Grade 10

Students calculate Euclidean distances between points and determine midpoints or missing endpoints of line segments using coordinate formulas, including integer, decimal, fractional, and radical values. They use slopes to identify parallel and perpendicular lines and write equations of lines through specified points, including horizontal and vertical cases. They apply coordinates, distance, midpoint, slope, and line equations to solve linear geometric problems involving collinearity, intersections, equal lengths, coordinate proofs, and classifications of triangles and quadrilaterals.

Students classify variables as categorical, discrete numerical, or continuous numerical, calculate mean, median, mode, range, and interquartile range from raw data or simple frequency tables, and construct and interpret box plots using five-number summaries to compare centre, spread, skewness, and possible outliers. They construct and interpret histograms with appropriately scaled contiguous intervals, describe distributions by centre, spread, clusters, gaps, skewness, and possible outliers, and use scatterplots and lines of best fit to describe linear correlation and make cautious predictions, distinguishing association from causation and interpolation from extrapolation. They evaluate random, stratified, systematic, convenience, and voluntary-response sampling for undercoverage, nonresponse, self-selection, and other bias, distinguishing systematic bias from random sampling variation when generalizing from samples to populations.

Apply the product, quotient, power-of-a-power, power-of-a-product, and power-of-a-quotient rules to simplify numerical and algebraic expressions with integer exponents, including expressions in scientific notation. Interpret zero exponents as 1 for nonzero bases and negative integer exponents as reciprocals, while preserving restrictions such as nonzero denominators and excluding $0^0$. Perform multiplication, division, addition, and subtraction in scientific notation by combining coefficients and powers of 10, using common powers when necessary, and renormalizing results to $1 \le |a| < 10$.

Students factor out the greatest common monomial factor from integer-coefficient binomials and trinomials and use grouping to reveal common binomial factors in standard low-degree polynomial expressions. They recognize and apply differences of squares, perfect-square trinomials, monic quadratic trinomials, and quadratic trinomials with non-unit integer leading coefficients, including appropriate sign choices. They select an efficient factoring strategy, factor expressions completely when possible, and verify equivalent factored forms by multiplication to support simplifying rational expressions and solving polynomial equations.

Calculate gross and net income from wages, salaries, commissions, bonuses, and fixed or percentage deductions, and create personal budgets by organizing income, fixed and variable expenses, savings, and discretionary spending across weekly, monthly, or annual periods. Calculate and interpret single percent increases and decreases, simple interest using $I=Prt$, and compound interest using $A=P(1+r)^n$ for whole-number periods, distinguishing simple from exponential growth and excluding successive changes, fractional periods, and advanced tax or investment models. Compare borrowing, saving, purchasing, and basic investment options by evaluating rates, fees, compounding periods, payment schedules, total costs, net gains, cost per unit, and effective rates using proportional reasoning, tables, graphs, and formulas.

Apply angle relationships for parallel lines cut by a transversal, including corresponding and alternate angles, vertical angles, linear pairs, and same-side interior angles, to find unknown measures, solve simple equations, and justify parallelism; determine polygon interior and exterior angles using $(n-2)180^\circ$, supplementary adjacent angles, and a $360^\circ$ exterior-angle sum for convex polygons. Identify congruent triangles using rigid motions and SSS, SAS, ASA, or AAS while rejecting SSA as sufficient, and identify similar triangles and figures using corresponding angles, proportional sides, and AA, SAS, or SSS, including rotated and reflected configurations. Use matching correspondence, scale factors, proportions, and consistent units to determine missing lengths in similar figures and triangles, solve straightforward indirect-measurement problems, and distinguish linear scale factors from additive changes.

Students solve one-variable linear equations with integer, decimal, and fractional coefficients by distributing, combining like terms, and applying inverse operations, including equations with variables on both sides. They distinguish one solution, no solution from contradictions such as $0=5$, and infinitely many solutions from identities such as $0=0$, while rearranging linear formulas to isolate a variable and solving one-variable linear inequalities, reversing the inequality when multiplying or dividing by a negative and representing solution sets on number lines or in interval notation. They model contextual relationships with linear equations and inequalities, preserving units and domain restrictions and interpreting solutions as exact values, bounds, or feasible ranges.

Students represent nonvertical linear relationships in equations, tables, graphs, and verbal descriptions, identifying slope as the constant rate of change and the y-intercept as the initial value. They determine slope from graphs, tables, or two points, interpret its sign, magnitude, and units, and write or graph equations in the form $y=mx+b$. They compare linear functions across representations by analyzing rates of change, initial values, outputs at given inputs, and intersections, recognizing that lines may change which function is greater after crossing.

Students calculate the surface areas of right prisms, closed cylinders, regular pyramids, and right circular cones using $SA=2B+Ph$, $SA=2\pi r^2+2\pi rh$, $SA=B+\tfrac12Pl$, and $SA=\pi r(r+\ell)$, distinguishing base, lateral, slant, and perpendicular heights and reporting square units. Students calculate volumes of prisms, cylinders, pyramids, and cones using $V=Bh$ and $V=\tfrac13Bh$, including $V=\pi r^2h$ and $V=\tfrac13\pi r^2h$, and determine how a scale factor $k$ changes lengths, areas, and volumes by $k$, $k^2$, and $k^3$, respectively. Students solve problems involving composite solids by decomposing figures into familiar solids, combining volumes and exposed surface areas while excluding shared internal faces, using consistent units and exact or appropriately rounded results.

Classify nonzero one-variable polynomials by degree and number of terms, and identify their degree and leading coefficient, including constants, missing powers, and negative or fractional coefficients. Evaluate one- or multivariable polynomial expressions for specified real values, and add or subtract polynomials by managing signs, distributing negatives, and combining like terms in standard form. Multiply monomials by polynomials and one-variable binomials using the distributive property and exponent laws, and expand the special products for binomial squares and conjugates.

Represent finite sample spaces for simple and two-stage experiments using exhaustive, non-overlapping organized lists, tables, and tree diagrams, identifying ordered outcomes and complete paths without omissions or repetitions. Calculate theoretical and experimental probabilities as fractions, decimals, or percentages, use complements with $P(A^c)=1-P(A)$, and interpret experimental results as estimates that generally stabilize with more trials while distinguishing them from theoretical values. Determine probabilities of compound events by adding mutually exclusive or appropriately adjusting overlapping cases and multiplying probabilities for independent events, using sample-space representations to avoid double-counting and excluding continuous models, advanced counting, formal inference, and complex conditional-probability theory.

Students recognize quadratic relations from equations in standard, factored, or vertex form, from vertical-parabola graphs, and from tables with equally spaced inputs and constant nonzero second differences, and graph them by plotting ordered pairs and connecting the pattern with a smooth curve. They identify the vertex, axis of symmetry, direction of opening, $y$-intercept, and real zeros, including two distinct zeros, one repeated zero, or no real zeros, from graphs or equations. They solve accessible quadratic equations with integer coefficients by factoring and applying the zero-product property, including repeated factors, without using the quadratic formula or complex solutions.

Students classify real numbers as natural, whole, integer, rational, or irrational, distinguish rational and irrational values, and represent, compare, and order positive and negative numbers, zero, fractions, decimals, and radicals on a number line. They evaluate square roots of nonnegative perfect squares, simplify square roots by extracting perfect-square factors, and estimate irrational square roots between rational benchmarks to the tenths or hundredths. They add and subtract like square radicals after simplifying them, combining coefficients while recognizing that unlike radicals cannot be combined.

Identify the hypotenuse, opposite side, and adjacent side relative to an acute angle in a right triangle, and determine sine, cosine, and tangent ratios using the appropriate side relationships. Select and apply a trigonometric ratio to find unknown side lengths or use inverse trigonometric functions to find acute angles, interpreting calculator results in degrees and rounding appropriately. Model and solve contextual problems involving right triangles, including height, distance, and angle situations, using trigonometry or the Pythagorean theorem and checking the reasonableness of results.

Students model contextual situations with systems of two linear equations, defining variables and interpreting coefficients, constants, units, and constraints. They determine whether an ordered pair satisfies both equations and solve systems efficiently by substitution, elimination, or graphing, verifying solutions in the original equations. They identify and interpret one solution, no solution, or infinitely many solutions as intersecting, parallel, or coincident lines, including feasible values and reasonable graph-based approximations in context.