### Clarifications

*Clarification 1:*Problem types include cases where two points are given to determine the slope.

*Clarification 2: *Instruction includes making connections of slope to the constant of proportionality and to similar triangles represented on the coordinate plane.

**Subject Area:**Mathematics (B.E.S.T.)

**Grade:**8

**Strand:**Algebraic Reasoning

**Date Adopted or Revised:**08/20

**Status:**State Board Approved

## Benchmark Instructional Guide

### Connecting Benchmarks/Horizontal Alignment

### Terms from the K-12 Glossary

- Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students determined the constant of proportionality in a proportional relationship. In grade 8, students are determining the slope of a linear relationship from a given table, graph, or written relationship. In Algebra 1, students will write a two-variable linear equation from a graph, written description, or a table to represent relationships between quantities in mathematical and real-world context.- Students identified the unit rate or the constant of proportionality in prior grade levels. This benchmark is the first one that references the slope, which represents a constant rate of change, and this is not the same as a constant of proportionality unless the relationship goes through the origin.
- Instruction includes interpreting the meaning and value of slope in real-world context.
- Understanding slope can be introduced through a graph and the change in value of the $y$ and the $x$
- To introduce the concept to students, use at least two points on a graph in quadrant one.
- Instruction includes using a variety of vocabulary to make connections to real-world concepts and future courses. To describe the slope, one can say either “the vertical change divided by the horizontal change” or “rise over run.”
- Students should have experience utilizing a slope formula to determine the slope between two points on a line.
- Slope of a line can be found by the expression $m$ = , where ($x$
_{1}, $y$_{1}) and ($x$_{2}, $y$_{2}) are two different points on the line.

- Slope of a line can be found by the expression $m$ = , where ($x$

### Common Misconceptions or Errors

- Students may invert the $x$-and $y$-values when calculating slope. To address this misconception, students should represent the relationship visually.

### Strategies to Support Tiered Instruction

- Teacher supports students who invert the $x$-and $y$-values when calculating slope by using real-world problems that students can relate to and helping students represent the relationship visually.
- Instruction includes providing students with graph paper with grid lengths larger than 1 centimeter and using appropriate scaling of the axes to allow for students to see the unit rate more easily.

### Instructional Tasks

*Instructional Task 1*

**(MTR.6.1)**Mr. Elliot needs to drain his above ground pool before the winter. The graph below represents the relationship between the number of gallons of water remaining in the pool and the number of hours that the pool has drained. Determine the slope and explain what it means in this situation.

*Instructional Task 2*

**(MTR.4.1, MTR.7.1)**Jack and Jill are selling gallons of water that are sold in different size pails. Jack charges $1.75 for every 2 gallons of water a pail holds. Additionally, he charges a $2 service fee. Jill's prices can be modeled with the graph shown.

- Part A. Identify the slope of Jack's relationship. Explain what it means.
- Part B. Identify the slope of Jill's graph and explain what it means.
- Part C. Graph Jack’s prices on the same graph as Jill.
- Part D. Whose has the better deal, Jack or Jill? Explain.

### Instructional Items

*Instructional Item 1*

A linear relationship is given in the table below. Determine the slope of the relationship.

**The strategies, tasks and items included in the B1G-M are examples and should not be considered comprehensive.*

## Related Courses

## Related Access Points

## Related Resources

## Formative Assessments

## Lesson Plans

## Original Student Tutorials

## Perspectives Video: Professional/Enthusiast

## Teaching Idea

## Video/Audio/Animation

## MFAS Formative Assessments

Students are asked to identify, describe and compare the slopes of two proportional relationships given the graph of one and the equation of the other.

Students are asked to construct a function to model a linear relationship between two quantities given two ordered pairs in context.

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

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

Students are asked to determine the rate of change and initial value of a linear function when given a graph, and to interpret the rate of change and initial value in terms of the situation it models.

Students are asked to graph a proportional relationship, given a table of values, and find and interpret the slope.

Students are asked to write a function to model a linear relationship given its graph.

Students are given a graph of a proportional relationship and asked to determine the unit rate of the relationship and compare it to the slope of the graph.

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

Students are asked to determine the rate of change and initial value of a linear function given a table of values, and interpret the rate of change and initial value in terms of the situation it models.

Students are asked to determine the rate of change of two functions presented in different forms (table and graph) and determine which is the greater rate of change within a real-world context.

## Original Student Tutorials Mathematics - Grades 6-8

Learn to construct a function to model a linear relationship between two quantities and determine the slope and y-intercept given two points that represent the function with this interactive tutorial.

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

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.

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

## Original Student Tutorials Mathematics - Grades 9-12

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

## Student Resources

## Original Student Tutorials

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.

Type: Original Student Tutorial

Learn to construct a function to model a linear relationship between two quantities and determine the slope and y-intercept given two points that represent the function with this interactive tutorial.

Type: Original Student Tutorial

See how sweet it can be to determine the slope of linear functions and compare them in this interactive tutorial. Determine and compare the slopes or the rates of change by using verbal descriptions, tables of values, equations and graphical forms.

Type: Original Student Tutorial

Learn what slope is in mathematics and how to calculate it on a graph and with the slope formula in this interactive tutorial.

Type: Original Student Tutorial

Learn how to write the equation of a linear trend line when fitted to bivariate data in a scatterplot in this interactive tutorial.

This is part 4 in 6-part series. Click below to open the other tutorials in the series.

- Scatterplots Part 1: Graphing
- Scatterplots Part 2: Patterns, Associations and Correlations
- Scatterplots Part 3: Trend Lines
- Scatterplots Part 5: Interpreting the Equation of the Trend Line
- Scatterplots Part 6: Using Linear Models

Type: Original Student Tutorial

## Video/Audio/Animation

"Slope" is a fundamental concept in mathematics. Slope of a linear function is often defined as " the rise over the run"....but why?

Type: Video/Audio/Animation