First Differences WS 5.1. Rate of Change WS 5.2. Slope/Rate of Change WS 5.3. Partial Variation WS 5.5. Mid Chapter Review & EQAO Practice
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1 UNIT 5 - LINEAR RELATIONS Date Lesson Topic HW Nov First Differences WS 5.1 Nov Rate of Change WS 5.2 Nov Slope/Rate of Change WS 5.3 Nov Direct Variation QUIZ ( ) WS 5.4 Nov Partial Variation WS 5.5 OPT. Mid Chapter Review & EQAO Practice WS MCR Nov MORE Partial & Direct Variation WS 5.6 Dec Solving Equations QUIZ ( ) WS 5.7 Dec Determining Values in a Linear Relation WS 5.8 Dec Solving Problems involving Linear Relations QUIZ ( ) WS 5.9 Dec Two Linear Relations WS 5.10 Dec Review for Unit 5 Test WS 5.11 Dec UNIT 5 TEST WS
2 MFM 1P Lesson 5.1 Recognizing Linear Relations Here is a pattern of squares. The pattern continues. Here are tables for the perimeter and area of each frame. Here are the graphs of the data. The graph of Perimeter against Frame number shows a linear relation. The points lie on a straight line. We join the points with a broken line to show the trend. The graph of Area against Frame number shows a non-linear relation. The points lie along a curve. We join the points with a broken curve to show the trend. The broken line and broken curve indicate that only plotted points are data points.
3 Add a third column to the Perimeter table. Record the changes in perimeter. Frame Number Perimeter Change in perimeter The numbers in the third column are called: First Differences. The first differences for a linear relation are equal. For every increase of 1 in the frame number,the perimeter increases by 4 units. All linear relations have first differences that are constant. We can use this to identify a linear relation from its table of values Add a third column to the Area table. Record the changes in area. Frame Number Area Change in area For a non-linear relation, the first differences are not equal. For every increase of 1 in the frame number, the area increases by a different amount each time
4 Account Balanace Ex. This table shows the money in a bank account when it was opened, and at the end of each following week. Number of Weeks Account Balance ($) First Differences ( 1 ) a) Determine the first differences. What do the first differences represent? b) Is this a linear or non-linear relation? Explain c) Graph this relationship. Describe how the patterns in the table are shown in the graph d) How much money is in the account after 6 weeks? Number of Weeks WS 5.1
5 MFM 1P Lesson 5.2 Speed as a Rate of Change This graph shows how the distance changes over time during a trip. The graph is linear. Recall that: Average speed distance time Choose any two points on the line, such as (1, 75) and (3, 225). The distance travelled is the vertical distance between the points on the graph, or the rise. The rise is: 225 km - 75 km = 150 km The time taken is the horizontal distance between the points, or the run. The run is: 3 h - 1 h = 2 h distancetravelled Average Speed time taken rise run 150 km 2 h 75km/ h rise is called the rate of change. run the average speed is 75 km/h. It is a measure of the steepness of the line. The rate of change tells you how many units to move up or down for every unit you move to the right on the graph. The rate of change of distance over time is the average speed.
6 Distance from downtown (km) Ex. Nathalie leaves downtown Ottawa to meet a friend at Dow s Lake, 6 km away. She skates along the Rideau Canal. It takes Nathalie 5 min to skate 1 km. a) Assume Nathalie keeps skating at this rate. Make a table to show her distance from downtown at 5-min intervals and graph the data. Time (min) Distance from downtown (km) 6 Nathalie's Skate to Dow's Lake Time (min) b) Determine the rate of change from the graph. What does the rate of change represent? Choose 2 points on the line: The rise is: The run is: rise Rate of change = run c) How long does it take Nathalie to reach Dow s Lake? d) What is Nathalie s average speed How did you determine this? in kilometres per hour? WS 5.2
7 MFM 1P Lesson 5.3 Other Rates of Change Using Tables and Graphs to Explore Rates of Change The graph shows the amount of Pentothal required for dogs of different masses. Record the data from the graph in a table. Include masses up to 6 kg. Add a third column to the table to show the first differences in the drug dose. Mass (kg) Dosage (mg) First Differences ( 1 ) a) What do you notice about the first differences? Explain. b) Select 2 points on the graph. Determine the rate of change. What does this rate of change represent? c) Compare the first differences and the rate of change. What do you notice?
8 Ex. The graph shows the average fuel efficiency for an older sport utility vehicle (SUV) and a new Smart car. The rate of change can be found for each vehicle. For the Smart car: Choose any two points on the line: For the sport utility vehicle: Choose any two points on the line: The rise is: The rise is: The run is: The run is: Rate of change = Rate of change = If you are taking a trip of 800 km and gas costs $1.25/L, how much would the trip cost for each car?
9 The rate of change of a linear relationship is the steepness of the line. The larger the rate of change, the steeper the line. The steepness of a line or its rate of change is known as the slope. Ex. The graph shows how the height of a small plane changes with time. a) What is the plane s initial height? b) Determine the rate of change. The rise is: The run is: Rate of change = c) What does the rate of change tell you about the plane s flight? WS 5.3
10 MFM 1P Lesson 5.4 Direct Variation When two variable quantities have a constant (unchanged) ratio, their relationship is called a direct variation. It is said that one variable "varies directly" as the other. The graph of a direct variation is a straight line that passes through the origin (0, 0). Ex. Rhonda bought sesame snacks at the bulk food store. The cost was $1.10 per 100 g. Here is a table of costs for different masses and a graph representing the relationship. We say that the cost varies directly as the mass. When two quantities vary directly, they are proportional. The cost is directly proportional to the mass. Dependent Varaible = Rate of Change x Independent Variable Use the rule to determine how much Rhonda has to pay for 350 g of sesame snacks.
11 Cost ($) Ex. Gas for a car is sold by the litre. Here are the costs of gas for 5 customers at a gas station. a) Graph the data b) Does the graph represent direct variation? Explain Volume of Gas (L) c) Determine the rate of change. Explain what it represents. d) Write an equation for this relation. e) Use the equation to determine the cost of 25.8 L of gas. WS 5.4
12 Cost ($) MFM 1P Lesson 5.5 Partial Variation PARTIAL VARIATION A relationship between two variables in which one variable is a constant multiple of the other plus a constant value. Graph DOES NOT pass through the origin. Represented by y = mx + b form y = mx + b, where m is the variable cost and b is the fixed cost Ex. The cost of a pizza with tomato sauce and cheese is $9.00. It costs $1.00 for each additional topping. This table shows the cost of a pizza with up to 8 additional toppings. # of Toppings Cost All points lie on a straight line. The graph does not pass through the origin. This illustrates partial variation. The cost of a pizza is the sum of a variable cost and a fixed cost.. The points are joined with a broken line since we cannot order a fraction of toppings Number of Toppings The rise is: The rate of change is: The run is:
13 You have determined values by using a graph. This example shows how to determine values using an equation. Ex. Jolanda has a window cleaning service. She charges a $12 fixed cost, plus $1.50 per window. a) Write an equation to determine the cost when the number of windows cleaned is known. b) What does Jolanda charge to clean 11 windows? WS 5.5
14 MFM 1P Lesson 5.6 Changing Direct and Partial Variation Situations Determine whether each graph represents direct variation, partial variation, or neither. Give reasons for your choices. A. B. Match each graph with its description below. Label the graphs with as much detail as possible. The Creature Comforts dog kennel costs $28 per day. Rana rents a moped while on vacation. She is charged an initial fee of $35 plus $2 for each litre of fuel consumed. Write an equation for each graph. Suppose the daily cost at the dog kennel increases to $30. How does the graph change? Sketch the new graph on the grid. How does the equation change? Write the new equation.
15 Earnings ($) When Rana rented a moped the next year, the prices had changed. The initial fee had been reduced to $28. The price for fuel had increased to $3 per litre. How does the graph change? Sketch the new graph on the grid. How does the equation change? Write the new equation. Ex. Jocelyn works full time selling newspaper ads. She is paid $32 a day, plus $8 for every ad she sells. a) Make a table of Jocelyn s earnings b) Graph the data. for daily sales up to 60 ads. Number of Ads Earnings ($) E n Number of Ads Sold 60 c) Write an equation to determine Jocelyn s earnings, E dollars, when she sells n ads.
16 d) There is a change in management. Jocelyn is now paid $30 per day and $10 for every ad she sells. i) Graph Jocelyn s earnings for daily sales up to 60 ads. How is the graph different from the graph in part b? ii) Write the new equation for Jocelyn s earnings. WS 5.6
17 MFM 1P Lesson 5.7 Solving Equations A one-step equation is as straightforward as it sounds. You will only need to perform one step in order to solve the equation. One goal in solving an equation is to have only variables on one side of the equal sign and numbers on the other side of the equal sign. The other goal is to have the number in front of the variable equal to one. The strategy for getting the variable by itself with a coefficient of 1 involves using opposite operations. The most important thing to remember in solving a linear equation is that whatever you do to one side of the equation, you MUST do to the other side. So if you subtract a number from one side, you MUST subtract the same value from the other side. Ex. 1 Solve. x 5 17 Remember the goal is to have the variable by itself on one side of the equation. In this problem, that means eliminating the + 5 from the left side of the equation. Since the 5 is added to the variable, we eliminate it by subtracting 5. However, if we subtract 5 from the left side of the equation, we MUST also subtract 5 from the right side. Ex. 2 Solve. 15 x 3 It does not matter that the variable in this equation is on the right side of the equation. The position of the variable is not an issue. Remember that the goal is to have the variable on one side by itself. It does not matter which side. To get the variable by itself, we need to add 3 to both sides.
18 Ex. 3 Solve. 3x 18 Check your answer to show that it is correct. The variable in this equation is already on one side of the equation by itself. There is no need to add or subtract anything to both sides. However, the number in front of the variable is not 1. The -3 that is in front of the variable indicates multiplication of -3 by x. The opposite operation of multiplication is division. So we will divide both sides by -3. y Ex. 4 Solve. 7 3 Check your answer to show that it is correct. The variable in this equation is already on one side by itself, but it is divided by 3. To get rid of the 3 that is attached to the variable by division, we will perform the opposite operation which is multiplication. Notice that our variable can be any letter. It does not always have to be x.
19 Solving Two-Step Equations A two-step equation is as straightforward as it sounds. You will need to perform two steps in order to solve the equation. In solving two-step equations you will make use of the same techniques used in solving onestep equation only you will perform two operations rather than just one. When you rearrange an equation, you must do it in the reverse of the order of operations. In other words, rearrange in the order of SAMDEB. Ex. 1 Solve. 3x 4 13 This problem does not have the variable by itself on one side. We need to get rid of the 4 that is added, so we ll need to move that to the other side and making it negative. Even after doing that, there is still a 3 multiplied by the variable, so division will be necessary to eliminate it. Ex. 2 Solve. 2a 5 21 The variable is not by itself on one side. We will get rid of the -5 by moving it to the other side and making it positive. This still leaves a -2 in front of the variable so we will have to divide both sides by -2. Ex. 3 Solve for x. WS 5.7
20 Earnings ($) MFM 1P Lesson 5.8 Determining Values in a Linear Relation Ex. Janice is a piano teacher. She earns $23 per hour. Janice wants to earn $200. There are three ways she can determine the time she has to work. Continue the table until the Earnings is $200 or greater. Time Worked (h) Earnings ($) E n Time Worked (h)
21 Ex. A candle is 13 cm long. It is lit and it burns at the rate of 0.2 cm/min. a) Write an equation for the height, H centimetres, of the candle after time, t minutes. b) How long does it take until the candle is 10 cm high? WS 5.8
22 MFM 1P Lesson 5.9 Solving Problems Involving Linear Relations Vimy Ridge, in Northern France, was the site of one of the most infamous battles of World War I.In 2007, groups of Canadian high school students visited the site to commemorate the 90th anniversary of the battle. When they travelled to France, students converted their money from dollars to Euros. The price of one currency in terms of another is called the exchange rate. Exchange rates vary from day to day. This graph is based on the average exchange rate in January a) Does this graph represent direct variation or partial variation? Explain how you know. b) Use the graph. i) Determine the value of $70, in Euros. ii) What is the value of 100 Euros, in dollars? c) Determine the rate of change. d) Write an equation for the value, d What does it represent? Canadian dollars, of E Euros.
23 e) Use the equation to determine the value of $100, in Euros. What is the value of 20 Euros, in dollars? Ex. Alexei went on a ski trip that cost $1700. He borrowed the money from his parents to pay for the trip. Every month, he pays them back the same amount of money.
24 Distance (m) To find out when Alexei will have paid off his loan, use the graph or use the equation. Ex. Ashok and Katie each recorded a distance for a ball rolling over a period of time. Ashok found that the ball rolled 9 m in 3 s. Katie found that the ball rolled 6 m in 2 s. a) Draw a distance-time graph for the ball. Assume the ball continues at the same average speed. D t Time (s) b) Use the graph to find when will the ball have rolled 10 m? c) Write an equation that relates the distance, d metres, and the time, t seconds. d) Use the equation. How far did the ball roll in 8 s? WS 5.9
25 Distance (m) MFM 1P Lesson 5.10 Two Linear Relations Ex. Diana and Kim live 2.5 km apart. They leave their homes at the same time. They travel toward each other s house. Diana is on rollerblades. She skates at an average speed of 250 m/min. Kim is walking at an average speed of 60 m/min. Make a table of values that represents each girls distance from Kim s house. Time (min) 0 Kim s distance (m) Diana s distance (m) Graph the data for each person on the same grid Distance from Kim s House At what point do they lines intersect? What does this point represent? Time (min) Ex. Julie started a business making bracelets. Her costs are $180 for tools and advertising, plus $2 in materials per bracelet. Each bracelet is sold for $17. a) Write an equation for the total cost, C dollars, to make n bracelets. b) Write an equation for the revenue, R dollars, from the sale of n bracelets.
26 Total Cost ($); Revenue ($) c) Graph both equations on the same grid Number of Braclets d) Where do the lines intersect? What does this point represent? WS 5.10
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