In this lesson, students compare two functions represented in different ways (graph and table, graph and equation, and table and verbal description). In each case, students use the different representations to find outputs for different inputs. Even though they use different representations, students are looking for the same information about the contexts and need to interpret each representation appropriately.
In a graph, students identify the input on the horizontal axis, then find the corresponding coordinate point on the graph, which lets them read the associated output. In a table, they find the input value in the first row (or column) and read the output value in the second. For functions given by equations, students substitute the input value into the expression on the right side of the equation and compute the corresponding output value on the left. Students also look for inputs corresponding to a given output by trying to reverse these procedures.
Each representation gives us the ability to find input-output pairs. However, each representation has strengths and weaknesses. 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. Comparing the different strengths of these representations helps students make decisions about how to use these tools strategically in the future.
Note that this lesson specifically avoids comparisons of linear functions to other linear functions, in order to avoid students associating “function” with only linear relationships. In a later lesson, students revisit some of these ideas and compare linear functions.
- Compare and contrast (orally) representations of functions, and describe (orally) the strengths and weaknesses of each type of representation.
- Interpret multiple representations of functions, including graphs, tables, and equations, and explain (orally) how to find information in each type of representation.
Let’s connect tables, equations, graphs, and stories of functions.
- I can compare inputs and outputs of functions that are represented in different ways.
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 \(O\).
Volume is the number of cubic units that fill a three-dimensional region, without any gaps or overlaps.
For example, the volume of this rectangular prism is 60 units3, because it is composed of 3 layers that are each 20 units3.