# Mathematics 8th Grade

Grade 8 Students begin grade 8 with transformational geometry. They study rigid transformations and congruence, then dilations and similarity (this provides background for understanding the slope of a line in the coordinate plane). Next, they build on their understanding of proportional relationships from grade 7 to study linear relationships. They express linear relationships using equations, tables, and graphs, and make connections across these representations. They expand their ability to work with linear equations in one and two variables. Building on their understanding of a solution to an equation in one or two variables, they understand what is meant by a solution to a system of equations in two variables. They learn that linear relationships are an example of a special kind of relationship called a function. They apply their understanding of linear relationships and functions to contexts involving data with variability. They extend the definition of exponents to include all integers, and in the process codify the properties of exponents. They learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities. They encounter irrational numbers for the first time and informally extend the rational number system to the real number system, motivated by their work with the Pythagorean Theorem.

**Estimated Completion Time:** 2 semesters/18-36 weeks

**State Course Number: **115800

The course-level objectives for 8th grade Math come from Missouri Learning Standards. The competencies are divided by unit below; separate module-level objectives are located at the beginning of each assignment.

**Unit 1 – RIGID TRANSFORMATION & CONGRUENCE**

In this unit, you learn to understand and use the terms “reflection,” “rotation,” “translation,” recognizing what determines each type of transformation, e.g., two points determine a translation. You learn to understand and use the terms “transformation” and “rigid transformation.” You identify and describe translations, rotations, and reflections, and sequences of these, using the terms “corresponding sides” and “corresponding angles,” and recognizing that lengths and angle measures are preserved. You draw images of figures under rigid transformations on and off square grids and the coordinate plane. You use rigid transformations to generate shapes and to reason about measurements of figures. You learn to understand the congruence of plane figures in terms of rigid transformations. You recognize when one plane figure is congruent or not congruent to another. You use the definition of “congruent” and properties of congruent figures to justify claims of congruence or non-congruence.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Verify experimentally the congruence properties of rigid transformations. (8.GM.A.1)

- Verify that angle measure, betweenness, collinearity and distance are preserved under rigid transformations.
- Investigate if orientation is preserved under rigid transformations.

- Understand that two-dimensional figures are congruent if a series of rigid transformations can be performed to map the pre-image to the image. (8.GM.A.2)

- Describe a possible sequence of rigid transformations between two congruent figures.

- Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. (8.GM.A.3)
- Understand that two-dimensional figures are similar if a series of transformations (rotations, reflections, translations and dilations) can be performed to map the pre-image to the image. (8.GM.A.4)

- Describe a possible sequence of transformations between two similar figures.

- Explore angle relationships and establish informal arguments. (8.GM.A.5)

- Derive the sum of the interior angles of a triangle.
- Explore the relationship between the interior and exterior angles of a triangle.
- Construct and explore the angles created when parallel lines are cut by a transversal.
- Use the properties of similar figures to solve problems.

**Unit 2 – DILATIONS, SIMILARITY, INTRO TO SLOPE**

In this unit, you learn to understand and use the term “dilation,” and to recognize that a dilation is determined by a point called the “center” and a number called the “scale factor.” You learn that under a dilation, the image of a circle is a circle and the image of a line is a line parallel to the original. You draw images of figures under dilations on and off the coordinate plane. You use the terms “corresponding sides” and “corresponding angles” to describe correspondences between a figure and its dilated image, and recognizing that angle measures are preserved, but lengths are multiplied by the scale factor. You learn to understand the similarity of plane figures in terms of rigid transformations and dilations. You learn to recognize when one plane figure is similar or not similar to another. You use the definition of “similar” and properties of similar figures to justify claims of similarity or non-similarity. You learn the terms “slope” and “slope triangle,” and use the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Understand that two-dimensional figures are congruent if a series of rigid transformations can be performed to map the pre-image to the image. (8.GM.A.2)

- Describe a possible sequence of rigid transformations between two congruent figures.

- Describe the effect of dilations, translations, rotations and reflections on two-dimensional figures using coordinates. (8.GM.A.3)
- Understand that two-dimensional figures are similar if a series of transformations (rotations, reflections, translations and dilations) can be performed to map the pre-image to the image. (8.GM.A.4)

- Describe a possible sequence of transformations between two similar figures.

- Explore angle relationships and establish informal arguments. (8.GM.A.5)

- Derive the sum of the interior angles of a triangle.
- Explore the relationship between the interior and exterior angles of a triangle.
- Construct and explore the angles created when parallel lines are cut by a transversal.
- Use the properties of similar figures to solve problems.

- Apply concepts of slope and y-intercept to graphs, equations and proportional relationships. (8.EEI.B.6)

- Explain why the slope (m) is the same between any two distinct points on a non-vertical line in the Cartesian 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.

**Unit 3 – LINEAR RELATIONSHIPS**

In this unit, you learn to understand and use the terms “rate of change,” “linear relationship,” and “vertical intercept.” You deepen their understanding of slope, and they learn to recognize connections among rate of change, slope, and constant of proportionality, and between linear and proportional relationships. You learn to understand that lines with the same slope are translations of each other. You represent linear relationships with tables, equations, and graphs that include lines with negative slopes or vertical intercepts, and horizontal and vertical lines. You learn to use the term “solution of an equation” when working with one or two linear equations in two variables, and learn to understand the graph of a linear equation as the set of its solutions. Students use these terms and representations in reasoning about situations involving one or two constant rates.

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Graph proportional relationships. (8.EEI.B.5)

- Interpret the unit rate as the slope of the graph.
- Compare two different proportional relationships.

- Apply concepts of slope and y-intercept to graphs, equations and proportional relationships. (8.EEI.B.6)

- Explain why the slope (m) is the same between any two distinct points on a non-vertical line in the Cartesian 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.

- Verify experimentally the congruence properties of rigid transformations. (8.GM.A.1)

- Verify that angle measure, betweenness, collinearity and distance are preserved under rigid transformations.
- Investigate if orientation is preserved under rigid transformations.

- Analyze and solve systems of linear equations. (8.EEI.C.8)

- Graph systems of linear equations and recognize the intersection as the solution to the system.

**Unit 4 – LINEAR EQUATIONS/SYSTEMS**

In this unit, you write and solve linear equations in one variable. These include equations in which the variable occurs on both sides of the equal sign, and equations with no solutions, exactly one solution, and infinitely many solutions. You learn that any one such equation is false, true for one value of the variable, or true for all values of the variable. You interpret solutions in the contexts from which the equations arose. You write and solve systems of linear equations in two variables and interpret the solutions in the contexts from which the equations arose. You learn what is meant by a solution for a system of equations, namely that a solution of the system is a solution for each equation in the system. You use the understanding that each pair of values that make an equation true are coordinates of a point on the graph of the equation and conversely that the coordinates of each point on the graph of an equation make the equation true. Thus, a pair of values that satisfies a system of equations are coordinates of a point that lies on the graphs of all the equations in the system, and, conversely, a point that lies on the graphs of all the equations in the system has coordinates that satisfy all the equations in the system. You learn to understand and use the terms “system of equations,” “solution for the system of equations,” “zero solutions,” “no solution,” “one solution,” and “infinitely many solutions.”

**Missouri Learning Standards:** You will know you have achieved the learning goal when you can:

- Solve linear equations and inequalities in one variable. (8.EEI.C.7)

- Create and identify linear equations with one solution, infinitely many solutions or no solutions.
- Solve linear equations and inequalities with rational number coefficients, including equations and inequalities whose solutions require expanding expressions using the distributive property and combining like terms.

- Analyze and solve systems of linear equations. (8.EEI.C.8)

- Graph systems of linear equations and recognize the intersection as the solution to the system.
- Explain why solution(s) to a system of two linear equations in two variables corresponds to point(s) of intersection of the graphs.
- Explain why systems of linear equations can have one solution, no solution or infinitely many solutions.
- Solve systems of two linear equations.

**Unit 5 – FUNCTIONS AND VOLUME**

In this unit, you are introduced to the concept of a function. You learn to understand and use the terms “input,” “output,” and “function,” e.g., “temperature is a function of time.” You describe functions as increasing or decreasing between specific numerical inputs, and they consider the inputs of a function to be values of its independent variable and its outputs to be values of its dependent variable. (The terms “Independent variable” and “dependent variable” were introduced in grade 6.) You use tables, equations, and graphs to represent functions, and describe information presented in tables, equations, or graphs in terms of functions. In working with linear functions, you coordinate and synthesize your understanding of “constant of proportionality” (which was introduced in grade 7), “rate of change” and “slope” (which were introduced earlier in grade 8), and increasing and decreasing. You perceive similarities in structure between pairs of known and new volume formulas: for a rectangular prism and a cylinder; and for a cylinder and a cone. You rearrange these formulas to show functional relationships and use them to reason about how the volume of a figure changes as another measurement changes, e.g., the height of a cylinder is proportional to its volume; if the radius of a cylinder triples, its volume becomes nine times larger.

**Missouri Learning Standard:** You will know you have achieved the learning goal when you can:

- Explore the concept of functions. (The use of function notation is not required.) (8.F.A.1)

- Understand that a function assigns to each input exactly one output.
- Determine if a relation is a function.
- Graph a function.

- Compare characteristics of two functions each represented in a different way. (8.F.A.2)
- Investigate the differences between linear and nonlinear functions. (8.F.A.3)

- Interpret the equation y = mx + b as defining a linear function, whose parameters are the slope (m) and the y-intercept (b).
- Recognize that the graph of a linear function has a constant rate of change
- Give examples of nonlinear functions.

- Use functions to model linear relationships between quantities. (8.F.B.4)

- Explain the parameters of a linear function based on the context of a problem.
- Determine the parameters of a linear function.
- Determine the x-intercept of a linear function.

- Describe the functional relationship between two quantities from a graph or a verbal description. (8.F.B.5)
- Solve problems involving surface area and volume. (8.GM.C.9)

- Understand the concept of surface area and find surface area of pyramids.
- Understand the concepts of volume and find the volume of pyramids, cones and spheres.

**Unit 6 – ASSOCIATIONS IN DATA**

In this unit, you generate and work with bivariate data sets that has more variability than in previous units. You learn to understand and use the terms “scatter plot” and “association,” and describe associations as “positive” or “negative” and “linear” or “non-linear.” You describe scatter plots, using a term previously used to describe univariate data “cluster,” and the new term “outlier.” You fit lines to scatter plots and informally assess their goodness of fit by judging the closeness of the data points to the lines, and compare predicted and actual values. You learn to understand and use the terms “two-way table,” “bar graph,” and “segmented bar graph,” using two-way tables to investigate categorical data.

**Major Instructional Goals: **You will know you have achieved the learning goal when you can:

- Construct and interpret scatter plots of bivariate measurement data to investigate patterns of association between two quantities. (8.DSP.A.1)
- Generate and use a trend line for bivariate data, and informally assess the fit of the line. (8.DSP.A.2)
- Interpret the parameters of a linear model of bivariate measurement data to solve problems. (8.DSP.A.3)
- Understand the patterns of association in bivariate categorical data displayed in a two-way table. (8.DSP.A.4)

- Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.
- Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

**Unit 7 – EXPONENTS & SCIENTIFIC NOTATION**

In grade 6, you studied whole-number exponents. In this unit, you extend the definition of exponents to include all integers, and in the process codify the properties of exponents. You apply these concepts to the base-ten system, and learn about orders of magnitude and scientific notation in order to represent and compute with very large and very small quantities.

**Major Instructional Goals: **You will know you have achieved the learning goal when you can:

- Know and apply the properties of integer exponents to generate equivalent expressions. (8.EEI.A.1)
- Express very large and very small quantities in scientific notation and approximate how many times larger one is than the other. (8.EEI.A.3)
- Use scientific notation to solve problems. (8.EEI.A.4)

- Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.
- Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

**Unit 8 – PYTHAGOREAN THEOREM & IRRATIONAL NUMBERS**

In this unit, you work with geometric and symbolic representations of square and cube roots. You understand and use notation such as 2 and 53 for square and cube roots. You understand the terms “rational number” and “irrational number,” using long division to express fractions as decimals. You use their understanding of fractions to plot rational numbers on the number line and their understanding of approximation of irrationals by rationals to approximate the number-line location of a given irrational. You learn (without proof) that 2 is irrational. You understand two proofs of the Pythagorean Theorem—an algebraic proof that involves manipulation of two expressions for the same area and a geometric proof that involves decomposing and rearranging two squares. You use the Pythagorean Theorem in two and three dimensions, e.g., to determine lengths of diagonals of rectangles and right rectangular prisms, and to estimate distances between points in the coordinate plane.

**Major Instructional Goals: **You will know you have achieved the learning goal when you can:

- Explore the real number system. (8.NS.A.1)

- Know the differences between rational and irrational numbers.
- Understand that all rational numbers have a decimal expansion that terminates or repeats.
- Convert decimals which repeat into fractions and fractions into repeating decimals.
- Generate equivalent representations of rational numbers.

- Estimate the value and compare the size of irrational numbers and approximate their locations on a number line. (8.NS.A.2)
- Investigate concepts of square and cube roots. (8.EE.A.2)

- Solve equations of the form x2 = p and x3 = p, where p is a positive rational number.
- Evaluate square roots of perfect squares less than or equal to 625 and cube roots of perfect cubes less than or equal to 1000.
- Recognize that square roots of non-perfect squares are irrational.

- Explore the concept of functions. (The use of function notation is not required.) (8.F.A.1)

- Understand that a function assigns to each input exactly one output.
- Determine if a relation is a function.
- Graph a function.

- Use functions to model relationships between quantities. (8.F.B)
- Explore angle relationships and establish informal arguments. (8.G.A.5)

- Derive the sum of the interior angles of a triangle.
- Explore the relationship between the interior and exterior angles of a triangle.
- Construct and explore the angles created when parallel lines are cut by a transversal.
- Use the properties of similar figures to solve problems.

- Use models to demonstrate a proof of the Pythagorean Theorem and its converse. (8.G.B.6)
- Use the Pythagorean Theorem to determine unknown side lengths in right triangles in problems in two- and three-dimensional contexts. (8.G.B.7)
- Use the Pythagorean Theorem to find the distance between points in a Cartesian coordinate system. (8.G.B.8)
- Investigate patterns of association in bivariate data. (8.SP.A)