Topic outline

  • Helpful Tools

    Instructor:  Mr. Roed

    Phone - 989-828-6601

    Email: - Preferred

    Classroom location: Shepherd Middle School Room 308

    Office Hours: After school Monday through Thursday by appointment.

    Questions: Please post questions about the course in the Q&A room under the helpful tools section. If it is a personal matter, please email me. I will typically respond within one business day of receiving your message. (Business days are Monday thru Friday, not including holidays)

  • Thinking with Mathematical Models

    Unit Goals 

    Students will review, extend their understanding of, and improve their skills in working with linear functions and equations. The Unit also introduces concepts associated with nonlinear functions.

    Algebraic functions that represent patterns in experimental data are called mathematical models. Students will use these functions to estimate answers to questions about relationships in the data.

    This Unit also introduces inverse variation. Students work with inverse variations in several real-world contexts. The Unit also develops student understanding of associations between variables using basic ideas of correlation and two-way tables. 

    Link to Parent Letter


    Pages: 23Quizzes: 2
  • Looking for Pythagoras

    In Looking for Pythagoras, students explore two big ideas: the Pythagorean Theorem and real numbers. In the process of solving the Problems in this Unit, students also review and make connections among the concepts of area, distance, and irrational numbers.

    Students begin the Unit by finding the distance between points on a coordinate grid. They explore the areas of figures drawn on a dot grid, and they learn that the positive square root of a number is the side length of a square with an area equal to that number. Students also learn that the cube root of a number is the side length of a cube with a volume equal to that number. Then students discover the Pythagorean relationship through an exploration of squares drawn on the sides of a right triangle. The application of the Pythagorean Theorem often leads to square roots, which are not rational numbers. This is the motivation for students to sort the numbers they know, the set of real numbers, into rational numbers and irrational numbers. In the last Investigation of the Unit, students apply the Pythagorean Theorem to a variety of Problems.

    Link to Textbook

    Pages: 25Quizzes: 2
  • Say it with Symbols

    Traditionally, the goal of Algebra instruction has been to develop of students’ proficiency in working with expressions and equations. This work includes simplifying, factoring, expanding, evaluating, or solving expressions and equations. In addition to these traditional tasks, Say It With Symbols develops the understanding of using symbolic expressions to represent and reason about relationships. The students will write and interpret equivalent expressions, combine expressions to form new expressions, predict patterns of change represented by an equation or expression, and solve equations. This manipulation of symbolic expressions is explored using the properties of equality and the Distributive and Commutative Properties. They will see that the properties reveal new information about a given context by critically examining each part of an expression and explaining how each part relates to the original expression.

    This Unit also puts emphasis on multiple representations, such as graphic, tabular, and symbolic representations. Students examine the graph and table of an expression as well as the context the expression models. Having access to graphing calculators and computers will provide students with a natural focus on functions and modeling patterns of quantitative change. If you are teaching this Unit for Grade 8 content only, you may skip Problems 3.3, 3.4, 4.2, 4.3, 4.4 and 5.3. However, parts of Problem 4.4 should be assigned as students need experience in recognizing which function is needed to solve the problem.



    • Develop understanding of equivalent expressions and equations
    • Model situations with symbolic statements
    • Recognize when two or more symbolic statements represent the same context
    • Use the properties of real numbers, such as the Distributive Property, to write equivalent expressions
    • Determine if different symbolic expressions are mathematically equivalent
    • Interpret the information that equivalent expressions represent in a given context
    • Determine the equivalent expression or equation that is most helpful in answering a particular question about a relationship
    • Use algebraic equations to describe the relationship among the volumes of cylinders, cones and spheres that have the same height and radius
    • Solve linear equations involving parentheses
    • Determine if a linear equation has a finite number of solutions, an infinite number of solutions, or no solution
    • Develop understanding and some fluency with factoring quadratic expressions
    • Solve quadratic equations by factoring
    • Recognize how and when to use symbols, rather than tables or graphs, to display relationships, generalizations, and proofs


    • Develop an understanding of specific functions such as linear, exponential and quadratic functions
    • Develop proficiency in identifying and representing relationships expressed in problem contexts with appropriate functions and use these relationships to solve the problem
    • Analyze equations to determine the patterns of change in the tables and graphs that the equations represent
    • Relate parts of a symbolic statement or expression to the underlying properties of the relationship they represent and to the context of the problem
    • Determine characteristics of a graph (intercepts, maxima and minima, shape, etc.) of an equation by looking at its symbolic representation


    Parent Letter

    Pages: 26Quizzes: 2
  • Growing, Growing, Growing

    This Unit continues the discussion of functions by examining exponential functions. Models of exponential growth and decay are numerous such as growth or decay of populations—from bacteria, amoebas, radioactive material and money, to mammals (including people). Doubling, tripling, halving, and so on, are all intuitive situations for students to help them make sense of exponential functions.

    The growth pattern in exponential functions is multiplicative. That is, for each additive change in the independent variable, there is a multiplicative change in the dependent variable. For example, in Problem 1.1, students look at the number of ballots created by repeatedly cutting in half a sheet of paper. As the number of cuts increases by one, the number of ballots increases by a factor of two. This factor is called the growth factor.

    Investigation 1 continues to look at doubling, tripling, and quadrupling patterns. It ends by contrasting linear and exponential growth factors. Investigation 2 introduces the y‑intercept, or initial value, which it is sometimes called in exponential growth situations. Investigation 3 introduces growth rates that are not whole numbers and leads to growth rates, usually expressed as percents. Investigation 4 introduces growth factors that are less than one, but greater than zero. These are exponential decay situations. In Investigation 5, patterns with exponents are explored.

    The Unit ends by looking at the effects of growth factors and y‑intercepts on graphs of exponential functions. Since exponential growth patterns can grow rapidly, students may encounter answers on their calculators expressed in scientific notation. Therefore, scientific notation is introduced in Investigation 1 and used throughout the Unit.

    Student Textbook

    Parent Letter

    Pages: 23Quizzes: 2
  • It's in the System

    There are two overarching goals of this Unit: to develop student understanding of the methods in which systems of equations and inequalities with two variables can be used to model problem situations; and to develop skills in the graphic and symbolic methods needed to solve those systems. Progress toward these goals is supported by the Problems of the four Investigations. The first Investigation focuses on graphic methods for solving equations in the form Ax+By=C. The second Investigation develops graphic and symbolic methods for solving systems of two such linear equations. The third Investigation develops methods for solving linear and quadratic inequalities in one variable. The fourth Investigation develops concepts and solution methods for solving systems of linear inequalities in the standard form Ax+By<C.



    Parent Letter

    Pages: 22Quizzes: 2
  • Butterflies, Pinwheels, and Wallpaper

    The overarching goal of Butterflies, Pinwheels, and Wallpaper is to develop student understanding of congruence and similarity of geometric figures, and the mathematical techniques for finding and applying those relationships of shapes. The basic idea of congruence is that two figures have the same shape and size if it is possible to perform one or more transformations that “move” one figure onto the other. The basic idea of similarity is that two figures have the same shape if it is possible to perform a dilation, and perhaps one or more rigid motions, to transform one figure onto the other.

    The two main topics of this Unit that highlight congruence and similarity are rigid motions and dilation. Investigations 1–3 develop and apply properties of line reflections, rotations, and translations. These rigid motions are used to transform figures for creating symmetric designs and to compare the size and shape of congruent figures. Investigation 4 extends the transformation concept to include dilations of similar figures.

    Butterflies, Pinwheels, and Wallpaper builds on important prior work in the Grade 7 Units Shapes and Designs and Stretching and Shrinking. It also makes significant connections to two prior Units on measurement, Grade 6 Covering and Surrounding and Grade 7 Filling and Wrapping.

    Student Edition

    Parent Letter

    Pages: 21Quizzes: 2