Linear Functions
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to reason about the values that can be assigned to graphs depending on which feature of the graph, such as slope or y-intercept, the viewer focuses on. Since there are no numbers or other labels on the graph, it is important for students to explain how they know the sign of the slope and y-intercept based on the position of the graph.
Launch
Display the three graphs for all to see. Tell students that all three graphs have the same scale. Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Student Task
Diego said that these graphs are ordered from smallest to largest. Mai said they are ordered from largest to smallest. But these are graphs, not numbers! What do you think Diego and Mai are thinking?
Sample Response
Sample reasoning:
Diego:
-
In the first graph, the line decreases when we read left to right. That means it has a negative slope. The second line stays horizontal the entire time, so it must have a slope of zero. The third graph increases as we read left to right, so it has a positive slope. That means the slopes are ordered from least to greatest.
-
Diego is looking at the right side of each graph where the line “ends up.”
Mai:
-
The y-intercept of the first graph is positive and higher than the second graph. They-intercept of the last graph is negative, so the y-intercepts are ordered from greatest to least.
-
Mai is looking at the left side of each graph where the line “starts.”
Activity Synthesis (Teacher Notes)
Display the three graphs for all to see. Invite students to share what they think Diego and Mai are thinking, and record student responses next to the graphs. Encourage students to reference the graphs in their explanation and to use precise language, like “y-intercept” and “slope.” Emphasize that even though there are no numbers shown, we can tell the sign of the slope and the sign of the y-intercept by looking at the position of the line.
Standards
- 8.F.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.F.B.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two <span class="math">\((x, y)\)</span> values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
15 min
15 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Linear Functions
5 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to reason about the values that can be assigned to graphs depending on which feature of the graph, such as slope or y-intercept, the viewer focuses on. Since there are no numbers or other labels on the graph, it is important for students to explain how they know the sign of the slope and y-intercept based on the position of the graph.
Launch
Display the three graphs for all to see. Tell students that all three graphs have the same scale. Give students 1–2 minutes of quiet work time, and follow with a whole-class discussion.
Student Task
Diego said that these graphs are ordered from smallest to largest. Mai said they are ordered from largest to smallest. But these are graphs, not numbers! What do you think Diego and Mai are thinking?
Sample Response
Sample reasoning:
Diego:
-
In the first graph, the line decreases when we read left to right. That means it has a negative slope. The second line stays horizontal the entire time, so it must have a slope of zero. The third graph increases as we read left to right, so it has a positive slope. That means the slopes are ordered from least to greatest.
-
Diego is looking at the right side of each graph where the line “ends up.”
Mai:
-
The y-intercept of the first graph is positive and higher than the second graph. They-intercept of the last graph is negative, so the y-intercepts are ordered from greatest to least.
-
Mai is looking at the left side of each graph where the line “starts.”
Activity Synthesis (Teacher Notes)
Display the three graphs for all to see. Invite students to share what they think Diego and Mai are thinking, and record student responses next to the graphs. Encourage students to reference the graphs in their explanation and to use precise language, like “y-intercept” and “slope.” Emphasize that even though there are no numbers shown, we can tell the sign of the slope and the sign of the y-intercept by looking at the position of the line.
Standards
- 8.F.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.
- 8.F.B.4·Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two <span class="math">\((x, y)\)</span> values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models, and in terms of its graph or a table of values.