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Grade 8 Mathematics Module 5, Topic A: Functions

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Title:

 Grade 8 Mathematics Module 5, Topic A: Functions

URL:

 https://www.engageny.org/resource/grade-8-mathematics-module-5-topic-overview

Content Source:

 EngageNY
Type: Lesson/Unit Plan

Overview:

In Topic A of Module 5, students learn the concept of a function and why functions are necessary for describing geometric concepts and occurrences in everyday life. The module begins by explaining the important role functions play in making predictions. For example, if an object is dropped, a function allows us to determine its height at a specific time. To this point, our work has relied on assumptions of constant rates; here, students are given data that shows that objects do not always travel at a constant speed. Once we explain the concept of a function, we then provide a formal definition of a function. A function is defined as an assignment to each input, exactly one output (8.F.A.1). Students learn that the assignment of some functions can be described by a mathematical rule or formula. With the concept and definition firmly in place, students begin to work with functions in real-world contexts. For example, students relate constant speed and other proportional relationships (8.EE.B.5) to linear functions. Next, students consider functions of discrete and continuous rates and understand the difference between the two.  For example, we ask students to explain why they can write a cost function for a book, but they cannot input 2.6 into the function and get an accurate cost as the output. 

Students apply their knowledge of linear equations and their graphs from Module 4 (8.EE.B.5, 8.EE.B.6) to graphs of linear functions. Students know that the definition of a graph of a function is the set of ordered pairs consisting of an input and the corresponding output (8.F.A.1). Students relate a function to an input-output machine:  a number or piece of data, goes into the machine, known as the input, and a number or piece of data comes out of the machine, known as the output. In Module 4, students learned that a linear equation graphs as a line and that all lines are graphs of linear equations. In Module 5, students inspect the rate of change of linear functions and conclude that the rate of change is the slope of the graph of a line. They learn to interpret the equation y = mx + b (8.EE.B.6) as defining a linear function whose graph is a line (8.F.A.3). Students will also gain some experience with non-linear functions, specifically by compiling and graphing a set of ordered pairs, and then by identifying the graph as something other than a straight line.

Once students understand the graph of a function, they begin comparing two functions represented in different ways (8.EE.C.8), similar to comparing proportional relationships in Module 4. For example, students are presented with the graph of a function and a table of values that represent a function and are then asked to determine which function has the greater rate of change (8.F.A.2). Students are also presented with functions in the form of an algebraic equation or written description. In each case, students examine the average rate of change and know that the one with the greater rate of change must overtake the other at some point.  

Content Standard(s):
Mathematics
MA2015 (2016)
Grade: 8
13 ) 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. [8-F3]

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.

NAEP Framework
NAEP Statement::
8A1f: Interpret the meaning of slope or intercepts in linear functions.

NAEP Statement::
8A2b: Analyze or interpret linear relationships expressed in symbols, graphs, tables, diagrams, or written descriptions.

Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.13- Given a set of graphs, identify which graph is linear.
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
9. Interpret y = mx + b as defining a linear equation whose graph is a line with m as the slope and b as the y-intercept.

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

b. Given two distinct points in a coordinate plane, find the slope of the line containing the two points and explain why it will be the same for any two distinct points on the line.

c. Graph linear relationships, interpreting the slope as the rate of change of the graph and the y-intercept as the initial value.

d. Given that the slopes for two different sets of points are equal, demonstrate that the linear equations that include those two sets of points may have different y-intercepts.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Analyze linear equations in the form y=mx + b as representing a line where m represents the rate of change, called the slope of the line when graphed. and b is the initial value, called the y-intercept when graphed.
  • Create similar right triangles by connecting the "rise over run" between any two points on a given line and use them to show why their slopes are the same.
  • Explain why any two points on a given line will have the same slope.
  • Graph linear relationships on a coordinate plane when given in multiple contexts.
Teacher Vocabulary:
  • Slope
  • Rate of change
  • Initial Value
  • Y-intercept
Knowledge:
Students know:
  • how to graph points on a coordinate plane.
  • Where to graph the initial value/y-intercept.
  • Understand how/why triangles are similar.
  • how to interpret y=mx equations.
Skills:
Students are able to:
  • create a graph of linear equations in the form y = mx + b and recognize m as the slope and b as the y-intercept.
  • point out similar triangles formed between pairs of points and know that they have the same slope between any pairs of those points.
  • Show that lines may share the same slope but can have different y-intercepts.
  • Interpret a rate of change as the slope and the initial value as the y-intercept.
Understanding:
Students understand that:
  • Slope is a graphic representation of the rate of change in linear relationships and the y-intercept is a graphic representation of an initial value in a linear relationship.
  • When given an equation in the form y = mx + b it generally symbolizes that there will be lines with varying y-intercepts. even when the slope is the same.
  • Use of the visual of right triangles created between points on a line to explain why the slope is a constant rate of change.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.9.1: Define linear functions, nonlinear functions, slope, and y-intercept.
M.8.9.2: Recall how to solve problems using the distributive property.
M.8.9.3: Recognize linear equations.
M.8.9.4: Identify ordered pairs.
M.8.9.5: Recognize ordered pairs.
M.8.9.6: Define similar triangles, intercept, slope, vertical, horizontal, and origin.
M.8.9.7: Recognize similar triangles.
M.8.9.8: Generate the slope of a line using given ordered pairs.
M.8.9.9: Analyze the graph to determine the rate of change.
M.8.9.10: Demonstrate how to plot points on a coordinate plane using ordered pairs from table.
M.8.9.11: Identify the slope-intercept form (y=mx+b) of an equation where m is the slope and y is the y-intercept.
M.8.9.12: Graph a function given the slope-intercept form of an equation.
M.8.9.13: Recognize that two sets of points with the same slope may have different y-intercepts.
M.8.9.14: Graph a linear equation given the slope-intercept form of an equation.

Prior Knowledge Skills:
  • Define ordered pairs.
  • Name the pairs of integers and/or rational numbers of a point on a coordinate plane.
  • Demonstrate when two ordered pairs differ only by signs, the locations of the points are related by reflections across one or both axes.
  • Identify which signs indicate the location of a point in a coordinate plane.
  • Recall how to plot ordered pairs on a coordinate plane.
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.
Mathematics
MA2019 (2019)
Grade: 8
13. Determine whether a relation is a function, defining a function as a rule that assigns to each input (independent value) exactly one output (dependent value), and given a graph, table, mapping, or set of ordered pairs.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Define a function as a rule assigning each input exactly one output.
  • Identify functions when given a relation as graph, table of values, mapping, or set of ordered pairs.
Teacher Vocabulary:
  • Relation
  • Function
  • Input
  • Output
Knowledge:
Students know:
  • how to interpret a graph, table, mapping, and ordered pairs.
Skills:
Students are able to:
  • give an accurate definition of a function.
  • Analyze graphs, tables, mappings, and sets of ordered pairs to determine if a relation is a function.
Understanding:
Students understand that:
  • Functions assign every input one output, but they may see outputs repeat.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.13.1: Define function, ordered pairs, input, output.
M.8.13.2: Demonstrate how to plot points on a Cartesian plane using ordered pairs.
M.8.13.3: Recall how to complete input/output tables.
M.8.13.4: Recognize numeric patterns.
M.8.13.5: Given a function, create a rule.

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.8.13 Determine whether a relation is a function given a graph or a table.

Mathematics
MA2019 (2019)
Grade: 8
15. Compare properties of functions represented algebraically, graphically, numerically in tables, or by verbal descriptions.

a. Distinguish between linear and non-linear functions.
Unpacked Content
Evidence Of Student Attainment:
Students:
  • Describe the comparison of linear functions qualitatively and quantitatively by discussing and analyzing the rates of change (slopes), initial values (y-intercepts), and any points of intersection.
  • Tell the difference between functions that are linear and those that are non-linear by analyzing information in a variety of contexts.
Teacher Vocabulary:
  • Function
  • Linear
  • Non-linear
  • Slope
Knowledge:
Students know:
  • how to find rates of change and initial values for function represented multiple ways.
  • how to graph functions when given an equation, table, or verbal description.
Skills:
Students are able to:
  • identify the differences between functions represented in multiple contexts.
  • Tell the differences between linear and nonlinear functions.
Understanding:
Students understand that:
  • Converting to different representations of functions can assist in their comparisons of linear functions qualitatively and quantitatively.
Diverse Learning Needs:
Essential Skills:
Learning Objectives:
M.8.15.1: Define rate of change.
M.8.15.2: Recognize linear and nonlinear functions.
M.8.15.3: Recall how to read/interpret information from a table.
M.8.15.4: Identify algebraic expressions.
M.8.15.5: Recall how to name points on a Cartesian plane using ordered pairs.
M.8.15.6: Compare and contrast the differences between linear and nonlinear functions.

Prior Knowledge Skills:
  • Define expression, equivalent, and equivalent expressions.
  • Recall mathematical terms such as sum, difference, etc.
  • Recognize that a variable without a written coefficient is understood to have a coefficient of one.
  • Recall how to convert mathematical terms to mathematical symbols and numbers and vice versa.
  • Restate numerical expressions with words.
Alabama Alternate Achievement Standards
AAS Standard:
M.AAS.8.15 Identify linear and nonlinear functions graphically.
Tags: algebra, coordinate, formula, function, graph, input, linear equations, ordered pairs, output, proportional, slope, triangles, unit rate, variables, vertical axis
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There are eight lessons in this topic.

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Author: Hannah Bradley
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