## Learning Math: Patterns, Functions, and Algebra

# Linear Functions and Slope

## Explore linear relationships by looking at lines and slopes. Using computer spreadsheets, examine dynamic dependence and linear relationships and learn to recognize linear relationships expressed in tables, equations, and graphs. Also, explore the role of slope and dependent and independent variables in graphs of linear relationships, and the relationship of rates to slopes and equations.

### In This Session

**Part A:** Linear Relationships in Patterns

**Part B:** Slope

**Part C:** Rates

**Part D:** Putting It Together

**Part E:** Thinking About Technology

**Homework**

In the previous session, you developed proportional reasoning skills by making absolute and relative comparisons, comparing ratios, making scale drawings, and looking at graphs of proportional relationships. In this session, we’ll explore linear relationships by looking at lines and slopes. **Note 1**

We’ll also use spreadsheets to aid us in our exploration. If you feel comfortable using a spreadsheet program, feel free to start the lesson now. If you’ve never used a spreadsheet before, or if you have but would like a review of the basics, click here for a short tutorial.

### Learning Objectives

In this session, we’ll use spreadsheets to explore dynamic dependence and linear relationships. We will:

- Recognize linear relationships as they are expressed in tables, equations, and graphs
- Understand that linearity is expressed in constant differences
- Use spreadsheets to work with tables, equations, and graphs
- Explore the role of slope and dependent and independent variables in graphs of linear relationships
- Develop an understanding of rates and how they are related to slopes and equations

### Key Terms

**Previously Introduced:**

**Closed-Form Description: **A closed-form description of a pattern tells how to get from any input to its output, without having to know any previous outputs. A rule such as “take the input, triple it, and add two” is a closed-form description of a pattern.

**Recursive Description: **A recursive description of a pattern tells you how to proceed from one step to the next. For example, a recursive description might be, “Add two to the value of the output each time the input goes up by one.” The Fibonacci sequence, where each output is the sum of the two numbers before it, is a recursive description of a pattern.

**Origin: **The origin of a coordinate system is the point (0, 0).

**New in This Session:**

**Independent Variable: **In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.

**Dependent Variable: **In a function where the value of variable A depends on the value of variable B, variable A is referred to as the dependent variable and variable B is referred to as the independent variable. In a table with inputs and outputs, the input is the independent variable and the output is the dependent variable.

**Slope: **The slope of a line is often described as the ratio of rise to run. Slope is also the amount that the dependent variable changes for each increase by one in the independent variable. The formula for slope is: slope = (change in y) / (change in x).

**Rate: **A rate describes how much one variable changes in relation to another.

**Linear Relationship: **A constant rate produces a linear relationship between two variables.

### Notes

**Note 1**

In this session we’ll explore the concepts of linear function and slope. Another goal is to examine how the use of computer spreadsheets can affect our understanding of these two ideas.

In mathematics, technology is often introduced for its own sake, with little or no consideration for how it can enhance students’ understanding of mathematical concepts. In this session, spreadsheets will allow us to observe changes in dependent and independent variables for multiple data points. We’ll also use spreadsheets to produce graphs that illustrate the connection between rate and slope.

To complete the session, you’ll need to know how to perform a few basic procedures. If you’re not already familiar with spreadsheets, you may want to read the spreadsheet tutorial in the course material.

**Groups:** Those with more computer experience can provide a quick demonstration to others, or pair up with those who have less computer experience.

**Materials Needed:** Computer with spreadsheet program for individuals working alone or for each pair or small group, graph paper, rulers (optional), toothpicks (optional).

**Review**

**Groups:** Discuss the homework, particularly Problem H3 and the differences between the “proportional” equations and the “linear” equations.