Unit 1: Equations, Transformations, Congruence, and Similarity
Students develop the concept of transformations and the effects that each type of transformation has on an object by exploring the relationship between the original figure and its image in regards to their corresponding parts being moved an equal distance which leads to concept of congruence of figures. They learn to describe transformations with both words and numbers. Students relate rigid motions to the concept of symmetry and use them to prove congruence or similarity of two figures. They physically manipulate figures to discover properties of similar and congruent figures; and focus on the sum of the angles of a triangle and use it to find the measures of angles formed by transversals (especially with parallel lines), find the measures of exterior angles of triangles, and to informally prove congruence.
Unit 2: Exponents
Students deepen their understanding of real numbers to include irrational numbers by exploring and applying exponents and square roots. Students will investigate algorithms for computing with exponents and square roots, as well as scientific notation.
Unit 3: Geometric Applications of Exponents
Students understand the statement of the Pythagorean Theorem and its converse, and can explain why the Pythagorean Theorem holds, for example, by decomposing a square in two different ways. They apply the Pythagorean Theorem to find distances between points in a coordinate system, to find lengths, and to analyze polygons. Students continue their understanding of volume by solving problems involving cones, cylinders, and spheres.
Unit 4: Functions
Students recognize a relationship as a function when each input is assigned to exactly one output, and reason from a context, a graph, or a table, after first being clear which quantity is considered the input and which is the output. They produce a counterexample: an "input value" with at least two "output
values" when a relationship is not a function, explain how to verify that for each input there is exactly one output; and translate functions numerically, graphically, verbally, and algebraically.
Unit 5: Linear Functions
This unit focuses on extending the understanding of ratios and proportions. Unit rates have been explored in Grade 6 as the comparison of two different quantities with the second unit a unit of one, (unit rate). In seventh grade unit rates were expanded to complex fractions and percents through solving
multi-step problems such as: discounts, interest, taxes, tips, and percent of
increase or decrease. Proportional relationships were applied in scale drawings, and students should have developed an informal understanding that the steepness of the graph is the slope or unit rate. Now unit rates are addressed formally in graphical representations, algebraic equations, and geometry through similar triangles. The first part of this unit will focus on compare two different proportional relationships represented in different ways, graphing proportional relationships, and interpreting unit rate as the slope. This unit continues when students will build on their knowledge of proportional relationships using similar triangles to explain why the slope is the same between any two points on a non-vertical line. They will derive the equation y = mx for a line through the origin; and y = mx + b for a line intercepting the vertical axis at b; and interpret equations in y = mx + b form as linear functions.
Unit 6: Linear Models & Tables
Students are given opportunities and examples to figure out the meaning of
y = mx + b. They will be able to "see" m and b in graphs, tables, and formulas or equations, and they will interpret those values in contexts. They identify the rate of change and the initial value from tables, graphs, equations, or verbal descriptions, write a model for a linear function, and sketch a graph when given a verbal description of a situation. They will analyze scatter plots, informally develop a line of best fit, use bivariate data to create graphs and linear models; and recognize patterns and interpret bivariate data.
Unit 7: Solving Systems of Equations
This unit extends solving equations to understanding solving systems of equations, or a set of two or more linear equations that contain one or both of the same two variables. Once again the focus is on a solution to the system. Student experiences are with numerical and graphical representations of
solutions. Beginning work involves systems of equations with solutions that are ordered pairs of integers, making it easier to locate the point of intersection, simplify the computation, and hone in on finding a solution. More complex systems are investigated and solved by using graphing technology. Students understand the solution to a system of equations is the point of intersection when the equations are graphed, and contains the values that satisfy both equations. They find the solution to a system of equations algebraically, and estimate the solution by graphing. Students understand that parallel lines have the same slope but never intersect; therefore, have no solution; and lines that are co-linear share all of the same points; therefore, they have infinitely many solutions. They apply knowledge of systems of equations to real-world situations.