LessonTitle: Zooming in on Linear Equations Alg 5.2
UtahState Core Algebra Content Standard 2 Process Standards 1-5
Summary
In this lesson, students use graphing calculators or computer software. As they program in and then analyze patterns of equations, they develop understanding of the slope intercept form of a linear equation. They apply this understanding to write equations to match various graphs.
Enduring Understanding
The slope intercept form of a linear equation, y = mx + b, reveals the rate of change occurring in a mathematical story and an orientation point (the y intercept) on the graph of the story. / Essential Questions
What story does the slope intercept equation form
(y = mx + b) tell? How do the equation and the graph stories compare?
Skill Focus
  • The slope intercept form of a linear equation
/ Vocabulary Focus
slope, parallel, perpendicular, origin, calculator vocabulary such as max, min, windows, trace
Materials Graphing Calculators or Computers, LCD projector, graph paper.
Launch ideas:
“Since we have not used the calculators to graph linear equations, the introduction to using the calculators for graphing will peak the students interest enough for a launch. We will also discuss the essential questions”
Explore ideas:
“Teachers will do a little bit of direct instruction on functions of the calculators then the students will use that knowledge to do the activity in small groups. The students are responsible to learn calculator functions, apply prior knowledge of linear equations, participate and work cooperatively in their small groups.”
Summarize ideas:
“Groups will randomly be selected to share answers.”
“The worksheet ‘Reasoning about Linear Equations’ (5.2b) is a great summary. Some teachers had the students work in groups and some teachers made transparencies and had the students work in front of the class and explain their answers. This worksheet clearly shows the difficulty most students have in analyzing. There needs to be no-slope equations added on the worksheet and more equations to fill the worksheet.”
Application:
“Lesson 5.2b could be done as independent practice, but we are completing it in small groups. Since some of us are using this activity as an introduction to slope-intercept those classes will do a lot of guided practice. The others are doing this lesson as a follow-up to ensure the student have a complete understanding. They will include practice if the students do not understand slope-intercept.”
Assess
  • Students write a ‘Show All You Know’ assessment about y = mx + b equations. They should be sure to answer all the essential questions and include numeric data, equations, graphs and stories. Evaluate using a teacher/class created rubric.

Information

Activity 1 in the TI Interactive book covers the same concepts. Exploring with Geometer’s Sketchpad activities, “The Slope of a Line” and “Equations of Lines” could also be used.

Zooming in on a Linear Equation utilizes TI graphing calculators. If you choose to use TI interactive instead, then

  • Open a TI Interactive document and record students’ names. Then open a graph.
  • Have students enter in several equations with positive slopes and observe what happens. Label the graph “Steepness of Slopes” and label the x axis “run” and the y axis “rise.” Save to document.
  • Create a second graph and enter in equations with negative slopes. Label the graph “Negative Slopes.” Save to document.
  • Create a third graph. Enter in several equations with the same slope and different y intercepts. Label the graph “Parallel Slopes.” Save to document.
  • Ask students to write a conclusion about what they observed about “y = mx + b.” Explain that this is one form of an equation and that the m stands for the slope and the b stands for the y intercept.

Students will need help making up stories to fit equations. Suggestions: 1) Their story could be one about walking. The slope could be feet per second. The y intercept could be head starts or penalties. 2) They could use rate of pay and use a bonus pay or a parking fee for the positive or negative y intercepts. 3) They could use the mountain hiking idea. Give suggestions and let them try to come up with other creative and fun contexts.

Alg 5.2a Zooming in on a Linear Equation (y = mx + b)

Name______

Practice Graphing in Your Calculator

  1. To Graph: Hit the Y = key on the calculator. Type in the following equation: y = 3x + 2. The x key is near the alpha key: X/T. Then hit the graph button. If you cannot see a line hit zoom 6.
  2. To Find Points on the Line: Hit Zoom 8. Then hit Enter. Then hit the Trace button. At the bottom of the screen you will see x and y coordinates on the graph. Move the right and left arrows to find different points along the line.
  1. Put the equation y = 3x + 2 on the graph below using the points displayed when you trace the line. Make sure you use a ruler to connect the points and try to be as accurate as you can.
  1. Using the graphing calculator, graph the following equations. Use the trace button to help you graph the lines on the graph below. Be sure to label each line appropriately.

Line a: y = –2x + 1

Line b: y = 3x + 1

Line c: y = x + 1

  1. What is the same about each graph? ______
  1. What is different about each graph? ______
  1. Find the slope of each line from question 4 by counting the rise over the run on the graph.
  1. slope of Line a: y = –2x + 1 ______
  2. slope of Line b: y = 3x + 1 ______
  3. slope of Line c: y = x + 1 ______
  1. Compare each equation to its slope, what do you notice?
  1. Find the y-intercept of each line from question 4.
  1. y-intercept of Line a: ______
  2. y-intercept of Line b: ______
  3. y-intercept of Line c: ______
  1. Compare each equation to its y-intercept, what do you notice?
  1. Using the graphing calculator, graph the following equations. Use the trace button to help you graph the lines on the graph below. Be sure to label each line appropriately.

Line a: y = 2x + 3

Line b: y = 2x – 4

Line c: y = 2x

  1. What is the same about each graph? ______
  1. What is different about each graph? ______
  1. Find the slope of each line from question 11 by counting the rise over the run on the graph.
  1. slope of Line a: y = 2x + 3 ______
  2. slope of Line b: y = 2x – 4 ______
  3. slope of Line c: y = 2x ______
  1. Compare each equation to its slope, what do you notice?
  1. Find the y-intercept of each line from question 11.
  1. y-intercept of Line a: ______
  2. y-intercept of Line b: ______
  3. y-intercept of Line c: ______
  1. Compare each equation to its y-intercept, what do you notice?
  1. If an equation is in function form (solved for y), what does the number in front of the x tell you?
  1. If an equation is in function form (solved for y), what does the constant (term without a variable – the number) tell you?
  1. Without graphing the following equations, predict their slope (m) and y-intercept (b).
  1. y = 4x + 6m = ______b = ______
  2. y = 2x – 3m = ______b = ______
  3. y = x + 2m = ______b = ______
  4. y = –6x – 8 m = ______b = ______
  5. y = –5xm = ______b = ______
  6. y = x – 10m = ______b = ______
  7. y = –x + 4m = ______b = ______
  1. Create some data points for the equation y = 2x. Graph the points and make the graph line. Label it line G.

x y

-2

0

2

4

  1. Move line G up 3. Label the new line H.
  1. Move line G down 2. Label it J.
  1. What is the equation of line H?
  1. What is the equation of line J?
  1. Move line H up 2. Label it K. What is the equation of line K?
  1. What is the relationship of lines G, H, J and K?
  1. Explain what we know about equations where the slope is the same and the y intercept is different (m is the same and b is different).
  1. I graphed the following equation in my calculator: y = 2x + 4. Write a new equation that would show the following changes:
  2. I want the slope to stay the same but I want the line to be shifted up 2 units.
  1. I want the line to have the same steepness but to be pointed in the opposite direction.
  1. I want the slope to be the same but I want the line to be shifted down 4 units.
  1. Describe how the graphs of the following equations would be the same and how they would be different.
  2. y = 4x – 7 and y = 4x + 9
  1. y = 2x and y = 3x
  1. Using the slope-intercept form of an equation: y = mx + b, tell what would happen to the graph if each of the following changes were made:
  2. b is increased
  1. b is decreased
  1. m is increased
  1. m is decreased

Alg 5.2b Revisiting Slope

Write the equation of the line. Then record the points used to find the slope.

1) Equation: y = _3x______2) Equation: y = ______

Point used for rise/run _(2,6)___(0,0)_Point used for rise/run ______

3) Equation: y = ______4) Equation: y = ______

Point used for rise/run ______Point used for rise/run ______

5) Equation: y = ______6) Equation: y = ______

Point used for rise/run ______Point used for rise/run ______

7) Equation: y = ______8) Equation: y = ______

Point used for rise/run ______Point used for rise/run ______

Why is this slope undefined?

Another way to think about rise/run (slope) is to use this formula:

Change in y/ Change in x or ( y1 - y2)/(x1 - x2)

Use this formula to redo the slopes from the above graphs. Use the rise/run points and show all work.

  1. 2.
  1. 4.
  1. 6.

7. 8.