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Grade 10 Academic Math Chapter 1 – Linear Systems

Grade 10 Academic Math Chapter 1 – Linear Systems. Modelling Word Problems Days 4 through Days 9. Day 4 Agenda. Warm-up Types of Modelling Problems Mixture Problems Relative Value Problems Practice. Learning Goal. By the end of the lesson…

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Grade 10 Academic Math Chapter 1 – Linear Systems

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  1. Grade 10 Academic Math Chapter 1 – Linear Systems Modelling Word Problems Days 4 through Days 9

  2. Day 4 Agenda • Warm-up • Types of Modelling Problems • Mixture Problems • Relative Value Problems • Practice

  3. Learning Goal By the end of the lesson… … students will be able to read and interpret a mixture or relative value word problems and create a pair of linear relation equations, resulting in a linear system

  4. Curriculum Expectations • Solve problems that arise from realistic situations described in words… by choosing an appropriate algebraic… method • Ontario Catholic School Graduate Expectations: The graduate is expected to be… a self-directed life long learner who CGE4f applies effective… problem solving… skills

  5. Mathematical Process Expectations • Connecting – make connections among mathematical concepts and procedures; and relate mathematical ideas to situations or phenomena drawn from other contexts

  6. Modelling Types • 1. Break-Even Problems • 2. Mixture Problems • 3. Relative Value Problems • 4. Rate Problems

  7. Mixture Problems • 2 things come together to give a total number or amount • 2 things come together to form a total cost, weight, points, etc. • Equations are usually in form Ax + By = C

  8. Mixture Problems • Ex. 1 Henry sharpens figures skates for $3 a pair and hockey skates for $2.50 per pair. If he earns $240 and sharpens 94 pairs of skates, how many pairs of each type of skate does he sharpen?

  9. Example 1 Mixture (Cont’d)

  10. Mixture Problems Let x represent # of figure skates Let y represent # of hockey skates x + y = 94 (# of skates eq’n) 3x + 2.5y = 240 (earnings eq’n)

  11. Mixture Problems • Ex. 2 Joe has 38 loonies and toonies totalling $55. How many of each type of coin does he have?

  12. Example 2 Mixture (Cont’d)

  13. Mixture Problems Let l represent # of loonies Let t represent # of toonies l + t = 38 (# of coins equation) l + 2t = 55 (value equation)

  14. Mixture Problems • Ex. 3 (p.44, #11e) • Benoit invested some money at 8% and some at 10%. He earned a total of $235 in interest.

  15. Example 3 Mixture (Cont’d)

  16. Mixture Problems Let x represent Amount of $ invested at 8% Let y represent Amount of $ invested at 10% 0.08x + 0.1y = 235 (interest equation) Note: In order to do a $ invested eq’n, we need the amount invested

  17. Mixture Problems • Ex. 4, p.51, #4c • The total value of nickels and dimes is 75¢

  18. Example 4 Mixture (Cont’d)

  19. Mixture Problems Let n represent # of nickels Let d represent # of dimes 0.05n + 0.10d = 0.75 (value equation) Note: In order to do a # of coins eq’n, we need to know the # coins

  20. Relative Value Problems • Usually 2 unknown numbers, ages, etc. • No set form to the equations • Must follow the directional words such as more than, less, times, is, twice, sum, difference, etc.

  21. Relative Value Problems • Ex. 1, p.51, #7 • The sum of two numbers is 72. Their difference is 48. Find the numbers.

  22. Example 1 Relative Value (Cont’d)

  23. Relative Value Problems Let x represent the first number Let y represent the other number x + y = 72 (sum equation) x – y = 48 (difference equation)

  24. Relative Value Problems • Ex. 2, p.51, #8) • A number is four times another number. Six times the smaller number plus half of the larger number equals 212. Find the numbers.

  25. Example 2 Relative Value (Cont’d)

  26. Relative Value Problems Let x represent the first number Let y represent the other number x = 4y (multiplication eq’n) 0.5x + 6y = 212 (difference equation)

  27. Relative Value Problems • Ex. 3, p.24, #7 • At the December concert, 209 tickets were sold. There were 23 more student tickets sold than twice the number of adult tickets. How many of each were sold?

  28. Example 3 Relative Value (Cont’d)

  29. Relative Value Problems Let x represent # of student tickets Let y represent # of adult tickets x - 23 = 2y (relative # of tickets) x + y = 209 (# of tickets)

  30. Relative Value Problems • Ex. 4, p.24, #8 • A rectangle with a perimeter of 54cm is 3m longer than it is wide. What are its length and width?

  31. Relative Value Problems Let l represent width of the rect. Let w represent length of the rect. 2x + 2y = 54 (perimeter eq’n) l – 3 = w (relative length to width eq’n)

  32. Humour Break

  33. Break-Even Problems • Usually look for the point at which two things cost the same • Can refer to the point at which cost and number of things are equal • Equations usually take the form of y = mx + b

  34. Break-Even Problems • Ex. 1. Barney’s Banquet Hall charges $500 to rent the room, plus $15 for each meal and Patrick’s Party Palace charges $400 for the hall plus $18 for each meal. When will both places cost the same amount?

  35. Example 1 Break-Even (Cont’d)

  36. Break-Even Problems Let x represent # meals Let y represent the cost y = 15x + 500 (Barney’s BH) y = 18x + 400 (Patrick’s PP)

  37. Break-Even Problems • Ex. 2. The Millennium Wheelchair Co. has just started its business. It costs them $125 to make each wheelchair plus $15,000 in start-up costs. They plan to sell the chairs for $500 each. How many chairs do they have to sell in order to break even?

  38. Example 2 Break-Even (Cont’d)

  39. Break-Even Problems Let x represent # of wheelchairs Let y represent cost or revenue y = 125x + 15000 (Cost eq’n) y = 500x (Revenue eq’n)

  40. Break-Even Problems • Ex. 3. p.44, #11c • It costs $135 to rent the car, based on $25 per day, plus $0.15/km

  41. Example 3 Break-Even (Cont’d)

  42. Break-Even Problems Let x represent # of days Let y represent # of km driven 25x + 0.15y = 135 (Cost eq’n) Note: This is not a usual example. Usually if you are dealing with car rental, you have an eq’n like y = 0.15x + 25

  43. Humour Break

  44. Rate (Speed Distance Time) Problems (Copy) • Usually looking for time, speed or distance • Distance = Speed x Time (from science – can be rearranged for speed and time also) • Easiest to use a chart to help develop the equations

  45. Rate (Speed Distance Time) Problems • But first, we have the Distance = Speed x Time (equation) Or... D = S x T

  46. Rate (Speed Distance Time) Problems • We can also rearrange this eq’n to solve for speed... Speed = Distance ------------ Time Or...

  47. Rate (Speed Distance Time) Problems • We can also rearrange this eq’n to solve for Time... Time = Distance ------------ Speed

  48. Rate (Speed Distance Time) Problems • Ex. 1 Fred travelled 95 km by car and train. The car averaged 60 km/h and the train averaged 90 km/hr. If the trip took 1.5 hours, how long did he travel by car? • Let’s use a speed distance time chart to organize our information...

  49. Example 1 Rate (Cont’d)

  50. Rate (Speed Distance Time) Problems Let x represent the time in the car Let y represent the time on the train

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