Lesson 7

Connecting Representations of Functions

7.1: Which are the Same? Which are Different?

CCSS Standards


Addressing

  • 8.F.A.2

Warm-up: 5 minutes

The purpose of this warm-up 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.

Launch

Give students 1–2 minutes of quiet work time followed by a whole-class discussion.

Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

Student Facing

Here are three different ways of representing functions. How are they alike? How are they different?

  1. $y = 2x$
  2. Embedded alt text:
  3. $p$ -2 -1 0 1 2 3
    $q$ 4 2 0 -2 -4 -6

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

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 that relationship in each representation.

7.2: Comparing Temperatures

CCSS Standards


Addressing

  • 8.F.A.2
  • 8.F.A.3

Activity: 10 minutes

This is the first of three activities where 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.

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.

Visual-Spatial Processing: Visual Aids. Provide handouts of the representations for students to draw on or highlight.

Student Facing

The graph shows the temperature between noon and midnight in City A on a certain day.

Embedded alt text:

The table shows the temperature, $T$, in degrees Fahrenheit, for $h$ hours after noon, in City B. 

$h$ 1 2 3 4 5 6
$T$ 82 78 75 62 58 59
  1. Which city was warmer at 4:00 p.m.?
  2. Which city had a bigger change in temperature between 1:00 p.m. and 5:00 p.m.?
  3. How much greater was the highest recorded temperature in City B than the highest recorded temperature in City A during this time?
  4. Compare the outputs of the functions when the input is 3.

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

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 was quicker since the numbers are given and I only have to subtract. In the graph I had to figure out the temperature values for both times before I could subtract.)

7.3: Comparing Volumes

CCSS Standards


Addressing

  • 8.F.A.2
  • 8.F.A.3

Activity: 10 minutes

This is the second of three activities where 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.

Launch

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.

If using the digital activity, students will have an interactive version of the graph that the print statement uses. Using this version, students can click on the graph to determine coordinates, which might be helpful. The focus of the discussion should remain on how and why students used the graph and equation.

Conceptual Processing: Processing Time. Review an image or video of a cube in order to activate prior knowledge of the context of the problem.

Student Facing

The volume, $V$, of a cube with side length $s$ is given by the equation $V = s^3$. The graph of the volume of a sphere as a function of its radius is shown.

  1. Is the volume of a cube with side length $s=3$ greater or less than a sphere with radius 3?

  2. Estimate the radius of a sphere that has the same volume as a cube with side length 5.

  3. Compare the outputs of the two volume functions when the inputs are 2.

Here is an applet to use if you choose. Note: If you want to graph an equation with this applet, it expects you to enter $y$ as a function of $x$, so you need to use $y$ instead of $V$ and $x$ instead of $s$. 

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

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 around the context.

Display the equation and graph for all to see. Invite groups to share how they used the representations to answer the questions. Consider asking the following questions to have students 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.”

7.4: It’s Not a Race

CCSS Standards


Addressing

  • 8.F.A.2
  • 8.F.A.3

Optional activity: 10 minutes

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, followed with whole-class discussion.

Conceptual Processing: Eliminate Barriers. Allow students to use calculators to ensure inclusive participation in the activity. Also, assist students to see the connections between new problems and prior work. Students may benefit from a review of different representations to activate prior knowledge.

Student Facing

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, $d$, in miles, every minute for 10 minutes.

$t$ 1 2 3 4 5 6 7 8 9 10
$d$ 0.9 1.9 3.0 4.1 5.1 6.2 6.8 7.4 8 9.1
  1. How many miles per minute is 55 miles per hour?
  2. Who had traveled farther after 5 minutes? After 10 minutes?
  3. How long did it take Elena’s family to travel as far as Andre’s family had traveled after 8 minutes?
  4. 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

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

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?”

7.5: Comparing Different Areas

CCSS Standards


Addressing

  • 8.F.A.2

Cool-down: 5 minutes

Launch

Student Facing

The table shows the area of a square for specific side lengths.

side length (inches) 0.5 1 2 3
area (square inches) 0.25 1 4 9

The area $A$ of a circle with radius $r$ is given by the equation $A = \pi \boldcdot r^2$.

Is the area of a square with side length 2 inches greater than or less than the area of a circle with radius 1.2 inches?

Student Response

Details about student responses to this activity are available at IM Certified Partner LearnZillion (requires a district subscription) or without a subscription here (requires registration).

Student Lesson Summary

Student Facing

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 $y = 3x +2$ where $y$ is the dependent variable and $x$ is the independent variable. If we wanted to find the output that goes with 2, we can input 2 into the equation for $x$ and finding the corresponding value of $y$. In this case, when $x$ is 2, $y$ is 8 since $3\boldcdot 2 + 2=8$.

If we had a graph of this function instead, then the coordinates of points on the graph are the input-output pairs. So we would read the $y$-coordinate of the point on the graph that corresponds to a value of 2 for $x$. Looking at the graph of this function here, we can see the point $(2,8)$ on it, so the output is 8 when the input is 2.

Embedded alt text:

A table representing this function shows the input-output pairs directly (although only for select inputs).

$x$ -1 0 1 2 3
$y$ -1 2 5 8 11

Again, the table shows that if the input is 2, the output is 8.