In this video lesson, students work with graphs and tables that represent functions. They learn the conventions of graphing the independent variable (input) on the horizontal axis and the dependent variable (output) on the vertical axis and that each coordinate point represents an input-output pair of the function. By matching contexts and graphs and reading information about functions from graphs and tables, students become familiar with the different representations and draw connections between them.
Grade 8, Episode 8: Unit 5, Lesson 4 | Illustrative Math
In this Cyberchase lesson, students will watch a video clip in which Hacker creates a cyberfrog with numeric buttons that produce different numbers of hops. The relationship between input and output values is used to teach students how to use algebraic expressions and, subsequently, equations. The entire lesson can be found using the Content Source link or each individual activity can be found in the Content Section.
In this video lesson, students begin to analyze graphs of functions and use them to answer questions about a context. Students look at what happens over intervals of input values and learn that graphs can be viewed as dynamic objects that tell stories. They connect specific features of the graph with specific features of the contextual situation and investigate what happens over ranges of input values. As students learn to interpret graphs in terms of a context and use them to answer questions, they learn an important skill in mathematical modeling (MP4).
Grade 8, Episode 9: Unit 5, Lesson 5 | Illustrative Math
In this video lesson, students learn the term function for a rule that produces a single output for a given input. This lesson introduces and connects the different ways in which we represent functions in mathematics: verbal descriptions, equations, tables, and graphs. Students transition from input-output diagrams and descriptions of rules to equations. The lesson also introduces the use of independent and dependent variables in the context of functions.
Grade 8, Episode 7: Unit 5, Lessons 2 & 3 | Illustrative Math
In this video lesson, students use linear functions to model real-world situations. Students are given data for an almost linear relationship and develop a linear model. They use their model to make predictions and discuss the reasonableness of the model. Sometimes it is difficult to tell from the information given if a linear model is appropriate. When none of the given data perfectly fit by a linear function, students have to determine whether a linear approximation is reasonable and for which values it would be reasonable. Students also use piecewise linear graphs to find information about the real-life situation they represent. In situations where a quantity changes at different constant rates over different time intervals, students model the situation with a piecewise linear function.
In this program, students learn about finding values for elements in the domain of quadratic functions, defined as the group of all x values or all inputs.
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.
