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Curriculum (United States)

Mathematics — Grade 10

10th grade math according to the Common Core State Standards, usually Geometry: similarity, right triangle trigonometry, the laws of sines and cosines, circles, coordinate geometry, volume and surface area, and conditional probability. Every topic comes with printable worksheets.

Create a worksheet for this curriculum

What do students learn in 10th grade math?

In 10th grade students use similarity and right triangle trigonometry, apply the laws of sines and cosines and work with inscribed and central angles, tangents and chords. In the coordinate plane they find distances, partition segments and compute areas. They find the volume and surface area of cylinders, cones and spheres, model with density, and count outcomes to compute conditional and compound probabilities.

About the standards: the Common Core high school standards are organized by conceptual category (Number and Quantity, Algebra, Functions, Geometry, Statistics and Probability), not by grade. Assigning a topic to a grade is our decision, following the common course sequence Algebra I, Geometry, Algebra II and Precalculus. Standards marked (+) go beyond the college and career readiness core.

Page status: the topics of this grade are ready as plain exercises (instruction and formula); problems with diagrams and word problems are being added next.

Curriculum scope

  1. Curriculum point HSA-SSE · Teaching content · checked against the act

    Seeing Structure in Expressions

    CCSS Mathematics (2010), High School, domain Seeing Structure in Expressions (HSA-SSE)

  2. Curriculum point HSA-SSE.A · Teaching content · checked against the act

    Interpret the structure of expressions

    CCSS Mathematics (2010), High School, Seeing Structure in Expressions, cluster A

  3. Curriculum point HSA-SSE.B · Teaching content · checked against the act

    Write expressions in equivalent forms to solve problems

    CCSS Mathematics (2010), High School, Seeing Structure in Expressions, cluster B

  4. Curriculum point HSA-SSE.A.1 · Teaching content · checked against the act

    Interpret expressions that represent a quantity in terms of its context.★ a. Interpret parts of an expression, such as terms, factors, and coefficients. b. Interpret complicated expressions by viewing one or more of their parts as a single entity. For example, interpret P(1+r)n as the product of P and a factor not depending on P.

    CCSS Mathematics (2010), High School, Seeing Structure in Expressions, standard HSA-SSE.A.1

  5. Curriculum point HSA-SSE.A.2 · Teaching content · checked against the act

    Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – ( y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

    CCSS Mathematics (2010), High School, Seeing Structure in Expressions, standard HSA-SSE.A.2

    Using Special Product Formulas (we teach in grade 9-11) · Factoring Polynomials with Grouping and Cubes (we teach in grade 9-11)

  6. Curriculum point HSA-APR · Teaching content · checked against the act

    Arithmetic with Polynomials and Rational Expressions

    CCSS Mathematics (2010), High School, domain Arithmetic with Polynomials and Rational Expressions (HSA-APR)

  7. Curriculum point HSA-APR.A · Teaching content · checked against the act

    Perform arithmetic operations on polynomials

    CCSS Mathematics (2010), High School, Arithmetic with Polynomials and Rational Expressions, cluster A

  8. Curriculum point HSA-APR.A.1 · Teaching content · checked against the act

    Understand that polynomials form a system analogous to the integers, namely, they are closed under the operations of addition, subtraction, and multiplication; add, subtract, and multiply polynomials.

    CCSS Mathematics (2010), High School, Arithmetic with Polynomials and Rational Expressions, standard HSA-APR.A.1

  9. Curriculum point HSA-CED · Teaching content · checked against the act

    Creating Equations★

    CCSS Mathematics (2010), High School, domain Creating Equations★ (HSA-CED)

  10. Curriculum point HSA-CED.A · Teaching content · checked against the act

    Create equations that describe numbers or relationships

    CCSS Mathematics (2010), High School, Creating Equations★, cluster A

  11. Curriculum point HSA-CED.A.1 · Teaching content · checked against the act

    Create equations and inequalities in one variable and use them to solve problems. Include equations arising from linear and quadratic functions, and simple rational and exponential functions.

    CCSS Mathematics (2010), High School, Creating Equations★, standard HSA-CED.A.1

  12. Curriculum point HSA-CED.A.2 · Teaching content · checked against the act

    Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales.

    CCSS Mathematics (2010), High School, Creating Equations★, standard HSA-CED.A.2

  13. Curriculum point HSA-CED.A.3 · Teaching content · checked against the act

    Represent constraints by equations or inequalities, and by systems of equations and/or inequalities, and interpret solutions as viable or non-viable options in a modeling context. For example, represent inequalities describing nutritional and cost constraints on combinations of different foods.

    CCSS Mathematics (2010), High School, Creating Equations★, standard HSA-CED.A.3

  14. Curriculum point HSA-CED.A.4 · Teaching content · checked against the act

    Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm’s law V = IR to highlight resistance R.

    CCSS Mathematics (2010), High School, Creating Equations★, standard HSA-CED.A.4

  15. Curriculum point HSA-REI · Teaching content · checked against the act

    Reasoning with Equations and Inequalities

    CCSS Mathematics (2010), High School, domain Reasoning with Equations and Inequalities (HSA-REI)

  16. Curriculum point HSA-REI.A · Teaching content · checked against the act

    Understand solving equations as a process of reasoning and explain the reasoning

    CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, cluster A

  17. Curriculum point HSA-REI.C · Teaching content · checked against the act

    Solve systems of equations

    CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, cluster C

  18. Curriculum point HSA-REI.D · Teaching content · checked against the act

    Represent and solve equations and inequalities graphically

    CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, cluster D

  19. Curriculum point HSA-REI.C.5 · Teaching content · checked against the act

    Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.

    CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, standard HSA-REI.C.5

  20. Curriculum point HSA-REI.D.11 · Teaching content · checked against the act

    Explain why the x-coordinates of the points where the graphs of the equations y = f(x) and y = g(x) intersect are the solutions of the equation f(x) = g(x); find the solutions appr oximately, e.g., using technology to graph the functions, make tables of values, or find successive approximations. Include cases where f(x) and/or g(x ) are linear, polynomial, rational, absolute value, exponential, and logarithmic functions.★

    CCSS Mathematics (2010), High School, Reasoning with Equations and Inequalities, standard HSA-REI.D.11

  21. Curriculum point HSF-IF · Teaching content · checked against the act

    Interpreting Functions

    CCSS Mathematics (2010), High School, domain Interpreting Functions (HSF-IF)

  22. Curriculum point HSF-IF.B · Teaching content · checked against the act

    Interpret functions that arise in applications in terms of the context

    CCSS Mathematics (2010), High School, Interpreting Functions, cluster B

  23. Curriculum point HSF-IF.C · Teaching content · checked against the act

    Analyze functions using different representations

    CCSS Mathematics (2010), High School, Interpreting Functions, cluster C

  24. Curriculum point HSF-IF.B.4 · Teaching content · checked against the act

    For a function that models a relationship between two quantities, interpret key features of graphs and tables in terms of the quantities, and sketch graphs showing key features given a verbal description of the relationship. Key features include: intercepts; intervals where the function is increasing, decreasing, positive, or negat ive; relative maximums and minimums; symmetries; end behavior; and periodicity .★

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.B.4

  25. Curriculum point HSF-IF.B.5 · Teaching content · checked against the act

    Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. For example, if the function h(n) gives the number of person-hours it takes to assemble n engines in a factory, then the positive integers would be an appropriate domain for the function. ★

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.B.5

  26. Curriculum point HSF-IF.B.6 · Teaching content · checked against the act

    Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. ★

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.B.6

    Finding Average Rate of Change (we teach in grade 9-11)

  27. Curriculum point HSF-IF.C.7 · Teaching content · checked against the act

    Graph functions expressed symbolically and show key features of the graph, by hand in simple cases and using technology for more complicated cases. ★ a. Graph linear and quadratic functions and show intercepts, maxima, and minima. b. Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. c. Graph polynomial functions, identifying zeros when suitable factorizations are available, and showing end behavior. d. (+) Graph rational functions, identifying zeros and asymptotes when suitable f actorizations are available, and showing end behavior. e. Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude.

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.C.7

  28. Curriculum point HSF-IF.C.8 · Teaching content · checked against the act

    Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.C.8

    Finding the Vertex and Using Vertex Form (we teach in grade 9-11)

  29. Curriculum point HSF-IF.C.9 · Teaching content · checked against the act

    Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum.

    CCSS Mathematics (2010), High School, Interpreting Functions, standard HSF-IF.C.9

  30. Curriculum point HSF-BF · Teaching content · checked against the act

    Building Functions

    CCSS Mathematics (2010), High School, domain Building Functions (HSF-BF)

  31. Curriculum point HSF-BF.A · Teaching content · checked against the act

    Build a function that models a relationship between two quantities

    CCSS Mathematics (2010), High School, Building Functions, cluster A

  32. Curriculum point HSF-BF.B · Teaching content · checked against the act

    Build new functions from existing functions

    CCSS Mathematics (2010), High School, Building Functions, cluster B

  33. Curriculum point HSF-BF.A.1 · Teaching content · checked against the act

    Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

    CCSS Mathematics (2010), High School, Building Functions, standard HSF-BF.A.1

    Writing Equations of Lines and Parallel Lines (we teach in grade 9-11) · Combining Functions Using Composition (we teach in grade 9-11)

  34. Curriculum point HSF-BF.B.3 · Teaching content · checked against the act

    Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

    CCSS Mathematics (2010), High School, Building Functions, standard HSF-BF.B.3

    Translating Function Graphs (we teach in grade 9-11)

  35. Curriculum point HSF-BF.B.4 · Teaching content · checked against the act

    Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

    CCSS Mathematics (2010), High School, Building Functions, standard HSF-BF.B.4

    Finding the Inverse of a Linear Function (we teach in grade 9-11)

  36. Curriculum point HSF-LE · Teaching content · checked against the act

    Linear, Quadratic, and Exponential Models★

    CCSS Mathematics (2010), High School, domain Linear, Quadratic, and Exponential Models★ (HSF-LE)

  37. Curriculum point HSF-LE.A · Teaching content · checked against the act

    Construct and compare linear, quadratic, and exponential models and solve problems

    CCSS Mathematics (2010), High School, Linear, Quadratic, and Exponential Models★, cluster A

  38. Curriculum point HSG-CO · Teaching content · checked against the act

    Congruence

    CCSS Mathematics (2010), High School, domain Congruence (HSG-CO)

  39. Curriculum point HSG-CO.A · Teaching content · checked against the act

    Experiment with transformations in the plane

    CCSS Mathematics (2010), High School, Congruence, cluster A

  40. Curriculum point HSG-CO.B · Teaching content · checked against the act

    Understand congruence in terms of rigid motions

    CCSS Mathematics (2010), High School, Congruence, cluster B

  41. Curriculum point HSG-CO.C · Teaching content · checked against the act

    Prove geometric theorems

    CCSS Mathematics (2010), High School, Congruence, cluster C

  42. Curriculum point HSG-CO.D · Teaching content · checked against the act

    Make geometric constructions

    CCSS Mathematics (2010), High School, Congruence, cluster D

  43. Curriculum point HSG-CO.A.1 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.A.1

  44. Curriculum point HSG-CO.A.2 · Teaching content · checked against the act

    Represent transformations in the plane using, e.g., transparencies and geometry software; describe transformations as functions that take points in the plane as inputs and give other points as outputs. Compare transformations that preserve distance and angle to those that do not (e.g., translation versus horizontal stretch).

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.A.2

  45. Curriculum point HSG-CO.A.3 · Teaching content · checked against the act

    Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.A.3

  46. Curriculum point HSG-CO.A.4 · Teaching content · checked against the act

    Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.A.4

  47. Curriculum point HSG-CO.A.5 · Teaching content · checked against the act

    Given a geometric figure and a rotation, reflection, or translation, draw the transformed figure using, e.g., graph paper, tracing paper, or geometry software. Specify a sequence of transformations that will carry a given figure onto another.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.A.5

  48. Curriculum point HSG-CO.B.6 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.B.6

  49. Curriculum point HSG-CO.B.7 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.B.7

  50. Curriculum point HSG-CO.B.8 · Teaching content · checked against the act

    Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.B.8

  51. Curriculum point HSG-CO.C.9 · Teaching content · checked against the act

    Prove theorems about lines and angles. Theorems include: vertical angles are congruent; when a transversal crosses parallel lines, alternate interior angles are congruent and corresponding angles are congruent; points on a perpendicular bisector of a line segment are exactly those equidistant from the segment’s endpoints.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.C.9

  52. Curriculum point HSG-CO.C.10 · Teaching content · checked against the act

    Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180°; base angles of isosceles triangles are congruent; the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; the medians of a triangle meet at a point.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.C.10

  53. Curriculum point HSG-CO.C.11 · Teaching content · checked against the act

    Prove theorems about parallelograms. Theorems include: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.C.11

  54. Curriculum point HSG-CO.D.12 · Teaching content · checked against the act

    Make formal geometric constructions with a variety of tools and methods (compass and straightedge, string, reflective devices, paper folding, dynamic geometric software, etc.). Copying a segment; copying an angle; bisecting a segment; bisecting an angle; constructing perpendicular lines, including the perpendicular bisector of a line segment; and constructing a line parallel to a given line through a point not on the line.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.D.12

  55. Curriculum point HSG-CO.D.13 · Teaching content · checked against the act

    Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle.

    CCSS Mathematics (2010), High School, Congruence, standard HSG-CO.D.13

  56. Curriculum point HSG-SRT · Teaching content · checked against the act

    Similarity, Right Triangles, and Trigonometry

    CCSS Mathematics (2010), High School, domain Similarity, Right Triangles, and Trigonometry (HSG-SRT)

  57. Curriculum point HSG-SRT.A · Teaching content · checked against the act

    Understand similarity in terms of similarity transformations

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, cluster A

  58. Curriculum point HSG-SRT.B · Teaching content · checked against the act

    Prove theorems involving similarity

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, cluster B

  59. Curriculum point HSG-SRT.C · Teaching content · checked against the act

    Define trigonometric ratios and solve problems involving right triangles

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, cluster C

  60. Curriculum point HSG-SRT.D · Teaching content · checked against the act

    Apply trigonometry to general triangles

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, cluster D

  61. Curriculum point HSG-SRT.A.1 · Teaching content · checked against the act

    Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.A.1

    Finding the Area of Similar Figures (we teach in grade 10)

  62. Curriculum point HSG-SRT.A.2 · Teaching content · checked against the act

    Given two figures, use the definition of similarity in terms of similarity transformations to decide if the y 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.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.A.2

  63. Curriculum point HSG-SRT.A.3 · Teaching content · checked against the act

    Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.A.3

  64. Curriculum point HSG-SRT.B.4 · Teaching content · checked against the act

    Prove theorems about triangles. Theorems include: a line parallel to one side of a triangle divides the other two proportionally, and conversely; the Pythagorean Theorem proved using triangle similarity.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.B.4

  65. Curriculum point HSG-SRT.B.5 · Teaching content · checked against the act

    Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.B.5

    Using the Intercept Theorem to Find Lengths (we teach in grade 10)

  66. Curriculum point HSG-SRT.C.6 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.C.6

  67. Curriculum point HSG-SRT.C.7 · Teaching content · checked against the act

    Explain and use the relationship between the sine and cosine of complementary angles.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.C.7

  68. Curriculum point HSG-SRT.C.8 · Teaching content · checked against the act

    Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.C.8

    Finding Missing Sides of Right Triangles (we teach in grade 10)

  69. Curriculum point HSG-SRT.D.9 · Teaching content · checked against the act

    (+) Derive the formula A = 1/2 ab sin(C) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.D.9

  70. Curriculum point HSG-SRT.D.10 · Teaching content · checked against the act

    (+) Prove the Laws of Sines and Cosines and use them to solve problems.

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.D.10

  71. Curriculum point HSG-SRT.D.11 · Teaching content · checked against the act

    (+) Understand and apply the Law of Sines and the Law of Cosines to find unkno wn measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

    CCSS Mathematics (2010), High School, Similarity, Right Triangles, and Trigonometry, standard HSG-SRT.D.11

    Using the Laws of Sines and Cosines (we teach in grade 10)

  72. Curriculum point HSG-C · Teaching content · checked against the act

    Circles

    CCSS Mathematics (2010), High School, domain Circles (HSG-C)

  73. Curriculum point HSG-C.A · Teaching content · checked against the act

    Understand and apply theorems about circles

    CCSS Mathematics (2010), High School, Circles, cluster A

  74. Curriculum point HSG-C.B · Teaching content · checked against the act

    Find arc lengths and areas of sectors of circles

    CCSS Mathematics (2010), High School, Circles, cluster B

  75. Curriculum point HSG-C.A.1 · Teaching content · checked against the act

    Prove that all circles are similar.

    CCSS Mathematics (2010), High School, Circles, standard HSG-C.A.1

  76. Curriculum point HSG-C.A.2 · Teaching content · checked against the act

    Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

    CCSS Mathematics (2010), High School, Circles, standard HSG-C.A.2

    Finding Central and Inscribed Angles in Circles (we teach in grade 10) · Finding Lengths of Chords and Tangent Segments (we teach in grade 10)

  77. Curriculum point HSG-C.A.3 · Teaching content · checked against the act

    Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle.

    CCSS Mathematics (2010), High School, Circles, standard HSG-C.A.3

  78. Curriculum point HSG-C.A.4 · Teaching content · checked against the act

    (+) Construct a tangent line from a point outside a given circle to the circle.

    CCSS Mathematics (2010), High School, Circles, standard HSG-C.A.4

  79. Curriculum point HSG-C.B.5 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Circles, standard HSG-C.B.5

  80. Curriculum point HSG-GPE · Teaching content · checked against the act

    Expressing Geometric Properties with Equations

    CCSS Mathematics (2010), High School, domain Expressing Geometric Properties with Equations (HSG-GPE)

  81. Curriculum point HSG-GPE.A · Teaching content · checked against the act

    Translate between the geometric description and the equation for a conic section

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, cluster A

  82. Curriculum point HSG-GPE.B · Teaching content · checked against the act

    Use coordinates to prove simple geometric theorems algebraically

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, cluster B

  83. Curriculum point HSG-GPE.A.1 · Teaching content · checked against the act

    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.

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.A.1

  84. Curriculum point HSG-GPE.A.2 · Teaching content · checked against the act

    Derive the equation of a parabola given a focus and directrix.

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.A.2

  85. Curriculum point HSG-GPE.B.4 · Teaching content · checked against the act

    Use coordinates to prove simple geometric theorems algebraically. For example, prove or disprove that a figure defined by four given points in the coordinate plane is a rectangle; prove or disprove that the point (1, √3) lies on the circle centered at the origin and containing the point (0, 2).

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.B.4

  86. Curriculum point HSG-GPE.B.5 · Teaching content · checked against the act

    Prove the slope criteria for parallel and perpendicular lines and use them to solve geometric problems (e.g., find the equation of a line parallel or perpendicular to a given line that passes through a given point).

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.B.5

  87. Curriculum point HSG-GPE.B.6 · Teaching content · checked against the act

    Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.B.6

    Dividing a Line Segment in a Given Ratio (we teach in grade 10)

  88. Curriculum point HSG-GPE.B.7 · Teaching content · checked against the act

    Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★

    CCSS Mathematics (2010), High School, Expressing Geometric Properties with Equations, standard HSG-GPE.B.7

    Finding the Area of a Triangle Using Coordinates (we teach in grade 10) · Finding Distance Between Two Points Using Pythagoras (we teach in grade 8)

  89. Curriculum point HSG-GMD · Teaching content · checked against the act

    Geometric Measurement and Dimension

    CCSS Mathematics (2010), High School, domain Geometric Measurement and Dimension (HSG-GMD)

  90. Curriculum point HSG-GMD.A · Teaching content · checked against the act

    Explain volume formulas and use them to solve problems

    CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, cluster A

  91. Curriculum point HSG-GMD.B · Teaching content · checked against the act

    Visualize relationships between two-dimensional and three-dimensional objects

    CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, cluster B

  92. Curriculum point HSG-GMD.A.1 · Teaching content · checked against the act

    Give an informal argument for the formulas for the circumference of a circle, area of a circle, volume of a cylinder, pyramid, and cone. Use dissection arguments, Cavalieri’s principle, and informal limit arguments.

    CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, standard HSG-GMD.A.1

  93. Curriculum point HSG-GMD.A.3 · Teaching content · checked against the act

    Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

    CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, standard HSG-GMD.A.3

    Finding the Volume of Cylinders, Cones, and Spheres (we teach in grade 8) · Surface Area of Cylinders, Cones, and Spheres (we teach in grade 10)

  94. Curriculum point HSG-GMD.B.4 · Teaching content · checked against the act

    Identify the shapes of two-dimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects.

    CCSS Mathematics (2010), High School, Geometric Measurement and Dimension, standard HSG-GMD.B.4

  95. Curriculum point HSG-MG · Teaching content · checked against the act

    Modeling with Geometry

    CCSS Mathematics (2010), High School, domain Modeling with Geometry (HSG-MG)

  96. Curriculum point HSG-MG.A · Teaching content · checked against the act

    Apply geometric concepts in modeling situations

    CCSS Mathematics (2010), High School, Modeling with Geometry, cluster A

  97. Curriculum point HSG-MG.A.1 · Teaching content · checked against the act

    Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). ★

    CCSS Mathematics (2010), High School, Modeling with Geometry, standard HSG-MG.A.1

  98. Curriculum point HSG-MG.A.2 · Teaching content · checked against the act

    Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★

    CCSS Mathematics (2010), High School, Modeling with Geometry, standard HSG-MG.A.2

    Finding Mass Using Density and Volume (we teach in grade 10)

  99. Curriculum point HSG-MG.A.3 · Teaching content · checked against the act

    Apply geometric methods to solve design problems (e.g., designing an object or structure to satisfy physical constraints or minimize cost; working with typographic grid systems based on ratios). ★

    CCSS Mathematics (2010), High School, Modeling with Geometry, standard HSG-MG.A.3

  100. Curriculum point HSS-ID · Teaching content · checked against the act

    Interpreting Categorical and Quantitative Data

    CCSS Mathematics (2010), High School, domain Interpreting Categorical and Quantitative Data (HSS-ID)

  101. Curriculum point HSS-ID.A · Teaching content · checked against the act

    Summarize, represent, and interpret data on a single count or measurement variable

    CCSS Mathematics (2010), High School, Interpreting Categorical and Quantitative Data, cluster A

  102. Curriculum point HSS-ID.B · Teaching content · checked against the act

    Summarize, represent, and interpret data on two categorical and quantitative variables

    CCSS Mathematics (2010), High School, Interpreting Categorical and Quantitative Data, cluster B

  103. Curriculum point HSS-ID.B.5 · Teaching content · checked against the act

    Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

    CCSS Mathematics (2010), High School, Interpreting Categorical and Quantitative Data, standard HSS-ID.B.5

    Finding Conditional Probability from a Table (we teach in grade 9-10)

  104. Curriculum point HSS-CP · Teaching content · checked against the act

    Conditional Probability and the Rules of Probability

    CCSS Mathematics (2010), High School, domain Conditional Probability and the Rules of Probability (HSS-CP)

  105. Curriculum point HSS-CP.A · Teaching content · checked against the act

    Understand independence and conditional probability and use them to interpret data

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, cluster A

  106. Curriculum point HSS-CP.B · Teaching content · checked against the act

    Use the rules of probability to compute probabilities of compound events in a uniform probability model

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, cluster B

  107. Curriculum point HSS-CP.A.1 · Teaching content · checked against the act

    Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.A.1

    Finding the Probability of a Union of Events (we teach in grade 10)

  108. Curriculum point HSS-CP.A.2 · Teaching content · checked against the act

    Understand that two events A and B are independent if the probability of A and B occurring together is the product of their probabilities, and use this characterization to determine if they are independent.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.A.2

  109. Curriculum point HSS-CP.A.3 · Teaching content · checked against the act

    Understand the conditional probability of A given B as P(A and B)/P(B), and interpret independence of A and B as saying that the conditional probability of A given B is the same as the probability of A, and the conditional probability of B given A is the same as the probability of B.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.A.3

  110. Curriculum point HSS-CP.A.4 · Teaching content · checked against the act

    Construct and interpret two-way frequency tables of data when two categories are associated with each object being classified. Use the two-way table as a sample space to decide if events are independent and to approximate conditional probabilities. For example, collect data from a random sample of students in your school on their favorite subject among math, science, and English. Estimate the probability that a randomly selected student from your school will favor science given that the student is in tenth grade. Do the same for other subjects and compare the results.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.A.4

  111. Curriculum point HSS-CP.A.5 · Teaching content · checked against the act

    Recognize and explain the concepts of conditional probability and independence in everyday language and everyday situations. For example, compare the chance of having lung cancer if you are a smoker with the chance of being a smoker if you have lung cancer.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.A.5

  112. Curriculum point HSS-CP.B.6 · Teaching content · checked against the act

    Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.B.6

    Finding Conditional Probability from a Table (we teach in grade 9-10)

  113. Curriculum point HSS-CP.B.7 · Teaching content · checked against the act

    Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.B.7

    Finding the Probability of a Union of Events (we teach in grade 10)

  114. Curriculum point HSS-CP.B.8 · Teaching content · checked against the act

    (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.B.8

    Using Total Probability and Bayes' Formula (we teach in grade 10)

  115. Curriculum point HSS-CP.B.9 · Teaching content · checked against the act

    (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

    CCSS Mathematics (2010), High School, Conditional Probability and the Rules of Probability, standard HSS-CP.B.9

    Finding Permutations and Counting Arrangements (we teach in grade 10) · Calculating Probabilities Using Combinations (we teach in grade 10)

  116. Curriculum point HSS-MD · Teaching content · checked against the act

    Using Probability to Make Decisions

    CCSS Mathematics (2010), High School, domain Using Probability to Make Decisions (HSS-MD)

  117. Curriculum point HSS-MD.B · Teaching content · checked against the act

    Use probability to evaluate outcomes of decisions

    CCSS Mathematics (2010), High School, Using Probability to Make Decisions, cluster B

  118. Curriculum point HSS-MD.B.6 · Teaching content · checked against the act

    (+) Use probabilities to make fair decisions (e.g., drawing by lots, using a random number gener ator).

    CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.B.6

  119. Curriculum point HSS-MD.B.7 · Teaching content · checked against the act

    (+) Analyze decisions and strategies using probability concepts (e.g., product t esting, medical testing, pulling a hockey goalie at the end of a game).

    CCSS Mathematics (2010), High School, Using Probability to Make Decisions, standard HSS-MD.B.7

Skills step by step

Finding Permutations and Counting Arrangements

Students learn to use multiplication rules and permutations to calculate the number of possible ordered arrangements and outcomes for everyday scenarios.

Curriculum point HSS-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

Students learn to determine the total number of ordered arrangements for a set of items or people using multiplication counting rules and permutations. In this grade, students calculate how many ways distinct positions can be filled when order matters and each choice reduces the options available for the next selection.

A common error is confusing when to add versus when to multiply possibilities across consecutive steps. Students also frequently forget to account for restrictions first; for instance, when building odd or even numbers without repeated digits, they may pick the starting digits before securing the required ending digit, leading to incorrect counts.

Practice tasks on this skill ask students to solve concrete arrangement problems, such as:

  • Calculating how many ways a set of students can stand in a single-file queue.
  • Finding the number of possible outcomes when awarding first-, second-, and third-place prizes to a group of contestants.
  • Determining how many unique even or odd multi-digit numbers can be created using a given set of digits without repetition.
Create a worksheet for this skill

Task types: Permutations — standing in a queue · Arrangements — awarding prizes · Even numbers containing a digit once · Odd numbers containing a digit once

Surface Area of Cylinders, Cones, and Spheres

Students learn to calculate the total surface area of curved 3D solids—cylinders, cones, and spheres—and analyze the flat nets used to build them.

Curriculum point HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

Students learn to calculate the total outer area of three-dimensional curved solids, including cylinders, cones, and spheres. They apply geometric formulas that combine circular bases with curved lateral surfaces, and they analyze two-dimensional nets to understand how flat shapes fold into three-dimensional cones.

A common mistake is confusing a cone's vertical height with its slant height, which leads to incorrect lateral area calculations. Students also frequently forget to add the circular bases to the lateral area, such as omitting the top and bottom circles of a cylinder or struggling to connect the circumference of a cone's base to the arc length of its unrolled side.

Worksheet tasks provide diagrams of solids with labeled dimensions such as radii, heights, or slant heights. Typical exercises ask students to find the surface area of a cylinder, the surface area of a cone, or the surface area of a sphere, as well as calculate the central angle of a cone's net from its circular dimensions.

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Task types: Surface area of a cylinder · Surface area of a sphere · Central angle of a cone's net · Surface area of a cone

Finding the Volume of Cylinders, Cones, and Spheres

Students learn to use formulas to calculate the volumes of cylinders, cones, and spheres in geometric and real-world problems.

Curriculum point 8.G.C.9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.
Curriculum point HSG-GMD.A.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. ★

In 10th grade geometry, students apply specific geometric formulas to calculate the volume of curved three-dimensional solids: cylinders, cones, and spheres. They identify key dimensions from diagrams or descriptions—such as radius, diameter, and perpendicular height—and evaluate the formulas to determine total capacity.

A frequent error involves mixing up the radius and the diameter. Students often substitute a given diameter directly into the formula without dividing it by two first. Another common mistake is applying the wrong exponent or omitting fractional coefficients, such as squaring the radius instead of cubing it in a sphere calculation, or forgetting to multiply by 1/3 when working with a cone.

Worksheet tasks focus directly on three formats: Volume of a cylinder, Volume of a cone, and Volume of a sphere. A typical problem provides a shape with given dimensions, such as a cylinder with a radius of 4 units and a height of 10 units, and asks students to compute the volume either in terms of pi or rounded to a specified decimal place.

Factoring Polynomials with Grouping and Cubes

Students learn to rewrite polynomial expressions by factoring out common monomials, grouping terms, and applying sum and difference of cubes patterns.

Curriculum point HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – ( y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

In 10th grade, students use the structure of polynomial expressions to identify ways to rewrite them into factored forms. Working with multi-term expressions and higher powers, students recognize when terms share an algebraic factor, when a four-term expression can be split into pairs, and how to spot the specific structure of a sum or difference of two perfect cubes.

A frequent stumbling block is managing negative signs. When factoring by grouping, students often forget to distribute a negative sign when pulling a negative factor out of the second pair of terms, leading to mismatched binomials. With the sum and difference of cubes, students frequently confuse the signs inside the linear and quadratic factors, especially the sign of the middle term in the resulting trinomial.

Worksheet tasks focus on three distinct structures:

  • Factoring out a monomial: identifying and extracting the greatest common numerical and variable factor from every term in an expression.
  • Factoring by grouping: partitioning a four-term polynomial into two pairs to extract common factors and reveal a shared binomial factor.
  • Sum and difference of cubes: identifying expressions built from two cubic terms and rewriting them into their corresponding linear and quadratic factors.
Create a worksheet for this skill

Task types: Factoring out a monomial · Factoring by grouping · Sum and difference of cubes

Using the Laws of Sines and Cosines

Students find missing sides, angles, and areas in non-right triangles using the Law of Sines, the Law of Cosines, and sine area formulas.

Curriculum point HSG-SRT.D.11 (+) Understand and apply the Law of Sines and the Law of Cosines to find unkno wn measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

At this level, students extend triangle trigonometry beyond right-angled triangles to any general triangle. They use the Law of Sines and the Law of Cosines to calculate unknown side lengths and angle measures, and they use the sine ratio to calculate a triangle's total area when given two sides and the included angle.

A frequent stumbling block is selecting the incorrect rule for the information provided. For example, students often attempt to use the Law of Sines on a triangle where only three side lengths or two sides and the included angle are known, which does not provide a complete side-angle pair. Another common mistake occurs during the algebra of the Law of Cosines, where students mistakenly subtract terms before multiplying by the cosine of the angle.

Practice exercises fall into three main formats: Law of sines tasks where students solve for a missing measurement given angle-side pairings, Law of cosines problems focused on side-angle-side or side-side-side arrangements, and Area of a triangle with sine problems where students compute the area using two side lengths and the sine of the angle between them.

Create a worksheet for this skill

Task types: Area of a triangle with sine · Law of cosines · Law of sines

Finding Distance Between Two Points Using Pythagoras

Students learn to find the straight-line distance between two points on a coordinate grid by treating the segment as the hypotenuse of a right triangle.

Curriculum point 8.G.B.8 Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.
Curriculum point HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★

Students use coordinates on a grid to calculate the straight-line distance between two points. By finding the horizontal difference between the x-coordinates and the vertical difference between the y-coordinates, they identify the two leg lengths of a right triangle. They then apply the Pythagorean Theorem—squaring both lengths, adding them together, and taking the square root—to find the unknown distance.

A frequent mistake involves subtracting negative coordinates when calculating the leg lengths. For instance, a student finding the horizontal distance between -3 and 4 might calculate 4 - 3 = 1 instead of 4 - (-3) = 7. Another common slip is calculating the sum of the squared legs but forgetting the final step of taking the square root, leaving the answer as c2 rather than c.

Practice tasks generally appear in two formats:

  • Distance between points with Pythagoras: Students look at two plotted points on a coordinate grid, draw or visualize the horizontal and vertical legs of a right triangle, and use the theorem to determine the length of the connecting segment.
  • Distance between two points: Students are given two coordinate pairs as numbers without a drawn triangle, requiring them to compute the leg lengths algebraically and solve for the distance.

Finding Terms and Sums of Geometric Sequences

Students learn to identify the common ratio in a geometric sequence, find any specific term using a formula, and calculate the total sum of a given number of terms.

Curriculum point HSA-SSE.B.4 Derive the formula for the sum of a finite geometric series (when the common ratio is not 1), and use the formula to solve problems. For example, calculate mortgage payments.★
Curriculum point HSF-BF.A.2 Write arithmetic and geometric sequences both recursively and with an explicit formula, use them to model situations, and translate between the two forms. ★
Curriculum point HSF-LE.A.2 Construct linear and exponential functions, including arithmetic and geometric sequences, given a graph, a description of a relationship, or two input-output pairs (include reading these from a table).

Students learn to analyze sequences where each term is found by multiplying the previous term by a constant common ratio (where the ratio is not 1). They write explicit formulas to find any specific nth term without having to calculate every value in between. Additionally, they use the finite geometric series formula to calculate the total sum of a specified number of terms.

A common mistake occurs with the order of operations when calculating the nth term. Students often multiply the initial term by the common ratio before applying the exponent, rather than calculating the power first. Confusion also arises with powers in the formula, such as using an exponent of n instead of n − 1, or mismanaging signs when subtracting a negative ratio in the denominator of the sum formula.

Practice tasks generally fall into two formats:

  • Geometric sequence — nth term: Given the first few numbers of a sequence or a real-world scenario, students identify the starting term and common ratio, write the formula, and solve for a specific term, such as the 8th or 12th term.
  • Geometric sequence — sum of terms: Students determine the sum of a designated number of terms by substituting the first term, the common ratio, and the number of terms directly into the finite series formula.
Create a worksheet for this skill

Task types: Geometric sequence — nth term · Geometric sequence — sum of terms

Using Total Probability and Bayes' Formula

Students learn to calculate overall event chances across multiple stages and find reversed conditional probabilities using two-urn models.

Curriculum point HSS-CP.B.8 (+) Apply the general Multiplication Rule in a uniform probability model, P(A and B) = P(A)P(B|A) = P(B)P(A|B), and interpret the answer in terms of the model.

At this level, students apply the general multiplication rule to find the probability of combined events. Working with multi-step experiments, they calculate the total probability of an outcome that can happen through different paths, and use Bayes' formula to determine conditional probabilities in reverse—figuring out which initial path was taken after observing the final outcome.

A common mistake is confusing the condition with the outcome, treating the probability of drawing a specific item from an urn, P(ball | urn), as identical to the probability that a specific urn was picked, P(urn | ball). Students also frequently forget to multiply by the probability of selecting the first container itself, focusing only on the ratio of colored balls inside it.

Practice tasks center on two-urn experiments. For example, a student is given two urns containing different mixtures of colored marbles, such as red and green. In a total probability task, they determine the overall likelihood of drawing a red marble after randomly picking an urn. In a Bayes' formula task, they are told that a red marble was drawn and must calculate the probability that it came from the first urn.

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Task types: Bayes' formula — two urns · Total probability — two urns

Using Special Product Formulas

Students learn to use algebraic patterns to expand the square of a binomial and the product of a sum and a difference without multiplying term by term.

Curriculum point HSA-SSE.A.2 Use the structure of an expression to identify ways to rewrite it. For example, see x4 – y4 as (x2)2 – ( y2)2, thus recognizing it as a difference of squares that can be factored as (x2 – y2)(x2 + y2).

Students learn to identify the underlying structure of algebraic expressions to rewrite and expand them quickly. Rather than multiplying out terms individually, they apply standard shortcut patterns to expand binomial products involving addition and subtraction.

A common error happens when expanding the square of a sum or difference. Students frequently write (x + 4)^2 as x^2 + 16, mistakenly distributing the exponent to each term and dropping the middle term that comes from multiplying the two terms together and doubling the result.

Practice tasks focus directly on two standard formats:

  • Product of a sum and a difference: recognizing matching terms with opposite signs, such as (x + 4)(x - 4), and rewriting them directly as a difference of squares.
  • Square of a sum or difference: expanding expressions such as (x + 3)^2 or (x - 5)^2 into complete three-term expressions.
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Task types: Product of a sum and a difference · Square of a sum or difference

Writing Equations of Lines and Parallel Lines

Students learn to write the equation of a line passing through two given points or passing through a specific point parallel to another line.

Curriculum point HSF-BF.A.1 Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Students learn to write an explicit linear equation that describes the relationship between two coordinate values. In this grade, they determine the slope and intercept to define a line, using either two given coordinate points or a single point paired with an existing line that runs parallel to it.

A frequent error occurs when handling parallel lines: students sometimes confuse parallel and perpendicular relationships, mistakenly flipping or negating the slope rather than keeping it identical. Another common mistake is reversing the coordinate order when calculating slope, accidentally placing the change in x over the change in y or mixing up positive and negative signs during subtraction.

Worksheet tasks typically take one of two forms:

  • Linear function through two points: Students are given two coordinate pairs and must find the slope and solve for the equation.
  • Parallel line through a point: Students identify the slope from a given line's equation and use it alongside a new coordinate point to write the parallel line's equation.
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Task types: Linear function through two points · Parallel line through a point

Finding the Vertex and Using Vertex Form

Students learn to rewrite quadratic equations into vertex form to identify the vertex, axis of symmetry, and maximum or minimum values of a parabola.

Curriculum point HSF-IF.C.8 Write a function defined by an expression in different but equivalent forms to reveal and explain different properties of the function. a. Use the process of factoring and completing the square in a quadratic function to show zeros, extreme values, and symmetry of the graph, and interpret these in terms of a context. b. Use the properties of exponents to interpret expressions for exponential functions. For example, identify percent rate of change in functions such as y = (1.02)t, y = (0.97)t, y = (1.01)12t, y = (1.2)t/10, and classify them as representing exponential growth or decay.

Students work with quadratic equations to reveal key geometric features of parabolas. By completing the square, they transform standard quadratic expressions into vertex form to identify the graph's line of symmetry and its extreme values—specifically whether the parabola reaches a maximum or minimum point at its vertex.

A frequent error involves the signs when extracting the horizontal coordinate from vertex form, y = a(x - h)^2 + k. Students often take the sign inside the parentheses at face value, mistakenly naming the vertex of y = (x + 4)^2 + 1 as (4, 1) rather than (-4, 1), because they forget that the standard formula uses subtraction for the horizontal shift.

In practice, students complete tasks aligned with Vertex form of a quadratic function and Vertex of a parabola. These exercises ask them to convert standard quadratic expressions into vertex form through algebraic steps, read the coordinates of the turning point directly from the rewritten equation, and state whether that point represents a highest or lowest value.

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Task types: Vertex of a parabola · Vertex form of a quadratic function

Finding Missing Sides of Right Triangles

Students learn to use trigonometric ratios to find the unknown length of a leg opposite or adjacent to a given angle in a right triangle.

Curriculum point HSG-SRT.C.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. ★

At this stage, students use basic trigonometric ratios to calculate an unknown side length in a right-angled triangle. Given one acute angle measure and one side length, they identify the appropriate trigonometric ratio—sine, cosine, or tangent—to solve for an unknown leg length.

A frequent stumbling block is mixing up the sides relative to the reference angle. Students often confuse the opposite leg (the side facing across from the angle) with the adjacent leg (the side next to the angle, between it and the right angle). This leads to setting up the ratio upside down or choosing the wrong trigonometric function entirely.

Worksheet tasks provide a labeled right triangle diagram and focus on two concrete problem types:

  • Leg opposite an angle: using the given angle and another side to calculate the length of the side directly across from that angle.
  • Leg adjacent to an angle: using the given angle and another side to find the length of the leg bordering the angle.
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Task types: Leg adjacent to an angle · Leg opposite an angle

Finding Central and Inscribed Angles in Circles

Students learn to find missing angle measures in circles using the rule that a central angle is twice the measure of an inscribed angle that intercepts the same arc.

Curriculum point HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

In 10th grade geometry, students explore the relationship between angles inside a circle that open up to the same arc. They learn that a central angle, which has its vertex at the center of the circle, is always twice as large as an inscribed angle, which has its vertex on the circle itself. Using this relationship, students calculate unknown angles using simple doubling and halving.

A common error occurs when students confuse which angle is larger. Because both angles share the exact same endpoints on the circle, students often assume the angles are equal, or they accidentally double a central angle rather than dividing it by two. Visualizing that the inscribed angle sits farther back on the circle's edge helps them remember that it must be the narrower angle.

Practice tasks provide labeled circle diagrams based on two formats: finding a central angle from the inscribed angle (for example, taking an inscribed angle of 35 degrees and multiplying by 2 to get 70 degrees) and finding an inscribed angle from the central angle (such as dividing a central angle of 110 degrees by 2 to find 55 degrees).

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Task types: Central angle from the inscribed angle · Inscribed angle from the central angle

Finding Lengths of Chords and Tangent Segments

Students learn to calculate the lengths of chords and tangent segments in circles using properties of radii, tangents, and right triangles.

Curriculum point HSG-C.A.2 Identify and describe relationships among inscribed angles, radii, and chords. Include the relationship between central, inscribed, and circumscribed angles; inscribed angles on a diameter are right angles; the radius of a circle is perpendicular to the tangent where the radius intersects the circle.

In tenth-grade geometry, students apply circle properties to solve for unknown side lengths. They use the rule that a circle’s radius is perpendicular to a tangent line at the point of contact, as well as the geometric relationships connecting radii to chords. By combining these geometric theorems with right-triangle relationships, students calculate missing lengths directly from geometric diagrams.

A common error involves misidentifying the right angle in tangent problems. Students often assume the right angle lies at an external point rather than where the radius meets the tangent line, which causes them to mislabel the hypotenuse. When solving chord problems, students also frequently forget that a perpendicular line from the center cuts the chord into two equal halves, leading them to report half the segment instead of the full chord length.

Practice tasks center on two main formats:

  • Length of a tangent segment: Students look at a circle with a radius and an external tangent line, using the given radius and the distance from the center to find the tangent segment’s length.
  • Length of a chord: Students work from a diagram showing a circle with a chord, using the radius and the perpendicular distance from the circle's center to determine the total length of the chord.
Create a worksheet for this skill

Task types: Length of a chord · Length of a tangent segment

Finding Conditional Probability from a Table

Students learn to find the probability of an event given another condition using data from a table, expressing the answer as a fraction and explaining its meaning.

Curriculum point HSS-CP.B.6 Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A, and interpret the answer in terms of the model.
Curriculum point HSS-ID.B.5 Summarize categorical data for two categories in two-way frequency tables. Interpret relative frequencies in the context of the data (including joint, marginal, and conditional relative frequencies). Recognize possible associations and trends in the data.

Students learn to calculate the conditional probability of an event A given that another event B has occurred. They use data to identify the subset of outcomes that satisfy condition B, make that count their denominator, and find the fraction of those specific outcomes that also belong to A. Students then interpret what this fraction represents within the context of the problem.

A frequent error is dividing by the total population rather than the restricted condition. When asked for the probability of A given B, students often pull the grand total from the entire data set instead of restricting their focus to the row or column representing only condition B.

Worksheet tasks feature two-way frequency tables categorizing survey results or experimental data. Students read the table to locate the given condition, identify the overlapping category, and write the resulting conditional probability as a simplified fraction.

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Finding the Probability of a Union of Events

Students learn to use the Addition Rule to calculate the probability that at least one of two events occurs, accounting for any outcomes they share.

Curriculum point HSS-CP.A.1 Describe events as subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or complements of other events (“or,” “and,” “not”).
Curriculum point HSS-CP.B.7 Apply the Addition Rule, P(A or B) = P(A) + P(B) – P(A and B), and interpret the answer in terms of the model.

At this level, students calculate the probability that event A, event B, or both occur within a sample space. They apply the Addition Rule, represented as P(A or B) = P(A) + P(B) – P(A and B), interpreting outcomes described with "or," "and," and "not." Students work with probabilities expressed as fractions, decimals, or percentages to model real-world situations.

The most common error is forgetting to subtract the intersection, P(A and B). When two events can happen at the same time, simply adding their individual probabilities double-counts the shared outcomes, which frequently results in an inaccurate answer or a probability greater than 1.

Tasks based on the Probability of a union of events template typically give students a scenario—such as drawing a card with specific characteristics or selecting a person from survey categories. Students identify the probability of each separate event, determine the probability of the overlap where both conditions are met, and apply the formula to find the combined probability.

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Calculating Probabilities Using Combinations

Students learn to use permutations and combinations to calculate the exact probability of compound events, such as picking items from a group without replacement.

Curriculum point HSS-CP.B.9 (+) Use permutations and combinations to compute probabilities of compound events and solve problems.

In 10th grade, students learn to use permutations and combinations to calculate probabilities for compound events. Rather than trying to write out every possible result by hand, they apply combinatorial counting methods to determine both the total number of favorable outcomes and the total possible outcomes when selecting groups of items.

A common hurdle is confusing whether the order of selection matters. When picking a group where order is irrelevant, students often mistakenly use permutations instead of combinations. Another frequent error in multi-step selections is forgetting that each pick changes the pool, treating draws as independent events rather than accounting for the items removed from the total.

Practice problems typically feature scenarios like drawing colored balls from an urn or bag without replacement. For instance, given a set number of red and blue balls, a student might be asked to find the probability of drawing exactly two red balls and one blue ball in a sample of three, using combinations to evaluate both the successful subsets and the overall sample space.

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Translating Function Graphs

Students learn how adding or subtracting numbers shifts a graph up, down, left, or right, and how to identify these movements between equations and graphs.

Curriculum point HSF-BF.B.3 Identify the effect on the graph of replacing f(x) by f(x) + k, k f(x), f(kx), and f(x + k) for specific values of k (both positive and negative); find the value of k given the graphs. Experiment with cases and illustrate an explanation of the effects on the graph using technology. Include recognizing even and odd functions from their graphs and algebraic expressions for them.

Students learn to connect algebraic changes in a function to physical shifts of its graph on the coordinate plane. Specifically, they examine how replacing f(x) with f(x) + k or f(x + k) moves the curve vertically or horizontally using both positive and negative values of k. Given an original graph and its shifted image, they also find the specific value of k that caused the translation.

A frequent stumbling block is the direction of horizontal shifts. Because addition is associated with the positive direction on a number line, students often assume that f(x + 4) moves the graph four units to the right. In practice, adding inside the input shifts the graph to the left, while subtracting moves it to the right. Students also frequently confuse these horizontal adjustments inside the parentheses with vertical shifts added outside the function.

Worksheet tasks based on Translating a graph present students with a plotted curve and ask them to either graph the result of a transformation or identify the shift from an image. Students might take a given function, apply a rule such as f(x) − 3 or f(x + 2), and plot the translated points, or determine the numerical value of k by measuring the distance between key features on two displayed graphs.

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Finding Average Rate of Change

Students learn to calculate and interpret the average rate of change of a function across an interval using formulas, tables, and graphs.

Curriculum point HSF-IF.B.6 Calculate and interpret the average rate of change of a function (presented symbolically or as a table) over a specified interval. Estimate the rate of change from a graph. ★

Students learn to measure how much a function's output changes relative to its input over a given interval. They calculate this rate using functions presented as algebraic equations or tables of values, and they estimate the rate by reading coordinates off a graph. Beyond calculating the numerical value, students learn to interpret what that rate means in real-world scenarios.

A common error is reversing the calculation by placing the change in input over the change in output, mixing up the independent and dependent variables. Students also frequently subtract values in mismatched directions—such as subtracting the first input from the second, but subtracting the second output from the first—which leads to an incorrect sign.

In the Average rate of change practice tasks, students typically receive a function rule, a data table, or a plotted curve alongside a specified interval. They locate or calculate the function values at both boundaries of the interval, set up the ratio of the vertical change to the horizontal change, and compute or estimate the average rate over that span.

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Combining Functions Using Composition

Students learn to evaluate and write composite functions by using the output of one function as the input for another.

Curriculum point HSF-BF.A.1 Write a function that describes a relationship between two quantities.★ a. Determine an explicit expression, a recursive process, or steps for calculation from a context. b. Combine standard function types using arithmetic operations. For example, build a function that models the temperature of a cooling body by adding a constant function to a decaying exponential, and relate these functions to the model. c. (+) Compose functions. For example, if T(y) is the temperature in the atmosphere as a function of height, and h(t) is the height of a weather balloon as a function of time, then T(h(t)) is the temperature at the location of the weather balloon as a function of time.

Students learn to compose functions by taking the output of an inner function and feeding it directly as the input into an outer function. At this level, students write composite rules such as f(g(x)) and apply them to contexts where quantities are linked across steps, such as finding atmospheric temperature at a given time when temperature depends on height and height depends on time.

A frequent stumbling block is confusing composition with multiplication. Instead of replacing the variable in the outside function with the entire inside expression, students often multiply the two formulas together. Another common mistake is reversing the order of operations, applying the outer function first instead of working from the inside out.

Practice with the Composition of functions template presents two distinct function definitions and prompts students to write a simplified rule for the combined function or compute the final numerical value for a specific input.

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Finding the Inverse of a Linear Function

Students learn to write the inverse rule for a linear function by reversing its algebraic operations and solving for the new output.

Curriculum point HSF-BF.B.4 Find inverse functions. a. Solve an equation of the form f(x) = c for a simple function f that has an inverse and write an expression for the inverse. For example, f(x) =2 x3 or f(x) = (x+1)/(x–1) for x ≠ 1. b. (+) Verify by composition that one function is the inverse of another. c. (+) Read values of an inverse function from a graph or a table, given that the function has an inverse. d. (+) Produce an invertible function from a non-invertible function by restricting the domain.

At this level, students work with linear equations to determine their inverse functions. Given a linear rule such as f(x) = ax + b, students solve the equation for the input variable by systematically undoing each operation, producing a new formula that maps outputs back to their original inputs.

A common error is confusing the inverse notation f-1(x) with a negative exponent. Students often mistakenly write the reciprocal 1/f(x) instead of finding the inverse function. Another frequent issue is reversing the order of operations incorrectly, such as dividing by the slope before subtracting the constant term.

Practice tasks based on the Inverse of a linear function template present students with a single linear equation—often involving integers or simple fractions—and ask them to write the algebraic expression for its inverse.

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Finding the Area of Similar Figures

Students learn how side length scale factors affect two-dimensional space and calculate the unknown area of a similar figure.

Curriculum point HSG-SRT.A.1 Verify experimentally the properties of dilations given by a center and a scale factor: a. A dilation takes a line not passing through the center of the dilation to a parallel line, and leaves a line passing through the center unchanged. b. The dilation of a line segment is longer or shorter in the ratio given by the scale factor.

Students use the properties of dilations to determine how resizing a shape changes its measurements. When a figure is enlarged or reduced by a scale factor, segment lengths scale proportionally, and students apply this relationship to calculate the unknown area of a similar figure.

A common error is treating area like perimeter or side length. When two figures have a scale factor such as 3, students often multiply the original area directly by 3. They forget that area is two-dimensional and scales by the square of the scale factor, meaning the area actually increases by a factor of 9.

In practice, tasks like Area of a similar figure provide two similar polygons with a pair of corresponding side lengths and the area of one shape. Students identify the ratio between the matching sides, square that ratio, and use it to solve for the missing area.

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Using the Intercept Theorem to Find Lengths

Students learn to use the intercept theorem and triangle similarity to set up proportions and calculate missing side lengths in geometric figures.

Curriculum point HSG-SRT.B.5 Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

In tenth-grade geometry, students apply similarity criteria to triangles crossed by parallel lines. Using the intercept theorem, they identify that parallel lines cut transversal lines into proportional segments. Students set up ratios comparing corresponding side lengths and solve algebraic proportions using multiplication and division to determine unknown segment lengths.

A common error occurs when students compare the parallel cross-segments to side segments. Instead of setting up a ratio between the small triangle's side and the full side of the larger triangle, students often pair a parallel base with only the bottom segment of a side. This happens because they treat every visually separated segment as an independent side of similar triangles rather than accounting for the full side length.

Worksheet tasks from the Intercept theorem template present geometric figures such as a triangle intersected by a line parallel to its base. Given three segment lengths as whole numbers or decimals, students write a proportion matching the corresponding parts and solve for a missing value labeled with a variable.

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Dividing a Line Segment in a Given Ratio

Students learn to find the coordinates of a point on a directed line segment that divides the segment in a given ratio.

Curriculum point HSG-GPE.B.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio.

In 10th grade coordinate geometry, students learn to locate a specific point along a directed line segment between two known endpoints. Given the coordinates of the endpoints on the coordinate plane, they determine the exact x- and y-coordinates that partition the segment according to a specified ratio.

A common mistake occurs when students confuse a part-to-part ratio with a part-to-whole fraction. For example, when asked to partition a segment in a 1:3 ratio, students often calculate one-third of the horizontal and vertical distances instead of one-fourth. Another frequent error is running the calculation in the wrong direction, measuring from the second endpoint rather than the designated starting point of the directed segment.

Practice tasks based on the Dividing a segment in a ratio template present two coordinate points and a target ratio, such as partitioning directed segment AB from point A to point B in a ratio of 2:1. Students calculate the horizontal and vertical distance traveled, apply the appropriate fractional step, and name the resulting coordinate point.

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Finding the Area of a Triangle Using Coordinates

Students learn to find the area of a triangle on the coordinate plane by using vertex coordinates to determine side lengths and heights.

Curriculum point HSG-GPE.B.7 Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. ★

Students use the coordinates of a triangle's three vertices on a coordinate plane to compute its area. In tenth grade, they apply the distance formula to find lengths of perpendicular bases and heights, or they enclose the slanted figure inside a bounding rectangle and subtract the area of the surrounding right triangles.

A common mistake occurs when a triangle has no horizontal or vertical edges. Students often treat an arbitrary side as the height without checking that it meets the chosen base at a right angle. Sign errors are also frequent when subtracting negative coordinate values during distance calculations.

Worksheet tasks based on the Area of a triangle from coordinates template present three vertex points, either plotted on a grid or written as coordinate pairs. Students identify or construct the required measurements and solve for the total area in square units.

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Finding Mass Using Density and Volume

Students learn to find the mass of a solid block by calculating its volume and multiplying by its given density.

Curriculum point HSG-MG.A.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). ★

In 10th grade geometry, students connect three-dimensional geometric measurement to real-world objects. At this level, students find the volume of a solid rectangular block from its given dimensions and multiply that volume by a given density to determine the block's total mass.

A common error is confusing the algebraic relationship between mass, volume, and density. Because density is defined using division (mass divided by volume), students often mistakenly divide the volume by the density—or the density by the volume—rather than multiplying volume by density to isolate the mass.

In practice, tasks like mass of a block from density provide the length, width, and height of a block along with the density of the material. Students calculate the volume of the block and then multiply that volume by the density to find the final mass.

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Task templates to print

Every task type, grouped by skill. Clicking one opens the worksheet generator with it already selected.

Finding Permutations and Counting Arrangements

Surface Area of Cylinders, Cones, and Spheres

Factoring Polynomials with Grouping and Cubes

Using the Laws of Sines and Cosines

Finding Terms and Sums of Geometric Sequences

Using Total Probability and Bayes' Formula

Using Special Product Formulas

Writing Equations of Lines and Parallel Lines

Finding the Vertex and Using Vertex Form

Finding Missing Sides of Right Triangles

Finding Central and Inscribed Angles in Circles

Finding Lengths of Chords and Tangent Segments

Finding Conditional Probability from a Table

Finding the Probability of a Union of Events

Calculating Probabilities Using Combinations

Translating Function Graphs

Finding Average Rate of Change

Combining Functions Using Composition

Finding the Inverse of a Linear Function

Finding the Area of Similar Figures

Using the Intercept Theorem to Find Lengths

Dividing a Line Segment in a Given Ratio

Finding the Area of a Triangle Using Coordinates

Finding Mass Using Density and Volume

Other task types

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