Grade 10 · secular · Common Core-aligned
This is the year your child learns to prove things about shapes instead of just measuring them. They'll figure out what actually makes two shapes "the same" — not eyeballing it, but showing a slide, flip, turn, or resize connects them. From there they build up to writing formal proofs about triangles, then reuse that same logic for circles, coordinate geometry, and 3D shapes. Along the way, trig and geometric probability show up not as brand-new topics but as the same ideas from earlier in the year wearing new names. By June they should be able to look at a diagram and tell the difference between "this looks true" and "this is proven."
The whole year, in plain English. Tap any unit to see every skill inside, nothing is hidden.
This unit sets the rule for the whole year: two shapes are congruent only if you can show a slide, flip, or turn maps one exactly onto the other — not because they look the same size. Similar works the same way but allows a resize. Your child will build coordinate rules for each move, then learn to tell them apart just by looking at what changed.
Given a figure and a stated translation vector or reflection line or rotation angle/center, execute the correct coordinate rule to plot the image.
Given a preimage and image pair with a transformation not named, determine which single rigid motion (translation, reflection, or rotation) produced it by testing for a fixed point, distance preservation, and orientation.
Explain why a rigid motion mapping a preimage onto an image is sufficient to prove the two figures are congruent, in contrast to citing matching measured angles and sides as proof.
Given a scale factor and center of dilation, execute the coordinate rule to produce the image of a polygon.
Compare a rigid-motion mapping and a dilation mapping applied to the same figure and identify which measurable properties (distance, angle measure, orientation) each preserves or fails to preserve.
Given two figures where one is a scaled, reoriented, and repositioned version of the other, determine whether they are congruent, similar, or neither, and justify the determination by describing a composite mapping (dilation plus rigid motion, or rigid motion alone).
Given a real-world image (e.g., a floor tile pattern or a logo) with no coordinate grid and no stated transformation, determine whether a repeated design element is related by a rigid motion or a dilation, and justify the choice using the general preserved-property argument rather than any coordinate rule.
Given an entirely unfamiliar transformation-like rule described in words (e.g., a 'shear' or a non-uniform stretch that scales x and y by different factors) never named or taught in this unit, determine whether it preserves distance, preserves angle measure, or neither, by applying the same general reasoning strategy (test specific point-pairs, check ratios) used for rigid motions and dilations.
This is where 'same shape because of a mapping' turns into an actual system of rules — SSS, SAS, ASA, AAS — and where proof-writing itself gets taught as a skill: every line needs a reason, not just a claim. It ends with isosceles triangle theorems, which require chaining two proven facts together.
Given a rigid-motion mapping between two triangles (translation + rotation + reflection composition described in words or coordinates), state which sides and angles are corresponding without being told directly.
Explain why checking three corresponding parts (in an SSS, SAS, or ASA arrangement) is logically sufficient to guarantee a rigid motion exists mapping one triangle onto the other, using the construction argument (a unique triangle is determined by that information).
Construct a counterexample (two non-congruent triangles sharing an SSA or AAA correspondence) to demonstrate that SSA and AAA are not valid congruence criteria.
Select the correct congruence criterion (SSS, SAS, ASA, or AAS) that applies to a given diagram with marked parts, including diagrams with shared sides, vertical angles, or parallel-line angle relationships requiring an inferred additional pair.
Write a two-column proof of a stated theorem about lines and angles (vertical angles are congruent, or alternate interior angles are congruent given parallel lines) using only definitions and previously given facts as justifications.
Construct a complete two-column or flowchart proof of triangle congruence for a diagram containing overlapping or shared triangles, choosing and justifying the correct criterion and applying CPCTC to draw a conclusion beyond the congruence itself.
Prove the isosceles triangle theorem (base angles of an isosceles triangle are congruent) by constructing an auxiliary segment (the angle bisector from the apex) and chaining SAS congruence with CPCTC.
Given a real-world stability or design context (e.g., a triangulated bridge truss or a bracing structure) with no diagram provided, determine what minimal set of measurements would need to be verified equal to certify two triangular sections are congruent, and justify the choice.
Given a diagram with a marked SSA correspondence and a claimed congruence, write an explanation of specifically why the SSA information fails to guarantee congruence in that diagram (not merely state that SSA is invalid), identifying the specific ambiguous configuration.
Given a novel geometric claim outside the taught theorem list (e.g., the diagonal of a kite bisects the angles it connects), decide which combination of congruence criteria and prior theorems would need to be chained to prove it, without executing the full proof.
This unit builds the similarity version of Unit 2's rules — AA, SAS~, SSS~ — and makes clear these are not just the congruence rules with a symbol added. It also proves the Pythagorean theorem a new way (through similar triangles formed by an altitude) and ends with indirect measurement: shadows, mirrors, and scale drawings, where the whole trick is spotting a hidden similar triangle.
Given the dilation/scale-factor definition of similarity from Unit 1, state and use the AA similarity criterion to determine whether two triangles are similar, without checking side lengths.
Classify a pair of triangles as similar by AA, SAS~, SSS~, or neither, given a diagram with a mix of marked angles and side lengths.
Explain why one pair of congruent angles is sufficient evidence for triangle similarity (AA) while one pair of congruent sides is never sufficient evidence for triangle congruence.
Apply the triangle proportionality theorem to find an unknown segment length when a line is drawn parallel to one side of a triangle.
Prove that a line divides two sides of a triangle proportionally if and only if it is parallel to the third side, using the AA similarity criterion as the justification engine.
Derive the geometric-mean relationships in a right triangle with an altitude drawn to the hypotenuse, by identifying the three similar right triangles created.
Prove the Pythagorean theorem using the similar triangles formed by the altitude to the hypotenuse, without reference to an area-based proof.
Identify and justify the pair of similar triangles implicitly created by an indirect-measurement setup (shadow, mirror, or scale drawing) before computing an unknown length.
Given a mixed set of triangle pairs, decide whether the pair is congruent, similar-but-not-congruent, or neither, and name the specific criterion or counterexample that justifies the decision.
This unit moves proof off the diagram and onto the coordinate plane, using slope and the distance formula — both from Algebra I — to prove facts about quadrilaterals. It includes a strong focus on placing figures strategically (like putting a vertex at the origin) to make the arithmetic easier without changing what's being proved.
Derive the distance formula from the Pythagorean theorem applied to a right triangle formed by horizontal and vertical legs between two coordinate points, and use it to compute the length of a segment.
Apply the slope criteria (equal slope for parallel, negative reciprocal slope for perpendicular) to determine the relationship between two lines given as coordinates or equations.
Compute the midpoint of a segment and the point that partitions a directed segment in a given non-1:1 ratio, connecting the ratio to a weighted average of coordinates.
Explain why establishing that opposite sides of a quadrilateral are both parallel (equal slope) and equal in length (distance formula) is sufficient to conclude the figure is a parallelogram, connecting the coordinate evidence to the definition of a parallelogram.
Construct a coordinate proof that a given quadrilateral is a rectangle or rhombus by selecting and sequencing the correct combination of slope and distance computations.
Justify a strategic placement of a figure's vertices on the coordinate axes (e.g., a vertex at the origin, a side on an axis) that simplifies the arithmetic of an upcoming proof, without altering the shape or the property being proved.
Given only a verbal description of a quadrilateral (no diagram, no coordinates supplied), generate an original coordinate placement and complete proof of a stated property, without a template to match against.
Critique a proposed coordinate argument that relies on the visual appearance of a figure on a grid (e.g., 'the sides look equal so it's a rhombus') rather than on computed slope/distance values, identifying exactly where the argument fails to establish the claim.
Compare a synthetic (Unit 2 triangle-congruence-based) proof and a coordinate proof of the same parallelogram property, identifying what each method takes as given and what work each does to reach the same conclusion.
Recall the formulas for distance, midpoint, and slope criteria without reference materials.
This unit takes the fixed similarity ratio from Unit 3 and gives it names: sine, cosine, tangent. Your child first confirms with actual measurement that the ratio doesn't change with triangle size, before any trig vocabulary appears. From there it's solving for missing sides, missing angles, and real angle-of-elevation/depression problems.
Given several right triangles with a shared acute angle but different side lengths, a student measures corresponding sides and computes the ratio opposite/hypotenuse for each, and states that the ratio is constant across triangle size because the triangles are similar.
Student states the definitions of sine, cosine, and tangent as ratios of specific side pairs (opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent) relative to a named acute angle in a right triangle.
Given an acute angle measure and one side length of a right triangle, student sets up and solves the correct trig equation to find a missing side length, selecting sine, cosine, or tangent based on which sides are known and unknown relative to the given angle.
Given two side lengths of a right triangle, student uses an inverse trig function to find the measure of an acute angle, and explains why the inverse operation undoes the ratio rather than treating it as an unrelated calculator button.
Given a real-world description involving a line of sight to or from a horizontal, student draws and labels a right triangle diagram correctly distinguishing angle of elevation from angle of depression before selecting a trig ratio.
Student proves that sin(theta) = cos(90 - theta) for any acute angle theta by relating the two acute angles of a right triangle as complements and showing that the side labeled 'opposite' for one angle is the side labeled 'adjacent' for the other.
Given an indirect-measurement scenario with no diagram provided and multiple plausible right-triangle setups (e.g., a shadow problem where the observer's own height matters or doesn't), student selects and justifies which right triangle and which trig ratio actually models the situation, stating the similarity relationship that licenses the ratio's use.
Given a novel context outside right triangles entirely (e.g., a non-right triangle broken into two right triangles by an altitude, or a problem set in an unfamiliar unit system requiring unit conversion mid-solution), student generalizes right-triangle trig methods to solve the new problem without being told to split the triangle or convert units first.
Student compares the similarity-proportion method (Unit 3) and the trig-ratio method (this unit) applied to the same indirect-measurement problem and explains why both produce the same answer.
Every circle fact in this unit comes from one starting point: all radii of a circle are equal, so any triangle made of two radii is isosceles. From there come inscribed angles, tangent lines, and chord relationships — all proved with Unit 2's congruence or Unit 3's similarity, not new rules. The unit ends with the circle's equation, which turns out to just be the distance formula from Unit 4 wearing a disguise.
Given a circle diagram with two radii drawn, students state that the two radii are congruent by definition and identify the resulting triangle as isosceles.
Students derive that an inscribed angle measures half its intercepted arc, using the isosceles-triangle argument applied to two central-angle triangles formed by a radius to the inscribed angle's vertex.
Students prove, citing the HL or SSS congruence criterion, that two tangent segments drawn from the same external point to a circle are congruent.
Given a diagram of two intersecting chords pre-partitioned into two triangles with angle markings, students derive the equal-products relationship without being shown the rule, citing AA similarity.
Students distinguish, from an unlabeled diagram, whether a segment-intersection problem is the chord-chord, secant-secant, or secant-tangent case, and apply the correct product relationship.
Students calculate arc length and sector area by setting up a proportion of the central angle to 360 degrees, in either direction (solve for arc/area given angle, or angle given arc/area).
Students derive the standard-form equation of a circle from a given center and radius using the distance formula, before being shown the general formula.
Students convert a circle's equation from general form to standard form by completing the square on both variables, and identify the resulting center and radius.
Given an unlabeled coordinate-geometry problem, students identify whether the distance formula, slope, or circle-equation method applies and execute the correct procedure.
Given a novel composite figure (e.g., an inscribed regular hexagon and its circumscribed circle) never presented during instruction, students combine sector-area, triangle-area, and segment reasoning to find an unrequested region's area.
Students critique a peer's (or provided sample) circle proof to identify whether each line is justified by a previously established criterion or merely restates the diagram's appearance.
This unit moves from flat shapes to solids. Surface area turns out to be flat-shape area applied to the faces of a net; volume is built by stacking up flat cross-sections (Cavalieri's principle). The unit's other big idea — that doubling every dimension of a solid multiplies its volume by 8, not 2 — deliberately fights a strong, wrong intuition most kids have.
Given two solids with congruent bases and equal heights (one right, one oblique), state and apply Cavalieri's principle to conclude their volumes are equal without computing either volume.
Execute the volume formula for a prism, cylinder, pyramid, or cone given labeled dimensions, selecting the correct formula for the solid shown.
Compute the surface area of a prism, cylinder, cone, or pyramid by decomposing its net into 2D shapes with previously justified area formulas.
Derive the volume formula for a sphere by comparing a hemisphere's cross-sectional area at height h to the cross-sectional area of a cylinder-minus-cone at the same height, and explain why Cavalieri's principle guarantees equal volumes.
Given a linear scale factor k applied to a solid, predict the resulting factor by which surface area and volume change (k^2 and k^3 respectively), justifying the exponent from the number of linear dimensions each measure is built from.
Given a composite real-world object description with no formula provided, decompose it into known solids, identify the measurements needed, and compute total volume and surface area, correctly excluding internal seams from the surface area total.
Convert a computed volume into a real-world mass or vice versa using a given density value, and identify which unit conversions are required.
Classify a given real-world object (e.g., a silo, a lampshade, a piece of furniture) by which basic and composite solids best model it, distinguishing cases where a frustum or hemisphere is a better model than a full cone or sphere.
This closing unit reframes probability as a ratio of measurements — length, area, or volume — instead of counting outcomes. There's almost no new math technique here; it's entirely about the judgment calls: is this situation actually uniform (every point equally likely), and does the ratio you built actually match dimension to dimension.
Given a scenario described in words (e.g., 'a bus arrives at a random time in a 20-minute window'), state whether the sample space is best modeled as discrete (countable) or continuous (measurable) and justify the choice in one sentence.
Construct the ratio P(event) = (favorable length)/(total length) for a point landing on a specified sub-segment of a number line interval, given the interval and sub-segment endpoints.
Construct the ratio P(event) = (favorable area)/(total area) for a single target region matching the exact structure of the Day 2 worked example (one shape inscribed in another, e.g., a circle inscribed in a square, target = land in the inscribed shape), reusing the area formulas established in Unit 6 (circles) and earlier units (polygons) without any added composite-subtraction step.
Construct the ratio P(event) = (favorable area)/(total area) for a target/dartboard-style region composed of shapes whose areas were established in Unit 6 (circles) and earlier units (polygons), including regions requiring subtraction of an inner region from an outer one.
Explain, using the ratio-of-measures definition, why the numerator and denominator of a geometric probability must be measured in the same dimension (length/length, area/area, volume/volume), and identify the dimensional error in a flawed ratio (e.g., area over length).
Given a real or described random process, identify whether the uniform-distribution assumption underlying a geometric probability model plausibly holds, and state a specific reason the assumption could fail (e.g., a dart-thrower more likely to hit center, a bus more likely to arrive near the scheduled time).
Design a geometric probability model for a self-selected or teacher-assigned real scenario: define the sample space geometrically, construct the correct measure ratio, compute the probability, and write a justification of the uniform-distribution assumption the model requires, including a stated limitation.
Given a scenario about population or resource distribution (e.g., persons per square mile, trees per acre, bacteria per cubic centimeter), compute a density value as a ratio of a count to an area or volume measure, and use that density to estimate a count in a differently-sized region.
Compare a geometric (area/volume-based) probability model to a classical (counting-based) probability model for the same underlying random process, and explain what feature of the process determines which model is legitimate.
From the parent guide
This is the year your child learns to prove things about shapes instead of just measuring them. They'll figure out what actually makes two shapes "the same" — not eyeballing it, but showing a slide, flip, turn, or resize connects them. From there they build up to writing formal proofs about triangles, then reuse that same logic for circles, coordinate geometry, and 3D shapes. Along the way, trig and geometric probability show up not as brand-new topics but as the same ideas from earlier in the year wearing new names. By June they should be able to look at a diagram and tell the difference between "this looks true" and "this is proven."
Unit 1 · what to expect
This unit sets the rule for the whole year: two shapes are congruent only if you can show a slide, flip, or turn maps one exactly onto the other — not because they look the same size. Similar works the same way but allows a resize. Your child will build coordinate rules for each move, then learn to tell them apart just by looking at what changed.
The full guide covers all 8 units: where kids get stuck, what to say, and how to tell it's working. Included with the course.
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