Activity
Linear functions and problem solving 1: y-intercept and slope on graph board
Discover how to teach algebraic linear functions using tactile tools to build foundational concepts for exploring graphs with Desmos and JAWS.
Previous activities in this series introduced the graph board and the Desmos Graphing Calculator separately. Activity Set 1: Graphing Concepts used the graph board to teach key features of the coordinate plane, locating ordered pairs, plotting points, and graphing lines. Activity Set 2: Graph Board to Desmos focused on using Desmos to plot points, use audio trace, input functions, navigate tables, edit expressions and share graphs.
This third set of activities combines the graph board or pegboard with the Desmos Graphing Calculator. The first activity reviews (or introduces) the characteristics of linear functions using tactile tools, while the second and third activities use Desmos to explore linear functions and solve related problems.
It may be helpful for students to have a graph board or pegboard available throughout the process for reading or creating examples that match those graphed using the Desmos graphing calculator.
Activity 1: y-intercept and slope on the graph board
Objective
- Determine y-intercept and slope using a graph board or pegboard.
Materials needed
- APH Graphic Aid for Mathematics (graph board)
- Materials for plotting points and creating lines on the graph board
- See Graphing Concepts Activity 1 for details regarding setting up the graph board.
Lesson/activity sequence
This activity will require a line graphed on the graph board or pegboard. You may recognize the line from Graphing Concepts Activity 3. If time allows, the student can graph the line to reinforce skills, or it can be graphed in advance.
- Plot the points (5, 13) and (−5, −7) using push pins.
- Graph a line between the two points using rubber band or similar material.
Image 1: y=2×3
Note: The equation of the line and the point (0, 3) are indicated in the image, but they should not be introduced at this time. If a student points out where the line crosses the x- or y-axis, that is great!
Identifying the y-intercept
The y-intercept is the location where the graphed line crosses the y-axis. The y-intercept can be represented by the variable, b.
The y-intercept can be located in three steps.
- Step 1: Locate the Origin at (0, 0).
- Step 2: Follow the y-axis up or down.
- Step 3: Identify the point where the graphed line crosses the y-axis.
Your student should put a finger on the point (0, 3), to show the y-intercept, the point where the line crosses the y-axis. A push pin can be placed at that location.
Image 2: y-intercept
Determining slope
Slope is the ratio of change in y-coordinates to the change in x-coordinates. Slope:
- Describes a rate of change
- Describes how steep a line is
- Is positive when the line goes up from left to right
- Is negative when the line goes down from left to right
- Is represented by the expression “rise over run” where rise is the change in y-coordinates and run is the change in x-coordinates.
Slope also is steeper when its absolute value is greater and is represented by the variable m in common formulas. For the purpose of this activity, we will focus on “rise over run” to determine slope on the graph board.
To determine slope:
- Step 1: The student must identify two points on the line. The description and image below use the points (0, 3) and (5, 13), but a student can use any two points as long as she knows their coordinates.
- Step 2: Count straight up from (0, 3) until they come to the horizontal line the (5, 13) point is on. In doing this, they will count up 10 lines, for “rise.”
- Step 3: Count to the right until they come to the point (5, 13). They will count 5 lines to the right for “run.”
- Step 4: Our “rise over run” gives the ratio, 10/5. This can be reduced to 2/1, or just 2.
Image 3: Slope
The slope of this line is 2. No matter which points they used, the student determine the slope is 2. Finally, because the line slants up, the slope is positive 2.
Notes:
- Students may benefit from using a ruler or straight edge as they are getting accustomed to locating points and following lines on the graph board. This strategy also helps with reading tactile graphics.
- For example, a ruler can be placed at the horizontal line where y = 13 to help a student recognize where to stop counting up the x-axis. Gradually, the student should be able to determine the horizontal line just by using the second point.
Slope-intercept form
With two points and a determined slope, you can write the equation of the graphed line in slope-intercept form. The slope-intercept form of a linear equation is:
y = mx + b
The b-value is the y-intercept. We determined the y-intercept was at the point (0, 3). The y-value equals 3 where x equals 0. Let’s consider the math of that:
y = mx + b Original expression
3 = m(0) + b Substitute x- and y-values of the point (0, 3)
3 = b b = 3, the y-intercept
Image 4: y-intercept and equation
The m-value is the slope. We determined the slope of the line is 2, so m = 2.
Image 5: Slope and equation
The equation can be written: y = 2x + 3. We can confirm the equation using any other point on the line. For example, consider (−5, −7):
y = mx + b
−7 = 2(−5) + 3
−7 = −10 + 3
−7 = −7
True
The great thing about the slope-intercept form of a linear equation is that it gives you one point, the y-intercept, and it tells you the slope. With this information, students can graph a line by plotting the y-intercept then counting “rise over run” to get a second point.
Note: For lines with a positive slope, to determine points to the left of the y-axis, you have to reverse the “rise over run.” So, instead of going up and to the right, you go down and to the left.
Practice problems
The following are common types of problems that students may see.
Use the given points to graph a line. Determine the y-intercept and slope. Bonus for determining the x-intercept (where the line crosses the x-axis) also known as the zero.
1. (−2, −6) and (4, 6)
2. (−5, −5) and (7, 1)
Note: this problem requires students to determine a y-intercept that falls between two lines (y = −2.5).
Write an equation of a line in slope-intercept form with the given slope and y-intercept. Then graph the equations.
3. slope: 2, y-intercept: 5
4. slope: 1/2, y-intercept: −6
Graph each equation. Write the slope and y-intercept.
5. y = 3/4x − 3
6. −2x + 5y = 10
In Activity 2 of Linear Functions and Problem Solving, students will use the Desmos Graphing Calculator to explore how changes in the equation affect the slope and y-intercept of a linear function.
Resources
Activity set 1, Graphing Concepts series:
- Graphing concepts activity 1: Attributes and coordinates
- Graphing concepts activity 2: Plotting points
- Graphing concepts activity 3: Graphing a line
Activity set 2, Graph Board to Desmos series:
- Post 1: Setting up Jaws for Desmos
- Post 2: Plotting points and navigating Audio Trace
- Post 3: Representing functions in Desmos
- Input functions into the expression list
- Create tables from functions
- Navigate tables
- Editing expressions
- Sharing a graph
Activity set 3, Linear Functions and Problem Solving series (y-intercept and slope) series:
- Linear functions and problem solving 1: y-intercept and slope on a graph board
- Linear functions and problem solving 2: y-intercept and slope using Desmos
- Linear functions and problem solving 3: y-intercept and slope problem solving
Activity set 4, Undetermined points and systems of linear equations series:
- Undetermined points and systems of linear equations: What’s the Point activity 1
- Undetermined points and systems of linear equations 2: Finding the intersection
- Undetermined points and systems of linear equations 3: Solving a linear equation
Activity set 5: Inequalities and systems of inequalities series:
- Inequalities and systems of inequalities part 1: Simple inequalities
- Inequalities and systems of inequalities part 2: Linear inequalities
- Inequalities and systems of inequalities part 3: Systems of inequalities
Additional math resources by TEAM Initiative:
- TEAM Initiative web page
- Digital math: Introducing skill level checklists
- Digital math: Workflows from teacher to student and student to teacher
Pegboard post:
This algebra post was created as part of the TEAM Initiative to assist Teachers of Students with Visual Impairments (TSVIs) in teaching students digital math tools and to use tactile resources to support digital concepts. Written by John Rose. If you would like more information about the TEAM Initiative, contact Leslie Thatcher at [email protected].
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