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Curriculum Standards:


Describe the shapes of two-dimensional cross-sections of three-dimensional objects and use those cross-sections to solve mathematical and real-world problems. - G.GGMD.4
Mathematically proficient students make sense of quantities and their relationships in problem situations. They bring two complementary abilities to bear on problems involving quantitative relationships: the ability to decontextualize-to abstract a given situation and represent it symbolically and manipulate the representing symbols as if they have a life of their own, without necessarily attending to their referents-and the ability to contextualize, to pause as needed during the manipulation process in order to probe into the referents for the symbols involved. Quantitative reasoning entails habits of creating a coherent representation of the problem at hand; considering the units involved; attending to the meaning of quantities, not just how to compute them; and knowing and flexibly using different properties of operations and objects. - CCSS.Math.Practice.MP2.a
Explain the derivation of the formulas for the volume of a sphere and other solid figures using Cavalieri’s principle. - G.GGMD.2
Apply surface area and volume formulas for prisms, cylinders, pyramids, cones, and spheres to solve problems and justify results. Include problems that involve algebraic expressions, composite figures, geometric probability, and real-world applications. - G.GGMD.3
Explain the derivations of the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid, and cone. Apply these formulas to solve mathematical and real-world problems. - G.GGMD.1
Fit a linear function for a scatter plot that suggests a linear association. - CCSS.Math.Content.HSS-ID.B.6c
Fit a function to the data; use functions fitted to data to solve problems in the context of the data. Use given functions or choose a function suggested by the context. Emphasize linear, quadratic, and exponential models. - CCSS.Math.Content.HSS-ID.B.6a
Informally assess the fit of a function by plotting and analyzing residuals. - CCSS.Math.Content.HSS-ID.B.6b
Mathematically proficient students try to communicate precisely to others. They try to use clear definitions in discussion with others and in their own reasoning. They state the meaning of the symbols they choose, including using the equal sign consistently and appropriately. They are careful about specifying units of measure, and labeling axes to clarify the correspondence with quantities in a problem. They calculate accurately and efficiently, express numerical answers with a degree of precision appropriate for the problem context. In the elementary grades, students give carefully formulated explanations to each other. By the time they reach high school they have learned to examine claims and make explicit use of definitions. - CCSS.Math.Practice.MP6.a
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). - CCSS.Math.Content.HSF-LE.A.2
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.Math.Content.HSG-CO.B.6
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.Math.Content.HSG-CO.B.7
Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. - CCSS.Math.Content.HSG-CO.B.8
Graph the solutions to a linear inequality in two variables as a half-plane (excluding the boundary in the case of a strict inequality), and graph the solution set to a system of linear inequalities in two variables as the intersection of the corresponding half-planes. - CCSS.Math.Content.HSA-REI.D.12
Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. - CCSS.Math.Content.HSG-CO.D.13
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.Math.Content.HSG-CO.D.12
Compare properties of two functions each represented in a different way (algebraically, graphically, numerically in tables, or by verbal descriptions). Example: For example, given a graph of one quadratic function and an algebraic expression for another, say which has the larger maximum. - CCSS.Math.Content.HSF-IF.C.9
Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. - CCSS.Math.Content.HSG-SRT.C.8
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.Math.Content.HSG-SRT.C.6
Explain and use the relationship between the sine and cosine of complementary angles. - CCSS.Math.Content.HSG-SRT.C.7
Mathematically proficient students start by explaining to themselves the meaning of a problem and looking for entry points to its solution. They analyze givens, constraints, relationships, and goals. They make conjectures about the form and meaning of the solution and plan a solution pathway rather than simply jumping into a solution attempt. They consider analogous problems, and try special cases and simpler forms of the original problem in order to gain insight into its solution. They monitor and evaluate their progress and change course if necessary. Older students might, depending on the context of the problem, transform algebraic expressions or change the viewing window on their graphing calculator to get the information they need. Mathematically proficient students can explain correspondences between equations, verbal descriptions, tables, and graphs or draw diagrams of important features and relationships, graph data, and search for regularity or trends. Younger students might rely on using concrete objects or pictures to help conceptualize and solve a problem. Mathematically proficient students check their answers to problems using a different method, and they continually ask themselves, 'Does this make sense?' They can understand the approaches of others to solving complex problems and identify correspondences between different approaches. - CCSS.Math.Practice.MP1.a
Graph linear and quadratic functions and show intercepts, maxima, and minima. - CCSS.Math.Content.HSF-IF.C.7a
Prove that two triangles are similar using the Angle-Angle criterion and apply the proportionality of corresponding sides to solve problems and justify results. - G.GSRT.3
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. Example: For example, rearrange Ohm’s law 𝘝 = 𝘝𝘭𝘝𝘭𝘙 to highlight resistance 𝘝𝘭𝘙𝘙. - CCSS.Math.Content.HSA-CED.A.4
Graph exponential and logarithmic functions, showing intercepts and end behavior, and trigonometric functions, showing period, midline, and amplitude. - CCSS.Math.Content.HSF-IF.C.7e
Prove, and apply in mathematical and real-world contexts, theorems involving similarity about triangles, including the following: a) A line drawn parallel to one side of a triangle divides the other two sides into parts of equal proportion. b) If a line divides two sides of a triangle proportionally, then it is parallel to the third side. c) The square of the hypotenuse of a right triangle is equal to the sum of squares of the other two sides. - G.GSRT.4
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. Example: For example, represent inequalities describing nutritional and cost constraints on combinations of different foods. - CCSS.Math.Content.HSA-CED.A.3
Understand 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. Verify experimentally the properties of dilations given by a center and a scale factor. Understand the dilation of a line segment is longer or shorter in the ratio given by the scale factor. - G.GSRT.1
Create equations in two or more variables to represent relationships between quantities; graph equations on coordinate axes with labels and scales. - CCSS.Math.Content.HSA-CED.A.2
Use the definition of similarity to decide if figures are similar and justify decision. Demonstrate that two figures are similar by identifying a combination of translations, rotations, reflections, and dilations in various representations that move one figure onto the other. - G.GSRT.2
Explain each step in solving a simple equation as following from the equality of numbers asserted at the previous step, starting from the assumption that the original equation has a solution. Construct a viable argument to justify a solution method. - CCSS.Math.Content.HSA-REI.A.1
Graph square root, cube root, and piecewise-defined functions, including step functions and absolute value functions. - CCSS.Math.Content.HSF-IF.C.7b
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.Math.Content.HSA-CED.A.1
Construct geometric figures using a variety of tools, including a compass, a straightedge, dynamic geometry software, and paper folding, and use these constructions to make conjectures about geometric relationships. - G.GCO.11
Explain and use the relationship between the sine and cosine of complementary angles. - G.GSRT.7
Solve right triangles in applied problems using trigonometric ratios and the Pythagorean Theorem. - G.GSRT.8
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - G.GSRT.5
Interpret the parameters in a linear or exponential function in terms of a context. - CCSS.Math.Content.HSF-LE.B.5
Mathematically proficient students consider the available tools when solving a mathematical problem. These tools might include pencil and paper, concrete models, a ruler, a protractor, a calculator, a spreadsheet, a computer algebra system, a statistical package, or dynamic geometry software. Proficient students are sufficiently familiar with tools appropriate for their grade or course to make sound decisions about when each of these tools might be helpful, recognizing both the insight to be gained and their limitations. For example, mathematically proficient high school students analyze graphs of functions and solutions generated using a graphing calculator. They detect possible errors by strategically using estimation and other mathematical knowledge. When making mathematical models, they know that technology can enable them to visualize the results of varying assumptions, explore consequences, and compare predictions with data. Mathematically proficient students at various grade levels are able to identify relevant external mathematical resources, such as digital content located on a website, and use them to pose or solve problems. They are able to use technological tools to explore and deepen their understanding of concepts. - CCSS.Math.Practice.MP5.a
Prove, and apply in mathematical and real-world contexts, theorems about parallelograms, including the following: a) opposite sides of a parallelogram are congruent; b) opposite angles of a parallelogram are congruent; c) diagonals of a parallelogram bisect each other; d) rectangles are parallelograms with congruent diagonals; e) a parallelograms is a rhombus if and only if the diagonals are perpendicular. - G.GCO.10
Understand how the properties of similar right triangles allow the trigonometric ratios to be defined and determine the sine, cosine, and tangent of an acute angle in a right triangle. - G.GSRT.6
Apply the Addition Rule, 𝘝𝘭𝘙𝘙𝘗(𝘝𝘭𝘙𝘙𝘗𝘈 or 𝘝𝘭𝘙𝘙𝘗𝘈𝘉) = 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈) + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉) – 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.7
Apply the general Multiplication Rule in a uniform probability model, 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉) = 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈)𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉|𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈) = 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉)𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈|𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉), and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.8
Use permutations and combinations to compute probabilities of compound events and solve problems. - CCSS.Math.Content.HSS-CP.B.9
Evaluate and compare strategies on the basis of expected values. Example: For example, compare a high-deductible versus a low-deductible automobile insurance policy using various, but reasonable, chances of having a minor or a major accident. - CCSS.Math.Content.HSS-MD.B.5b
Find the expected payoff for a game of chance. Example: For example, find the expected winnings from a state lottery ticket or a game at a fast-food restaurant. - CCSS.Math.Content.HSS-MD.B.5a
Find the conditional probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉 as the fraction of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉’s outcomes that also belong to 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈, and interpret the answer in terms of the model. - CCSS.Math.Content.HSS-CP.B.6
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.Math.Content.HSG-CO.C.9
Find the point on a directed line segment between two given points that partitions the segment in a given ratio. - CCSS.Math.Content.HSG-GPE.B.6
Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. - CCSS.Math.Content.HSG-GPE.B.7
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.Math.Content.HSG-C.B.5
Use coordinates to prove simple geometric theorems algebraically. Example: 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.Math.Content.HSG-GPE.B.4
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.Math.Content.HSG-GPE.B.5
Construct a tangent line from a point outside a given circle to the circle. - CCSS.Math.Content.HSG-C.A.4
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.Math.Content.HSG-C.A.2
Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. - CCSS.Math.Content.HSG-C.A.3
Prove that all circles are similar. - CCSS.Math.Content.HSG-C.A.1
Prove that all circles are similar. - G.GCI.1
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.Math.Content.HSA-APR.A.1
Construct the inscribed and circumscribed circles of a triangle using a variety of tools, including a compass, a straightedge, and dynamic geometry software, and prove properties of angles for a quadrilateral inscribed in a circle. - G.GCI.3
Identify and describe relationships among inscribed angles, radii, and chords; among inscribed angles, central angles, and circumscribed angles; and between radii and tangents to circles. Use those relationships to solve mathematical and real-world problems. - G.GCI.2
Derive the formulas for the length of an arc and the area of a sector in a circle and apply these formulas to solve mathematical and real-world problems. - G.GCI.5
Compute (using technology) and interpret the correlation coefficient of a linear fit. - CCSS.Math.Content.HSS-ID.C.8
Construct a tangent line to a circle through a point on the circle, and construct a tangent line from a point outside a given circle to the circle; justify the process used for each construction. - G.GCI.4
Explain why the sum or product of two rational numbers is rational; that the sum of a rational number and an irrational number is irrational; and that the product of a nonzero rational number and an irrational number is irrational. - CCSS.Math.Content.HSN-RN.B.3
Calculate the distance between numbers in the complex plane as the modulus of the difference, and the midpoint of a segment as the average of the numbers at its endpoints. - CCSS.Math.Content.HSN-CN.B.6
Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. - CCSS.Math.Content.HSG-SRT.B.5
Use statistics appropriate to the shape of the data distribution to compare center and spread of two or more different data sets that include all real numbers. - G.SPID.2
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.Math.Content.HSG-SRT.B.4
Mathematically proficient students notice if calculations are repeated, and look both for general methods and for shortcuts. Upper elementary students might notice when dividing 25 by 11 that they are repeating the same calculations over and over again, and conclude they have a repeating decimal. By paying attention to the calculation of slope as they repeatedly check whether points are on the line through (1, 2) with slope 3, middle school students might abstract the equation (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦 – 2)/(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥 – 1) = 3. Noticing the regularity in the way terms cancel when expanding (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥 – 1)(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥 + 1), (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥 – 1)(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥² + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥 + 1), and (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥 – 1)(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥³ + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥² + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥 + 1) might lead them to the general formula for the sum of a geometric series. As they work to solve a problem, mathematically proficient students maintain oversight of the process, while attending to the details. They continually evaluate the reasonableness of their intermediate results. - CCSS.Math.Practice.MP8.a
Solve linear equations and inequalities in one variable, including equations with coefficients represented by letters. - CCSS.Math.Content.HSA-REI.B.3
Mathematically proficient students can apply the mathematics they know to solve problems arising in everyday life, society, and the workplace. In early grades, this might be as simple as writing an addition equation to describe a situation. In middle grades, a student might apply proportional reasoning to plan a school event or analyze a problem in the community. By high school, a student might use geometry to solve a design problem or use a function to describe how one quantity of interest depends on another. Mathematically proficient students who can apply what they know are comfortable making assumptions and approximations to simplify a complicated situation, realizing that these may need revision later. They are able to identify important quantities in a practical situation and map their relationships using such tools as diagrams, two-way tables, graphs, flowcharts and formulas. They can analyze those relationships mathematically to draw conclusions. They routinely interpret their mathematical results in the context of the situation and reflect on whether the results make sense, possibly improving the model if it has not served its purpose. - CCSS.Math.Practice.MP4.a
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.Math.Content.HSG-GPE.A.1
Derive the equation of a parabola given a focus and directrix. - CCSS.Math.Content.HSG-GPE.A.2
Understand that two events 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉 are independent if the probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉 occurring together is the product of their probabilities, and use this characterization to determine if they are independent. - CCSS.Math.Content.HSS-CP.A.2
Understand the conditional probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉 as 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉)/𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉), and interpret independence of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉 as saying that the conditional probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈 given 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉 is the same as the probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈, and the conditional probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉 given 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈 is the same as the probability of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉. - CCSS.Math.Content.HSS-CP.A.3
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. - CCSS.Math.Content.HSG-SRT.A.2
Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. - CCSS.Math.Content.HSG-GMD.A.3
Use geometric shapes, their measures, and their properties to describe real-world objects. - G.GM.1
Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. - CCSS.Math.Content.HSG-SRT.A.3
Rewrite expressions involving radicals and rational exponents using the properties of exponents. - CCSS.Math.Content.HSN-RN.A.2
Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. - CCSS.Math.Content.HSG-GMD.A.2
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.Math.Content.HSG-GMD.A.1
Develop a probability distribution for a random variable defined for a sample space in which probabilities are assigned empirically; find the expected value. Example: For example, find a current data distribution on the number of TV sets per household in the United States, and calculate the expected number of sets per household. How many TV sets would you expect to find in 100 randomly selected households? - CCSS.Math.Content.HSS-MD.A.4
Develop a probability distribution for a random variable defined for a sample space in which theoretical probabilities can be calculated; find the expected value. Example: For example, find the theoretical probability distribution for the number of correct answers obtained by guessing on all five questions of a multiple-choice test where each question has four choices, and find the expected grade under various grading schemes. - CCSS.Math.Content.HSS-MD.A.3
Calculate the expected value of a random variable; interpret it as the mean of the probability distribution. - CCSS.Math.Content.HSS-MD.A.2
Recognize that sequences are functions, sometimes defined recursively, whose domain is a subset of the integers. Example: For example, the Fibonacci sequence is defined recursively by 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧(0) = 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧(1) = 1, 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯+1) = 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯) + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯-1) for 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯 greater than or equal to 1. - CCSS.Math.Content.HSF-IF.A.3
Use function notation, evaluate functions for inputs in their domains, and interpret statements that use function notation in terms of a context. - CCSS.Math.Content.HSF-IF.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. - CCSS.Math.Content.HSF-BF.A.2
Mathematically proficient students look closely to discern a pattern or structure. Young students, for example, might notice that three and seven more is the same amount as seven and three more, or they may sort a collection of shapes according to how many sides the shapes have. Later, students will see 7 × 8 equals the well remembered 7 × 5 + 7 × 3, in preparation for learning about the distributive property. In the expression 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥² + 9𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥 + 14, older students can see the 14 as 2 × 7 and the 9 as 2 + 7. They recognize the significance of an existing line in a geometric figure and can use the strategy of drawing an auxiliary line for solving problems. They also can step back for an overview and shift perspective. They can see complicated things, such as some algebraic expressions, as single objects or as being composed of several objects. For example, they can see 5 – 3(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥 – 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦)² as 5 minus a positive number times a square and use that to realize that its value cannot be more than 5 for any real numbers 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦. - CCSS.Math.Practice.MP7.a
Define a random variable for a quantity of interest by assigning a numerical value to each event in a sample space; graph the corresponding probability distribution using the same graphical displays as for data distributions. - CCSS.Math.Content.HSS-MD.A.1
Mathematically proficient students understand and use stated assumptions, definitions, and previously established results in constructing arguments. They make conjectures and build a logical progression of statements to explore the truth of their conjectures. They are able to analyze situations by breaking them into cases, and can recognize and use counterexamples. They justify their conclusions, communicate them to others, and respond to the arguments of others. They reason inductively about data, making plausible arguments that take into account the context from which the data arose. Mathematically proficient students are also able to compare the effectiveness of two plausible arguments, distinguish correct logic or reasoning from that which is flawed, and-if there is a flaw in an argument-explain what it is. Elementary students can construct arguments using concrete referents such as objects, drawings, diagrams, and actions. Such arguments can make sense and be correct, even though they are not generalized or made formal until later grades. Later, students learn to determine domains to which an argument applies. Students at all grades can listen or read the arguments of others, decide whether they make sense, and ask useful questions to clarify or improve the arguments. - CCSS.Math.Practice.MP3.a
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.Math.Content.HSA-REI.C.5
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.Math.Content.HSG-CO.A.1
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.Math.Content.HSG-CO.A.2
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.Math.Content.HSG-CO.C.11
Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. - CCSS.Math.Content.HSG-CO.A.3
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.Math.Content.HSG-CO.C.10
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. - CCSS.Math.Content.HSG-CO.A.4
Solve a simple system consisting of a linear equation and a quadratic equation in two variables algebraically and graphically. Example: For example, find the points of intersection between the line 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺 = –3𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹 and the circle 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹² + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺² = 3. - CCSS.Math.Content.HSA-REI.C.7
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.Math.Content.HSG-CO.A.5
Understand that the standard equation of a circle is derived from the definition of a circle and the distance formula. - G.GGPE.1
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.Math.Content.HSG-MG.A.1
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.Math.Content.HSG-GMD.B.4
Use statistics appropriate to the shape of the data distribution to compare center (median, mean) and spread (interquartile range, standard deviation) of two or more different data sets. - CCSS.Math.Content.HSS-ID.A.2
Know and apply the Binomial Theorem for the expansion of (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹 + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺)ⁿ in powers of 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹 and y for a positive integer 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯, where 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺 are any numbers, with coefficients determined for example by Pascal’s Triangle. The Binomial Theorem can be proved by mathematical induction or by a combinatorial argument. - CCSS.Math.Content.HSA-APR.C.5
Use coordinates to prove simple geometric theorems algebraically. - G.GGPE.4
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. - CCSS.Math.Content.HSG-SRT.A.1a
The dilation of a line segment is longer or shorter in the ratio given by the scale factor. - CCSS.Math.Content.HSG-SRT.A.1b
Analyze slopes of lines to determine whether lines are parallel, perpendicular, or neither. Write the equation of a line passing through a given point that is parallel or perpendicular to a given line. Solve geometric and real-world problems involving lines and slope. - G.GGPE.5
Given two points, find the point on the line segment between the two points that divides the segment into a given ratio. - G.GGPE.6
Use the distance and midpoint formulas to determine distance and midpoint in a coordinate plane, as well as areas of triangles and rectangles, when given coordinates. - G.GGPE.7
Use the structure of an expression to identify ways to rewrite it. Example: For example, see 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹⁴ – 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺⁴ as (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹²)² – (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺²)², thus recognizing it as a difference of squares that can be factored as (𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹² – 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²)(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² + 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺²). - CCSS.Math.Content.HSA-SSE.A.2
Define angle, perpendicular line, parallel line, line segment, ray, circle, and skew in terms of the undefined notions of point, line, and plane. Use geometric figures to represent and describe real-world objects. - G.GCO.1
Describe rotations and reflections that carry a regular polygon onto itself and identify types of symmetry of polygons, including line, point, rotational, and self-congruence, and use symmetry to analyze mathematical situations. - G.GCO.3
Represent translations, reflections, rotations, and dilations of objects in the plane by using paper folding, sketches, coordinates, function notation, and dynamic geometry software, and use various representations to help understand the effects of simple transformations and their compositions. - G.GCO.2
Predict and describe the results of transformations on a given figure using geometric terminology from the definitions of the transformations, and describe a sequence of transformations that maps a figure onto its image. - G.GCO.5
Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segme - G.GCO.4
Prove two triangles are congruent by applying the Side-Angle-Side, Angle-Side-Angle, Angle-Angle-Side, and Hypotenuse-Leg congruence c - G.GCO.7
Analyze decisions and strategies using probability concepts (e.g., product testing, medical testing, pulling a hockey goalie at the end of a game). - CCSS.Math.Content.HSS-MD.B.7
Demonstrate that triangles and quadrilaterals are congruent by identifying a combination of translations, rotations, and reflections in various representations that move one figure onto the other. - G.GCO.6
Prove, and apply in mathematical and real-world contexts, theorems about the relationships within and among triangles, including the following: a) measures of interior angles of a triangle sum to 180°; b) base angles of isosceles triangles are congruent; c) the segment joining midpoints of two sides of a triangle is parallel to the third side and half the length; d) the medians of a triangle meet at a point. - G.GCO.9
Use probabilities to make fair decisions (e.g., drawing by lots, using a random number generator). - CCSS.Math.Content.HSS-MD.B.6
Prove, and apply in mathematical and real-world contexts, theorems about lines and angles, including the following: a vertical angles are congruent; b) when a transversal crosses parallel lines, alternate interior angles are congruent, alternate exterior angles are congruent, and consecutive interior angles are supplementary; c) any point on a perpendicular bisector of a line segment is equidistant from the endpoints of the segment; d) perpendicular lines form four right angles. - G.GCO.8
Solve quadratic equations by inspection (e.g., for 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹² = 49), taking square roots, completing the square, the quadratic formula and factoring, as appropriate to the initial form of the equation. Recognize when the quadratic formula gives complex solutions and write them as 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢 ± 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪 for real numbers 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢 and 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣. - CCSS.Math.Content.HSA-REI.B.4b
Relate the domain of a function to its graph and, where applicable, to the quantitative relationship it describes. Example: For example, if the function 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩(𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯) gives the number of person-hours it takes to assemble 𝘝𝘭𝘙𝘙𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘗𝘈𝘉𝘗𝘈𝘉𝘗𝘈𝘗𝘉𝘈𝘗𝘉𝘗𝘈𝘉𝘈𝘉𝘉𝘈𝑦𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝑥𝘈𝘉𝘈𝘉𝘈𝘉𝘗𝘈𝘉𝘗𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘈𝘉𝘧𝘧𝘧𝘯𝘧𝘯𝘧𝘯𝘯𝑥𝑥𝑥𝑦𝑥𝑦𝘺𝘹𝘹𝘺𝘹𝘺𝘹𝘯𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘺𝘹𝘢𝘣𝘪𝘢𝘣𝘩𝘯𝘯 engines in a factory, then the positive integers would be an appropriate domain for the function. - CCSS.Math.Content.HSF-IF.B.5
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 negative; relative maximums and minimums; symmetries; end behavior; and periodicity. - CCSS.Math.Content.HSF-IF.B.4
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces). - CCSS.Math.Content.HSG-SRT.D.11
Prove the Laws of Sines and Cosines and use them to solve problems. - CCSS.Math.Content.HSG-SRT.D.10
Combine standard function types using arithmetic operations. Example: 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. - CCSS.Math.Content.HSF-BF.A.1b




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