This year we will be adopting the Desmos middle school math program. It's part computer and part paper, but all amazing, thought provoking mathematics with opportunities for manipulatives and thinking through conversation. All math will be done with a mix of synchronous and asynchronous methods.
Scope and Sequence for the Year (subject to change)
Unit 1: Rigid Transformations and Congruence Transformations: In this section, students learn to visualize, describe, and perform translations, rotations, and reflections on a grid. Students will use properties of reflections, rotations, and translations to prove the criteria for congruence in later grades.
Defining Congruence: In Grade 7, students were asked to sketch geometric shapes that follow specific conditions, and noticed that certain conditions (e.g., knowing all three sides of a triangle) make only one possible triangle. This and the work with transformations lay the groundwork for thinking about congruence. Students determine if two figures are congruent using rigid transformations. In high school, students will describe sequences of transformations more precisely and prove shortcuts for determining congruent triangles.
Applying Congruence: The unit ends as students use transformations to determine missing angle measurements and discover new angle relationships. This builds on students’ work with supplementary, complementary, vertical, and adjacent angles from Grade 7. Students use transformations to informally argue that vertical angles and corresponding angles on parallel lines are congruent. These two angle relationships are used to observe that the sum of the angles in a triangle is `180` degrees. Finally, students use all of these relationships to find missing angle measurements.
Unit 2: Dilations, Similarity, and Introducing Slope DilationsStudents learn to understand and use the term dilation and recognize that a dilation is determined by a point called the center of the dilation and a number called the scale factor. This builds on the work that students did with scaled copies in Grade 7. Students also apply dilations to figures on a grid, which supports them in describing and applying dilations precisely on the coordinate plane.
SimilarityDilations help students make sense of similar figures and properties of similar figures. Students apply all the transformations they know in a new round of Transformation Golf and are introduced to the word similar. Students use a scavenger hunt to examine properties of similar figures, recognizing that angle measures are congruent, but lengths of corresponding sides of a pre-image are multiplied by the scale factor to get the side lengths of the image. This work with similar triangles lays the foundation for trigonometry and other triangle relationships in later grades.
Slope Finally, students learn the terms slope and slope triangle. Students recognize that all slope triangles on the same line are similar and use this fact to find the slopes of water slides. This is then abstracted later where students use slope to decide whether or not a point is on a specific line. The work with slope here will support students in their understanding on linear relationships.
Unit 3: Proportional and Linear Relationships Proportionality Revisited The first section focuses on representations of proportional relationships, building on students’ understanding of rates and proportional relationships in Grade 7. They get informally introduced to concepts used throughout the unit by analyzing different representations of proportional and non-proportional relationships. They consider the many ways to set up and scale axes in order to graph a proportional relationship. This will support students for later where they create graphical and other representations to answer questions about a context.
Slope-Intercept FormStudents build on the work in the first section to identify slope and start writing linear equations.At the end of the previous unit on dilations, students learned the terms slope and slope triangle relating to similar triangles on the same line. The next few lessons build on that knowledge to introduce lines that do not begin at `(0,\ 0)`. Students use the new terms vertical intercept or `y`-intercept where they construct graphs, tables, and equations for non-proportional linear relationships.
Solutions and Standard FormIn this section, students encounter equations of the form `Ax+By=C`, where both variables have to satisfy a constraint. Students explore solutions to linear equations of this form and `y=mx+b` (which they studied earlier). Students solve real-world problems using all of the representations of linear relationships they have studied and explain how real-world constraints affect the limitations of their representations.
Unit 4: Linear Equations and Linear Systems In this unit, students solve linear equations with rational coefficients and determine the number of solutions (i.e., whether there is one solution, no solution, or infinitely many solutions). They also explore and solve systems of two linear equations algebraically and by graphing.
Solving Linear Equations: In Grade 6, students used the distributive property and collected like terms to generate equivalent expressions, as well as reasoned about what it means to solve an equation. In Grade 7, students solved equations with variables on only one side of the equation (e.g., `3(x+8)=30`) and multiplied and divided rational numbers. Here, students extend and apply this knowledge to solve linear equations with variables on both sides of the equation (e.g., `3(x+8)=30+x`). In high school, students will rearrange formulas to highlight a quantity of interest and solve equations with coefficients represented by letters.
Systems of Linear Equations: This section focuses on systems of linear equations in two variables. This builds on students’ work in solving linear equations in the first section and graphing linear relationships in Unit 3. We need to develop the conceptual understanding of what it means to be a solution to a system of two linear equations. This work prepares students to solve systems of more than two equations and those involving non-linear equations in high school.
Unit 5: Functions and Volume In this unit, students learn about functions for the first time, analyze representations of functions, and examine functions in the context of the volume of cylinders, cones, and spheres.
Introduction to Functions: First we build on learning from Units 1 and 3 about the coordinate plane to deepen student intuition about scenarios and the graphs that represent them. Students learn the definition of a function by making predictions about rules, and use the definition to determine whether or not graphs, tables, or rules represent functions. The words independent variable and dependent variable help students describe these relationships.
Representing and Interpreting Functions: The unit shifts to analyzing functions. Students create and interpret graphs of functions that represent stories, examining both individual points and rates of change. They compare information from different representations of functions and consider the advantages and disadvantages of each. Students build on their learning from Unit 3 about linear functions to decide if functions are linear or not. The function section of the unit ends with a brief introduction to piecewise linear functions, where one line is not enough to capture the complexity of a relationship. In future years, students will use their learning from this unit to explore nonlinear functions in more depth.
Volume: Students calculate and compare the volumes of cylinders, cones, and spheres. This builds on students’ knowledge from Grade 7, where they explored the volume of prisms. Students begin with finding the volume of cylinders, then build and use a formula for the volume of a cone and a sphere. We will connect back to Lessons 7 and 8 as students create and analyze height vs. volume functions and radius vs. volume functions. Students use the relationships between height, radius, and volume to calculate missing dimensions. The work in this unit will support students in solving problems and exploring cross sections of solids in high school.
Unit 6: Associations in Data In this unit, students analyze bivariate data. Students use scatter plots and fitted lines to analyze numerical data. They use two-way tables, bar graphs, and segmented bar graphs to analyze categorical data.
Organizing Numerical Data: In Grade 6, students analyzed data in one variable, looking for its shape, center, spread, and outliers. This unit builds on students’ experience in earlier units with the coordinate plane and linear functions to focus on data in two variables. In this section, students examine different ways to organize bivariate data, including scatter plots.
Analyzing Numerical Data: The second section focuses on using scatter plots and fitted lines to analyze numerical data and identify associations. Students interpret the meaning of individual points on a scatter plot, which builds to using scatter plots and models to make predictions about unknown data. This section ends with students creating and analyzing their own model of the relationship between brain weight and body weight of animals. In high school, students will describe associations more precisely using correlation coefficients, and they will model bivariate data with nonlinear functions.
Categorical Data: The third section focuses on using two-way tables, frequency tables, and segmented bar graphs to identify associations in categorical data. I introduce two-way tables as a way to organize categorical data. Students compare data represented this way to scatter plots from earlier in the unit and to bar graphs, which are familiar to students from elementary school. Students look for associations between categories by considering the advantages of a relative frequency table that is based on percentages. Students also examine and create segmented bar graphs to visualize any associations. The unit ends with students using all of their skills to analyze data about federal budgets. Students will continue to summarize categorical data and recognize associations or trends in data in high school.
Unit 7: Exponents and Scientific Notation In this unit, students gain fluency with expressions involving exponents, powers of ten, and scientific notation. They also perform operations on numbers written in scientific notation.
Exponent Properties: Students were introduced to exponents in Grade 6, where they worked with expressions involving positive whole number exponents. Students build on this understanding and use exponents to describe repeated multiplication. Students identify and write equivalent exponential expressions involving the product of powers and powers of powers, with a focus on justifying why two expressions are equivalent. Students continue to consider whether two exponent expressions are equivalent, rewriting expressions using a single power as a strategy to compare numbers. They gain fluency with multiplying powers, dividing powers, using powers of powers, using zero and negative exponents, and combining bases.
Scientific Notation: In this section, students explore powers of `10` and are introduced to scientific notation. Students use non-integer multiples of powers of `10` to write numbers in equivalent forms (e.g., `75x105` can be written as `7.5x106`), placing these numbers on the number line later. Students will apply this learning and notice that some forms may be more helpful than others. They are introduced to the term “scientific notation”, where they practice distinguishing scientific notation from other ways of writing numbers. Students add, subtract, multiply, and divide numbers written in scientific notation to compare quantities. The unit ends where students use all of their skills to estimate quantities about celebrities’ net worths. Students will continue their work with exponents by investigating properties of rational exponents in high school.
Unit 8: The Pythagorean Theorem and Irrational Numbers In this unit, students examine square root and cube roots, and explore the Pythagorean theorem. They also classify rational and irrational numbers.
Square Roots and Cube Roots: In this section, students work with geometric and symbolic representations of square and cube roots. Students build on strategies from Grade 6 to find the area of any tilted square on a grid. The relationship between the area of a square or volume of a cube and its side length is the basis for the concept of square and cube roots. In addition to the concept and notation, students use various strategies to approximate the value of square roots and cube roots. In future grades, students will use square and cube roots in more complex functions and in equations that use exponents (e.g., `16=x^{2}`).
The Pythagorean Theorem: The unit builds on strategies students developed with tilted squares to explore side lengths of right triangles. Students explore the relationships between areas of squares attached to triangles, which serves as an informal introduction to the Pythagorean theorem. Students formalize this understanding by learning a proof of the Pythagorean theorem. Students use the Pythagorean theorem and its converse to reason about right triangles. The section ends with students applying their learning to find distances on the coordinate plane. In high school, students will use the Pythagorean theorem and other relationships among angles and sides of right triangles to solve right triangles in applied problems.
Rational and Irrational Numbers: Students have been using square roots and cube roots. Students broaden their scope by classifying numbers: rational and irrational. In Grade 7, students used long division in order to write fractions as decimals and learned that such decimals either repeat or terminate. In this unit, students convert between fractions and decimal representations of rational numbers. The unit ends with an investigation of rational numbers (which can be written as fractions) and irrational numbers (which cannot). Students will investigate imaginary numbers in high school.