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.

A lesson plan for grades 8–12 Mathematics

By Misty Jarman

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Related topics

• Learn more about algebra, graphing, graphing calculators, linear equations, mathematics, and slope.

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Learning outcomes

Students will compare slopes and y-intercepts of lines using a graphing calculator to get an understanding of the effect the slope and y-intercept have on a line.

Teacher planning

Time required for lesson

1 hour

Materials/resources

• pencil
• graph paper
• handout

Technology resources

TI-81 graphing calculator

Pre-activities

• Students should know how to clear the memory of graphing calculators so there will not be any confusing data.
• Students should be familiar with the basic keys on the calculator and know how to enter a linear equation.
• Students should be familiar with the concept of a line’s slope — its “angle” or “steepness.”

Activities

Part 1: Slope of the line

1. Have students graph the line y=x on their graphing calculator (This is the line which all others are variations of. Line cuts through the middle of quadrant I and III)
   • Have students identify the following information:
     • What is the slope of the line? (1)
     • Where does the line cross the y-axis? (origin (0))
     • Which direction does the top of the line aim? (right)
2. Have students keep the equation y=x as Y1 and enter the equation y=4x as Y2
   • Have students answer the same questions as above
   • slope = 4
   • crosses y-axis at origin (0)
   • line aims to the right
3. Have students enter y=(1/4)x as Y3 (**make sure they use parentheses**)
   • Answer questions in part 1
   • slope = 1/4
   • crosses y-axis at origin (0)
   • line aims right
4. Ask these questions and draw conclusions:
   • What does the coefficient of x(slope) in the equation do to the line? (changes the angle of the line- if m>1 line will be very steep (above y=x), if m<1 line will be less steep (below y=x))
5. Have students graph the following on their calculator:
   • Y1 as y=-x
   • Y2 as y=-4x
   • Y3 as y=(-1/4)x
6. Ask students these questions:
   • To which direction do all these lines aim? (left)
   • What part of the equation makes them aim left? (-)
   • Is the steepness of y=4x and y=-4x the same? (yes, they just aim in different directions. y=4x aims to the right and y=-4x aims to the left)

***Students should now understand that the coefficient of the x term in slope-intercept form (y=mx+b) is the slope of the line and it tells the direction the top of the line will aim as well as giving an idea of the steepness of the line.

Part 2: Y-Intercept of the line

1. Have students clear all equations in their calculator to begin part 2
2. Have students Graph the line y=x for Y1 on the calculator, y=x+5 as Y2, and y=x-3 as Y3.
3. Have students identify the following:
   • Where does the line y=x cross the y-axis? (origin (0))
   • Where does the line y=x+5 cross the y-axis? (5)
   • Where does the line y=x-3 cross the y-axis? (-3)
4. Have students identify from the equation where 0 and 5 appear. (1) y=x+0,(2) y=x+5, and (3) y=x-3 *This constant is the Y-intercept (point where the line crosses the y-axis)

Part 3: Testing their linear ability

1. Have students look at these equations and tell you what the graph will look like before they verify it on the calculator.
   • y=2x-6 (slope- line will be steeper than y=x, aim right and y-intercept- line will cross y-axis at -6)
   • y=(-1/2)x+2 (line will be less steep compared to y=x and will aim to the left and it will cross the y-axis at 2)

*You may continue to add practice as you see fit*

Assessment

Have each student work together with a partner and go through the handout provided.

Supplemental information

Comments

This lesson is very helpful to me with my Algebra 1/1B students. After this lesson, they seem to have a better understanding of what the slope of a line actually is and what it does to the line.