Grade 9 Mathematics Unit 4 - Linear Relations PDF

Grade 9 Mathematics Unit 4 – Linear Relations Section 4.1 – Writing Equations to Describe Patterns A variable is typically a letter that is used to a represent an unknown value and acts as a place holder. There are two variables we will focu...

Grade 9 Mathematics Unit 4 – Linear Relations Section 4.1 – Writing Equations to Describe Patterns A variable is typically a letter that is used to a represent an unknown value and acts as a place holder. There are two variables we will focus on in this unit, the independent and dependent variables. An independent variable is a variable whose value is not affected by the other variable. It is the value we “plug in” to an equation. In a graph the independent variable is always plotted on the horizontal (↔), or 𝑥-axis. A dependent variable is a variable whose value is determined by the value entered for the independent variable. In a graph the dependent variable is always plotted on the vertical (↕) or 𝑦-axis. Example 1: Luke wants to earn money this winter shoveling driveways. He will charge $9/h. What two quantities are being compared in this problem? 1. 2. Which quantity is the independent variable and which is the dependent variable? Explain. Independent: Dependent: L. Brenton Page |1 Grade 9 Mathematics Unit 4 – Linear Relations An operation is a math process. The operations we typically use are +, −,× and ÷. An expression is a mathematical statement that is made up of numbers and/or variables connected by operations. It does NOT contain an equals sign. For example, 3𝑝 is an expression, whereas 3𝑝 = 15 is an equation because it contains an equals sign. In this unit, we will use various representations to describe relations. We will describe in words and use expressions and equations to represent patterns from tables, graphs, charts, pictures and problem situations. When a relation is represented using a picture, we can use patterns to derive the expression or equation as seen below. Example 1: Observe the pattern below. a. Complete the table to show the relationship between the number of squares and the number of triangles. Number of Number of Independent variable: Squares (𝒔) triangles (𝒕) 1 4 2 3 Dependent variable: 4 5 L. Brenton Page |2 Grade 9 Mathematics Unit 4 – Linear Relations b. Describe the pattern in words by finishing the following sentence: If I know the number of squares, I can find the number of triangles by… c. Let 𝑠 represent the number of squares. Write an expression that represents the pattern. (Remember, expressions DO NOT have an equals sign) d. Let 𝑠 represent the number of squares and t represent the number of triangles. Write an equation relating 𝑠 and 𝑡 that represents the pattern. (Remember, equations DO have an equal sign) e. Determine the number of triangles that will be in the diagram with 23 squares. L. Brenton Page |3 Grade 9 Mathematics Unit 4 – Linear Relations Observe the chart above. Notice that each side increases by a constant value. Number of Number of Squares (s) triangles (t) 1 4 +1 +2 2 6 +1 +2 3 8 +1 +2 4 10 +1 +2 5 12 As the independent value increases by 1, we can see that the dependent value increases by 2 each time. Repeated addition also means multiplication. Therefore, we can use this information to help write the equation. _________________________________________________________________ Example2: Observe the pattern below. Figure 1 Figure 2 Figure 3 Figure 4 a. Complete the table to show the relationship between the figure number and the number of circles. Figure Number, f 1 2 3 4 5 Number of Dots, d b. Write an equation to describe the relationship. _____________________________________________ c. How many dots are there in the 25th figure. Use the equation. L. Brenton Page |4 Grade 9 Mathematics Unit 4 – Linear Relations Example 3: Observe the pattern below. Figure 1 Figure 2 Figure 3 a. Complete the table to show the relationship between the number of shaded tiles and the number of unshaded tiles. Number of Black 1 2 3 Tiles, 𝒃 Number of White Tiles, 𝒘 Independent variable: Dependent variable: b. Describe the pattern in words by finishing the following sentence: If I know the number of black squares, I can find the number of white squares by… c. Write an equation that represents the pattern. d. Determine the number of white squares when there are 103 black squares. L. Brenton Page |5 Grade 9 Mathematics Unit 4 – Linear Relations We can also write equations to represent situations. Example 1: Teegan goes to a carnival. The cost per ride is $2.00 and the entrance fee is $5.00. a. Write an equation that relates the total cost, C dollars, to the number of rides, r. b. Teegan goes on 8 rides. What is his total cost? Use the equation. Example 2: A taxi cab company charges a flat fee of $4.50 plus $2.25 for each kilometer travelled. a. Write an equation to represent the total cost, C, in terms of the number of kilometers travelled, k. b. If Sheldon travelled 52 km, how much was the cab ride? L. Brenton Page |6 Grade 9 Mathematics Unit 4 – Linear Relations Example 3: For each equation, find the value of C when 𝑘 = 3 a. 𝐶 = 6𝑘 b. 𝐶 = 2𝑘 + 1 c. 𝐶 = 4𝑘 − 5 Example 4: Use the pattern to complete the rest of the table. Write an equation for each table. a. b. 𝒙 𝒚 𝒕 𝒏 1 3 1 7 2 7 2 5 3 11 3 3 4 4 5 5 c. d. 𝒓 𝒒 𝒏 𝒑 1 5 1 12 2 7 2 9 3 9 3 6 4 4 5 5 L. Brenton Page |7 Grade 9 Mathematics Unit 4 – Linear Relations 4.1 Extra Practice 1. Marcel takes a summer job at a book packaging plant. He gets paid $50 a day plus $2 for every box packed. a. Write an equation that relates the number of boxes packed, b, to Marcel’s pay, P, for the day. b. Marcel packed 120 boxes in one day. How much did he get paid? Use the equation. 2. Anabelle is part of the yearbook committee. This year, the set-up cost to print yearbooks is $400 plus $3 for each yearbook printed. a. Write an equation to represent the total cost, C, in terms of the number of yearbooks printed, n. b. Anabelle takes 200 orders for yearbooks this year. What is the total cost to the yearbook committee? Use the equation. L. Brenton Page |8 Grade 9 Mathematics Unit 4 – Linear Relations 4. A party planning company charges $50 for a magician and $1.50 for each child attending. a. Write an equation to represent the total cost, C, for a birthday party with n children attending. b. What is the cost of a party if there are 18 children attending? 5. Ms. O’Brien is ordering St. Paul’s t-shirts. There is a set up fee of $50 and each t-shirt costs $8. a. Write an equation to represent the total cost, C, for an order of x t-shirts. b. If Ms. O’Brien orders 120 t-shirts, how much will it cost her? L. Brenton Page |9 Grade 9 Mathematics Unit 4 – Linear Relations Section 4.2: Linear Relations The graph we use to plot points is called the Cartesian Coordinate System. It consists of: 𝑥-axis, which goes horizontal (left/right) 𝑦-axis, which goes vertical (up/down) origin, the point where the 𝑥 and 𝑦 axes meet in the middle. The coordinates are (0,0). 4 quadrants. Every point on a coordinate grid has coordinates (𝑥, 𝑦). Points are also known as ordered pairs. The first number represents number of spaces you move left or right. The second number represents the number of spaces you move up or down. Always start at (0,0) and follow the directions of the coordinates. To plot the point (3,4): The first number is the 𝑥 coordinate. We find this number on the 𝑥 axis (left to right). In this case 𝑥 = 3. The second number is the 𝑦 coordinate. We find this number on the 𝑦 axis (up or down). In this case 𝑦 = 4. L. Brenton P a g e | 10 Grade 9 Mathematics Unit 4 – Linear Relations Example 1: Plot the following points on the coordinate grid below. 8– 8 1 2 3 4 5 6 7 1 2 3 4 5 6 7 1– 8 2 3 4 5 6 7 8 1 2 3 4 5 6 7 A(−4, 6) B(3, 7) C(5, −4) D(0, 8)E(−2, −1) F(−5, 0) y 8 7 6 5 4 3 2 1 – 8– 7– 6– 5– 4– 3– 2– 1 1 2 3 4 5 6 7 8 – 1 x – 2 – 3 – 4 – 5 – 6 – 7 – 8 Remember these important things about graphing: The interval should be consistent on each axis. For example, the 𝑥-axis could increase by 1 and the 𝑦-axis could increase by 10, but it must remain the same on a given axis. Label each axis. Title the graph Independent variable goes on the 𝑥-axis, dependent variable goes on the 𝑦-axis. L. Brenton P a g e | 11 Grade 9 Mathematics Unit 4 – Linear Relations When the relationship between the independent and dependent variables can be represented in a straight line, we have a linear relation. Relations that are not linear are called non-linear. In a linear relation, a constant change in one variable produces a constant change in the other variable. Example 1: Which graph(s) represents a linear relation? Dependent Dependent Dependent Independent Independent Independent Example 2: Use the equation 𝑦 = 2𝑥 − 3 to complete the corresponding table and graph. Describe the relation. 𝑥 𝑦 -2 -1 0 1 2 L. Brenton P a g e | 12 Grade 9 Mathematics Unit 4 – Linear Relations Example 3: Simon purchases songs to download to his phone. Draw a graph to represent the table. Is the relation linear, or nonlinear? Number of Songs Cost ($), 𝑪 Purchased, 𝒏 1 3.50 2 7.00 3 10.50 4 14.00 5 6 7 Does it make sense to join the points on the graph? We can not purchase part of a DVD so it does not make sense to join the points on the graph. The points on the graph are not joined or connected with a line since the values between the plotted points are not possible. This data is discrete. If we were able to purchase part of a song, then we could connect the points and the data would be continuous. Joining the points indicates that all the values between the plotted points are possible. If an equation is given without the context of a situation, we assume it is continuous. L. Brenton P a g e | 13 Grade 9 Mathematics Unit 4 – Linear Relations Example 4: Rebecca hikes up a hill at 2 km/h. Complete the table and graph the data showing the distance she has hiked. Number of Distance, 𝒅 km, 𝒌 1 2 3 4 5 6 Is this a linear or non-linear relation? Should we connect the points? Why/why not? Example 5: Observe the pattern below. Complete the table and draw a graph for the data. Figure 1 Figure 2 Figure 3 Number of Number of Squares (𝒏) Dashes (𝒅) 1 4 2 3 4 Is this a linear or non-linear relation? Should we connect the points? Why/why not? L. Brenton P a g e | 14 Grade 9 Mathematics Unit 4 – Linear Relations Example 6: Tell which graphs and tables represent linear relations. Explain. a. 𝒙 𝒚 b. Time 𝐓𝐞𝐦𝐩𝐞𝐫𝐚𝐭𝐮𝐫𝐞 0 4 9 11 1 9 10 15 2 14 11 18 3 19 12 17 c. d. y 10 8 6 4 2 -4 -2 2 4 x e. f. m 3 4 5 6 7 n 2 6 18 54 162 L. Brenton P a g e | 15 Grade 9 Mathematics Unit 4 – Linear Relations 4.2 Extra Practice 1. Complete the table of values then graph the relation. 𝑥 5−𝑥 -2 -1 0 1 2 2. Complete the table of values and then graph the relation. 𝑥 3𝑥 − 2 -2 -1 0 1 2 L. Brenton P a g e | 16 Grade 9 Mathematics Unit 4 – Linear Relations 3. Complete the table of values and then graph the relation. 𝑥 2𝑥 − 5 -2 -1 0 1 2 4. Which graph(s) represents a linear relation? y y y 30 60 5 25 50 4 20 D 40 3 15 30 2 10 B 20 5 1 10 1 2 3 4 5 x 1 2 3 4 5 x 1 2 3 x 5. Without graphing, determine which table of values are a linear relation. a. x -3 -2 -1 0 y 6 5 4 3 b. x 0 2 4 6 y 1 4 7 10 L. Brenton P a g e | 17 Grade 9 Mathematics Unit 4 – Linear Relations c. x 1 2 3 4 y 1 3 7 13 d. x 1 2 4 5 y 2 3 4 5 e. x 1 3 5 7 y 2 4 6 8 6. Complete the table of values for each linear relation, then graph it. a. 𝑦 = 4𝑥 b. 𝑦 = −3𝑥 c. 𝑦 = 1−𝑥 𝒙 𝒚 𝒙 𝒚 𝒙 𝒚 -1 -1 0 0 0 1 1 2 L. Brenton P a g e | 18 Grade 9 Mathematics Unit 4 – Linear Relations Section 4.3 – Another Form of the Equation for a Linear Relation Observe the following graphs and corresponding tables. A B C 𝒙 𝒚 𝒙 𝒚 𝒙 𝒚 -2 0 3 -2 -2 -2 -1 2 3 -1 -1 -2 0 4 3 0 0 -2 1 6 3 1 1 -2 2 8 3 2 2 -2 What do you notice about each graph and table? L. Brenton P a g e | 19