This post explores some ideas for reviewing concepts for TEKS A.2I.
write systems of two linear equations given a table of values, a graph, and a verbal description
Staar Performance
On recent STAAR tests, here is how students across the state of Texas have performed.
2023 #37 - 39% correct
Active, Playful Learning
The activities shared in this post are designed to follow the six principles of Active, Playful Learning:
Active
Engaging
Meaningful
Social
Iterative
Joyful
These six principles, together with a clear learning goal, help students learn.
Students learn through active, engaged, meaningful, socially interactive, iterative and joyful experiences in the classroom and out. When we add a learning goal or engage in guided play we achieve Active Playful Learning.
In other words, math review doesn't have to be boring STAAR prep or mindless worksheets. Instead, students' learning is enhanced when playing with numeracy and algebraic concepts in a guided context. Who says math can't be fun? You can watch a video to learn more about Active, Playful learning here.
Activities
Here's a walkthrough of all the activities on this blog post.
Prove It!
Learning objective: Students will use a table of values to write a linear equation and use substitution to check its validity.
This activity focuses on writing a single linear equation from a table of values and checking it's validity. When students master these skills, they transfer wholly to writing systems of linear equations.
Give students a table of values (at least four pairs of coordinates) that lie along the graph of a linear equation. [example]
Have students use the formula for slope [m=(y2-y1)/(x2-x1)] to find the slope using three combinations of x- and y-coordinates. Each correct solution is worth 1 point.
Have students use the point-slope formula [y-y1=m(x-x1)] to find the equation in slope-intercept form using three combinations of x- and y-coordinates. Each correct solution is worth 1 point.
Have students use substitution and the slope-intercept form [y=mx+b] to verify the linear equation using four coordinate pairs from the table. Each correct solution is worth 1 point for a grand total of 10 points.
Variations
Provide a larger tables of values for students to draw from and increase the points possible.
If one step of the process (i.e., using point-slope formula to find slope-intercept form) is more challenging for students, increase its relative score for extra practice.
In the table, add three extra columns and have students, in addition to everything else, generate extra coordinate pairs that are a part of the same linear equation.
Have students draw a graph of the linear function and verify its accuracy with a graphing calculator.
Real-life Systems of Equations
Learning objective: Students will generate equations from real-life scenarios that involve systems of equations.
Provide students with examples of real-life systems of equations.
Have students solve three of the problems, writing the system of equations that represent the scenarios.
After the majority of the class has written systems of equations for three problems, allow students to check their work with various partners.
Ask students to check only one problem per partner. This will increase activity and communication.
Variations
Using a graphing tool, have students graph the systems of equations and plot the solution.
Have students create tables to represent several points along the graph of each linear equation.
Ask students to manipulate the constants (in brackets []), utilizing the same scenario but different numbers. Students should solve the new problems to ensure the equations still generate fairly reasonable numbers (i.e., integers).
Using the examples as a starting point, have students generate their own problem scenarios that describe real-world systems of equations.
Table Race
Learning objective: Students will take derive equations from tables of values.
Put students into groups of three.
Either display or print out ten tables and the ten matching equations in random order. A sample set (with the correct equation written underneath) can be found here.
Have teams race to see which group can match the ten tables with the ten correct equations first.
Ask students to share their strategies for finding the matching equations.
Variations
To begin with, start with just five tables and equations. After the class shares their strategies, highlight different ways to match tables with equations. Then give the next set of five, allowing teams to try out new strategies.
Change some of the equations from the sample to slope-intercept form.
To raise the difficulty, withhold the equations for a set amount of time (e.g., five minutes). This allows students practice deriving equations by finding the slope and using the point-slope formula instead of simply using substitution.
Use the same equations but change all the x-values on the tables to be identical (e.g., 1, 2, 3). You will need to change the y-values so they still match the equation.
Graph to Equation Foursquare
Learning objective: Students will generate a table of values and an equation from the graph of a linear function.
This activity focuses on writing a single linear equation from a graph. When students master this skill, it transfers wholly to writing systems of linear equations.
Put students into groups of four.
Give each member in the group (4 total) a different 2 x 2 table with a graph of a linear function in the top-left cell (see example).
Each member uses the graph to put at least three coordinate pairs into the table in the top-right cell.
Each student should then rotate their paper one student (clockwise). On their new paper, students use the graph (top-left cell) and the table of values (top-right cell) to calculate the slope of the function (bottom-left cell).
Each student should then rotate their paper one student (clockwise). On their new paper, students use the graph (top-left cell), the table of values (top-right cell), and the slope of the function (bottom-left cell) when using the point-slope formula to write the formula of the function in slope-intercept form (bottom-right cell).
Each student should then rotate their paper one student (clockwise). On their new paper, students should use the slope-intercept form (bottom-right cell) to rewrite the equation in standard form [Ax + By= C] at the top of the table (see example).
Each student should then rotate their paper one student (clockwise). Students should have their original paper in front of them, now complete. Give them a few moments to check the work of their group and make corrections as needed.
Variations
For smaller groups, put students into trios. The third group member can write both the slope-intercept form and the standard form before rotating the paper back to its original owner.
For additional scaffolding, mark points on the graph that students should use to create the table of values.
To add difficulty, have the student mark out the x- and y-values on the table (top-right cell) used to find slope (bottom-left cell). The student using the point-slope formula (bottom-right cell) would then need to use the remaining coordinate pair from the table (top-right cell) to find the slope-intercept form (bottom-right cell).
Interactive Systems of Equations
Learning objective: Students will use an online graphing tool to manipulate systems of linear equations.
Have students use a graphing tool such as Desmos.
Have students enter two equations: y=mx+b and y=ax+c. As they enter each equation, they should select all when the site asks to create sliders for the constants m, b, a, and c.
Give students challenges to meet by manipulating the sliders, such as, "Generate a system of equations whose solution is (1, 0)," or, "Generate a system of equations whose solution is in quadrant III."
When students have generated the correct equations, they should write down the system of equations and generate a table of values for each equation. This can be done on paper or a digital recording sheet (example).
Variations
Show students how to use the gear icon in Desmos (appears when equation is highlighted on left) to generate a table of values.
Give students general guidelines for the slope (e.g., one should be negative and one should be positive) and have them meet the challenge (e.g., Generate a system of equations whose solution has a y-value less than -5).
Give students two slopes (e.g., 1.6 and -9) and have them manipulate the y-intercept to meet the challenge (e.g., Generate a system of equations whose solution is on the y-axis).
Instead of entering equations in slope-intercept form, have students use Desmos sliders with equations in standard form [Ax + By = C].
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