**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

- Intercept
- Linear Equation
- Slope

### Vertical Alignment

Previous Benchmarks

Next Benchmarks

### Purpose and Instructional Strategies

In grade 7, students graphed proportional relationships from a table, equation or a written description. In grade 8, students graph an equation from slope-intercept form from a written description, a table, a graph or an equation. In Algebra 1, students will graph a linear function when given a table, equation or written description.- Point-slope form and standard forms are not expectations at this grade level.
- Review the concept of slope from MA.8.AR.3.2 for students who may need additional work to determine the slope and understand the meaning of slope.
- The instruction includes examples where the slope is positive or negative and the $y$-intercept is given as a positive or a negative in the equation.
- When introducing the benchmark, review graphing on the coordinate plane and determining appropriate scales for the graph.
- Instruction includes the understanding that a real-world context can be represented by a linear two-variable equation even though it only has meaning for discrete values. Discussing discrete values will prepare students to represent domain and range of real-world contexts in later courses.
- For example, if a gym membership cost $10.00 plus $6.00 for each class, this can be represented as $y$ = 10 + 6$c$. When represented on the coordinate plane, the relationship is graphed using the points (0,10), (1,16), (2,22), and so on.

- For mastery of this benchmark, students should be given flexibility to represent real-world contexts with discrete values as a line or as a set of points.

### Common Misconceptions or Errors

- Students may incorrectly identify the slope and $y$-intercept.
- When graphing, students may incorrectly graph the line by inverting the directions of the slope values.
- For example, if the slope is $\frac{\text{2}}{\text{3}}$, a student may think that 2 represents the change in the horizontal direction rather than the vertical direction.

### Strategies to Support Tiered Instruction

- Teacher supports students who incorrectly identify the values for the slope and $y$-intercept by providing opportunities to notice patterns between a given value for $b$, a line graphed on the coordinate plane, and a given equation of the same line.
- 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 supporting students who incorrectly graph the line by inverting the directions of the slope values. Students may incorrectly calculate the slope with a common error of inverting the change in $y$ and the change in $x$. Teachers can support students using error analysis tasks, in which the expression is incorrectly written as .
- Instruction includes having students find the error and make corrections.
- Teacher supports students who incorrectly graph the slope of a given line through error analysis tasks, in which a line is incorrectly graphed by inverting the change in $y$ and the change in $x$ and then have students find and correct the error.
- Teacher co-creates an anchor chart naming the slope and $y$-intercept of a given line and then discusses where to start when graphing the line.
- Teacher provides graphs and equations of several linear equations then co-illustrates connections between the slopes and $y$-intercepts of each line to the corresponding parts of each equation using the same color highlights.
- Teacher co-creates a graphic organizer with students to include examples of positive and negative slope; the meaning of each variable in slope intercept form; and how to determine the slope and $y$-intercept in a table, graph and verbal description.
- Teacher provides instruction on creating an equation table to clear up the misconception of incorrectly graphing an equation on a coordinate plane.
- Teacher provides instruction on determining the slope and $y$-intercept when reading verbal description.

### Instructional Tasks

*Instructional Task 1*

**(MTR.6.1, MTR.7.1)**Brent wants to buy a 60" LED Smart TV. He opened a savings account and added money to the account every month. The table below shows the relationship between the number of months Brent has been saving and the total amount of money in his account.

- Part A. Graph the relationship on a coordinate plane.
- Part B. If the new Smart TV costs $1500 and tax will be $110, approximately how many more months does he need to save money in order to make the purchase?

*Instructional Task 2*

**(MTR.3.1, MTR.4.1)**Part B. Discuss with a partner your method of graphing the equation of the line.

### Instructional Items

*Instructional Item 1*

Graph $y$ = $x$ − 2 on the coordinate plane.

*Instructional Item 2*

Supplies for the car wash cost $25. The booster club is charging $10 per car. Graph the relationship between the amount of money earned and the number of cars washed.

*Instructional Item 3*

The table shown represents a linear relationship. Use the table to graph 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 Assessment

## Lesson Plans

## Problem-Solving Tasks

## Teaching Idea

## MFAS Formative Assessments

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

## Student Resources

## Problem-Solving Tasks

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Type: Problem-Solving Task

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Type: Problem-Solving Task

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

Type: Problem-Solving Task

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary and illustrated solutions are included.

Type: Problem-Solving Task

## Parent Resources

## Problem-Solving Tasks

Students graph proportional relationships and understand the unit rate as a measure of the steepness of the related line, called the slope. Students will also treat slopes more formally when they graph proportional relationships and interpret the unit rate as the slope of the graph.

Type: Problem-Solving Task

This task asks the student to graph and compare two proportional relationships and interpret the unit rate as the slope of the graph. Students are also asked to write an equation and graph each scenario.

Type: Problem-Solving Task

Students will answer questions about unit price of coffee, make a graph of the information, and explain the meaning of constant of proportionality/slope in the given context.

Type: Problem-Solving Task

This task has students engaging in a simple modeling exercise, taking verbal and numerical descriptions of battery life as a function of time and writing down linear models for these quantities. To draw conclusions about the quantities, students have to find a common way of describing them.

Type: Problem-Solving Task

This task lets students explore the differences between linear and non-linear functions. By contrasting the two, it reinforces properties of linear functions.

Type: Problem-Solving Task

This task provides the opportunity for students to reason about graphs, slopes, and rates without having a scale on the axes or an equation to represent the graphs. Students who prefer to work with specific numbers can write in scales on the axes to help them get started.

Type: Problem-Solving Task

In this problem-solving task students are challenged to apply their understanding of linear relationships to determine the amount of chicken and steak needed for a barbecue, which will include creating an equation, sketching a graph, and interpreting both. This resource also includes annotated solutions.

Type: Problem-Solving Task

Students are asked to create and graph linear equations to compare the savings of two individuals. The purpose of the table in (a) is to help students complete (b) by noticing regularity in the repeated reasoning required to complete the table.

Type: Problem-Solving Task

This problem-solving task asks students to find a linear function that models something in the real world. After finding the equation of the linear relationship between the depth of the water and the distance across the channel, students have to verbalize the meaning of the slope and intercept of the line in the context of this situation. Commentary and illustrated solutions are included.

Type: Problem-Solving Task