1. Quiz 1. Factor 2. Factor

3. 1-1: Modeling and Equation Solving

4. Those @#$%#$!!! Word Problems.

5. English  Math Translation EnglishMath IS= 3 more thanx+3 2 less thanx-2 Miles per hourmi/hr

6. English  Math Translation The width of a rectangle is 3 inches greater than its length. The area of the rectangle is 250 square inches. Write an equation for the area of the rectangle in terms of its length.

7. Your Turn: The length of a rectangle is twice its width. The perimeter of the rectangle is 66 feet. The perimeter of the rectangle is 66 feet. 1.Write an equation relating its perimeter to its width. to its width. 2. What is the rectangle’s area ?

8. A Ratio: Miles per hour = miles/hr $ per gallon = $/gal Apples per person = Apples/person A to B A : B A / B

9. Rate (“____ per ____”)  Miles per hour  $ per gallon  Apples per person  Televisions per household These are all slopes of graphs.

10. Slopetime distance What are the “units” of the x-axis? What are the “units” of the y-axis? Slope =

11. Slope Slope is a ratio. ‘Mph’ (speed) means what?

12. Slope Slope is a ratio. ‘Dollars per pound mean? “the ‘price’ for each “unit of weight”.

13. Slopeyx weight Price= What are the “units” of the x-axis? What are the “units” of the y-axis?

14. Slope Slope is a ratio. ‘candy bars per person “the number of candy bars each person gets.

15. Your turn: Find the “rate”  ratio change of one quantity to the change in another quantity for the following in another quantity for the following include the units!!! include the units!!! 3. The level in the tank changed from 20 gallons to 40 gallons over a 5 minute period. over a 5 minute period. 4. The number of people in the stadium changed from 200 people to 100 people during a 2 minute period. 200 people to 100 people during a 2 minute period. 5. The temperature of the solution changed from 75° F. to 100° F during a 10 minute period. 75° F. to 100° F during a 10 minute period.

16. Word Problems modeled by a line: A fixed amount PLUS an amount based upon the number of items used/bought/etc. items used/bought/etc. Example #1: The cost of membership to the club is $50 plus $5 per visit. per visit. (1) Write the equation for this relation. b = $50 b = $50 (2) What would be the membership cost for 40 visits?

17. Word Problems modeled by a line: A fixed amount PLUS an amount based upon the number of items used/bought/etc. items used/bought/etc. b = $30 b = $30 Example #2: A cell phone company charges $30 plus $0.03 per minute of usage. (1)Write the equation for this relation. (2) What will it cost this month if you used 500 minutes?

18. Your Turn: 6. A membership to a health club costs $25 plus $1 per visit. a.Write an equation to represent the cost of membership. b.How much would the membership cost if you visited 30 times? 7. A cell phone plan costs $0.02 per minute. a.Write an equation to represent the cost of the phone. b.How much would the phone cost if you didn’t use it? c.How much would the phone cost if you used it 120 minutes?

19. Conversion Factor: a number that equals ‘1’ (always) Comes from equilvalent measurements Conversion factor

20. Easy Unit Conversions Convert 37 feet into inches Convert 42 inches into feet

21. Your Turn: 8. Convert 14.7 psi to kpa (kilopascals) 1 psi = 6894.757 pascal 1 kpa = 1000 pascal (14.7 pounds per square inch—atmospheric pressure at sea level) 9. Convert 365 days into seconds

22. More challenging: converting units in the numerator and denominator!! Using properties of equality, you can decide which version of the conversion factor to use (the unit you version of the conversion factor to use (the unit you want to keep should be in the numerator). want to keep should be in the numerator).

23. Your Turn: 10. Convert: 62 lbs/cubic foot into ounces per cubic inch.

24. EXAMPLE 3 You are selling homemade candles at a craft fair for $ 3 each. You spend $ 120 to rent the booth and buy materials for the candles. Write an equation that shows your profit from Write an equation that shows your profit from selling c candles. Verbal Models to solve problems. Evaluate: What does profit mean? Profit is $ from sales subtract costs Verbal model $ from sales = ? $3 per candle * # candles P = 3c - 120 Write the equation:

25. Standard Form Linear word problems These type usually are total cost of two items. Pizzas cost $5 each. Drinks cost $1.25 each. 1. Write an equation that shows the total cost of buying pizzas and drinks. 2. What is the total cost if you buy 10 pizzas and 30 drinks?

26. Standard Form Equations Nails cost $5 per pound. Screws cost $10 per pound. Write an equation showing how many pounds of nails and screws you can buy if you can spend is $30. you can buy if you can spend is $30. 5x + 10y = 30 $10 per pound $5 per pound If there are two “____ per ____” phrases;  standard form equation.  standard form equation.

27. Standard Form Equations Small pizzas cost $7 each. Large pizzas cost $10 each. Write an equation showing relating the number of large and small pizzas that can be bought. pizzas that can be bought. 7x + 5y = total cost If you buy 3 large pizzas, how many small pizzas can you buy if your budget is $50? can you buy if your budget is $50? 7(3) + 5y = 50 5y = 50 – 21 y = 29/5 y = 5.8 5 small pizzas

28. Your Turn: Large soda fountain drinks cost $2.50. Small drinks cost $1.50. Your budget for the party is $25 11. Write an equation for this relation. 12. If you buy 7 large drinks, how many small drinks can you buy?

29. Your turn 13. You buy 60 pizzas. Small pizzas cost $10 and large pizzas cost $15. If you spend $700. How many of each size of pizza did you buy? 14. Your total Profit from selling 100 baseball cards is $950. You paid $7.50 for each card. What did you charge for each one? 15. Your total Profit from selling baseball cards is $250. You paid $7.50 for each card. Your charged $10 for each card. How many did you sell ?

30. Total income Model A real estate agent made $65,000 last year. She had a base salary of $25,000. She earns 4% commission on her total sales. What was the value of the real estate she sold? How do you calculate commission (or tips) ? Total income = wages + tips 65,000 = 25,000 + 0.04(total sales) Total income = salary + commission 65,000= 25,000 + commission commission = a percentage of total sales total sales = $1,000, 000

31. Your Turn: 16. At a restaurant, during one shift, a waiter’s wages are $30. She gets an additional 15% of each of the customers’ bills from tips. If the total amount of money she earned was $105, what was the total amount of the customers’ food bills? Verbal Model: Total income = wages + 15%(food bills) 105 = 30 + 0.15x

32. Numerical Models Year Minimum Hourly Wage MHW Purchasing power in 1996 Dollars 19601.005.30 19701.606.47 19803.105.90 19903.804.56 20005.154.69 During what 10 yr period was the biggest increase purchasing power?

33. Numerical Models Year Minimum Hourly Wage MHW Purchasing power in 1996 Dollars 19601.005.30 19701.606.47 19803.105.90 19903.804.56 20005.154.69 In what year did a worker earning MHW have the greatest purchasing power?

34. Numerical Models Year Minimum Hourly Wage MHW Purchasing power in 1996 Dollars 19601.005.30 19701.606.47 19803.105.90 19903.804.56 20005.154.69 In 1980 a worker was earning nearly twice the MHW as a worker in 1970. Why was there pressure to raise the minimum wage?

35. Your Turn Year Male population (1000’s) Female population (1000’s) 198030412 198545921 199069941 1995102164 2000129092 This shows prison populations by year. 17. What would you have to do in order to answer this question: “Which proportion of total prison population is increasing at the fastest rate, male or female?” increasing at the fastest rate, male or female?”

36. Numerical Models Year Total Population% Male 198031696.2 198548095.6 199074094.5 1995108594.1 2000138293.3

37. Algebraic Models There are an endless number of formulas that compare real world quantities: area volume cost/weight mass/volume viscosity pressure mass/volume viscosity pressure energy/weight cost/time energy/time

38. Algebraic Models Models that use formulas to relate the quantities. Getting the best deal. Comparing pizzas. Rectangular: 18” x 24” cost $15 Circular: 24” in diameter cost $20. Which is the better deal? How can you compare them? Price per unit area Rectangular Circular

39. Your Turn 18. Company ‘X’ charges a fixed amount of $30 per month for phone service plus $30 per month for phone service plus $7.50 per 100 minutes of usage. $7.50 per 100 minutes of usage. Company ‘Y’ charges a fixed amount of $40 per month for phone service plus $4 per 100 minutes of usage. After how many minutes of phone usage Will company ‘Y’ be cheaper than company ‘X’?

40. Graphical Models YearIndex 19009.5 19063 191119 191473 191991 Numerical data Graphical data Regression: using a “best-fit” curve to approximate the data.

41. Regression using a “best-fit” curve to using a “best-fit” curve to approximate the data. y = 20x -10 “linear” regression “quadratic” regression

42. Regression Allows us to convert a Numerical model into an Algebraic model. Replaces discrete data with a continuous function. Using the Algebraic model we can predict output values where no data exists.

43. There are many Algebraic functions that can model discrete data. “Stat” p/b

44. Regression: Enter data into lists: “stat” p/b Selet “edit” Clear the data in the list: Move cursor to highlight “L1” Select “clear” (NOT “del”) then “enter” Enter data into List #1 by alternately typing in a number then hitting enter typing in a number then hitting enter

45. Regression: Notice that all the statistic plots are turned off  turn on the one that has your list numbers (L1, L2)  turn on the one that has your list numbers (L1, L2) Make sure your window will display window will display the range of data the range of data (Scroll down to the correct plot) “enter” Turn on the plot by highlighting “on” then hit “enter” then hit “enter” “2 nd ” + “stat plot” (“y=“) “graph” Looks linear  we’ll do linear regression. linear regression.

46. Regression: “stat” p/b The cursor is prompting you to enter the correct lists containing the domain and range (“x” and “y” data) (Scroll down to “LinReg (ax+b)” and enter (or just type “4”) and enter (or just type “4”) Type in (L1,L2) by hitting the following key strokes: “graph” Looks linear  we’ll do linear regression. linear regression. “calc” “(“ + “2 nd ” + “1” (L1) + “,” (comma) + “2nd” + “2” (L2) + “)” “(“ + “2 nd ” + “1” (L1) + “,” (comma) + “2nd” + “2” (L2) + “)”

47. Regression: The general equation plus the specific values plus the specific values for the equation are given. for the equation are given. “enter” p/b Enter this equation into your “y=“ “graph” It looks like a VERY good fit for the data We just completed problem #2 of the homework. We just completed problem #2 of the homework.

48. Your Turn: 19. Do problem #3 of the homework.

49. Grapher Failure As x  2.5 from above, y  + infinity As x  2.5 from below, y  - infinity Since the grapher takes discrete values of ‘x’ and inputs them to get a mating value for ‘y’, it connects those two points with a line or curve. (‘tblset”)

50. Grapher Failure Hidden behavior

51. What is the difference between these statements? If ‘a’ is a real number that solves f(x) = 0 1. ‘a’ is a root of f(x) = 0 2. ‘a’ is a solution of f(x) = 0 3. ‘a’ is a zero of f(x) – 0 4. ‘a’ is the ‘x-intercept’ of the graph of f(x) = 0

52. Your Turn: 12. Graph this equation (draw it’s shape). 13. Determine the ‘zeroes’