# Medical LakeMiddle School # Course Information and Syllabus

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. They wrap up the year with a study of the volumes of cones, cylinders, and spheres.

8th Grade Math Course Syllabus (Illustrative Mathematics)

Unit 1: Rigid Transformations and Congruence

Work with transformations of plane figures in grade 8 draws on earlier work with geometry and geometric measurement. Students began to learn about two- and three-dimensional shapes in kindergarten, and continued this work in grades 1 and 2, composing, decomposing, and identifying shapes. Students’ work with geometric measurement began with length and continued with area. Students learned to “structure two-dimensional space,” that is, to see a rectangle with whole-number side lengths as composed of an array of unit squares or composed of iterated rows or iterated columns of unit squares. In grade 3, students distinguished between perimeter and area. They connected rectangle area with multiplication, understanding why (for whole-number side lengths) multiplying the side lengths of a rectangle yields the number of unit squares that tile the rectangle. They used area diagrams to represent instances of the distributive property. In grade 4, students applied area and perimeter formulas for rectangles to solve real-world and mathematical problems, and learned to use protractors. In grade 5, students extended the formula for the area of rectangles to rectangles with fractional side lengths. In grade 6, students combined their knowledge of geometry and geometric measurement to produce formulas for the areas of parallelograms and triangles, using these formulas to find surface areas of polyhedra. In grade 7, students worked with scaled copies and scale drawings, learning that angle measures are preserved in scaled copies, but areas increase or decrease proportionally to the square of the scale factor. Their study of scaled copies was limited to pairs of figures with the same rotation and mirror orientation. Viewed from the perspective of grade 8, a scaled copy is a dilation and translation, not a rotation or reflection, of another figure.  In grade 8, students extend their reasoning to plane figures with different rotation and mirror orientations.

Unit 2: Dilations, Similarity, and Introducing Slope

Work with transformations of plane figures in grade 8 builds on earlier work with geometry and geometric measurement, using students’ familiarity with geometric figures, their knowledge of formulas for the areas of rectangles, parallelograms, and triangles, and their abilities to use rulers and protractors. Grade 7 work with scaled copies is especially relevant. This work was limited to pairs of figures with the same rotation and mirror orientations (i.e. that are not rotations or reflections of each other). In grade 8, students study pairs of scaled copies that have different rotation or mirror orientations, examining how one member of the pair can be transformed into the other, and describing these transformations. Initially, they view transformations as moving one figure in the plane onto another figure in the plane. As the unit progresses, they come to view transformations as moving the entire plane.

Unit 3: Linear Relationships

Work with linear relationships in grade 8 builds on earlier work with rates and proportional relationships in grade 7, and grade 8 work with geometry. At the end of the previous unit on dilations, students learned the terms “slope” and “slope triangle,” used the similarity of slope triangles on the same line to understand that any two distinct points on a line determine the same slope, and found an equation for a line with a positive slope and vertical intercept. In this unit, students gain experience with linear relationships and their representations as graphs, tables, and equations through activities designed and sequenced to allow them to make sense of problems and persevere in solving them (MP1). Because of this dependency, this unit and the previous one should be done in order.

Unit 4: Linear Equations and Linear Systems

In this unit, students build on their grades 6 and 7 work with equivalent expressions and equations with one occurrence of one variable, learning algebraic methods to solve linear equations with multiple occurrences of one variable. Students learn to use algebraic methods to solve systems of linear equations in two variables, building on their grades 7 and 8 work with graphs and equations of linear relationships. Understanding of linear relationships is, in turn, built on the understanding of proportional relationships developed in grade 7 that connected ratios and rates with lines and triangles.

Unit 5: Functions and Volume

In this unit, students are introduced to the concept of a function as a relationship between “inputs” and “outputs” in which each allowable input determines exactly one output. In the first three sections of the unit, students work with relationships that are familiar from previous grades or units (perimeter formulas, proportional relationships, linear relationships), expressing them as functions. In the remaining three sections of the unit, students build on their knowledge of the formula for the volume of a right rectangular prism from grade 7, learning formulas for volumes of cylinders, cones, and spheres. Students express functional relationships described by these formulas as equations. They use these relationships to reason about how the volume of a figure changes as another of its measurements changes, transforming algebraic expressions to get the information they need (MP1).

Unit 6: Associations in Data

In this unit, students analyze bivariate data—using scatter plots and fitted lines to analyze numerical data, and using two-way tables, bar graphs, and segmented bar graphs to analyze categorical data.

Unit 7: Exponents and Scientific Notation

Students were introduced to exponent notation in grade 6. They worked with expressions that included parentheses and positive whole-number exponents with whole-number, fraction, decimal, or variable bases, using properties of exponents strategically, but did not formulate rules for use of exponents.

Unit 8: Pythagorean Theorem and Irrational Numbers

Work in this unit is designed to build on and connect students’ understanding of geometry and numerical expressions. The unit begins by foreshadowing algebraic and geometric aspects of the Pythagorean Theorem and strategies for proving it. Students are shown three squares and asked to compare the area of the largest square with the sum of the areas of the other two squares. The comparison can be done by counting grid squares and comparing the counts—when the three squares are on a grid with their sides on grid lines and vertices on intersections of grid lines—using the understanding of area measurement established in grade 3. The comparison can also be done by showing that there is a shape that can be decomposed and rearranged to form the largest square or the two smallest squares. Students are provided with opportunities to use and discuss both strategies.

Unit 9: Putting it All Together

This section consists of two optional lessons in which students solve complex problems. In the first, they investigate relationships of temperature and latitude, climate, season, cloud cover, or time of day. In particular, they use scatter plots and lines of best fit to investigate the question of modeling temperature as a function of latitude. In the second, they consider tessellations of the plane, understanding and using the terms “tessellation” and “regular tessellation” in their work, and using properties of shapes (e.g., the sum of the interior angles of a quadrilateral is 360 degrees) to make inferences about regular tessellations.