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# Graph Dynamic Linear Equations

In this lesson, students learn how dynamic linear equations in algebra work by graphing them using Microsoft Math.

## ​Objectives

• Students will learn about dynamic linear equations by graphing them.
• Students will practice graphing algebraic functions using technology.

## Learning Outcomes

• Students will understand the effects of parameter changes on the graph of y= mx + b, and use them to solve problems.
• Students will understand the conditions that make two linear graphs parallel or perpendicular, be able to use them to solve problems, and be able to explain them.

## Lesson procedure

### ​Introduction

In this activity, you will use a technology tool to graph basic algebraic equations called dynamic linear equations. First, you will investigate what happens on a linear equation graph when you change one of the parameters of the equation.

After checking that you understand how graphing dynamic linear equations works, you will use your knowledge about graphing linear equations to solve a real-life problem and demonstrate your understanding of the conditions that make two linear graphs parallel. Finally, you will write two paragraphs explaining dynamic linear equations to your fellow students.

I will guide you through each of these activities.  However, the goal is for you to find  results on your own from your investigations and practice.

### Lesson extension activities

Ask more advanced students to design the survey form in Microsoft Office Excel. For help designing a form using a Microsoft Office Excel template, click the Microsoft Office button, select New, click Installed Templates, click More Categories, and then click Surveys.

### Teacher Tips

• #### Measure the rate of change

Linear equations are used to model a constant rate of change between two variables and there are linear relationships everywhere. Think about when you came to school this morning.

• #### Mobile phones and relationships

Your mobile phone bill can be modeled with a linear relationship. A monthly base fee and a fee per minute used. Jessica pays \$25 per month in basic fees and she pays \$0.10 per minute each month. Frank pays \$10 in basic fees and \$0.25 per minute.

### ​Student activity

Follow the steps below to guide your students through this lesson plan.

Note teachers: Please download the student activity handouts located in the sidebar under Software and Materials Needed, for additional details about the main activities for this lesson plan. Students who need more time to complete Activities 1-3 can skip Activity 4.

Activity 1: Investigate dynamic linear equations by graphing them

Graph y = mx +b in Microsoft Math. Animate b from -2 to 2.

For help graphing, read How to Graph a Function in Microsoft Math (.doc, 125 KB).

1. Describe what happens to the graph when b = -2. Describe what happens to the graph when b=0. And, when b = +2.
2. Animate m from -2 to 2. Describe what happens to the graph when m = -2. Describe what happens to the graph when m=0. And when m = +2.
3. Now graph y = -3x + 5 on the same axis. Set m and b for y = mx +b so that both graphs coincide. What are m and b when the graphs become the same line?
4. Now set the controls so that the two lines are parallel. What are m and b when the lines are parallel? Is there more than one correct answer for m or b?
5. Now set the controls so that the two lines appear to be perpendicular (be sure to click on the Proportional Display button). What are m and b when the lines appear to be perpendicular? Is there more than one correct answer for m or b?

Activity 2: Check for understanding.

Answer the following questions. Graph using Microsoft Math if you need to confirm your thinking.

1. The graph of y = 2x + 1 crosses the y axis at (0,2). (True or false).
2. The graphs of y = -x – 1 and y= -x+1 are parallel. (True or false).
3. The graph of y =3x -1 is rising from left to right. (True or false).
4. The graph of y = -3x + 1 is falling from left to right. (True or false).
5. What has to be true of y = mx + b so that its graph will fall from left to right?
6. Write the equation for a line that is horizontal.
7. Where does y = -x -3 cross the y axis?

Teacher note: the closest you can get on m is .37.

8. Write a function that has a graph that is steeper than the graph of y = x + 1.
9. Write a function that has a graph that is falling from left to right and crosses the y axis at (0, -4).
10. Write a function that has a graph that is parallel to y = x - 4.
11. Write a function that has a graph that is perpendicular to y = -2x + 1.

Activity 3: Use a dynamic linear equation to solve a problem in real life.

Read the following scenario and answer the questions:

Excellence Rental Car charges \$35 per day + \$0.19 per mile for its Premier Plan. Excellence Rental Car has other rental plans that charge different amounts. To show customers the differences in the plans, the owner wanted to create a graph of her plans and run the graph as part of an advertisement in the newspaper.

She tried the following things with the Premier Plan charges on a graph.

1. First, she reduced the amount of the per day charges. What was the effect on the graph of the Premier Plan when she did this?
2. Next, she increased the cost per mile. What was the effect on the graph of the Premier Plan?
3. When she looked at the graphs of her other car rental plans, one plan turned out to be parallel to the graph of the Premier Plan.
4. What can you conclude about this other plan?

Activity 4: Explain the conditions and effects of graphing linear equations.

1. Write a paragraph explaining what happens to the graph of y = mx + b when the parameters m and b are changed. Imagine you are explaining it to a classmate.
2. Write a paragraph explaining the requirements on y = mx + b for two lines that are parallel or perpendicular.

## Conclusion

• Observe students as they work individually and in groups.
• Evaluate the work in the student handout.
• Prepare similar questions for quizzes and tests.