In Grade 8, instructional time should focus on three critical areas: (1) formulating and reasoning about expressions and equations, including modeling an association in bivariate data with a linear equation, and solving linear equations and systems of linear equations; (2) grasping the concept of a function and using functions to describe quantitative relationships; (3) analyzing two- and three-dimensional space and figures using distance, angle, similarity, and congruence, and understanding and applying the Pythagorean Theorem.
Math 8 builds on what students have learned about proportional and geometric relationships in Math 7 to develop several key concepts in algebra and geometry. Students start the year with rigid transformations and congruence in Unit 1, which sets them up to learn about similarity and dilations in Unit 2. Students use what they know about similar triangles to explore slope as they study linear relationships in Unit 3. This work with linear relationships builds toward solving linear equations with variables on both sides of the equal sign, and systems of linear equations in Unit 4.
Unit 5 invites students to consider functions, specifically what makes a relationship a function. Unit 5 also explores the volumes of cylinders, cones, and spheres. Unit 6 returns to linear relationships as students explore bivariate data. Unit 7 builds on the exponent work from Math 6 to explore properties of exponents and scientific notation as a tool for representing very large and very small quantities. Math 8 ends with the Pythagorean theorem as students encounter square roots, cube roots, and irrational numbers for the first time.
As students encounter challenging problems, and practice asking for help and a willingness to learn from others, they build the social emotional learning competencies of self-management, self awareness, relationship skills, and social awareness.
Grade Level(s): Eighth grade
Related Priority Standards (State &/or National): K-12 Mathematics Missouri Learning Standards
Course Essential Questions
- How is mathematics used to measure, model and calculate change?
- How can we use graphing and algebra to find the solution to a system of equations?
- What does a solution represent?
- Why is it important to know which variable is the independent variable?
- How do you use patterns to understand data?
- How can I represent very small and large numbers using integer exponents and scientific notation?
- How can the Pythagorean Theorem be used to solve problems?
Enduring Understandings, Goals, & Objectives
- There is often more than one-way to write an equation.
- The rate of change in a linear relationship is represented by the slope of the line representing the relationship.
- Geometry occurs in many situations, from architecture to floor patterns.
- Functions can model relationships between quantities.
- Patterns of data can be used to inform decision making.
- Right triangles have a special relationship among the side lengths which can be represented by a model and a formula.
- Scientific notation is used to represent very large or very small numbers.
Course Scope & Sequence (Units &/or Skills)
Unit 1 - Rigid Transformations
- Students will describe and perform translations, rotations, and reflections on a grid.
- Students will determine whether two figures are congruent using rigid transformations.
- Students will use transformations to determine missing angle measurements and discover new angle relationships.
- Students will identify and describe sequences of transformations that take one figure to another in the context.
- Students will use the terms translation, rotation, and reflection to describe and draw transformations on a grid.
- Students will use coordinates to describe figures and their images under transformations in the coordinate plane.
- Students will communicate precisely about transformations of polygons on a coordinate grid.
- Students will see that parallel lines are taken to parallel lines under any rigid transformations.
- Students will justify that pairs of vertical angles and alternate interior angles on parallel lines are congruent.
- Students will observe that the sum of the interior angles of any triangle is always 180 degrees.
- Students will apply what they’ve learned about angle relationships to informally establish the triangle sum theorem.
- Students will examine and create different patterns of shapes, including tessellations and complex designs that exhibit rotational symmetry.
Unit 2 - Dilations, Similarity, and Introducing Slope
- Students will describe dilations precisely in terms of their center of dilation and scale factor.
- Students will apply dilations to figures on and off of a coordinate grid.
- Students will identify similar figures and properties of similar figures using transformations.
- Students will explain slope in terms of similar triangles on the same line and determine the slopes of lines.
- Students will perform dilations, moving from informal to formal ways of measuring to determine the distances between the center, pre-image, and image in any dilation.
- Students will discover that dilations map line segments to line segments, and polygons to polygons.
- Students will identify and describe sequences of transformations, including dilations, that take one figure to another.
- Students will examine the side lengths and angle measurements of polygons in order to understand similarity.
- Students will examine angle measurements in triangles to determine whether or not two triangles are similar.
- Students will discover that the quotient of two side lengths in one triangle is equal to the quotient of the corresponding side lengths in a similar triangle.
- Students will use slope to help determine if points lie on a particular line.
Unit 3 - Proportional and Linear Relationships
- Students will compare proportional relationships using their equations, tables, and graphs.
- Students will interpret the intercept and slope of the graph or equation of a linear relationship.
- Students will use the concept that a graph represents all solutions of an equation to solve problems.
- Students will scale axes in multiple ways in order to graph proportional relationships.
- Students will compare two different proportional relationships given different representations on them.
- Students will write and interpret equations of lines by considering the slope and vertical intercept.
- Students will write equations of lines using the idea that any line in a plane is a vertical translation of a line through the origin.
- Students will develop a method for calculating the slope of any line given the coordinates of two points on the line.
- Students will write equations of horizontal and vertical lines in addition to lines with positive and negative slopes.
- Students will interpret multiple representations of non-proportional linear relationships in context, including slopes, intercepts, and solutions.
Unit 4 - Linear Equations and Linear Systems
- Students will write and solve equations with multiple occurrences of one variable.
- Students will use graphs and algebraic methods to solve systems of linear equations with two variables.
- Students will create and solve number puzzles that can be represented by linear equations in one variable.
- Students will use hanger diagrams to calculate unknown weights of objects by adding and removing equal items from each side.
- Students will categorize linear equations in one variable based on their structure, and solve equations from each category.
- Students will determine what it means to set two expressions equal to each other and solve for the unknown variable.
- Students will interpret points that lie on one, both, or neither line on a graph of two simultaneous equations in context.
- Students will create and interpret a graph of two lines in context.
- Students will understand that solving a system of equations means finding values of the variables that make both equations true.
- Students will solve a system of equations using algebraic methods.
- Students will identify, describe, and employ strategies for solving linear systems of equations with different features or structures.
Unit 5 - Functions and Volume
- Students will determine whether or not graphs, tables, or rules represent functions.
- Students will create and interpret graphs of functions that represent stories.
- Students will calculate and compare the volumes of cylinders, cones, and spheres.
- Students will use the relationships between height, radius, and volume to calculate missing dimensions.
- Students will write rules based on input-output pairs represented in tables.
- Students will determine whether or not a graph represents a function and explain their reasoning.
- Students will explore the relationship between equations and functions, and understand that an equation may look different depending on how the independent and dependent variables are defined.
- Students will interpret qualitative features of a function and specific points in context.
- Students will draw the graph of a function that represents a real world situation.
- Students will use data points to model a linear function and decide when it is reasonable to model the relationship with a linear function.
- Students will use piecewise functions to model real-world data sets presented as graphs.
- Students will estimate the volume of cylinders, cones, cubes, and spheres.
- Students will explore and use a strategy to calculate the volume of a cylinder.
- Students will use functions to explore how changing a cylinder’s radius or height impacts its volume.
- Students will recognize that the volume of a cone is ⅓ the volume of a cylinder with the same radius and the same height.
- Students will calculate missing dimensions of cylinders and cones given their volume and another dimension.
- Students will develop and use a formula for the volume of a sphere.
Unit 6 - Associations in Data
- Students will examine different ways to organize bivariate data, including scatter plots.
- Students will use scatter plots and fitted lines to analyze numerical data and identify associations.
- Students will use two-way tables and bar graphs to identify associations in categorical data.
- Students will compare and contrast the same data represented by a dot plot and by a scatter plot
- Students will interpret points on a scatter plot in terms of a context and add points to a scatter plot given information about an individual in the population.
- Students will practice creating linear models that match the association of sets of data.
- Students will interpret the slope of a line of fit in context.
- Students will visualize clustering and associations in data.
- Students will analyze and interpret bivariate data in context.
- Students will study categorical data displayed in two-way tables and in bar graphs.
- Students will use relative frequencies displayed in tables and in segmented bar graphs to identify possible associations between variables in data.
Unit 7 - Exponents and Scientific Notation
- Students will identify and create equivalent expressions involving positive, negative, and zero exponents.
- Students will express and perform operations with very large or very small quantities using powers of `10` and scientific notation.
- Students will discover ways to write equivalent exponential expressions involving the product of powers and powers of powers.
- Students will look for and make use of structure to identify equivalent exponent expressions that use powers of powers and products of powers.
- Students will rewrite products of powers, quotients of powers, and powers of powers as single powers.
- Students will develop an understanding of the meaning of zero and negative exponents.
- Students will write rules for simplifying exponential expressions.
- Students will represent large and small numbers using multiples of powers of 10
- Students will apply powers of 10 and exponent rules to solve problems in context.
- Students will multiply and divide numbers expressed in scientific notation, and express how many times as much one quantity is as the other.
- Students will add and subtract numbers expressed in scientific notation and express the resulting sums and differences in scientific notation.
- Students will use scientific notation as a tool for comparing, combining, and operating.
Unit 8 - The Pythagorean Theorem and Irrational Numbers
- Students will understand that square roots and cube roots represent the edge length of squares and cubes, and approximate their values.
- Students will use the Pythagorean theorem and its converse to reason about right triangles and find unknown measurements.
- Students will determine fractions and decimal approximations for rational and irrational numbers.
- Students will estimate side lengths of squares with known areas and review strategies for calculating areas.
- Students will develop an understanding of square roots and square root notation.
- Students will approximate the value of square roots by determining the two integer values it lies between and by drawing a square.
- Students will explore the relationship between the edge length and the volume of a cube.
- Students will identify patterns in the relationship between the squares of side lengths of triangles and learn that the relationship between the side lengths of a right triangle is that Pythagorean theorem.
- Students will use the Pythagorean theorem to calculate side lengths of right triangles.
- Students will develop and apply their understanding of the converse of the Pythagorean theorem.
- Students will use the Pythagorean theorem as a tool to solve problems involving diagonal distances.
- Students will apply the Pythagorean theorem to find distances between points in the coordinate plane.
- Students will explore connections between unit fractions and their decimal representations using long division to convert fractions to decimals.
- Students will develop a strategy for rewriting repeating decimals as fractions.
Course Resources & Materials: DESMOS
Date Last Revised/Approved: 2023