Grade 9

Plot and interpret points in all four quadrants, including points on the axes and with negative coordinates, and calculate horizontal and vertical distances, midpoints, and missing endpoints using coordinate relationships. Calculate and interpret positive, negative, zero, and undefined slopes, determine whether lines are parallel, and write equations of lines from a slope and y-intercept or from two points. Use coordinate methods to solve geometric problems involving collinearity, lengths, areas, and properties of triangles and quadrilaterals.

Calculate and interpret mean, median, mode, range, quartiles, interquartile range, and outliers, and construct and interpret box-and-whisker plots to compare center and spread while considering the effect of extreme values. Distinguish categorical, numerical, discrete, and continuous data; identify populations, samples, and variables; evaluate sampling methods for bias; and recognize misleading data displays and statistical claims. Construct scatter plots, describe the direction and strength of associations, draw and use informal lines of best fit for predictions within or near the observed range, and distinguish interpolation from less reliable extrapolation without inferring causation.

Students calculate discounts, markups, sales taxes, tips, percent increases, and percent decreases using appropriate original amounts, decimal multipliers, sequential changes, and accurate money rounding. They determine gross earnings and pay from hourly wages, salaries, commissions, bonuses, and straightforward overtime, interpret payroll deductions, and calculate net income from fixed-dollar and percentage deductions. They create and analyze personal budgets by classifying expenses, calculating totals and savings, comparing financial options using rates and fees, and applying simple interest $I=Prt$ to interpret principal, rate, time, interest, and total amount.

Apply angle relationships at intersections and with parallel lines cut by a transversal, including vertical, corresponding, alternate, and same-side interior angles, linear pairs, perpendicular lines, and simple equations for unknown measures. Use triangle and quadrilateral properties, including angle sums, exterior-angle relationships, side and diagonal relationships, and specialized properties of rectangles, rhombi, squares, parallelograms, and isosceles trapezoids. Apply interior- and exterior-angle sums to convex polygons, find angles of regular polygons, and justify unknown-angle solutions from marked diagrams using symbolic and written geometric reasoning rather than visual appearance.

Students solve one-variable linear equations using inverse operations, the distributive property, and combining like terms, including equations with integers, terminating decimals, fractions, parentheses, and variables on both sides. They preserve equality, recognize when an equation has one solution, no solution, or infinitely many solutions, and verify proposed solutions by substitution into the original equation. Students translate real-world relationships into linear equations, identify variables, rates, initial values, and units, connect verbal, tabular, graphical, and symbolic representations, and interpret whether solutions are meaningful in context.

Students identify independent and dependent variables and represent linear relations using tables, graphs, and equations in two variables, including slope-intercept form $y=mx+b$. They determine and interpret constant rates of change from tables, graphs, equations, and contexts, including positive, negative, zero, and fractional slopes with appropriate units, and determine $y$-intercepts from graphs and equations. Students write equations from tables and graphs, graph slope-intercept equations, and compare multiple linear relations by analyzing rates of change, initial values, outputs for shared inputs, and points of intersection.

Calculate the circumference and area of circles using radius or diameter, exact or approximated $\pi$, and determine the perimeter and area of composite plane figures by partitioning shapes, inferring missing lengths, and distinguishing boundaries from regions. Calculate the surface area and volume of prisms and right circular cylinders, using nets and formulas, appropriate linear, square, or cubic units, and distinguish exposed surfaces from shared or internal boundaries when solving problems involving composite solids. Convert decimal measurements among common metric units of length, area, and volume, including millimetres, centimetres, metres, kilometres, and litres, applying squared conversion factors for area, cubed factors for volume, and $1\text{ L}=1\text{ dm}^3$.

Students identify terms, coefficients, constants, variable parts, and like terms in one-variable polynomials with numerical coefficients and nonnegative integer exponents, and classify polynomials by degree and number of terms. Students add, subtract, and simplify polynomials by combining like terms, apply the distributive property to expand products of monomials and manageable polynomials, multiply a polynomial by a monomial, and recognize equivalent expressions in standard form. Students factor out a common monomial by reversing the distributive property, using appropriate numerical factors and variable exponents while managing signs, parentheses, and integer or rational coefficients.

Evaluate powers with integer exponents, including zero and negative exponents, for integers, fractions, and terminating decimals, while attending to parentheses, signs, and order of operations. Apply exponent laws for products, quotients, powers of powers, products, and quotients to simplify numerical and algebraic expressions with integer exponents, including zero and negative exponents for nonzero bases. Determine exact principal square and cube roots of familiar perfect powers and estimate non-perfect square roots using nearby perfect squares, bounding values to the tenths or hundredths when appropriate.

Represent finite sample spaces for simple one- and multi-step experiments using organized lists, tables, and tree diagrams, and calculate theoretical probabilities by counting equally likely favorable outcomes out of all outcomes. Determine probabilities of complementary, mutually exclusive, and independent compound events using complements, addition, and multiplication, including simple “at least one” situations while excluding conditional probability and dependent events. Calculate experimental probabilities and use simulations to estimate them, then compare experimental and theoretical values in fractions, decimals, percentages, tables, or graphs while explaining random variation and the stabilizing effect of more trials.

Classify and interpret natural, whole, integer, rational, irrational, and real numbers, including their containment relationships, decimal forms, signs, magnitudes, units, and contextual meanings. Compare, order, and locate real numbers on a number line, including integers, fractions, terminating or repeating decimals, square roots, and familiar irrational numbers such as $\sqrt{2}$, $\sqrt{3}$, and $\pi$, using rational bounds and appropriate decimal approximations. Evaluate real-number expressions by applying grouping symbols, exponents, roots, and operations in the correct order, distinguish exact values from estimates, recognize undefined operations, and solve contextual problems while checking units, signs, magnitude, and reasonableness.

Students write and interpret systems of two linear equations in two variables, determining whether an ordered pair satisfies both equations simultaneously and representing the solution as the intersection of their graphs. They solve systems by graphing, estimate intersection coordinates when needed, and distinguish one solution, no solution from distinct parallel lines, and infinitely many solutions from coincident lines. They model comparison and other contextual situations with systems, interpret solutions using appropriate units and meanings, and apply practical restrictions such as nonnegative or whole-number quantities.

Apply coordinate rules to translate, reflect across the coordinate axes or horizontal and vertical lines, and rotate figures about the origin by $90^\circ$, $180^\circ$, or $270^\circ$, recognizing that rigid transformations preserve lengths, angle measures, shape, and congruence. Determine positive dilation scale factors from corresponding lengths or coordinates and dilate figures centered at the origin using $(x,y)\mapsto(kx,ky)$, distinguishing enlargements and reductions from rigid transformations. Perform ordered sequences of translations, reflections, rotations, and dilations to track figures and establish congruence through rigid motions or similarity through rigid motions followed by a dilation.