K-12 Teaching and Learning From the UNC School of Education

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  • Investigating linear equations: Using a graphic calculator to compare the slope and y-intercept of lines to understand the slope-intercept form (y = mx+b) and what effect each has on a line.
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Learning outcomes

The student will learn how to write the equation of a line given its graph or its slope and y-intercept. The student will also learn the effects of changing the slope and y-intercept on the graph and the equation of the line.

Teacher planning

Time required for lesson

60 minutes

Materials/resources

  • approximately 20 ft by 20 ft of floor space (floors with square tiles work best)
  • wide masking tape (or laminated adding machine tape)
  • magic marker
  • graphing calculator
  • handout

Pre-activities

The students should know the slope-intercept form of an equation and how to graph a line given its slope and y-intercept.

Activities

  1. The teacher will select 2 students to place a 15-20 ft strip of masking tape (or laminated adding machine tape) on the floor. Then the teacher will select 2 students to place another 15-20 ft strip of tape on the floor perpendicular to the first strip. Use the tiles on the floor as graph paper.
  2. Next the teacher will ask students, one at a time, to take the magic marker and write on the tape the following:
    1. x-axis
    2. y-axis
    3. 0 at the origin
    4. scale the positive x-axis
    5. scale the negative x-axis
    6. scale the positive y-axis
    7. scale the negative y-axis
  3. The teacher will choose a group of 5 students to stand at different points on the x-axis. Be sure one student stands on the origin. The teacher instructs the students to add 3 to the x-coordinate and move vertically to the result. For example, if Adam is standing on (-4, 0), he will move to (-4, -1) and if Beth is standing on (2, 0), she will move to (2, 5). The students will be points on a line. The teacher will ask the class “What is the equation of this line?” The answer is y = x + 3. The students return to their original starting point on the x-axis and the teacher chooses different values to add to x.
  4. Next the teacher chooses another group of 5 students to stand on the x-axis and instructs them to subtract 3 from the x-coordinate. For example, Adam moves from (-4, 0) to (-4, -7) and Beth moves from (2, 0) to (2, -1). The teacher asks “What is the equation of this line?” (Answer: y = x - 3) Repeat several times, subtracting different values from x.
  5. Then the teacher chooses another group of 5 students to stand on the x-axis and instructs them to multiply the x-coordinate by 2. “What is the equation of this line?” (Answer: y = 2x) Repeat using -2; (1/2); etc.
  6. Lastly, the teacher will choose another group of 5 students to stand on the x-axis and instructs them to multiply the x-coordinate by 2 and subtract 1. “What is the equation of this line?” (Answer: y = 2x -1) Repeat several times using different combinations of positive/negative multipliers with adding/subtracting different values.

Assessment

The students will complete the following activities and questions. Students may use a graphing calculator. (Note: the problems and questions below are identical to those on the handout.)

  1. Write the equations of the lines in slope-intercept form. Then draw the graph of the lines, labeling each line with its equation. Answer the questions that follow:
    • 1. slope = 1, y-intercept = 3
    • 2. slope = 1, y-intercept = -3
    • 3. slope = 1, y-intercept = 0
    • 4. slope = 1, y-intercept = -1
    • How are these lines alike?
    • How are these lines different?
    • What was the effect of changing the y-intercept?
  2. Write the equations of the lines in slope-intercept form. Then draw the graph of the lines, labeling each line with its equation. Answer the questions that follow.
    • 1. slope = 1/2, y-intercept = 0
    • 2. slope = 1, y-intercept = 0
    • 3. slope = 3/2, y-intercept = 0
    • 4. slope = 2, y-intercept = 0
    • How are these lines alike?
    • How are these lines different?
    • What was the effect of changing the slope?
  3. Write the equations of the lines in slope-intercept form. Then draw the graph of the lines, labeling each line with its equation. Answer the questions that follow:
    • 1. slope = 2, y-intercept = 3
    • 2. slope = -2, y-intercept = 3
    • 3. slope = 1/2, y-intercept = 3
    • 4. slope = -1/2, y-intercept = 3
    • How are these lines alike?
    • How are these lines different?
    • What was the effect of changing the slope?

  • Common Core State Standards
    • Mathematics (2010)
      • Grade 8

        • Functions
          • 8.F.3Interpret the equation y = mx + b as defining a linear function, whose graph is a straight line; give examples of functions that are not linear. For example, the function A = s2 giving the area of a square as a function of its side length is not linear...
          • 8.F.4Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x, y) values, including reading these from a table or from a graph....
      • High School: Algebra

        • Creating Equations
          • ALG.CE.2Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.
      • High School: Functions

        • Interpreting Functions
          • FUN.IF.7Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases.★ Graph linear and quadratic functions and show intercepts, maxima, and minima. Graph square root,...
        • Linear, Quadratic, & Exponential Models
          • FUN.LQE.2Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

North Carolina curriculum alignment

Mathematics (2004)

Grade 8

  • Goal 5: Algebra - The learner will understand and use linear relations and functions.
    • Objective 5.01: Develop an understanding of function.
      • Translate among verbal, tabular, graphic, and algebraic representations of functions.
      • Identify relations and functions as linear or nonlinear.
      • Find, identify, and interpret the slope (rate of change) and intercepts of a linear relation.
      • Interpret and compare properties of linear functions from tables, graphs, or equations.
    • Objective 5.02: Write an equation of a linear relationship given: two points, the slope and one point on the line, or the slope and y-intercept.

Grade 9–12 — Introductory Mathematics

  • Goal 4: Algebra - The learner will understand and use linear relations and functions.
    • Objective 4.01: Develop an understanding of function.
      • Translate among verbal, tabular, graphic, and algebraic representations of functions.
      • Identify relations and functions as linear or nonlinear.
      • Find, identify, and interpret the slope (rate of change) and intercepts of a linear relation.
      • Interpret and compare properties of linear functions from tables, graphs, or equations.
    • Objective 4.02: Write an equation of a linear relationship given: two points, the slope and one point on the line, or the slope and y-intercept.