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Lessons

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    (Also available in WeScheme)

    Students learn to distinguish functions from other relations.

    Lesson Goals

    Students will be able to:

    • Generate points for a graph from a rule.

    • Test whether or not a graph represents a function using the vertical line test.

    • Identify whether or not a table represents a function.

    Student-Facing Lesson Goals

    • Let’s identify which graphs are functions and which aren’t using the vertical line test.

    • Let’s identify whether or not a table represents a function.

    • Let’s learn how to generate points for a graph from a rule.

    Materials

    Preparation

    • Decide whether you want to have students interact directly with the Interactive Function Activity on their computers or whether it would be easier for your situation to project each graph from the peardeck on the white board and have students come up to add points that satisfy the rules or dictate points that satisfy the rules to a scribe who will add them to the graph.

    • Rulers or straight edges might be useful in supporting students during the section of the lesson that focuses on the vertical line test.
    Glossary
    dependent variable

    When modeling a relationship between an input and an output (e.g. - distance over time), we are curious about how a change in the input (typically graphed on the x-axis) impacts the output (y). When the output is entirely dependent on the input, we refer to the output as the "dependent variable".

    domain

    the type or set of inputs a function expects, i.e., the independent variable(s) that govern the output of the function

    function

    a relation from a set of inputs to a set of possible outputs, where each input is related to exactly one output

    graph

    a diagram (such as a series of one or more points, lines, line segments, curves, or areas) that describes how one variable changes as the value of another (independent) variable changes.

    independent variable

    When modeling a relationship between an input and an output (e.g. - distance over time), we are curious about how a change in the input (typically graphed on the x-axis) impacts the output (y). When the output is entirely dependent on the input, we refer to the input as the "independent variable".

    range

    the type or set of outputs that a function produces, i.e., the dependent variable(s)

    vertical line test

    a strategy for determining whether or not a graph represents a function by confirming that no vertical line can be drawn such that it would pass through 2 points on the plot. In a function, each x-value may only correspond to a single y-value.

    🔗Functions Are Reliable 10 minutes

    Overview

    Students work collaboratively to generate visual representations of functions, using rules to make coordinate pairs.

    Launch

    Functions are a way of describing a certain kind of relationship.

    Functions are reliable. If we give them what they need, they produce the same thing with that same collection of inputs every time.

    You are already familiar with lots of relationships. Time worked is related to money earned. The speed of a car is related to the gas it consumes per miles driven. Not all relationships are functions. What kind of relationships are functions, exactly? That’s what we’re going to find out…​

    Investigate

    Just as we can give the rectangle function inputs and it will produce an image, we can make graphs of functions by feeding inputs (x-values) into a rule that produces outputs (y-values). The resulting x-y pairs describe the location of points that will satisfy the rule. We can repeat the process with an infinite number of inputs, plotting the input-output pairs as a line or curve on a graph showing the relationship between the independent variable (x) and dependent variable (y).

    • On each slide of the Interactive Function Activity (Google), we’ll see a coordinate grid and a rule.

    • Each of you will choose an x-value that is within the domain of the graph, apply the rule to your x-value to get a y-value, and then place a dot on the graph whose coordinates are the 𝑥 you put in and the 𝑦 you got back.

    • When all of our dots appear on the screen together, we’ll end up with a visual representation of the function!

    You might just want to do a few of these slides, or you might do lots of them.

    Limitations of the Vertical Line Test

    Use of the vertical line test is contingent upon restricting ourselves to functions where the domain is composed of x-values and the range is composed of y-values. Once students are exploring questions about whether ‘x is a function of y’ (an important question for Algebra 2 students thinking about inverse functions), they will need to make use of other strategies.

    We recommend urging students to pay special attention to the domain and range of each function, noting that the domain is not necessarily the horizontal axis.

    Synthesize

    • How can we make a graph of a function from its rule?

    • Are there curves or lines that a function could not make? Why or why not?

    🔗Identifying Functions from Graphs

    Overview

    Students learn to use the vertical line test to determine whether or not a graph is a function. Students have already seen all kinds of functions! The subset of functions that are Number->Number can be graphed in 2 dimensions, with the Domain along the x-axis and the Range along the y-axis. The Vertical Line Test is a special trick that only works for this subset of functions.

    Launch

    We can write functions that take in an x-value and produce a y-value. Different inputs can yield the same outputs. But each input can only be associated with one output.

    We can test a graph to see whether or not it’s a function using the "vertical line test". Imagine overlaying a series of vertical lines on the graph. If the graph represents a function, none of the vertical lines will cross the graph at more than one point. If there is any vertical line that can be drawn that would intersect more than one point on the graph, it means the same input can have more than one output and the relationship is not a function.

    Passes the Vertical Line Test -> Is a Function

    Fails the Vertical Line Test -> Is Not a Function

    a v opening to the right with a vertical line that intersects the v twice

    Investigate

    As students work, circulate around the room and make sure that they are actually drawing vertical lines on the graphs. Some students may benefit from circling the point where each vertical line intersects the graph.

    Synthesize

    Confirm that students have correctly identified which graphs represent functions.

    • What did you Notice?

      • Functions could be lines, curves, v-shaped or scatterplots! Answers will vary.

    • What did you Wonder?

      • Why might some scatterplots represent functions and others not? Are there other forms that functions can take? How do you end up with a circle on a graph? Answers will vary.

    🔗Identifying Functions from Tables

    Overview

    Students apply their understanding of how to use the vertical line test on graphs to learn to recognize whether or not tables are functions.

    Launch

    Turn to How Tables Fail the Vertical Line Test and follow the directions.

    Circulate around the room verifying that students are remembering how to use the vertical line test and correctly identifying which tables represent functions.

    • How can we identify whether or not a table of values represents a function?

      • If a table has more than one y-value (or output) for the same x-value (or input), it cannot represent a function.

    Investigate

    As students work, circulate around the room and make sure that they are actually circling or highlighting the points on the tables that tell them that the table doesn’t represent a function.

    Synthesize

    Confirm that students have correctly identified which graphs represent functions, and then lead a discussion on the activities above.

    • What did you Notice?

      • Answers will vary. It can still be a function if y-values repeat. It didn’t matter whether or not the x-values followed a pattern. It was easier for me to read the tables when the x-values were in order.

    • What did you Wonder?

      • Answers will vary. Why weren’t the x-values always in order? If the points were on a graph, would they be connected? Can there ever be decimal values for x and y? What would these tables look like on a graph?

    These materials were developed partly through support of the National Science Foundation, (awards 1042210, 1535276, 1648684, 1738598, 2031479, and 1501927). CCbadge Bootstrap by the Bootstrap Community is licensed under a Creative Commons 4.0 Unported License. This license does not grant permission to run training or professional development. Offering training or professional development with materials substantially derived from Bootstrap must be approved in writing by a Bootstrap Director. Permissions beyond the scope of this license, such as to run training, may be available by contacting contact@BootstrapWorld.org.