Unit 2

Expressions and Linear Equations

Grade 8 Math 

 

Unit Length and Description:

 

35 days

 

In previous years, students have been working informally with one-variable linear equations. This important line of development culminates in Unit 2 with the solution of general one-variable linear equations, including cases with infinitely many solutions or no solutions as well as cases requiring algebraic manipulation using properties of operations. Coefficients and constants in these equations may be any rational numbers.  In Unit 2, students also build on previous work to make the connection between proportional relationships, lines, and linear equations as they develop ways to represent a line by different equations (y = mx + b, y y1 = m(x x1), etc.). Students use their knowledge of unit rates and graphing to understand that the points (x, y) on a non-vertical line are the solutions of the equation y = mx + b, where m is the slope of the line as well as the unit rate of a proportional relationship (in the case b = 0).  Students build a critical foundation, in preparation for future coursework, as they analyze and solve linear equations and pairs of simultaneous linear equations, while working with a variety of mathematical and real-life problems. The equation of a line provides a natural transition into the idea of a function explored in Unit 5.

 

Standards:

 

8.EE.B.5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways. For example, compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

 

8.EE.B.6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and the equation 𝑦= mx+𝑏 for a line intercepting the vertical axis at 𝑏.

 

  • Standards Clarification:  This standard is initially addressed in this unit as it focuses on modeling slope and the equation of a line.  The standard will be re-addressed through the concept of transformations in Unit 4.

 

8.EE.C.7 Solve linear equations in one variable.

a.   Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form 𝑥=𝑎, 𝑎=𝑎, or 𝑎=𝑏 results (where 𝑎 and 𝑏 are different numbers).

b.   Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

 

8.EE.C.8 Analyze and solve pairs of simultaneous linear equations.

a.   Understand that solutions to a system of two linear equations in two variables correspond to points of intersection of their graphs, because points of intersection satisfy both equations simultaneously.

b.   Solve systems of two linear equations in two variables algebraically, and estimate solutions by graphing the equations. Solve simple cases by inspection. For example, 3𝑥+2𝑦=5 and 3𝑥+2𝑦=6 have no solution because 3𝑥+2𝑦 cannot simultaneously be 5 and 6.

c.    Solve real-world and mathematical problems leading to two linear equations in two variables. For example, given coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

 

8.F.A.2 Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a linear function represented by a table of values and a linear function represented by an algebraic expression, determine which function has the greater rate of change.

 

  • Standards Clarification:  This standard is addressed in this unit as it pertains to the development and representation of linear equations. It is a supporting standard in this unit, but foundational to the development of linear equations. The standard will be re-addressed and fully developed through the concept of functions in Unit 5.

 

8.F.A.3 Interpret the equation 𝑦=𝑚𝑥+𝑏 as defining a linear function whose graph is a straight line; give examples of functions that are not linear. For example, the function 𝐴=𝑠2 giving the area of a square as a function of its side length is not linear because its graph contains the points (1,1), (2,4) and (3,9) which are not on a straight line.

 

  • Standards Clarification:  This standard is addressed in this unit as it pertains to the development and representation of linear equations. It is a supporting standard in this unit, but foundational to the development of linear equations.  The standard will be re-addressed and fully developed through the concept of functions in Unit 5.

 

8.F.B.4 Construct a function to model a linear relationship between two quantities. Determine the rate of change and initial value of the function from a description of a relationship or from two (x,y) values, including reading these from a table or from a graph. Interpret the rate of change and initial value of a linear function in terms of the situation it models and in terms of its graph or a table of values.

 

  • Standards Clarification:  This standard is addressed in this unit as it pertains to the development and representation of linear equations. It is a supporting standard in this unit, but foundational to the development of linear equations.  The standard will be re-addressed and fully developed through the concept of functions in Unit 5.

 

Focus Standards of Mathematical Practice:

 

MP.1 Make sense of problems and persevere in solving them.

MP.2 Reason abstractly and quantitatively.

MP.3 Construct viable arguments and critique the reasoning of others.

MP.4 Model with mathematics.

MP.5 Use appropriate tools strategically.

MP.6 Attend to precision.

MP.7 Look for and make use of structure.

MP.8 Look for and express regularity in repeated reasoning.

 

Instructional Outcomes:

Full Development of the Major Clusters, Supporting Clusters, Additional Clusters and Mathematical Practices for this unit could include the following instructional outcomes:

 

8.EE.B.5

·        I can determine the rate of change by the definition of slope.

·        I can graph proportional relationships.

·        I can compare two different proportional relationships represented in different ways. (For example, compare a distance-time graph to a distance-time equation to determine which of the two moving objects has greater speed).

·        I can interpret the unit rate of proportional relationships as the slope of a graph.

 

8.EE.B.6

·        I can find the slope of a line between a pair of distinct points.

·        I can determine the y-intercept of a line (interpreting unit rate as the slope of the graph is included in 8.EE).

·        I can analyze patterns for points on a line through the origin.

·        I can derive an equation of the form y=mx for a line through the origin.

·        I can analyze patterns for points on a line that does not pass through or include the origin.

·        I can derive an equation of the form y=mx + b for a line intercepting the vertical axis at b (the y-intercept).

·        I can use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane.

 

8.EE.C.7

·        I can give examples of linear equations in one variable with one solution and show that the given example equation has one solution by successively transforming the equation into an equivalent equation of the form x=a.

·        I can give examples of linear equations in one variable with infinitely many solutions and show that the given example has infinitely many solutions by successively transforming the equation into an equivalent equation of the form a=a.

·        I can give examples of linear equations in one variable with no solution and show that the given example has no solution by successively transforming the equation into an equivalent equation of the form b=a, where a and b are different numbers.

·        I can solve linear equations with rational number coefficients.

·        I can solve equations whose solutions require expanding expressions using the distributive property and/or collecting like terms.

 

8.EE.C.8

·        I can identify the solution(s) to a system of two linear equations in two variables as the point(s) of intersection of their graphs. 

·        I can describe the point(s) of intersection between two lines as the points that satisfy both equations simultaneously.

·        I can define “inspection”.

·        I can solve a system of two equations (linear) in two unknowns algebraically.

·        I can identify cases in which a system of two equations in two unknowns has no solution.

·        I can identify cases in which a system of two equations in two unknowns has an infinite number of solutions.

·        I can solve simple cases of systems of two linear equations in two variables by inspection.

 

8.F.A.2

·        I can represent a linear equation algebraically using slope and y-intercept.

·        I can represent a linear equation graphically.

·        I can represent a linear equation in a table.

·        I can represent a linear equation using a verbal description.

·        I can compare and contrast two linear equations with different representations.

·        I can draw conclusions based on different representations of a linear equation.

 

8.F.A.3

·        I can recognize that the graph of y=mx+b is a straight line.

·        I can recognize that the graph of y=mx+b is a straight line where m is the slope and b is the y-intercept.

·        I can recognize non-linear equations by analyzing tables, graphs, and equations.

·        I can compare the characteristics of a linear and non-linear equation using various representations.

 

8.F.B.4

·        I can recognize that slope is determined by the constant rate of change.

·        I can recognize that the y-intercept is the initial value where x=0.

·        I can determine the rate of change (slope) from two (x,y) values, a verbal description, values in a table, or graph.

·        I can determine the initial value (y-intercept) from two (x,y) values, a verbal description, values in a table, or graph.

·        I can construct a model of a linear relationship between two quantities.

·        I can relate the rate of change and initial value to real world quantities in a linear equation in terms of the situation modeled and in terms of its graph or a table of values.

 

Enduring Understandings:

 

·        Linear equations in one variable can have one solution, infinitely many solutions, or no solutions.

·        Equations offer an efficient way to organize and solve problems, and an effective way to express relationships.

·        Mathematical models are powerful tools we can use to represent real-world problems.

·        Mathematical situations and structures can be translated and represented abstractly using variables, expressions, and equations.

·        An equation can be written for two quantities that vary proportionally.

·        Unit rates can be explained in graphical representations, algebraic representations, and in geometry through similar triangles.

·        The unit rate for a data set that represents a proportional relationship can be interpreted as slope when the data is graphed on a coordinate plane.

·        The slope m is the same for any two distinct points on a non-vertical line graphed on the coordinate plane..

·        Graphs of linear equations that intersect the y-axis at any point other than the origin (0, 0) do not represent proportional relationships.

·        The points (x, y) on a non-vertical line are the solutions of the equation y = mx + b.

 

Essential Questions:

 

·       If two different graphs model the same information, will they have the same meaning?

·       What has to be present to produce the same results?

·       How can situations be expressed with symbols in different ways, but have the same meaning?

·       What are the conditions that make circumstances equal?

·       How can I communicate mathematical information and ideas more effectively?

·       How do we understand and represent linear relationships, inverse relationships, and various nonlinear relationships?

·       What is the meaning of slope?

·       How can we transfer data and information between multiple representations? (e.g. graphs, tables, equations, descriptions, etc.)

·       What is the difference between a ratio and a unit rate?

·       How can proportional relationships be used to represent authentic situations in life and solve actual problems?

·       In what way(s) do proportional relationships relate to functions and functional relationships?

·       What do the points on a line represent?

·       What does the slope of a line represent?

·       What does the point of intersection of two simultaneous equations represent?