CA CCSSM Standards:
8.EE.1 — Know and apply the properties of integer exponents to generate equivalent numerical expressions (e.g., 3² × 3⁻⁵ = 3⁻³ = 1/27).
8.EE.3 — Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantities, and express how many times as much one is than the other.
8.EE.4 — Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used; choose units of appropriate size; interpret scientific notation generated by technology.

Learning goal: Students develop a durable mental model of magnitude — understanding that exponent rules are shortcuts for counting factors of the base, and that scientific notation is the language scientists use to work across scales from atoms to galaxies. Rather than memorizing rules, students discover them through structured exploration and then apply them fluently to real quantities.

Students will be able to:

• Apply the product, quotient, power, zero-exponent, and negative-exponent rules to generate equivalent expressions, and explain why each rule works.
• Convert numbers between standard form and scientific notation in both directions, including very small decimals.
• Compare two quantities expressed as powers of 10 and state how many times as large one is than the other.
• Add, subtract, multiply, and divide numbers in scientific notation, including problems mixing decimal and scientific forms.
• Interpret calculator/computer notation (e.g., 4.5E8) and choose units of appropriate size for measurements of very large or very small quantities.

CA CCSSM Standards:
8.NS.1 — Know that numbers that are not rational are called irrational; understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert repeating decimals into fractions.
8.NS.2 — Use rational approximations of irrational numbers to compare their size, locate them approximately on a number line diagram, and estimate the value of expressions (e.g., π²).
8.EE.2 — Use square root and cube root symbols to represent solutions to equations of the form x² = p and x³ = p, where p is a positive rational number; evaluate square roots of small perfect squares and cube roots of small perfect cubes; know that √2 is irrational.

Learning goal: Students expand their concept of number beyond the rationals, understanding irrationality through decimal behavior (never terminating, never repeating) and learning that irrational numbers still have precise locations on the number line that can be trapped between rational approximations to any desired accuracy.

Students will be able to:

• Classify a given number as rational or irrational and justify the classification using its decimal expansion or form.
• Convert a repeating decimal (e.g., 0.4̄7̄) into a fraction using an algebraic argument.
• Approximate a square or cube root to the tenths and hundredths place by testing squares/cubes of successive values.
• Order a mixed set of rational and irrational numbers (e.g., π, √10, 3.5, 22/7) on a number line.
• Solve equations of the form x² = p and x³ = p using root symbols, and evaluate roots of perfect squares and cubes mentally.

CA CCSSM Standards:
8.EE.5 — Graph proportional relationships, interpreting the unit rate as the slope of the graph; compare two different proportional relationships represented in different ways.
8.EE.6 — Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane; derive the equation y = mx for a line through the origin and y = mx + b for a line intercepting the vertical axis at b.
8.EE.7 — Solve linear equations in one variable, including cases with one solution, infinitely many solutions, or no solutions (8.EE.7a), and equations requiring the distributive property and collecting like terms (8.EE.7b).
8.EE.8 — Analyze and solve pairs of simultaneous linear equations: understand solutions as intersection points (8.EE.8a), solve systems algebraically and estimate solutions by graphing (8.EE.8b), and solve real-world problems leading to two linear equations in two variables (8.EE.8c).

Learning goal: Students unify their understanding of proportionality, slope, and equations into a coherent picture of linearity: slope is a rate you can see, the equation y = mx + b describes every point on a line at once, and a system of equations asks where two linear stories intersect. Students also learn that equations can have one, no, or infinitely many solutions — and what each case means.

Students will be able to:

• Graph a proportional relationship and interpret its unit rate as the slope; compare two relationships given in different representations (graph vs. table vs. equation).
• Use similar slope triangles to explain why slope is constant along a line, and derive y = mx and y = mx + b.
• Solve multi-step linear equations in one variable, including those requiring the distributive property and collecting like terms on both sides.
• Classify a linear equation as having one solution, no solution, or infinitely many solutions, and justify the classification.
• Solve a system of two linear equations by graphing (identifying the intersection) and algebraically (substitution or elimination).
• Translate a real-world scenario (e.g., comparing phone plans, counting vehicles from total wheels) into a system of equations and interpret the solution in context.

CA CCSSM Standards:
8.F.1 — Understand that a function is a rule that assigns to each input exactly one output; the graph of a function is the set of ordered pairs consisting of an input and the corresponding output.
8.F.2 — Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions).
8.F.3 — Interpret the equation y = mx + b as defining a linear function whose graph is a straight line; give examples of functions that are not linear.
8.F.4 — Construct a function to model a linear relationship between two quantities; determine the rate of change and initial value from a description, two (x, y) values, a table, or a graph; interpret them in terms of the situation.
8.F.5 — Describe qualitatively the functional relationship between two quantities by analyzing a graph (increasing, decreasing, linear, nonlinear); sketch a graph that exhibits the qualitative features of a function described verbally.

Learning goal: Students meet the function concept — the single most important idea bridging middle school and high school mathematics — as a reliable input–output rule. They learn to move fluently among equations, tables, graphs, and verbal descriptions of the same function, and to read the story a graph tells about a real situation.

Students will be able to:

• Determine whether a relation (given as a table, mapping, set of pairs, or graph) is a function, and justify using the definition.
• Compare two functions presented in different forms (e.g., an equation vs. a table vs. a verbal description) by rate of change and initial value.
• Identify functions as linear or nonlinear from equations, tables, and graphs, giving examples of each.
• Construct a linear function y = mx + b from a story, a table, a graph, or two points, and interpret m and b in context.
• Describe a graph qualitatively (where it increases, decreases, stays constant, is linear or nonlinear) and sketch a graph from a verbal description of a real situation.

CA CCSSM Standards:
8.G.1 — Verify experimentally the properties of rotations, reflections, and translations: lines map to lines, segments to segments of the same length, angles to angles of the same measure, and parallel lines to parallel lines.
8.G.2 — Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe such a sequence.
8.G.3 — Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.
8.G.4 — Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar figures, describe such a sequence.
8.G.5 — Use informal arguments to establish facts about the angle sum and exterior angle of triangles, the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Learning goal: Students learn that geometry can be built on motion: congruence means one figure can be carried onto another by rigid motions, and similarity means the same with a dilation allowed. This transformational view — the foundation of high school geometry — is verified experimentally before being used to argue about angles and similar triangles.

Students will be able to:

• Perform and identify translations, reflections, rotations, and dilations of figures on the coordinate plane.
• Write and apply coordinate rules for transformations, such as (x, y) → (x + 3, −y) or (x, y) → (2x, 2y).
• Verify by measurement that rigid motions preserve lengths, angle measures, and parallelism, while dilations preserve angles but scale lengths.
• Prove two figures congruent or similar by exhibiting an explicit sequence of transformations carrying one onto the other.
• Construct informal arguments for the 180° triangle angle sum, the exterior angle relationship, transversal angle pairs (alternate interior/exterior, corresponding), and the AA criterion for triangle similarity.

CA CCSSM Standards:
8.G.6 — Explain a proof of the Pythagorean Theorem and its converse.
8.G.7 — Apply the Pythagorean Theorem to determine unknown side lengths in right triangles in real-world and mathematical problems in two and three dimensions.
8.G.8 — Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
8.G.9 — Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Learning goal: Students don’t just use the Pythagorean Theorem — they can explain why it is true and use its converse as a test. They apply it as a practical measuring tool (ladders, distances, diagonals) and extend measurement into three dimensions with the volume formulas for cylinders, cones, and spheres, understanding the ⅓ and 4/3 factors through direct comparison rather than memorization.

Students will be able to:

• Explain a visual or algebraic proof of the Pythagorean Theorem in their own words.
• Use the converse to determine whether three given side lengths form a right triangle.
• Find an unknown leg or hypotenuse in applied problems (ladders, sails, ramps, diagonals of rectangles and boxes), rounding appropriately.
• Compute the distance between any two points on the coordinate plane using the Pythagorean Theorem.
• Apply V = πr²h, V = ⅓πr²h, and V = 4/3πr³ to solve real-world volume problems, and explain how the cone and sphere formulas relate to the cylinder.

CA CCSSM Standards:
8.SP.1 — Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities; describe patterns such as clustering, outliers, positive or negative association, linear and nonlinear association.
8.SP.2 — Know that straight lines are widely used to model relationships between two quantitative variables; for scatter plots that suggest a linear association, informally fit a straight line and informally assess the model fit by judging the closeness of the data points to the line.
8.SP.3 — Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.
8.SP.4 — Understand that patterns of association can also be seen in bivariate categorical data by displaying frequencies and relative frequencies in a two-way table; use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Learning goal: Students learn to see and model relationships between two variables — the heart of data science. They construct scatter plots, describe patterns honestly (including outliers and the limits of a model), fit and use linear models with interpreted slope and intercept, and detect association in categorical data through two-way tables.

Students will be able to:

• Construct a scatter plot from bivariate data and describe its pattern: positive/negative/no association, linear/nonlinear, clusters, and outliers, with plausible explanations for outliers.
• Informally fit a trend line to a linear-looking scatter plot and assess its fit by how close the points lie to the line.
• Write the equation of a fitted line and use it to make predictions, interpreting the slope and intercept in the context of the data.
• Recognize the danger of extrapolating far beyond the data.
• Organize bivariate categorical data into a two-way table, compute row/column relative frequencies, and use them to describe possible association.