Standards are the floor. Mastery and understanding are the goal.

Explore how each subject becomes a coherent sequence of knowledge, reasoning, practice, and transfer—not a disconnected checklist.

Grade 10 Math

37 curriculum outcomes with explicit teaching progressions

Select a record to inspect its progression

Congruence · CCSS.Math.Content.HS.G-CO.A.1

Define geometric terms precisely

Independently know precise definitions of angle, circle, perpendicular line, parallel line, and line segment, based on the undefined notions of point, line, distance along a line, and distance around a circular arc. across representations and contexts.

Component progression

  1. 1I can explain why point, line, distance along a line, and distance around a circular arc serve as primitive notions, and evaluate whether a proposed geometric definition uses them precisely without becoming circular.
  2. 2I can state precise definitions of angle, circle, perpendicular line, parallel line, and line segment using the undefined terms.
  3. 3I can distinguish a precise definition (true in every case) from a picture or single example of the term.
Congruence · CCSS.Math.Content.HS.G-CO.A.2

Represent transformations as functions

Independently represent transformations in the plane using tools such as tracing paper and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs, and compare transformations that preserve distance and angle to those that do not. across representations and contexts.

Component progression

  1. 1I can apply a given transformation to specific points, using tracing paper or geometry software, and record the input-output pairs.
  2. 2I can describe a transformation as a rule that takes any point as input and produces an output point.
  3. 3I can determine whether a given transformation preserves distance and angle measure, and classify it accordingly.
Congruence · CCSS.Math.Content.HS.G-CO.A.3

Describe the symmetries of a figure

Independently given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. across representations and contexts.

Component progression

  1. 1I can identify all lines of reflection that carry a given rectangle, parallelogram, trapezoid, or regular polygon onto itself.
  2. 2I can identify all rotation angles (about the figure's center) that carry a given figure onto itself.
  3. 3I can compare the symmetries of rectangles, parallelograms, trapezoids, and regular polygons, explaining why they differ.
Congruence · CCSS.Math.Content.HS.G-CO.A.4

Define transformations formally

Independently develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. across representations and contexts.

Component progression

  1. 1I can define a translation using a directed line segment, explaining how every point moves the same distance in the same direction.
  2. 2I can define a rotation using a center point and angle, explaining the relationship between a point and its image in terms of the center and angle.
  3. 3I can define a reflection using a line of reflection, explaining the relationship between a point and its image in terms of perpendicular distance to the line.
Congruence · CCSS.Math.Content.HS.G-CO.A.5

Draw and sequence transformations

Independently given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using graph paper, tracing paper, or geometry software; specify a sequence of transformations that will carry a given figure onto another. across representations and contexts.

Component progression

  1. 1I can draw the image of a figure under a specified translation, rotation, or reflection, preserving exact distances and angles.
  2. 2I can draw the image of a figure after applying a specified sequence of transformations in order.
  3. 3I can determine and specify a sequence of transformations that carries a given figure onto a second, congruent figure.
Congruence · CCSS.Math.Content.HS.G-CO.B.6

Use rigid motions to determine congruence

Independently use geometric descriptions of rigid motions to transform figures and to predict the effect of a given rigid motion on a given figure; given two figures, use the definition of congruence in terms of rigid motions to decide if they are congruent. across representations and contexts.

Component progression

  1. 1I can predict the effect of a given rigid motion (translation, rotation, or reflection) on a given figure before drawing it.
  2. 2I can explain that two figures are congruent exactly when a sequence of rigid motions maps one onto the other.
  3. 3I can determine whether two given figures are congruent by finding (or showing there is no) sequence of rigid motions mapping one onto the other.
Congruence · CCSS.Math.Content.HS.G-CO.B.7

Connect triangle congruence to corresponding parts

Independently use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and corresponding pairs of angles are congruent. across representations and contexts.

Component progression

  1. 1I can identify corresponding sides and corresponding angles between two triangles given a stated or implied correspondence.
  2. 2I can explain why a rigid motion mapping one triangle onto another must map corresponding sides and angles onto congruent sides and angles.
  3. 3I can use congruence of all corresponding sides and angles to justify that two triangles are congruent, and vice versa.
Congruence · CCSS.Math.Content.HS.G-CO.B.8

Justify triangle congruence criteria from rigid motions

Independently explain how the criteria for triangle congruence (asa, sas, and sss) follow from the definition of congruence in terms of rigid motions. across representations and contexts.

Component progression

  1. 1I can explain why knowing two sides and the included angle of a triangle is enough to construct a rigid motion mapping it onto a triangle with the same measurements.
  2. 2I can extend the same reasoning to explain why asa and sss also guarantee a rigid motion exists mapping one triangle onto the other.
  3. 3I can explain why an arrangement like ssa does not reliably determine a unique rigid motion, and so is not a valid congruence criterion.
Congruence · CCSS.Math.Content.HS.G-CO.C.9

Prove theorems about lines and angles

Independently prove theorems about lines and angles, including that vertical angles are congruent, that when a transversal crosses parallel lines alternate interior angles are congruent and corresponding angles are congruent, and that points on a perpendicular bisector of a segment are exactly those equidistant from the segment's endpoints. across representations and contexts.

Component progression

  1. 1I can prove that vertical angles formed by two intersecting lines are congruent.
  2. 2I can prove that alternate interior angles and corresponding angles are congruent when a transversal crosses a pair of parallel lines.
  3. 3I can prove that a point lies on the perpendicular bisector of a segment if and only if it is equidistant from the segment's two endpoints.
Congruence · CCSS.Math.Content.HS.G-CO.C.10

Prove theorems about triangles

Independently prove theorems about triangles, including that the measures of interior angles sum to 180°, that base angles of an isosceles triangle are congruent, that the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length, and that the medians of a triangle meet at a point. across representations and contexts.

Component progression

  1. 1I can prove that the interior angles of a triangle sum to 180°, using a line through one vertex parallel to the opposite side.
  2. 2I can prove that the base angles of an isosceles triangle are congruent.
  3. 3I can prove that the segment joining the midpoints of two sides of a triangle is parallel to the third side and half its length.
  4. 4I can prove that the three medians of a triangle meet at a single point (the centroid).
Congruence · CCSS.Math.Content.HS.G-CO.C.11

Prove theorems about parallelograms

Independently prove theorems about parallelograms, including that opposite sides are congruent, opposite angles are congruent, the diagonals bisect each other, and that a parallelogram is a rectangle exactly when its diagonals are congruent. across representations and contexts.

Component progression

  1. 1I can prove that opposite sides and opposite angles of a parallelogram are congruent.
  2. 2I can prove that the diagonals of a parallelogram bisect each other.
  3. 3I can prove that a parallelogram is a rectangle if and only if its diagonals are congruent.
Congruence · CCSS.Math.Content.HS.G-CO.D.12

Make formal geometric constructions

Independently make formal geometric constructions with a variety of tools and methods, including copying a segment, copying an angle, bisecting a segment, bisecting an angle, constructing perpendicular lines including a perpendicular bisector, and constructing a line parallel to a given line through a point not on the line. across representations and contexts.

Component progression

  1. 1I can use a compass and straightedge (or equivalent tools) to construct a copy of a given segment and a copy of a given angle.
  2. 2I can construct the bisector of a given segment and the bisector of a given angle.
  3. 3I can construct a perpendicular line (including a perpendicular bisector) and a line parallel to a given line through a point not on it.
Congruence · CCSS.Math.Content.HS.G-CO.D.13

Construct inscribed regular figures

Independently construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. across representations and contexts.

Component progression

  1. 1I can construct an equilateral triangle inscribed in a given circle using compass-and-straightedge methods.
  2. 2I can construct a square inscribed in a given circle.
  3. 3I can construct a regular hexagon inscribed in a given circle, using the fact that the radius equals the side length.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.A.1

Verify properties of dilations

Independently verify experimentally the properties of dilations given by a center and a scale factor: a dilation takes a line not through the center to a parallel line and leaves a line through the center unchanged, and the dilation of a line segment is longer or shorter according to the ratio given by the scale factor. across representations and contexts.

Component progression

  1. 1I can apply a dilation with a given center and scale factor to points and segments, and record the results.
  2. 2I can verify experimentally that a dilation maps a line not through the center to a parallel line.
  3. 3I can verify experimentally that a dilation changes the length of a segment by exactly the scale factor.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.A.2

Define similarity using similarity transformations

Independently given two figures, use the definition of similarity in terms of similarity transformations to decide if they are similar; explain, using similarity transformations, the meaning of similarity for triangles as the equality of all corresponding pairs of angles and the proportionality of all corresponding pairs of sides. across representations and contexts.

Component progression

  1. 1I can explain that two figures are similar exactly when a sequence of rigid motions and dilations maps one onto the other.
  2. 2I can verify that similar triangles have all corresponding angles congruent and all corresponding sides in the same proportion.
  3. 3I can determine whether two given figures are similar by finding (or showing there is no) sequence of similarity transformations mapping one onto the other.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.A.3

Establish the AA triangle similarity criterion

Independently use the properties of similarity transformations to establish the aa (angle-angle) criterion for two triangles to be similar. across representations and contexts.

Component progression

  1. 1I can explain why, once two pairs of corresponding angles in two triangles are congruent, the third pair must be congruent as well.
  2. 2I can construct a sequence of a rigid motion and a dilation that maps one triangle onto another, given only that two pairs of corresponding angles are congruent.
  3. 3I can use the aa criterion to determine whether two triangles are similar.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.B.4

Prove theorems using similarity

Independently prove theorems about triangles using similarity, including that a line parallel to one side of a triangle divides the other two sides proportionally (and its converse), and the pythagorean theorem proved using triangle similarity. across representations and contexts.

Component progression

  1. 1I can prove that a line parallel to one side of a triangle divides the other two sides proportionally, using similar triangles.
  2. 2I can prove the pythagorean theorem by using the altitude to the hypotenuse of a right triangle to create similar triangles.
  3. 3I can apply the side-splitter theorem and the similarity proof of the pythagorean theorem to solve problems.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.B.5

Use congruence and similarity to solve problems

Independently use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. across representations and contexts.

Component progression

  1. 1I can determine whether a given problem calls for a congruence criterion or a similarity criterion, and select the appropriate one.
  2. 2I can use similar triangles to solve for unknown side lengths or angle measures in a geometric figure.
  3. 3I can prove a stated relationship between parts of a geometric figure using an appropriate congruence or similarity criterion.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.C.6

Define trigonometric ratios

Independently understand that, by similarity, side ratios in right triangles are properties of the angles in the triangle, leading to definitions of trigonometric ratios for acute angles. across representations and contexts.

Component progression

  1. 1I can use similar right triangles to show that the ratio of two given sides is the same for every right triangle with a given acute angle.
  2. 2I can define sine, cosine, and tangent of an acute angle as specific ratios of a right triangle's sides.
  3. 3I can compute sine, cosine, and tangent for an acute angle given the side lengths of a right triangle.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.C.7

Use the complementary-angle sine-cosine relationship

Independently explain and use the relationship between the sine and cosine of complementary angles. across representations and contexts.

Component progression

  1. 1I can identify the two acute angles of a right triangle as complementary.
  2. 2I can show, using the side ratios of a right triangle, that sin(a) = cos(90°-a) for an acute angle a.
  3. 3I can use the sine-cosine complementary relationship to find an unknown trigonometric value without a calculator.
Similarity Trigonometry · CCSS.Math.Content.HS.G-SRT.C.8

Solve right triangles using trigonometry

Independently use trigonometric ratios and the pythagorean theorem to solve right triangles in applied problems. across representations and contexts.

Component progression

  1. 1I can select the correct trigonometric ratio (sine, cosine, or tangent) to relate a known angle and side to an unknown side.
  2. 2I can solve for a missing side length or angle measure in a right triangle using trigonometric ratios or the pythagorean theorem.
  3. 3I can solve an applied problem (such as angle of elevation or depression) by modeling it as a right triangle and solving it.
Circles · CCSS.Math.Content.HS.G-C.A.1

Prove all circles are similar

Independently prove that all circles are similar. across representations and contexts.

Component progression

  1. 1I can identify the dilation (center and scale factor) that maps one given circle onto another.
  2. 2I can construct an argument that a dilation centered at one circle's center, with an appropriate scale factor, maps it onto any other circle.
  3. 3I can generalize the argument to conclude that any two circles are similar.
Circles · CCSS.Math.Content.HS.G-C.A.2

Identify angle and chord relationships in circles

Independently identify and describe relationships among inscribed angles, radii, and chords, including the relationship between central, inscribed, and circumscribed angles (such as a central angle being twice the inscribed angle subtending the same arc), that an inscribed angle on a diameter is a right angle, and that the radius of a circle is perpendicular to the tangent where the radius intersects the circle. across representations and contexts.

Component progression

  1. 1I can identify the central angle and an inscribed angle that subtend the same arc, and state their measure relationship.
  2. 2I can identify a circumscribed angle formed by two tangent lines drawn from an external point, and relate its measure to the arcs it intercepts.
  3. 3I can prove that an inscribed angle subtending a diameter is a right angle.
  4. 4I can explain and apply the fact that a circle's radius is perpendicular to the tangent line at the point of tangency.
Circles · CCSS.Math.Content.HS.G-C.A.3

Construct triangle circles and inscribed quadrilateral angles

Independently construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. across representations and contexts.

Component progression

  1. 1I can construct the inscribed circle of a triangle using angle bisectors to find the incenter.
  2. 2I can construct the circumscribed circle of a triangle using perpendicular bisectors to find the circumcenter.
  3. 3I can prove that opposite angles of a quadrilateral inscribed in a circle are supplementary.
Circles · CCSS.Math.Content.HS.G-C.B.5

Derive arc length and sector area formulas

Independently derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. across representations and contexts.

Component progression

  1. 1I can use the fact that all circles are similar to show that, for a fixed central angle, arc length is proportional to the circle's radius.
  2. 2I can define the radian measure of an angle as the constant of proportionality relating arc length to radius for that angle.
  3. 3I can derive the formula for the area of a sector from the arc-length relationship, and use the arc length and sector area formulas to solve problems.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.A.1

Derive and use the equation of a circle

Independently derive the equation of a circle of given center and radius using the pythagorean theorem; complete the square to find the center and radius of a circle given by an equation. across representations and contexts.

Component progression

  1. 1I can derive the equation of a circle with a given center and radius by applying the pythagorean theorem (distance formula) to a general point on the circle.
  2. 2I can write the equation of a circle given its center and radius.
  3. 3I can complete the square on an equation of a circle in general form to find its center and radius.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.A.2

Derive the equation of a parabola

Independently derive the equation of a parabola given a focus and directrix. across representations and contexts.

Component progression

  1. 1I can explain that a parabola is the set of points equidistant from a fixed focus point and a fixed directrix line.
  2. 2I can set up an equation setting the distance from a general point to the focus equal to its distance to the directrix.
  3. 3I can simplify the equidistance equation algebraically to derive the parabola's equation in standard form.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.B.4

Prove geometric theorems using coordinates

Independently use coordinates to prove simple geometric theorems algebraically. across representations and contexts.

Component progression

  1. 1I can assign general coordinates to a figure's vertices in a way that simplifies a coordinate proof without losing generality.
  2. 2I can use the distance formula and midpoint formula to establish side lengths and midpoints needed for a coordinate proof.
  3. 3I can complete an algebraic proof of a simple geometric theorem (such as that the diagonals of a specific quadrilateral bisect each other) using coordinates.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.B.5

Prove and use the slope criteria for parallel and perpendicular lines

Independently prove the slope criteria for parallel and perpendicular lines, and use them to solve geometric problems (for example, finding the equation of a line parallel or perpendicular to a given line that passes through a given point). across representations and contexts.

Component progression

  1. 1I can prove that two non-vertical lines are parallel if and only if they have the same slope.
  2. 2I can prove that two non-vertical lines are perpendicular if and only if their slopes are opposite reciprocals.
  3. 3I can find the equation of a line parallel or perpendicular to a given line and passing through a given point.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.B.6

Find a point that partitions a segment in a given ratio

Independently find the point on a directed line segment between two given points that partitions the segment in a given ratio. across representations and contexts.

Component progression

  1. 1I can interpret what it means for a point to partition a directed line segment in a given ratio, distinguishing direction from the segment's endpoints.
  2. 2I can apply the section formula to find the coordinates of a point partitioning a segment in a given ratio.
  3. 3I can verify that a found point actually divides the segment in the required ratio.
Coordinate Geometry · CCSS.Math.Content.HS.G-GPE.B.7

Compute perimeter and area using coordinates

Independently use coordinates to compute perimeters of polygons and areas of triangles and rectangles, for example using the distance formula. across representations and contexts.

Component progression

  1. 1I can use the distance formula to compute the side lengths of a polygon given its vertices' coordinates.
  2. 2I can compute the perimeter of a polygon given its vertices' coordinates.
  3. 3I can compute the area of a triangle or rectangle given its vertices' coordinates.
Measurement Dimension · CCSS.Math.Content.HS.G-GMD.A.1

Argue for circle and solid volume formulas

Independently give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone, using dissection arguments, cavalieri's principle, and informal limit arguments. across representations and contexts.

Component progression

  1. 1I can give an informal dissection or limit argument for why the circumference and area formulas for a circle are reasonable.
  2. 2I can use cavalieri's principle to argue that two solids with the same cross-sectional area at every height have the same volume.
  3. 3I can give an informal argument for the volume formulas of a cylinder, pyramid, and cone using cavalieri's principle or limit arguments.
Measurement Dimension · CCSS.Math.Content.HS.G-GMD.A.3

Use volume formulas to solve problems

Independently use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. across representations and contexts.

Component progression

  1. 1I can identify the correct volume formula for a given solid (cylinder, pyramid, cone, or sphere).
  2. 2I can compute the volume of a cylinder, pyramid, cone, or sphere given its dimensions.
  3. 3I can solve an applied problem requiring computation of, or solving for an unknown dimension from, a solid's volume.
Measurement Dimension · CCSS.Math.Content.HS.G-GMD.B.4

Relate cross-sections and rotations to 3D objects

Independently identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. across representations and contexts.

Component progression

  1. 1I can identify the shape produced by slicing a given three-dimensional object with a plane at a specified orientation.
  2. 2I can identify the three-dimensional solid generated by rotating a given two-dimensional shape around a specified axis.
  3. 3I can move fluently between a three-dimensional object's cross-sections and its generating two-dimensional shape and axis of rotation.
Modeling Geometry · CCSS.Math.Content.HS.G-MG.A.1

Model objects using geometric shapes

Independently use geometric shapes, their measures, and their properties to describe objects (for example, modeling a tree trunk or a human torso as a cylinder). across representations and contexts.

Component progression

  1. 1I can choose a geometric shape that reasonably approximates a given real-world object.
  2. 2I can use the chosen shape's measures and properties to describe or estimate a property of the real object.
  3. 3I can evaluate where a geometric model of a real object is likely to be inaccurate, and why.
Modeling Geometry · CCSS.Math.Content.HS.G-MG.A.2

Apply density in modeling situations

Independently apply concepts of density based on area and volume in modeling situations (for example, persons per square mile, btus per cubic foot). across representations and contexts.

Component progression

  1. 1I can compute a density rate (a quantity per unit area or per unit volume) from given data.
  2. 2I can interpret what a computed density rate means about the situation being modeled.
  3. 3I can use a density rate to solve a modeling problem, such as estimating a total quantity from an area or volume.
Modeling Geometry · CCSS.Math.Content.HS.G-MG.A.3

Solve design problems with geometric methods

Independently apply geometric methods to solve design problems (for example, designing an object or structure to satisfy physical constraints or minimize cost, or working with typographic grid systems based on ratios). across representations and contexts.

Component progression

  1. 1I can translate a design problem's physical constraints into geometric relationships or equations.
  2. 2I can solve the resulting geometric relationships to find dimensions or a design satisfying all stated constraints.
  3. 3I can evaluate whether a proposed design solution is realistic and justify that it satisfies (or optimizes) the original constraints.
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