Activity
Linear functions and problem solving 2: y-intercept and slope using Desmos
Explore slope, y-intercept and x-intercept—perfect for algebra students using JAWS, Desmos and tactile tools to deepen their understanding of linear relationships.
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—Linear Functions and Problem Solving—builds on previous lessons by integrating the graph board or pegboard with the Desmos Graphing Calculator. The first activity reviews or introduces key characteristics of linear functions using tactile tools, while this post focuses on using Desmos to explore the concepts of slope and y-intercept.
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 2: Determining the y-intercept and slope using the Desmos Graphing Calculator
In this activity, students will follow the same general process as in Activity 1: y-intercept and slope on the graph board to plot points, graph a line and determine slope and y-intercept. This activity will also give students opportunities to practice skills from previous activity sets.
Objectives
- Graph a line in the Desmos graphing calculator given two points.
- Determine y-intercept and slope using regression parameters in Desmos.
Materials needed
- PC laptop or desktop with JAWS
- Internet browser
- The graph board from Activity 1 may be helpful for students to use as a comparison
Lesson/activity sequence
Graph a line given two points
Like in the previous activity, we are going to plot two points to graph a line. Thus far, these activities have described:
- Plotting points on a graph board or pegboard
- Plotting points by inputting ordered pairs directly in the expression list
- Creating a table of points from an equation input into the expression list
In this activity, we will create a table to plot the points (5, 13) and (−5, −7).
Step 1: Create a table
Create a table in expression 1 (Ctrl + Alt + t). This is a good opportunity to practice navigating a table. Remember to listen to the information JAWS reads, especially if you are new to navigating Desmos.
- When you create the table, JAWS will give you information about the table. One important piece of information is that there are three columns and three rows.
- Upon creating the table, the computer focus should be in column 1, row 2. Column 1 is the x-values, and column 2 is the y-values. Shift-tab moves from column 1, row 2, up to column 3, row 1. This is the header row. The third column is a blank column available to use in the case that you are working with more than two variables. We are working with only two variables, so continue navigating.
- Shift-tab moves to column 2, row 1. This is the column header for the y-values. The key information JAWS reads is “y subscript 1”.
- The subscript indicates how many tables there are in the Desmos window. The first table you create will have “subscript 1” on the x- and y- headers. Each table created after that, in the same window, will be subscript 2, subscript 3, and so on.
- Shift-tab moves to the “Hide column button.” We are going to ignore that for now. Just be aware of its location.
- Shift-tab moves to column 1, row 1. This is the column header for the x-values. The key information JAWS reads is “x subscript 1,” or whatever subscript is associated with the table you are navigating.
- Tab four times. You should be in column 1, row 2. The important information JAWS should read is “x subscript 1” and “col 1 row 2.” You are in row 2, ready to input data for the x-value.
Step 2: Input points
Input the points (5, 13) and (−5, −7).
- Type the first x-value, 5. Press tab to navigate to column 2, row 2.
- Type the first y-value, 13. Press tab to navigate to column 1, row 3.
- Type the second x-value, −5. Press tab to navigate to column 2, row 3.
- Type the second y-value, −7. Press tab to navigate to column 1, row 4.
The two points are plotted!
Image: Plotted points
Troubleshooting: Navigating using tab can get you lost at this point. If you do press tab too many times and don’t recognize where you are:
- Try pressing shift-tab until you hear “x subscript 1” and “col 1 row 4.” Remember to listen to JAWS. These audio cues let you know that your focus is in the table.
- Pressing Ctrl + Alt + e should return the focus to the expression list. Up arrow and down arrow navigate between expressions. Page up and page down skip table cells, so you can navigate to the top or bottom of a table without having to navigate each cell.
Step 3: Adjust graph settings
The graph needs to include both the points that have been input. Adjust the graph settings so that the x-minimum value is −10, the x-maximum value is 10, the y-minimum value is −15, and the y-maximum value is 15.
- Open graph settings (Ctrl + Alt + g).
- Tab to “x-minimum.” It should already be −10.
- Tab to “x-maximum.” It should already be 10.
- Tab to “y-minimum.” When your focus is in the edit box, the text is selected. So, you can just type over it. If you feel more comfortable pressing backspace to delete the text first, that is fine. Input −15.
- Tab to “y-maximum.” Input 15.
- Press Escape to exit graph settings.
- Navigate to the expression list (Ctrl + Alt + e).
Image 2: Graph settings
Step 4: Input a regression using the table input
I use the term regression, because that is the term that Desmos uses. You will hear JAWS speak it, so you should know what it means. It may not appear in a student’s math curriculum until later. A regression is a function that graphs a line that fits the data that has been input. Another term that is used for it is line of best fit.
- Create a new expression (Ctrl + Alt + x). Instead of the shortcut, you can press down arrow until you hear “expression 2.”
- Input the regression function: y1 ~ mx1 + b. This is the slope-intercept form of a linear equation, with some differences for the graphing calculator. Inputting y1 and x1 tell Desmos to use the data from the table you have created. You can use data from another table by indicating the subscript of the data (e.g., y2, x2, y3, x3). Inputting the tilde (~) instead of the equals sign lets Desmos know that this is a regression.
Image 3: Graphed line
Your line is graphed! Audio trace the regression. You should hear a graphed line that goes up from left to right. That means the line has a positive slope.
Determine the y-intercept and slope
You may be surprised to learn that most of the work is done. This part is easy.
- Press escape to exit Audio Trace. Your focus should be back in the expression list in expression 2.
- Press tab to navigate to the regression parameters. Tab once gets to “delete expression.” Tab twice gets to the m-value. This is the slope. We learn that m = 2. That is the same as what we determined using the graph board. Hooray!
- Press tab to navigate to the b-value. This is the y-intercept. We learn that b = 3. That is also the same as what we determined using the graph board. Double hooray!
Image 4: Parameters
In Activity 3 of Linear Functions and Problem Solving, students will learn how to answer questions involving the y-intercept, slope, and x-intercept. The following activity will build on these skills while introducing additional Desmos features to deepen understanding of linear functions.
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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