# Mathematics

## What Do Students Learn in 8th Grade Math?

In eighth grade, students make several advances in their algebraic reasoning, particularly as it relates to linear equations. Students extend their understanding of proportional relationships to include all linear equations, and they consider what a “solution” looks like when it applies to a single linear equation as well as a system of linear equations. They learn that linear equations can be a useful representation to model bivariate data and to make predictions. Functions emerges as a new domain of study, laying a foundation for more in-depth study of functions in high school. Lastly, students study figures, lines, and angles in two-dimensional and three-dimensional space, investigating how these figures move and how they are measured.

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## Units

Unit 1

15 Lessons

### Exponents and Scientific Notation

Students learn to simplify complex-looking exponential expressions, and they learn efficient ways to describe, communicate, and operate with very large and very small numbers.

Unit 2

12 Lessons

### Solving One-Variable Equations

Students hone their skills of solving multi-step equations and inequalities, redefining their definition of "solution" to include cases such as infinite solutions, and interpreting solutions in context.

Unit 3

22 Lessons

### Transformations and Angle Relationships

Students investigate congruence and similarity by studying transformations of figures in the coordinate plane, and apply these transformations to discover new angle relationships.

Unit 4

12 Lessons

### Functions

Students learn how to represent, interpret, and analyze functions in various forms, leading to understanding features such as rates of change, initial values, and intervals of increase and decrease.

Unit 5

15 Lessons

### Linear Relationships

Students compare proportional relationships, define and identify slope from various representations, graph linear equations in the coordinate plane, and write equations for linear relationships.

Unit 6

11 Lessons

### Systems of Linear Equations

Students explore what happens when you consider two linear equations simultaneously, and explore the many rich applications that can be modeled with systems of linear equations in two variables.

Unit 7

16 Lessons

### Pythagorean Theorem and Volume

Students learn about irrational numbers, approximating square roots of non-perfect square numbers, and investigate the well-known Pythagorean Theorem to solve for missing measures in right triangles.

Unit 8

9 Lessons

### Bivariate Data

Students combine their knowledge of linear functions with knowledge of data representations and analysis to make the jump from univariate data in one variable to bivariate data in two variables.

## How Do Students Build 8th Grade Math Skills Across Units?

In Unit 1, Exponents and Scientific Notation, students start off the year with a study of patterns and structure, using this structure to formalize properties of exponents. They reach back to skills learned in sixth grade to simplify complex exponential expressions and to represent and operate with very large and very small numbers.

In Unit 2, Solving One-Variable Equations, students continue to hone their skill of solving equations. Students solved equations in sixth and seventh grades, and in eighth grade, students become more efficient and more strategic in how they approach and solve equations in one variable. Including this unit at this point in the year allows time for spiraling and incorporating these skills into future units.

In Unit 3, Transformations and Angle Relationships, students formalize their understanding of congruence and similarity as defined by specific movements of figures in the coordinate plane. They experiment with, manipulate, and verify hypotheses around how shapes move under different transformations. Studying similarity, students observe how ratios between similar triangles stay the same, which sets them up for understanding slope in Unit 5. Students also make informal arguments, which prepares them for more formal proofs in high school geometry.

Unit 4, Functions, introduces students to the concept of a function, which relates inputs and outputs. Students analyze and compare functions, developing appropriate vocabulary to use to describe these relationships. They investigate real-world examples of functions that are both linear and nonlinear, and use functions to model relationships between quantities. This introductory study of functions prepares students for Unit 5, in which they focus on a particular kind of function—linear equations.

Unit 5, Linear Relationships, and Unit 6, Systems of Linear Equations, are all about lines. Students make the connection between proportional relationships, functions, and linear equations. They deepen their understanding of slope, making the connection back to similar triangles in Unit 3. Students think critically about relationships between two quantities: how they are represented, how they compare to other relationships, and what happens when you consider more than one linear equation at a time. Throughout these two units, students utilize their skills from Unit 2 as they manipulate algebraic equations and expressions with precision.

In Unit 7, Pythagorean Theorem and Volume Applications, students discover the Pythagorean Theorem, which is supported by a study of irrational numbers. Students now have a full picture of the real number system.

Lastly, in Unit 8, Bivariate Data, students analyze data in two variables using linear equations and two-way tables. They use these structures to make sense of the data and to make justifiable predictions.

Note that this course follows the 2017 Massachusetts Curriculum Frameworks, which include the Common Core Standards for Mathematics.

### What math skills do students learn in 8th grade?

In eighth grade, students apply their understanding of algebra to solve linear equations and reason about their solutions. Additionally, students are introduced to functions and deepen their understanding of 2-and 3-dimensional shapes.

### What Common Core Math Standards do students focus on in 8th grade?

The major work of the course focuses on the following Common Core State Standards:

• 8.EE.A Work with radicals and integer exponents.
• 8.EE.B Understand the connections between proportional relationships, lines, and linear equations.
• 8.EE.C Analyze and solve linear equations and pairs of simultaneous linear equations.
• 8.F.A Define, evaluate, and compare functions.
• 8.F.B Use functions to model relationships between quantities.
• 8.G.A Understand congruence and similarity using physical models, transparencies, or geometry software.
• 8.G.B Understand and apply the Pythagorean Theorem.

### What are the fluency expectations and culminating standards for 8th grade math?

• 8.EE.7: Solve linear equations in one variable. (Culminating)
• 8.G.9: Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems. (Culminating)

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