The purpose of this activity is for students to identify connections between three different representations of functions: equation, graph, and table. Two of the functions displayed are the same but with different variable names. It is important for students to focus on comparing input-output pairs when deciding how two functions are the same or different.
Student Lesson in Spanish
Launch
Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Activity
None
Student Task Statement
Here are three different ways of representing functions. How are they alike? How are they different?
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Student Response
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Building on Student Thinking
Activity Synthesis
Ask students to share ways the representations are alike and different. Record and display the responses for all to see. To help students clarify their thinking, ask students to reference the equation, graph, or table when appropriate. If the relationship between the inputs and outputs in each representation does not arise, ask students what they notice about this relationship in each representation.
6.2
Activity
10 mins
Comparing Temperatures
Standards Alignment
Building On
Addressing
8.F.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
This is the first of three activities in which students make connections between different functions represented in different ways. In this activity, students are given a graph and a table of temperatures from two different cities and are asked to make sense of the representations in order to answer questions about the context (MP2).
Launch
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and then time to share responses with their partner. Follow with a whole-class discussion.
Representation: Internalize Comprehension. Demonstrate and encourage students to use color coding and annotations to highlight connections between representations in a problem. For example, ask students to use the same color to highlight the temperatures for each city that occurred at the same time, such as blue for 4:00 p.m. Supports accessibility for: Visual-spatial processing
MLR5 Co-Craft Questions. Keep books or devices closed. Display only the graph and the table, without revealing the questions. Give students 2–3 minutes to write a list of mathematical questions that could be asked about this situation, before comparing their questions with a partner. Invite each group to contribute one written question to a whole-class display. Ask the class to make comparisons among the shared questions and their own. Reveal the intended questions for this task, and invite additional connections. Advances: Reading, Writing
Activity
None
Student Task Statement
The graph shows the temperature between noon and midnight in City A on a certain day.
Graph in a coordinate plane, horizontal, time in hours after noon, 0 to 12 by ones, vertical, temperature in degrees Farenheit, 50 to 60 by ones. The graph starts at 0 comma 50 and climbs slowly, the more quickly as it moves right, going through 2 comma 52 and 4 comma 57 before peaking at 5 point 8 comma 59. The graph then steadily declines as it moves right until it reaches the point 12 comma 52 point 5.
The table shows the temperature, , in degrees Fahrenheit for hours after noon in City B.
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Which city was warmer at 4:00 p.m.?
Which city had a bigger change in temperature between 1:00 p.m. and 5:00 p.m.?
How much greater was the highest recorded temperature in City B than the highest recorded temperature in City A on this day?
Compare the outputs of the functions when the input is 3.
Student Response
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Building on Student Thinking
Activity Synthesis
Display the graph and table for all to see. Select groups to share how they used the two different representations to get their answers for each question. To further student thinking about the advantages and disadvantages of each representation, ask:
“Which representation do you think is better for identifying the highest recorded temperature in a city?” (The graph, since I just have to find the highest part. In the table I have to read all the values in order to find the highest temperature.)
“Which representation do you think is quicker for figuring out the change in temperature between 1:00 p.m. and 5:00 p.m.?” (The table is quicker since the numbers are given, and I only have to subtract. In the graph, I have to figure out the temperature values for both times before I can subtract.)
6.3
Activity
10 mins
Comparing Volumes
Standards Alignment
Building On
Addressing
8.F.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
This is the second of three activities in which students make connections between different functions represented in different ways. In this activity, students are given an equation and a graph of the volumes of two different objects. Students then compare inputs and outputs of both functions and what those values mean in the context of the shapes (MP2).
In the digital version of the activity, students use an applet to compare the two functions. The applet allows students to have an interactive version of the graph to identify coordinates. Use the digital version if it would benefit students identifying points more precisely.
This activity uses the Collect and Display math language routine to advance conversing and reading as students clarify, build on, or make connections to mathematical language.
Activity Synthesis
The purpose of this discussion is for students to think about how they used the information from the different representations to answer questions about the context.
Display the equation and graph for all to see. Direct students’ attention to the reference created using Collect and Display. Ask students to share how they used the representations to answer the questions. Invite students to borrow language from the display as needed. As they respond, update the reference to include additional phrases. Consider asking the following questions to prompt students to expand on their answers:
“How did you use the given representations to find an answer? How did you use the equation? The graph?”
“For which problems was it nicer to use the equation? The graph? Explain your reasoning.”
6.4
Activity
Optional
10 mins
It’s Not a Race
Standards Alignment
Building On
Addressing
8.F.2
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s² giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9), which are not on a straight line.
In this activity, students continue their work comparing properties of functions represented in different ways. Students are given a verbal description and a table to compare and decide whose family traveled farther over the same time intervals. The purpose of this activity is for students to continue building their skill interpreting and comparing functions.
Launch
Give students 3–5 minutes of quiet work time, and follow with a whole-class discussion.
Activity
None
Student Task Statement
Elena’s family is driving on the freeway at 55 miles per hour.
Andre’s family is driving on the same freeway, but not at a constant speed. The table shows how far Andre's family has traveled in miles, , every minute for 10 minutes.
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How many miles per minute is 55 miles per hour?
Who has traveled farther after 5 minutes? After 10 minutes?
How long did it take Elena’s family to travel as far as Andre’s family had traveled after 8 minutes?
For both families, the distance in miles is a function of time in minutes. Compare the outputs of these functions when the input is 3.
Student Response
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Building on Student Thinking
If students do not notice that Elena’s family’s speed has different units than Andre's family, ask
“What do you notice so far about the information you have for the two families?”
“What needs to be true to compare two rates?”
Activity Synthesis
The purpose of this discussion is for students to think about how they use a verbal description and table to answer questions related to the context. Ask students to share their solutions and how they used the equation and graph. Consider asking some of the following questions:
“How did you use the table to get information? How did you use the verbal description?”
“What did you prefer about using the description to solve the problem? What did you prefer about using the table to solve the problem?”
Lesson Synthesis
Conclude this lesson by inviting students to summarize some of the strengths and weaknesses of the representations students worked with during the lesson.
Arrange students in groups of 2–3. Give students 3–4 minutes to prepare to answer these question, calling on 1–3 groups for each question:
“What is something you did today with all the different representations?” (We used each representation to find outputs for different inputs.)
“What was your favorite representation? Why?” (I like equations because if I have an equation, I can make a graph or a table.)
“What are some strengths and weaknesses of the different representations?” (Graphs require estimation but easily let us identify important features such as highest point or steepest section. Tables immediately let us find output values but only for limited input values. Equations let us precisely compute outputs for all inputs but only one at a time.)
Student Lesson Summary
Functions are all about getting outputs from inputs. For each way of representing a function—equation, graph, table, or verbal description—we can determine the output for a given input.
Let’s say we have a function represented by the equation , where is the dependent variable and is the independent variable. If we wanted to find the output that goes with 2, we could input 2 into the equation for and find the corresponding value of . In this case, when is 2, is 8 since .
If we had a graph of this function instead, then the coordinates of points on the graph would be the input-output pairs.
So we would read the -coordinate of the point on the graph that corresponds to a value of 2 for . Looking at the following graph of a function, we can see the point on it, so the output is 8 when the input is 2.
The graph of a line in the coordinate plane with the origin labeled “O”. The horizontal axis has the numbers negative 1 through 2 indicated and there are vertical gridlines between each integer. The vertical axis has the numbers negative 2 through 8, in increments of 2, indicated, and there are horizontal grid lines in between each integer. The line begins to the right of the y axis and below the x axis. It slants upward and to the right passing through the point with coordinates negative 1 comma negative 1, crosses the y axis at 2, and passes through the indicated point labeled 2 comma 8.
A table representing this function shows the input-output pairs directly (although only for select inputs).
Again, the table shows that if the input is 2, the output is 8.
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Help us improve by sharing suggestions or reporting issues.
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.
Arrange students in groups of 2. Give students 3–5 minutes of quiet work time and then time to share their responses with their partner. Follow with a whole-class discussion.
As students work, use Collect and Display to create a shared reference that captures students’ developing mathematical language. Collect the language that students use to describe the connections across the two different representations. Display words and phrases, such as “If , then the volume of the cube is 27 cm3 since the equation is ,” “Going up from on the axis, the graph is at a vertical height of about 110 cm3,” “If the input is 2,then on the point on the graph where , I can trace to the left until I reach the vertical axis, where the value of is about 35 cm3,” and “If the input is 2, then in the equation, is equal to 2, so .”
Representation: Develop Language and Symbols. Use virtual or concrete manipulatives to connect symbols to concrete objects or values. For example, show or provide students with a cube and a sphere. Discuss the relationship between the side length or radius and the volume of the object. Supports accessibility for: Visual-Spatial Processing, Conceptual Processing
Activity
None
Student Task Statement
The volume, , of a cube with edge length cm is given by the equation .
The volume of a sphere is a function of its radius (in cm), and the graph of this relationship is shown here.
Is the volume of a cube with edge length greater or less than the volume of a sphere with radius 3 cm?
If a sphere has the same volume as a cube with edge length 5 cm, estimate the radius of the sphere.
Compare the outputs of the two volume functions when the inputs are 2.
Student Response
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Building on Student Thinking
Some students may struggle with the many parts of the second question. These two questions can help scaffold the question for students who need it:
“What information is given to you, and what can you do with it?”
“What information is the focus of the question, and what would you like to know to be able to answer that question.”
