In this lesson, students will determine whether a set of data has a functional relationship and represent this relationship as both a formula and a graph.
Preinstructional Planning
Objectives
Students will:
- Define a function as a rule that assigns exactly one output to each input
- Indicate that functions can be represented numerically with tables, algebraically with formulas, graphically with graphs made up of “(input, output)” ordered pairs, or through verbal description
- Plot ordered pairs when given the coordinates
- Define the terms dependent variable, independent variable, linear function, and y-intercept
Materials
- A Well-Functioning Research Mission: Representing Functions printable
- Answer Key: Think FUNCTIONally printable
- Standards Chart: Think FUNCTIONally printable
- Graph paper
- Whiteboard or large graph paper and markers
During Instruction
Set Up
1. Make a class set of the A Well-Functioning Research Mission: Representing Functions printable.
2. Print a copy of the Answer Key: Think FUNCTIONally printable for your use.
Lesson Directions
Introduction to New Material
Step 1: Tell the class a story involving a real-world, linear functional relationship. For example: An ice cream stand is selling large cones for $1.50 each. Ask, “If I bought 1 cone, how much would I pay?” Write 1 large cone = $1.50 on the board. Repeat for 2, 3, and 4 cones. Also ask, “What would I pay if I bought no cones?” Write 0 cones = $0 on the board.
Step 2: Ask the class if there’s an efficient way to organize this data, and suggest a table if no one from the class suggests it. Draw the following table on the board, (pointing out that it could also be presented horizontally):
Step 3: Ask the class if they could determine what the cost would be if you gave them a different number of cones you wanted to purchase, say 10 cones ($15.00). Ask how the answer was determined (10 cones at $1.50 each equals $15.00). Point out that each purchase of a different number of cones would result in a different cost, and that makes it a function relationship. Tell the class that a function relationship is one where a rule assigns exactly one output for each input. In this case, the rule is that each large cone costs $1.50, so if the input is 3 cones, the output is $4.50. If the stand charged both $1.50 and another value for one large cone, we would no longer have a function. It’s one and only one output for each input.
Step 4: Ask if this relationship could be presented as a formula (C = $1.50N, where C is the cost in dollars and N is the number of cones purchased).
Step 5: Also ask if this relationship could be presented as a graph. Assuming the class is familiar with ordered pairs, point out that the inputs and outputs (number of cones, cost) information on the table could be graphed on a coordinate grid. Using the table, graph the points as follows:
Step 6: Point out that the graph depicts a functional relationship because there is only one y value for each x value. Show the class the “vertical line test” by demonstrating that, when drawing a vertical line from each point on the horizontal axis, the line cones/cost line is only crossed once, a sign that this is a function.
This is a quick way to determine that there is only one output for each input. If the vertical line hit the line twice, that would be proof that there isn’t a functional relationship because there would be more than one output for an input.
Step 7: Point out that the graph shows a relationship between the number of cones purchased and the total cost, and that cost depends on the number of cones purchased. Tell the class that the number of cones is the independent variable in this case and that the label for the independent variable is on the x-axis while the label for the dependent variable is placed on the y-axis.
Mention that the line is straight and that it increases at a constant rate because the cost increases at a constant rate ($1.50 per cone), making it a linear function. While some functions are represented by curves, they are not linear functions because they don’t change at a constant rate.
Step 8: Make the point that, although for this particular function, the dependent variable is 0 when the independent variable is 0, this is not necessarily the case for all functions.
For example, if the ice cream stand started featuring live music and charged an admission fee of $5, if a person paid the admission fee but didn’t buy any ice cream, the cost would be $5.00. This would be represented by (0, 5) on the grid. Indicate that (0, 5) would be the initial value for the function and the line would hit the y-axis at that point, making it the y-intercept. In this function, the cost of 1 cone would be $6.50, 2 cones would be $8.00, and so on. The formula would be C = $1.50N + $5.00.
Step 9: Conclude by reiterating that there are four ways to represent a function:
- Verbal Description/Words
- Numerically (in table format)
- Algebraically (as a formula)
- Graphically
Guided Practice
Step 10: Group students in pairs, distribute graph paper, and ask them to solve the following problem:
Pencils come 12 to a pack.
a. Is there a functional relationship between the number of packs and the number of pencils? Explain your thinking. Answer: Yes, there is a unique output (the number of pencils) for each input (the number of packs).
b. Create a table that represents this relationship. Answer: See table below.
c. Develop a formula that represents this relationship. Answer: P = 12N, where P equals the number of pencils and N equals the number of packs.
d. Prepare a graph that represents this relationship. Be sure to first determine which variable is independent and which is dependent. Answer: See graph below.
Step 11: Checking for Understanding: Review answers as a class and respond to any questions.
Independent Practice
Step 12: Assign the A Well-Functioning Research Mission: Representing Functions printable for classwork or homework.
Step 13: Checking for Understanding: Review the answers to the A Well-Functioning Research Mission: Representing Functions printable, which are provided on page 1 of the Answer Key: Think FUNCTIONally printable. Make sure students explain their mathematical thinking. Address any misconceptions that may arise.
Post Instructional
Standards
- Grade 8: Functions and Representations of Functions (CCSS: 8.F.1, 8.F.2)
- Grades 6–8: Making Sense of Problems, Reasoning, Modeling, Using Appropriate Tools Strategically, and Attending to Precision (CCSS Practice MP1, MP2, and MP4–6); NCTM Algebra
For more information, download the comprehensive Standards Chart: Think FUNCTIONally printable.