This lesson focuses on qualitative aspects of graphs, so there are no units or scale on the axes. In the warm up, students analyze two different graphs that represent the same situation (based on a series of photos). Depending on which quantities are chosen as the dependent and independent variable, both graphs describe different aspects of the same story. The two functions represented have the same independent variable (time), but different dependent variables (distance from edge of lawn vs. distance from the camera).
In the following activity, students identify independent and dependent variables from contexts and select an appropriate graph to match their choices. Different choices are possible, so students must be precise about which choice they are making and explain how the choice relates to the graph (MP6). In the final activity, students create a graph from a story. In doing so, students have to make many choices about the aspects of a situation they want to represent with a mathematical object—this is an important part of modeling with mathematics (MP4). Depending on the variables chosen, graphs of the same situation can appear to be different but still tell the same story.
- Compare and contrast (orally) peers’ graphs that represent the same context.
- Comprehend that graphs representing the same context can appear different, depending on the variables chosen.
- Draw the graph of a function that represents a context, and explain (orally) which quantity is a function of which.
Let’s draw a graph from a story.
Students are asked to make displays of their work in groups of 2–3. Prepare materials for creating a visual display in this way such as markers, chart paper, board space, etc.
- I can draw the graph of a function that represents a real-world situation.
A dependent variable represents the output of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of bananas is the dependent variable because it depends on the number of apples.
An independent variable represents the input of a function.
For example, suppose we need to buy 20 pieces of fruit and decide to buy some apples and bananas. If we select the number of apples first, the equation \(b=20-a\) shows the number of bananas we can buy. The number of apples is the independent variable because we can choose any number for it.
A radius is a line segment that goes from the center to the edge of a circle. A radius can go in any direction. Every radius of the circle is the same length. We also use the word radius to mean the length of this segment.
For example, \(r\) is the radius of this circle with center \(O\).