In this video lesson, students continue to explore systems where the equations are both of the form y=mx+b. They connect algebraic and graphical representations of systems, first by matching graphs to systems, then by drawing their own graphs from given systems. Additionally, students see how to see the number of solutions from both the graphical and algebraic representations. Then they examine other types of systems with different structures and use the structure of a system of equations to reason about its lack of solutions. When students look at the structure of a system before starting to solve it in order to develop a good approach to solving, they engage in MP7.
Grade 8, Episode 5: Unit 4, Lessons 13 & 14 | Illustrative Math
Solve a number riddle by finding the point of intersection for two lines. This video focuses on finding the solution for a system of equations, they are represented as an algebraic expression and a set of linear equations that are graphed to find an intersection point.
This video was submitted through the Innovation Math Challenge, a contest open to professional and nonprofessional producers and is part of the Math at the Core: Middle School collection.
In this program, students learn the method of solving by substitution, which works by solving one of the equations, you choose which one, for one of the variables, you choose which one, and then plugging this back into the other equation, substituting for the chosen variable and solving for the other. Then you back-solve for the first variable.
Learn how algebra can quickly determine the point where two lines intersect. This video focuses on setting linear equations equal to each other to find a common solution.
Simultaneous equations and their solutions are the focus of Topic D. Students begin by comparing the constant speed of two individuals to determine which has greater speed (8.EE.C.8c). Students graph simultaneous linear equations to find the point of intersection and then verify that the point of intersection is in fact a solution to each equation in the system (8.EE.C.8a). To motivate the need to solve systems algebraically, students graph systems of linear equations whose solutions do not have integer coordinates. Students use an estimation of the solution from the graph to verify their algebraic solution is correct. Students learn to solve systems of linear equations by substitution and elimination (8.EE.C.8b). Students understand that a system can have a unique solution, no solution, or infinitely many solutions, as they did with linear equations in one variable. Finally, students apply their knowledge of systems to solve problems in real-world contexts, including converting temperatures from Celsius to Fahrenheit.
Optional Module 4, Topic E is an application of systems of linear equations (8.EE.C.8b). Specifically, a system that generates Pythagorean triples. First, students learn that a Pythagorean triple can be obtained by multiplying any known triple by a positive integer (8.G.B.7). Then, students are shown the Babylonian method for finding a triple that requires the understanding and use of a system of linear equations.
In Topic A of Module 5, students learn the concept of a function and why functions are necessary for describing geometric concepts and occurrences in everyday life. The module begins by explaining the important role functions play in making predictions. For example, if an object is dropped, a function allows us to determine its height at a specific time. To this point, our work has relied on assumptions of constant rates; here, students are given data that shows that objects do not always travel at a constant speed. Once we explain the concept of a function, we then provide a formal definition of a function. A function is defined as an assignment to each input, exactly one output (8.F.A.1). Students learn that the assignment of some functions can be described by a mathematical rule or formula. With the concept and definition firmly in place, students begin to work with functions in real-world contexts. For example, students relate constant speed and other proportional relationships (8.EE.B.5) to linear functions. Next, students consider functions of discrete and continuous rates and understand the difference between the two. For example, we ask students to explain why they can write a cost function for a book, but they cannot input 2.6 into the function and get an accurate cost as the output.
Students apply their knowledge of linear equations and their graphs from Module 4 (8.EE.B.5, 8.EE.B.6) to graphs of linear functions. Students know that the definition of a graph of a function is the set of ordered pairs consisting of an input and the corresponding output (8.F.A.1). Students relate a function to an input-output machine: a number or piece of data, goes into the machine, known as the input, and a number or piece of data comes out of the machine, known as the output. In Module 4, students learned that a linear equation graphs as a line and that all lines are graphs of linear equations. In Module 5, students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line. They learn to interpret the equation y = mx + b (8.EE.B.6) as defining a linear function whose graph is a line (8.F.A.3). Students will also gain some experience with non-linear functions, specifically by compiling and graphing a set of ordered pairs, and then by identifying the graph as something other than a straight line.
Once students understand the graph of a function, they begin comparing two functions represented in different ways (8.EE.C.8), similar to comparing proportional relationships in Module 4. For example, students are presented with the graph of a function and a table of values that represent a function and are then asked to determine which function has the greater rate of change (8.F.A.2). Students are also presented with functions in the form of an algebraic equation or written description. In each case, students examine the average rate of change and know that the one with the greater rate of change must overtake the other at some point.
In Module 4, Topic B, students work with constant speed, a concept learned in Grade 6 (6.RP.A.3), but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane (8.EE.C.8a). It is in Topic B that students begin to investigate the shape of a graph of a linear equation. Students predict that the graph of a linear equation is a line and select points on and off the line to verify their claim. Also in this topic is the standard form of a linear equation, ax + by = c, and when a, b ≠ 0, a non-vertical line is produced. Further, when a or b = 0, then a vertical or horizontal line is produced.
