Standard #: MAFS.8.EE.2.6 (Archived Standard)


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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.


General Information

Subject Area: Mathematics
Grade: 8
Domain-Subdomain: Expressions & Equations
Cluster: Understand the connections between proportional relationships, lines, and linear equations. (Major Cluster) -

Clusters should not be sorted from Major to Supporting and then taught in that order. To do so would strip the coherence of the mathematical ideas and miss the opportunity to enhance the major work of the grade with the supporting clusters.

Date Adopted or Revised: 02/14
Date of Last Rating: 02/14
Status: State Board Approved - Archived
Assessed: Yes

Test Item Specifications

    N/A

    Assessment Limits :
    All triangles must be right triangles and on a coordinate grid. Numbers in items must be rational numbers. Functions must be linear.
    Calculator :

    Yes

    Context :

    Allowable



Sample Test Items (2)

Test Item # Question Difficulty Type
Sample Item 1 Select all pairs of triangles that can be used to show the slope of a line is the same anywhere along the line.

 

 

N/A MS: Multiselect
Sample Item 2

Two collinear points are given in the table.

Give a third point that is also on this line.

 

N/A TI: Table Item


Related Courses

Course Number1111 Course Title222
1205050: M/J Accelerated Mathematics Grade 7 (Specifically in versions: 2014 - 2015, 2015 - 2020, 2020 - 2022, 2022 and beyond (current))
1205070: M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
1204000: M/J Foundational Skills in Mathematics 6-8 (Specifically in versions: 2014 - 2015, 2015 - 2022, 2022 and beyond (current))
7812030: Access M/J Grade 8 Pre-Algebra (Specifically in versions: 2014 - 2015, 2015 - 2018, 2018 - 2019, 2019 - 2022, 2022 and beyond (current))


Related Resources

Formative Assessments

Name Description
Deriving Lines - 2

Students are asked to derive the general equation of a line with a y-intercept of (0, b).

Slope Triangles

Students are asked to use similar triangles to explain why the slope is the same regardless of the points used to calculate it.

Deriving Lines - 1

Students are asked to derive the general equation of a line containing the origin.

Lesson Plans

Name Description
Slope Intercept - Lesson #3

This is lesson 3 of 3 in the Slope Intercept unit. This lesson introduces similar triangles to explain why slope is the same between any two points on a non-vertical line. In this lesson students perform an activity to determine that slope is constant throughout a line and students will discover the slope for vertical and horizontal lines.

Slope Intercept - Lesson #2

This is lesson 2 of 3 in the Slope Intercept unit. This lesson introduces graphing non-proportional linear relationships. In this lesson students will perform an activity to collect data to derive y = mx + b and will use a Scratch program to plot the graph of the data, as well as check for proportional and/or linear relationships.

Designing a Skateboard Kicker Ramp

In this lesson students will design a "Skateboard Kicker Ramp" to discover that slope of similar triangles is the same at any two distinct points.  Students will model with mathematics the concept of slope by looking at the pattern set by similar triangles.

Original Student Tutorial

Name Description
Hailey’s Treehouse: Similar Triangles & Slope

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Problem-Solving Task

Name Description
Find the Change

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Tutorials

Name Description
Finding the slope from two ordered pairs

This tutorial shows how to find the slope from two ordered pairs. Students will see what happens to the slope of a horizontal line.

Using Similar Triangles to Prove that Slope is Constant for a Line

In this tutorial, you will use your knowledge about similar triangles, as well as parallel lines and transversals, to prove that the slope of any given line is constant.

Finding the slope from two ordered pairs

This tutorial shows an example of finding the slope between two ordered pairs. Slope is presented as rise/run, the change in y divided by the change in x and also as m.

Student Resources

Original Student Tutorial

Name Description
Hailey’s Treehouse: Similar Triangles & Slope:

Learn how similar right triangles can show how the slope is the same between any two distinct points on a non-vertical line as you help Hailey build stairs to her tree house in this interactive tutorial.

Problem-Solving Task

Name Description
Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.

Tutorials

Name Description
Finding the slope from two ordered pairs:

This tutorial shows how to find the slope from two ordered pairs. Students will see what happens to the slope of a horizontal line.

Using Similar Triangles to Prove that Slope is Constant for a Line:

In this tutorial, you will use your knowledge about similar triangles, as well as parallel lines and transversals, to prove that the slope of any given line is constant.

Finding the slope from two ordered pairs:

This tutorial shows an example of finding the slope between two ordered pairs. Slope is presented as rise/run, the change in y divided by the change in x and also as m.



Parent Resources

Problem-Solving Task

Name Description
Find the Change:

This activity challenges students to recognize the relationship between slope and the difference in x- and y-values of a linear function. Help students solidify their understanding of linear functions and push them to be more fluent in their reasoning about slope and y-intercepts. This task has also produced a reasonable starting place for discussing point-slope form of a linear equation.



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