ALEX Classroom Resource

  

Grade 8 Mathematics Module 4, Topic B: Linear Equations in Two Variables and Their Graphs

  Classroom Resource Information  

Title:

 Grade 8 Mathematics Module 4, Topic B: Linear Equations in Two Variables and Their Graphs

URL:

 https://www.engageny.org/resource/grade-8-mathematics-module-4-topic-b-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Module 4, Topic B, students work with constant speed, a concept learned in Grade 6 (6.RP.A.3), but this time with proportional relationships related to average speed and constant speed. These relationships are expressed as linear equations in two variables. Students find solutions to linear equations in two variables, organize them in a table, and plot the solutions on a coordinate plane (8.EE.C.8a). It is in Topic B that students begin to investigate the shape of a graph of a linear equation.  Students predict that the graph of a linear equation is a line and select points on and off the line to verify their claim. Also in this topic is the standard form of a linear equation, ax + by = c, and when a, b ≠ 0, a non-vertical line is produced. Further, when a or b = 0, then a vertical or horizontal line is produced.

Content Standard(s):
Mathematics
MA2019 (2019)
Grade: 8
8. Graph proportional relationships.

a. Interpret the unit rate of a proportional relationship, describing the constant of proportionality as the slope of the graph which goes through the origin and has the equation y = mx where m is the slope.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Represent given proportional relationships with graphs.
  • Determine the characteristics that remain consistent in proportional relationships, such as the unit rate and inclusion of the origin.
  • Use a graphical representation of a proportional relationship in context to: explain the meaning of any point (x, y). explain the meaning of (0, 0). and why it is included.
Teacher Vocabulary:
  • Ratio
  • Proportion
  • Proportional
  • Independent variable
  • Dependent variable
  • y-intercept
  • origin
Knowledge:
Students know:
  • what a proportion is and how it is represented on a table or verbally.
  • how to graph coordinates and identify the origin and quadrants on the coordinate plane.
Skills:
Students are able to:
  • create graphs to visually verify a constant rate as a straight line through the corresponding coordinates and the origin.
  • Identify the unit rate (constant of proportionality) within two quantities in a proportional relationship shown on a graph and in the form y =mx.
Understanding:
Students understand that:
  • unit rate is sometimes referred to as the constant of proportionality.
  • proportional relationships are represented by a straight line that runs through the origin.
  • y=mx is the equation form that represents all proportions, where m is the rate of change/constant of proportionality which can now be called the slope.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.8.1: Define proportional relationships, unit rate, and slope.
M.8.8.2: Demonstrate how to write ratios.
M.8.8.3: Recall how to solve proportions using cross products.
M.8.8.4: Recall how to find the unit rate.
M.8.8.5: Demonstrate how to graph on a Cartesian plane.
M.8.8.6: Recall that for a relationship to be proportional, the graph must pass through the origin.
M.8.8.7: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.

Prior Knowledge Skills:
  • Define unit rate, proportion, and rate.
  • Create a ratio or proportion from a given word problem.
  • Calculate unit rate by using ratios or proportions.
  • Write a ratio as a fraction.
  • Define ratio, rate, proportion, percent, equivalent, input, output, ordered pairs, diagram, unit rate, and table.
  • Create a ratio or proportion from a given word problem, diagram, table, or equation.
  • Calculate unit rate or rate by using ratios or proportions with or without a calculator.
  • Restate real-world problems or mathematical problems.
  • Construct a graph from a set of ordered pairs given in the table of equivalent ratios.
  • Calculate missing input and/or output values in a table with or without a calculator.
  • Draw and label a table of equivalent ratios from given information.
  • Identify the parts of a table of equivalent ratios (input, output, etc.).
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.8 Using a real-world scenario, match a table with its graph. Identify proportional or nonproportional relationships.
Mathematics
MA2019 (2019)
Grade: 8
12. Solve systems of two linear equations in two variables by graphing and substitution.

a. Explain that the solution(s) of systems of two linear equations in two variables corresponds to points of intersection on their graphs because points of intersection satisfy both equations simultaneously.

b. Interpret and justify the results of systems of two linear equations in two variables (one solution, no solution, or infinitely many solutions) when applied to real-world and mathematical problems.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Graph a system of two linear equations, recognizing that the ordered pair for the point of intersection is the x-value that will generate the given y-value for both equations.
  • Recognize that graphed lines with one point of intersection (different slopes) will have one solution, parallel lines (same slope, different y-intercepts) have no solutions, and lines that are the same (same slope, same y-intercept) will have infinitely many solutions.
  • Use substitution to solve a system, given two linear equations in slope-intercept form or one equation in standard form and one in slope-intercept form.
  • Make sense of their solutions by making connections between algebraic and graphical solutions and the context of the system of linear equations.
Teacher Vocabulary:
  • System of linear equations
  • Point of intersection
  • One solution
  • No solution
  • Infinitely many solutions
  • Parallel lines
  • Slope-intercept form of a linear equation
  • Standard form of a linear equation
Knowledge:
Students know:
  • The properties of operations and equality and their appropriate application.
  • Graphing techniques for linear equations (using points, using slope-intercept form, using technology).
  • Substitution techniques for algebraically finding the solution to a system of linear equations.
Skills:
Students are able to:
  • generate a table from an equation.
  • Graph linear equations.
  • Identify the ordered pair for the point of intersection.
  • Explain the meaning of the point of intersection (or lack of intersection point) in context.
  • Solve a system algebraically using substitution when both equations are written in slope-intercept form or one is written in standard form and the other in slope-intercept form.
Understanding:
Students understand that:
  • any point on a line when substituted into the equation of the line, makes the equation true and therefore, the intersection point of two lines must make both equations true.
  • Graphs and equations of linear relationships are different representations of the same relationships, but reveal different information useful in solving problems, and allow different solution strategies leading to the same solutions.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.12.1: Define variables.
M.8.12.2: Recall how to estimate.
M.8.12.3: Recall how to solve linear equations.
M.8.12.4: Demonstrate how to graph solutions to linear equations.
M.8.12.5: Recall how to graph ordered pairs on a Cartesian plane.
M.8.12.6: Recall that linear equations can have one solution (intersecting), no solution (parallel lines), or infinitely many solutions (graph is simultaneous).
M.8.12.7: Define simultaneous.
M.8.12.8: Recall how to solve linear equations.
M.8.12.9: Recall properties of operations for addition and multiplication.
M.8.12.10: Discover that the intersection of two lines on a coordinate plane is the solution to both equations.
M.8.12.11: Define point of intersection.
M.8.12.12: Recall how to solve linear equations.
M.8.12.13: Demonstrate how to graph on the Cartesian plane.
M.8.12.14: Identify ordered pairs.
M.8.12.15: Recall how to solve linear equations in two variables by using substitution.
M.8.12.16: Create a word problem from given information.
M.8.12.17: Recall how to solve linear equations.
M.8.12.18: Explain how to write an equation to solve real-world mathematical problems.

Prior Knowledge Skills:
  • Define quadrant, coordinate plane, coordinate axes (x-axis and y-axis), horizontal, vertical, and reflection.
  • Demonstrate an understanding of an extended coordinate plane.
  • Draw a four-quadrant coordinate plane.
  • Draw and extend vertical and horizontal number lines.
  • Interpret graphing points in all four quadrants of the coordinate plane in real-world situations.
  • Recall how to graph points in all four quadrants of the coordinate plane.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.12 Solve two-step linear equations where coefficients are less than 10 and answers are integers.
Tags: algebra, coordinate, graph, linear equations, proportional, slope, variables
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There are five lessons in this topic.

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Author: Hannah Bradley
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