Activity
Linear functions and problem solving 3: y-intercept and slope problem solving
Can your algebra student answer questions about slope, y-intercept and x-intercept in using 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—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 the key characteristics of linear functions using tactile tools, while the second activity focuses on applying these concepts in Desmos to explore slope, y-intercept, and x-intercept.
In this third activity, students will combine skills learned in earlier lessons to input linear functions into the calculator and view key features of the graph. They will practice answering questions related to the y-intercept, slope and x-intercept.
Activity 3: Answer questions related to the graph of a Linear function
In this activity, we will combine the skills learned in previous activities to input linear functions in the calculator and view features of a graph. We will practice answering questions related to the y-intercept, slope and the x-intercept.
We have discussed the y-intercept, the point where the graphed line crosses the y-axis. The x-intercept is the point where the graphed line crosses the x-axis. It is also known as the zero, the root and the solution. A student may be asked for any of these (e.g., What is the solution of the linear function? What is the root of the given linear function?). The key is that any one of these terms may be used for the point where the graphed line crosses the x-axis.
Note: it may help students to pair a graph board with these activities to have a tactile model of the x-intercept, y-intercept and slope.
Graph a line given two points
Graph the linear function that includes the points (−4, −16) and (8, 20). Recall the steps from the previous activity:
- Step 1: Create a table (Ctrl + Alt + t)
- Step 2: Input the x-value and y-value for each point
- Step 3: Adjust the graph settings (Ctrl + Alt + g) to include the points on the graph paper. The x-minimum and maximum can remain at −10 and 10, but the y-minimum and maximum should be greater than −16 and 20. I set the y-minimum at −30 and the y-maximum to 30.
- Step 4: Input a linear regression into expression 2. Remember: y1 ~ mx1 + b.
Image 1: Problem 1, graph 1
Using points of interest
You can use points of interest in Desmos to determine the y-intercept and the x-intercept.
Determine the y-intercept and the x-intercept of the graphed function.
- Step 1: With the focus in expression 2, audio trace the linear regression (Alt + t).
- Step 2: Navigate forward by point of interest (i).
- Step 3: Navigate backward by point of interest (k).
- Each time you navigate to a new point of interest, JAWS will read the type of point and the ordered pair.
- At the last point of interest (moving forward or backward), JAWS will say “no more points of interest,” and then read the type of point and the ordered pair.
- Desmos defines the y-intercept as “intercept” and the x-intercept as “zero.” So, for this function, you should hear “zero at x: 1.33333 y: 0” and “intercept at x: 0 y: −4.”
Image 2: Problem 1, graph 2
Determine the slope.
Slope is not defined in Audio Trace Mode. As described in activity 2 of this set, slope is accessed in the regression parameters.
Ensure your focus is 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 = 3.
Given this information, you can answer the following:
The graph of a linear equation passes through the points (−4, −16) and (8, 20). What is the zero, the y-intercept and the slope?
Zero = (1.33, 0)
y-intercept = (0, −4)
slope = 3
Graph a line given an equation
Graph y = 5x−14 on the Desmos graphing calculator in a new window (F5 or Ctrl + r to reload the page). Determine the y-intercept, x-intercept, and slope.
- Step 1: Enter the equation in the expression list. Use the audio trace feature to hear the graph.
- Listening to this graph we notice that it goes up and to the right very quickly. That indicates a steep slope. Slow the playback down to ¼ speed (Alt + 1).
- Play the graph again. It starts below the x-axis (static) then goes above the x-axis (no static). This indicates that the x-intercept appears on the graph paper.
- When a graphed line crosses the y-axis, we hear a tone shift. We do not hear that for this graph, so that indicates that the y-intercept does not appear on the graph paper.
Image 3: Problem 2 graph 1
- Step 2: Use points of interest to determine the y-intercept and the x-intercept.
- While in the Audio Trace Mode, press k or i to determine the intercept (y-intercept) and the zero (x-intercept).
- The y-intercept should be at (0, −14). The x-intercept should be at (2.8, 0).Step 3: Edit the Graph Settings to change the viewing window. Ensure that the y-intercept at (0, −14) is included on the graph paper.
- Step 3: Edit the Graph Settings to change the viewing window.
- Ensure that the y-intercept at (0, −14) is included on the graph paper.
- Open graph settings (Ctrl + Alt + g).
- Tab to locate the y-axis minimum. Change the y-axis minimum so that the y-intercept is included in the graph (−20 is a good choice).
- Press Escape when finished.
- Navigate to the expression list (Ctrl + Alt + e).
- Audio trace (Alt + t) and play (h) the graph. You should hear the tone shift during playback, which indicates the y-intercept is visible on the graph paper.
Image 4: Problem 2, graph 2
- Step 4: Determine slope.
- Slope is not defined in Audio Trace Mode.
- This is not a regression, so you cannot tab from the equation to find the m-value.
- Slope is determined by analyzing the equation: y = 5x−14.
- Consider the slope-intercept form, y = mx+b, where m = slope and b = y-intercept.
- The m-value is 5, and that is the slope of the line.
In this activity set, we have determined y-intercept and slope on the graph board and using the Desmos graphing calculator. In addition, we have practiced graphing a line using a regression and determining points of interest in the audio trace mode.
In the next activity set, we will get into more problem solving with linear equations, including systems of equations!
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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