PRESENTED BY THE NUMEROLOGIST Mathematics, Grade 9. Welcome to today s topic Parts of Listen & Learn. Presentation, questions, Q&A

Transcription

1 Number Sense and Numeration PRESENTED BY THE NUMEROLOGIST Mathematics, Grade 9 Introduction Welcome to today s topic Parts of Presentation, questions, Q&A Housekeeping NOT the Chat Room Your questions Satisfaction Meter 1

2 What you will learn After viewing this presentation, you will be able to explain the rate of change concepts (constant rate of change, meaning of rate of change) explain the connection between rate of change and slope Agenda Background and importance Definition Exploring Constant Applying in an equation Recap 2

3 Agenda Background and importance Definition Exploring Constant Applying in an equation Recap Background Rates of Change are all around us. Slope of a hill Pitch of a roof Slant of a ramp Velocity of a car Price per unit Hourly wage 3

4 Importance Indicates how one variable changes with respect to another variable. Examples: Speed 10 km/h, Pitch of roof, Slope of a hill 6% grade Used in many different subjects: Physics, Chemistry, Biology, Economics, Computer Science Focus Question m 5 m/s You are in a plane, on your way to Cuba. Can t wait to get to the beach! The plane is at a height of m and is beginning its descent. It falls 5 m/s. To know when it s going to land, you need to know the rate of change and constant. Use the following lesson to help you solve this problem. 4

5 Agenda Background and importance Definition Exploring Constant Applying in an equation Recap Definition The change in one variable relative to the change in another. On a graph, the slope of a line represents rate of change. Example: Travel 10 m in 5 seconds : 5

6 Definition : Slope The steepness of a line can be determined by looking at the value of the rate of change. Example: 10 m/s is steeper than 2 m/s red line: 10 m/s The larger the blue line: 2 m/s magnitude of the rate of change, the steeper the line. Agenda Background and importance Definition Exploring Constant Applying in an equation Recap 6

7 Exploring Constant : The rate remains the same throughout the situation. Constant Rate = straight line Exploring Constant : To calculate the rate of change: Change in height Change in length Change in rise Change in run This is the slope. 7

8 Exploring Constant : Change in rise Change in run 2 s 6 m Measure the rise 14 8 = 6 m Measure the run 3 1 = 2 s Exploring Constant : Change in rise Change in run 2 s 6 m 6 m = 3 m/s 2 s 8

9 Exploring Constant : What is the rate of change (slope)? a) 1 m/s b) m/s c) m/s Exploring Constant : What is the rate of change (slope)? 4 m 3 s a) 1 m/s b) m/s c) m/s Positive slope: increasing line (rising from left to right) 9

10 Exploring Constant : What is the rate of change (slope)? a) 2 m/s b) 2 m/s c) m/s Exploring Constant : 2 s - 4 m What is the rate of change (slope)? a) 2 m/s b) 2 m/s c) m/s Negative slope: decreasing line (falling from left to right) 10

11 Agenda Background and importance Definition Exploring Constant Applying in an equation Recap Exploring Applying : Example: Shawn visits Rent-Me Car Rentals. The equation, C = 3k + 50, represents the cost for Shawn to rent a car. (k = distance in kilometres, C = total cost in dollars) a) What is the rate of change? b) What does the rate of change represent? 11

12 Exploring Applying : Example: Shawn visits Rent-Me Car Rentals. The equation, C = 3k + 50, represents the cost for Shawn to rent a car. (k = distance in kilometres, C = total cost in dollars) a) What is the rate of change? b) What does the rate of change represent? Exploring Applying : Example: Shawn visits Rent-Me Car Rentals. The equation, C = 3k + 50, represents the cost for Shawn to rent a car. (k = distance in kilometres, C = total cost in dollars) a) What is the rate of change? $3 per km b) What does the rate of change represent? 12

13 Exploring Applying : Slope= $3/km 6 ($) Look at the solution as shown on a graph. The rate of change (slope) is $3 per km. 2 km Exploring Applying : Example: Shawn visits Rent-Me Car Rentals. The equation, C = 3k + 50, represents the cost for Shawn to rent a car. (k = distance in kilometres, C = total cost in dollars) a) What is the rate of change? b) What does the rate of change represent? 13

14 Exploring Applying : Example: Shawn visits Rent-Me Car Rentals. The equation, C = 3k + 50, represents the cost for Shawn to rent a car. (k = distance in kilometres, C = total cost in dollars) b) What does the rate of change represent? The price per kilometre is $3. As the number of kilometres increases, the total cost changes. Agenda Background and importance Definition Exploring Constant Applying in an equation Recap 14

15 Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. An equation needs two parts; one that is constant, the other that varies. To determine the rate of change, find the part of the relationship that varies. Determine the constant (the part that does not change). Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. 1. To determine the rate of change, find the value that depends on a changing variable. a) $45 b) $20 per month 15

16 Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. 1. To determine the rate of change, find the value that depends on a changing variable. a) $45 b) $20 per month depends on number of months (m) Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. 2. Determine the constant (the part that does not change). a) $45 b) $20 per month 16

17 Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. 2. Determine the constant (the part that does not change). a) $45 b) $20 per month Exploring in an Equation: Savneet joins a gym. It costs $45 to join, and then $20 per month. Put everything together. The equation becomes C = 20m + 45, where m is the number of months and C is the total cost. 17

18 Agenda Background and importance Definition Exploring Constant Applying in an equation Recap Recap 1. The rate of change = slope 2. Decreasing slope negative Increasing slope positive 3. Rate of change depends on changing variables 4. Equation is made up of two parts: a) rate of change, and b) constant 18

19 Recap Answering the Focus Question Height (m) m Time (s) Slope = 5 m/s 2000 A plane began its descent at m. It fell 5 m/s. 1. Find the rate of change. a) m b) 5 m/s c) 5 m/s Recap Answering the Focus Question Height (m) m Slope = 5 m/s Time (s) 2000 A plane began its descent at m. It fell 5 m/s. 1. Find the rate of change. a) m b) 5 m/s c) 5 m/s Altitude of plane decreasing over time, therefore slope is negative. 19

20 Recap Answering the Focus Question Height (m) m Time (s) Slope = 5 m/s 2000 A plane began its descent at m. It fell 5 m/s. 2. Determine the constant. a) m b) 5 m/s c) 5 m/s Recap Answering the Focus Question Height (m) m Time (s) Slope = 5 m/s 2000 A plane began its descent at m. It fell 5 m/s. 2. Determine the constant. a) m b) 5 m/s c) 5 m/s Initial height is m. 20

21 Recap Answering the Focus Question A plane began its descent at m. It fell 5 m/s Height (m) Time (s) Slope = 5 m/s Create an equation to represent the plane s descent. a) D = 5s b) D = 5s c) D = 5s d) D = 5s Recap Answering the Focus Question A plane began its descent at m. It fell 5 m/s Height (m) Time (s) Slope = 5 m/s Create an equation to represent the plane s descent. a) D = 5s b) D = 5s c) D = 5s d) D = 5s Notice the sign on each value. Put together the slope and initial value. s represents time in seconds. 21

22 Resources NCTM examples/chap6/6.2/index.htm 22