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Grade 8 Mathematics Module 4, Topic C: Slope and Equation of Lines

Classroom Resource Information

Title:

Grade 8 Mathematics Module 4, Topic C: Slope and Equation of Lines

URL:

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

Content Source:

EngageNY

Type:

Lesson/Unit Plan

Overview:

In Module 4, Topic C, students know that the slope of a line describes the rate of change of a line. Students first encounter slope by interpreting the unit rate of a graph (8.EE.B.5). In general, students learn that slope can be determined using any two distinct points on a line by relying on their understanding of properties of similar triangles from Module 3 (8.EE.B.6). Students verify this fact by checking the slope using several pairs of points and comparing their answers. In this topic, students derive y = mx and y = mx + b for linear equations by examining similar triangles. Students generate graphs of linear equations in two variables first by completing a table of solutions, then using information about slope and y-intercept. Once students are sure that every linear equation graphs as a line and that every line is the graph of a linear equation, students graph equations using information about x- and y-intercepts. Next, students learn some basic facts about lines and equations, such as why two lines with the same slope and a common point are the same line, how to write equations of lines given slope and a point, and how to write an equation given two points. With the concepts of slope and lines firmly in place, students compare two different proportional relationships represented by graphs, tables, equations, or descriptions. Finally, students learn that multiple forms of an equation can define the same line.

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

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.

Tags:

coordinate, formula, graph, proportional, slope, triangles, unit rate, vertical axis

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Comments

There are nine lessons in this topic.

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This resource provided by:

Author:

Hannah Bradley

ALEX Classroom Resource

Grade 8 Mathematics Module 4, Topic C: Slope and Equation of Lines