Connecting Representations of Functions
5 min
Teacher Prep
Setup
Narrative
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.
Launch
Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Student Task
Here are three different ways of representing functions. How are they alike? How are they different?
y=2x
| p | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| q | 4 | 2 | 0 | -2 | -4 | -6 |
Sample Response
Sample responses:
- The equations for the first two would have the same form but different variables.
- The graphs for the first two are identical except for the labels on the axes.
- A table of values for both the equation and graph would have the same ordered pairs, but the variables names would be different.
- The third one has opposite outputs for the same input as the first two. The graph would be a line reflected across the y-axis as compared with the first two.
Activity Synthesis (Teacher Notes)
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.
Standards
- 8.F.2·Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <em>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.</em>
- 8.F.A.2·Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <span>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.</span>
10 min
10 min
Knowledge Components
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Connecting Representations of Functions
5 min
Teacher Prep
Setup
Narrative
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.
Launch
Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Student Task
Here are three different ways of representing functions. How are they alike? How are they different?
y=2x
| p | -2 | -1 | 0 | 1 | 2 | 3 |
|---|---|---|---|---|---|---|
| q | 4 | 2 | 0 | -2 | -4 | -6 |
Sample Response
Sample responses:
- The equations for the first two would have the same form but different variables.
- The graphs for the first two are identical except for the labels on the axes.
- A table of values for both the equation and graph would have the same ordered pairs, but the variables names would be different.
- The third one has opposite outputs for the same input as the first two. The graph would be a line reflected across the y-axis as compared with the first two.
Activity Synthesis (Teacher Notes)
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.
Standards
- 8.F.2·Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <em>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.</em>
- 8.F.A.2·Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). <span>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.</span>