did Bryan play? Joe = J, Bryan = B, Monses = M, Carlos = C How do we set this up? Bryan practices 4 times more than Monses B = 4M

32 LOTS OF VARIABLES TO PLAY WITH Joe, Carlos, Monses and Bryan all decide to go practice fútbol. Bryan practices 4 times as long as Monses. Carlos practiced half as long as Joe, and Joe played 30 minutes less than Bryan. The four guys played SOCCER for a total of 3 hours. How long did Carlos play fútbol? How long did Bryan play? B = 4M C = 1/2J J = B 30 B + C + J + M = 3 hours

33 LOTS OF VARIABLES TO PLAY WITH Our final setup is: B = 4M C = 1/2J J = B 30 B + C + J + M = 3 hours But WAIT- the answer we want is in MINUTES, not HOURS. Convert 3 hours to minutes 3 hours ( 60 minutes/hour) = 180 minutes B + C + J + M = 180

34 LOTS OF VARIABLES TO PLAY WITH B = 4M C = 1/2J J = B 30 B + C + J + M = 180 minutes What do we do with all this information? We are given the variables, so now all we need to do is substitute things in their place! We want to find what Carlos practiced So we are looking for C.

35 LOTS OF VARIABLES TO PLAY WITH B = 4M C = 1/2J J = B 30 B + C + J + M = 180 4M + C + (B 30) + M = 180 4M + C + (4M 30) + M = 180 4M + 4M + M + C 30 = 180 9M + C 30 = 180 9M + C = 210 9M + ½(B-30) = 210 9M + ½(4M 30) = 210 -substitute -substitute -combine M s -add 30 both sides -substitute for C

36 LOTS OF VARIABLES TO PLAY WITH B = 4M C = 1/2J J = B 30 B + C + J + M = 180 9M + ½(4M 30) = 210 9M + 2M 15 = 210 combine like terms 11M -15 = 210 add 15 to both sides 11M = 225 divide both sides by 11 M = 20.5

37 LOTS OF VARIABLES TO PLAY WITH B = 4M C = 1/2J J = B 30 B + C + J + M = 180 But we aren t LOOKING for M, we re looking for C M = 20.5 C = 1/2J = ½(B 30) = ½ (4M 30) = ½ (4* ) = ½ (82 30) = ½(52) = 26 minutes

38 LOTS OF VARIABLES TO PLAY WITH It s a very complicated problem BUT the key is to ORGANIZED Ways to organize: Name all your variables Make a table Replace what you can with 1 variable Choose the variable that is repeated MORE (EX: M)

39 AND NOW FOR SOMETHING COMPLETELY DIFFERENT

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41 . :. :. :. S C A T T E R P L O T S. :. :. :. Scatterplots COMPARE two sets of data EX: Sales at the grocery store versus pay day EX: Childhood diseases versus vaccines When scatterplot data line up they a special relation called a correlation When the data does not line up, it is said to have no correlation Positive correlation is when the data is INCREASING Negative correlation is when the data is DECREASING

42 . :. :. :. S C A T T E R P L O T S. :. :. :. Positive correlation -as x increases, y increases Negative correlation -as x increases, y decreasing

43 . :. :. :. S C A T T E R P L O T S. :. :. :. Which of the following best represents the approximate price of a Diamond at.25 carats? Is it asking for data from the x-axis or the y-axis? Here, it is asking for the data of the y-axis -look along the x-axis for.25 carats -go up until you hit the trend line -go left until you hit the y-axis

44 . :. :. :. S C A T T E R P L O T S. :. :. :. Which of the following best represents the approximate carats when the price is $800? Is it asking for data from the x-axis or the y-axis? Here, it is asking for the data of the Y-AXIS -look along the y-axis for $800 -go RIGHT until you hit the trend line -go DOWN until you hit the x-axis Your answer is on the x-axis

45 TABLES TABLES TABLES Main Event and Astro Bowl are both having summer promotions in order to increase their business. Main Event charges a flat rate fee of $10.00 and $0.50 per game to play. Astro Bowl charges $2.50 per game to play. At what dollar amount do both promotions match up? Number of Main Event Astro Bowl Games Played 0 $10.00 $0 1 $10.50 $ $11.00 $ $ $ 4 $ $ 5 $ $ 6 $ $

46 TABLES TABLES TABLES The MOST important thing about tables is that you need to look for PATTERNS. How slow is the left column going up? How fast is the right column going up? Number of Main Event Astro Bowl Games Played 0 $10.00 $0 1 $10.50 $ $11.00 $ $ $ 4 $ $ 5 $ $ 6 $ $ The left column is increasing by 0.50 and the right by 2.50

47 TABLES TABLES TABLES We are looking for when the two columns match up. Once we have our pattern, we just fill in the columns and pick the amount of games where the price matches up Number of Games Main Event Astro Bowl Played 0 $10.00 $0 1 $10.50 $ $11.00 $ $11.50 $ $12.00 $ $12.50 $ $ $ The prices match when the customer plays 5 games

48 READING INTO EQUATIONS Felicity is planning her quinceañera. The total cost c, is determined by a onetime fee of $500 for the banquet hall and $25 per person p that attends. If Felicity has a total budget of $3750, how many people can she invite? Write your equation in the form of an inequality.

49 READING INTO EQUATIONS What do we know? The total cost is c The banquet hall is $500 The amount per person p, is $25 The most she can spend is $3750 We know she can spend AT MOST $3750 We know that she can t spend MORE than $3750 Therefore, the inequality will be FACING the $ > p

50 USING 2 VARIABLES AND INEQUALITIES The Fiesta Oyster Bake charges $0.50 for ride tickets and $0.75 for food tickets. Dr. Todd and her Monkey have a total of $40 to spend on food and Monkey rides. Which inequality best represents the amount they have to spend on rides and food? Use the variable r for ride tickets and f for food tickets.

51 USING 2 VARIABLES AND INEQUALITIES First and foremost- this is an inequality situation. How do we know that? Inequalities ask about maximums and minimums -that, and it asks for an inequality that s a pretty good indicator about what they want, too

52 USING 2 VARIABLES AND INEQUALITIES What do we have? 0.50 cents for ride tickets 0.75 for food tickets $40 total to spend We know it s an inequality, and we know the MAX she can spend is $40 $40 >???

53 USING 2 VARIABLES AND INEQUALITIES ASSIGN VARIABLES TO MAKE IT EASIER r = 0.50 cents for ride tickets f = 0.75 for food tickets Add up the amount per ticket for rides and food Make sure they are LESS than $40!!! 40 >.50r +.75f

54 HOPE THIS HELPS Hope this helps to get you started on your exam If you have any questions, feel free to J melissa.todd@eisd.net