Activity
Undetermined points and systems of linear equations 2: Finding the intersection
Think one equation was fun? Try solving two at once! In this activity, students use pegboards and Desmos to dive into systems of linear equations—plotting, predicting, and uncovering the mystery point where two lines meet.
In this second installment of this series, students take their graphing skills to the next level by exploring systems of linear equations. Building on their experience with single lines, they now use both a tactile pegboard and the Desmos Graphing Calculator to investigate how two equations can intersect—and what that intersection reveals. Through hands-on activities and digital exploration, students practice identifying unknown points and solving real problems using systems of equations, all while reinforcing foundational algebra and accessibility skills.
Note: Some screen readers have trouble reading math equations in this websites. For a fully accessible version of this post, download the accessible Word document, Finding the Intersection.
Activity 2: Locating an unknown point using a system of linear equations
In this activity, students will find an unknown point using a system of linear equations. This is a good introduction to systems of linear equations and determining the solution to a system.
Objectives
- Graph a system of linear equations on a pegboard and in the Desmos graphing calculator.
- Determine the solution of a system of linear equations on a pegboard and using the audio trace feature in Desmos.
Materials needed
- Pegboard or graph board with materials for graphing a line
- Rubber mat, shelf liner, or Dycem non-slip matting
- PC laptop or desktop with JAWS
- Internet browser
Lesson/activity sequence
systems of linear equations: Pegboard
A system of linear equations is simply two or more linear equations. The solution to a system is the point where the linear equations intersect. Let’s look at the problem similar to the problem in Activity 1. Activity 1: What’s the Point? Link in Desmos
The point (n,-2) can be found on the graph of the equation y = 0.5x. What is the value of n?
In the last activity, we navigated the graph of the line using arrow keys to locate an unknown point. In this activity, we will use a system of linear equations. Let’s look at this on the pegboard.
Graph the equation y = 0.5x.
Step 1
- Place a peg at the origin. There is no b-value in the equation, which means that the y-intercept is at the point (0, 0).
- The slope or m-value is 0.5 or ½. From the origin, count up 1 and to the right 2. Continue this pattern until you run out of space and place a peg. It should be at the point (6, 3).
- Locate the origin again. Count down 1 and to the left 2 until you run out of space and place a peg. It should be at the point (−6, −3).
- Complete the graph with a rubber band.
Image 1:
Step 2
- Graph the equation y = −2. We only know the y-value from the unknown point, (n, −2), so we can use that information to graph a line.
- Locate the origin. There is no m-value or x-value in the equation y = −2; there is only a b-value. Count down two from the origin and place a peg at the y-intercept (0, −2).
- The slope or m-value is 0, so this is a horizontal line. Locate the far right end of the x-axis and count down two. Place a peg at the point (6, −2). Locate the far left end of the x-axis and count down two. Place a peg at the point (−6, −2).
- Complete the graph with a rubber band.
Image 2:
Step 3
- Locate the missing point.
- It may help to have a rubber mat or other non-slip material under the pegboard for this part of the activity.
- Using both hands and good tactile graphics reading skills, trace each of the graphed lines several times.
- Locate the place where the two lines cross each other. This is the intersection.
- Place a peg at the intersection. It should be on both lines at the point (−4, −2).
Image 3:
Congratulations, you have used a system of equations to locate the missing point at (−4, −2). So, n = −4.
Resource
Systems of linear equations: Desmos
Now, let’s locate the same point using Desmos and a system of equations. Open Desmos graphing calculator in an internet browser.
Step 1
- Input the first equation
- Input y = 0.5x in expression 1.
- Audio trace the equation. It is a good habit to listen and get a sense of the graphed line.
- Does it go up from left to right or down? Does it have a positive slope or negative slope?
- Do you notice any sonification clues? Do you hear static, meaning below the x-axis? Do you notice a tone shift?
Image 4:
Step 2
- Input the second equation.
- Input y = −2 in expression 2.
- Audio trace the equation. You should notice some differences between this graph and the previous graph.
- This is a horizontal line with a y-intercept of −2, so there will not be any apparent slope, up or down.
- Do you notice any sonification clues? Since the y-value is constant at −2, the static plays for the entirety of the graph.
- Do you hear a pop sound? That pop sound indicates that the line intersects with another graph, and that is a good thing.
Image 5:
Step 3
Check for any points of interest. Press i to navigate forward through points of interest or k to navigate backward through points of interest.
There should be two points of interest, (1) an intercept at (0, −2), and (2) the intersection with expression one at (−4, −2).
- The intercept is the y-intercept, where the line crosses the y-axis.
- The intersection gives us the answer to the problem. At the point (−4, −2), the x-value is −4, so n = −4.
Image 6:
Practice problems
Answer the following using a pegboard or graph board and the Desmos Graphing Calculator.
- The point (n, −4) can be found on the graph of the equation y = 2x. What is the value of n?
- The point (n, 1) can be found on the graph of the equation y = 1/3x + 2. What is the value of n?
- The point (n, 5) can be found on the graph of the equation y = −4x – 3. What is the value of n?
This activity has presented an introduction to systems of equations using a pegboard and the Desmos Graphing Calculator. In the next activity, we will use a system of equations to solve a linear equation.
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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