Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a variety of transformations (translations, rotations, reflections, and dilations), Represent the transformations in the plane using a variety of methods (e.g., technology, transparencies, mirrors (MIRAs), patty paper, compass), Describe transformations as functions that take points in the plane as inputs and give other points as outputs, explain why this satisfies the definition of a function, and adapt function notation to that of a mapping [e.g. F(x,y) → F(x+a, y+b)], Compare transformations that preserve distance and angle to those that do not. Students know: Characteristics of transformations (translations, rotations, reflections, and dilations), Methods for representing transformations, Characteristics of functions, Conventions of functions with mapping notation. Students understand/are able to: Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology, Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane, including when the transformation preserves distance and angle, Use the language and notation of functions as mappings to describe transformations. Mapping one point to another through a series of transformations can be recorded as a function, Some transformations (translations, rotations, and reflections) preserve distance and angle measure, and the image is then congruent to the pre-image, while dilations preserve EEG-CO.2. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a collection of figures that include rectangles, parallelograms, trapezoids, or regular polygons, Identify which figures that have rotations or reflections that carry the figure onto itself, Perform and communicate rotations and reflections that map the object to itself, Distinguish these transformations from those which do not carry the object back to itself, Describe the relationship of these findings to symmetry. Students know: Characteristics of transformations (translations, rotations, reflections, and dilations), Characteristics of rectangles, parallelograms, trapezoids, and regular polygons. Students understand/are able to: Accurately perform dilations, rotations, reflections, and translations on objects in the coordinate plane with and without technology, Communicate the results of performing transformations on objects and their corresponding coordinates in the coordinate plane. Mapping one point to another through a series of transformations can be recorded as a function, Since rotations and reflections preserve distance and angle measure, the image is then congruent. EEG-CO.3. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a triangle and its image under a sequence of rigid motions (translations, reflections, and translations), Verify that corresponding sides and corresponding angles are congruent. Given two triangles that have the same side lengths and angle measures, - Find a sequence of rigid motions that will map one onto the other. If and only if Students know: Characteristics of translations, rotations, and reflections including the definition of congruence, Techniques for producing images under transformations, Geometric terminology which describes the series of steps necessary to produce a rotation, reflection, or translation. Students understand/are able to: Use geometric descriptions of rigid motions to accurately perform these transformations on objects, Communicate the results of performing transformations on objects. If a series of translations, rotations, and reflections can be described that transforms one object exactly to a second object, the objects are congruent. Level IV Students will: EEG-CO.6-8. Demonstrate why shapes are congruent. Ex. Communicate why two given congruent shapes are congruent. Ex. Given two non-congruent shapes, communicate why the shapes are not congruent. Level III Students will: EEG-CO.6-8. Identify corresponding congruent (the same) parts of shapes. Ex. Given two congruent triangles, identify the corresponding sides. Ex. Given two congruent items (stars, squares, etc.), identify the corresponding parts. Ex. Given two shapes that are not congruent but are similar, identify the similar parts. Level II Students will: EEG-CO.6-8. Recognize congruent parts (angles and sides). Ex. Identify the congruent parts of a rectangle. Ex. Identify the congruent angles of an isosceles triangle. Level I Students will: EEG-CO.6-8. Recognize shapes that are congruent. Ex. Given a shape, match a congruent shape. Ex. Given three shapes, pick the two that are congruent.
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology, Explain the requirements of a mathematical proof, Present a complete mathematical proof of geometry theorems including the following: 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, Critique proposed proofs made by others. Prove Transversal Alternate interior angles Corresponding angles Students know: Requirements for a mathematical proof, Techniques for presenting a proof of geometric theorems. Students understand/are able to: Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems, Generate a conjecture about geometric relationships that calls for proof. Proof is necessary to establish that a conjecture about a relationship in mathematics is always true, and may also provide insight into the mathematics being addressed. EEG-CO.9-11. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Make, explain, and justify (or refute) conjectures about geometric relationships with and without technology, Explain the requirements of a mathematical proof, Present a complete mathematical proof of geometry theorems about parallelograms, including the following: opposite sides are congruent, opposite angles are congruent, the diagonals of a parallelogram bisect each other, and conversely, rectangles are parallelograms with congruent diagonals, Critique proposed proofs made by others. Students know: Requirements for a mathematical proof, Techniques for presenting a proof of geometric theorems. Students understand/are able to: Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems, Generate a conjecture about geometric relationships that calls for proof. Proof is necessary to establish that a conjecture about a relationship in mathematics is always true and may also provide insight into the mathematics being addressed. EEG-CO.9-11. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a center of dilation, a scale factor, and a polygonal image, Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex, Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line, Find the ratio of the length of the line segment from the center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor, Conjecture a generalization of these results for all dilations. Dilations Center Scale factor Students know: Methods for finding the length of line segments (both in a coordinate plane and through measurement), Dilations may be performed on polygons by drawing lines through the center of dilation and each vertex of the polygon then marking off a line segment changed from the original by the scale factor. Students understand/are able to: Accurately create a new image from a center of dilation, a scale factor, and an image, Accurately find the length of line segments and ratios of line segments, Communicate with logical reasoning a conjecture of generalization from experimental results. A dilation uses a center and line segments through vertex points to create an image which is similar to the original image but in a ratio specified by the scale factor, The ratio of the line segment formed from the center of dilation to a vertex in the new image and the corresponding vertex in the original image is equal to the scale factor. EEG-SRT.1-3. N/A (See EEG-CO.6-8.)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a center of dilation, a scale factor, and a polygonal image, Create a new image by extending a line segment from the center of dilation through each vertex of the original figure by the scale factor to find each new vertex, Present a convincing argument that line segments created by the dilation are parallel to their pre-images unless they pass through the center of dilation, in which case they remain on the same line, Find the ratio of the length of the line segment from the center of dilation to each vertex in the new image and the corresponding segment in the original image and compare that ratio to the scale factor, Conjecture a generalization of these results for all dilations. Dilations Center Scale factor Students know: Methods for finding the length of line segments (both in a coordinate plane and through measurement), Dilations may be performed on polygons by drawing lines through the center of dilation and each vertex of the polygon then marking off a line segment changed from the original by the scale factor. Students understand/are able to: Accurately create a new image from a center of dilation, a scale factor, and an image, Accurately find the length of line segments and ratios of line segments, Communicate with logical reasoning a conjecture of generalization from experimental results. A dilation uses a center and line segments through vertex points to create an image which is similar to the original image but in a ratio specified by the scale factor, The ratio of the line segment formed from the center of dilation to a vertex in the new image and the corresponding vertex in the original image is equal to the scale factor. EEG-SRT.1-3. N/A (See EEG-CO.6-8.)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation involving triangles, Determine solutions to problems by applying congruence and similarity criteria for triangles to assist in solving the problem, Justify solutions and critique the solutions of others. Given a geometric figure, Establish and justify relationships in the figure through the use of congruence and similarity criteria for triangles. Congruence and similarity criteria for triangles Students know: Criteria for congruent (SAS, ASA, S, SSS) and similar () triangles and transformation criteria, Techniques to apply criteria of congruent and similar triangles for solving a contextual problem. Students understand/are able to: Accurately solve a contextual problem by applying the criteria of congruent and similar triangles, Provide justification for the solution process, Analyze the solutions of others and explain why their solutions are valid or invalid, Justify relationships in geometric figures through the use of congruent and similar triangles. Congruence and similarity criteria for triangles may be used to find solutions of contextual problems, Relationships in geometric figures may be proven through the use of congruent and similar triangles. EEG-SRT.4-5. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a collection of right triangles, Construct similar right triangles of various sizes for each right triangle given, Compare the ratios of the sides of the original triangles to the ratios of the sides of the similar triangles, Communicate observations made about changes (or no change) to such ratios as the length of the side opposite an angle to the hypotenuse, or the side opposite the angle to the side adjacent, as the size of the angle changes or in the case of similar triangles, remains the same, Summarize these observations by defining the six trigonometric ratios. Side ratios Trigonometric ratios Students know: Techniques to construct similar triangles, Properties of similar triangles. Students understand/are able to: Accurately find the side ratios of triangles, Explain and justify relationships between the side ratios of a right triangle and the angles of a right triangle. The ratios of the sides of right triangles are dependent on the size of the angles of the triangle. EEG-SRT.6-8. N/A
Grade Level/ Course: Geometry Unit 2 Standard: G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Domain: Similarity, Right Triangles, and Trigonometry Cluster: Define trigonometric ratios and solve problems involving right triangles Quarter 1: Quarter 2: Recognize which methods could be used to solve right triangles in applied problems. Solve for an unknown angle or side of a right triangle using sine, cosine, and tangent. Apply right triangle trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. Quarter 3: Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Grade Level/ Course: Geometry Unit 2 Standard: G.MG.2 Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot).* (*Modeling Standard) Domain: Modeling with Geometry Cluster: Apply geometric concepts in modeling situations Quarter 1: Quarter 2: Quarter 3: Quarter 4: Define density. Apply concepts of density based on area and volume to model real-life situations (e.g., persons per square mile, BTUs per cubic foot). Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Grade Level/ Course: Geometry Unit 2 Standard: G.SRT.9 (+) 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. Domain: Similarity, right triangles, and trigonometry Cluster: Apply trigonometry to general triangles Quarter 1: Quarter 2: Recall right triangle trigonometry to solve mathematical problems. 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. Quarter 3: Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given any triangle, Derive the Law of Sines and the Law of Cosines. Given any triangle, Use the Law of Sines or the Law of Cosines to find unknown lengths of sides and measures of angles. Law of Sines Law of Cosines Students know: The auxiliary line drawn from the vertex perpendicular to the opposite side forms an altitude of the triangle. Properties of the Sine and Cosine ratios. Pythagorean Theorem. Pythagorean Identity. Students understand/are able to: Properly label a triangle according to convention. Perform algebraic manipulations. The given information will determine whether it is appropriate to use the Law of Sines or the Law of Cosines. Proof is necessary to establish that a conjecture about a relationship in mathematics is always true and may provide insight into the mathematics being addressed. EEG-SRT.9-11. N/A (+)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation, Choose the appropriate law and apply it to determine the measures of unknown quantities. Law of Sines Law of Cosines Students know: The Laws of Sines and Cosines can apply to any triangle, right or non-right. Laws of Sines and Cosines. Vector quantities can represent lengths of sides and angles in a triangle. Values of the sin(90°) and cos(90°). Students understand/are able to: Label triangles in context and by convention. Perform algebraic manipulations. Find inverse sine and cosine values. Proven laws allow us to solve problems in contextual situations. EEG-SRT.9-11. N/A (+)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given the focus and directrix of a parabola, Use the distance formula to write an equation for all points which are equidistant from a given focus and directory, Use logical repeated reasoning to generalize to a general point and general directrix, developing the formula for the equation of a parabola, Compare and contrast parabola equations for horizontal and vertical directrices. Parabola Focus Directrix Students know: The distance formula for points in the coordinate plane and methods of finding the distance between a point and a line, Properties of equality (Table 4). Students understand/are able to: Create and justify a formula through algebraic manipulation after setting the distances between two points and between a point and a line equal to each other. The points that satisfy the equation of a parabola represent all of the points that are equidistant from a given point and a given line. EEG-GPE.2-3. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a line, Create lines parallel to the given line and compare the slopes of parallel lines by examining the rise/run ratio of each line, Create lines perpendicular to the given line by rotating the line 90 degrees and compare the slopes by examining the rise/run ratio of each line, Use understandings of similar triangles and logical reasoning to prove that parallel lines have equal slopes and the slopes of perpendicular lines are negative reciprocals. Given a geometric problem involving parallel or perpendicular lines, Apply the appropriate slope criteria to solve the problem and justify the solution including finding equations of lines parallel or perpendicular to a given line. Slope criteria for parallel and perpendicular lines Students know: Techniques to find the slope of a line, Key features needed to solve geometric problems, Techniques for presenting a proof of geometric theorems. Students understand/are able to: Explain and justify conclusions reached regarding the slopes of parallel and perpendicular lines, Apply slope criteria for parallel and perpendicular lines to accurately find the solutions of geometric problems and justify the solutions, Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems. Modeling geometric figures or relationships on a coordinate graph assists in determining truth of a statement or theorem, Geometric theorems may be proven or disproven by examining the properties of the geometric shapes in the theorem through the use of appropriate algebraic techniques. EEG-GPE.5-6. N/A (See EEG.CO.1)
Grade Level/ Course (high school): Geometry Unit 4 Standard: G.GPE.6 Find the point on a directed line segment between two given points that partitions the segment in a given ratio. Domain: Expressing Geometric Properties with Equations Cluster: Use coordinates to prove simple geometric theorems algebraically Quarter 1: Quarter 2: Quarter 3: Recall the definition of ratio. Recall previous understandings of coordinate geometry. Given a line segment (including those with positive and negative slopes) and a ratio, find the point on the segment that partitions the segment into the given ratio. Quarter 4: Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given two points and a ratio that partitions the segment between the points, Construct a circle using one of the given points as the center and the distance between the points as the radius, Construct a dilation of the circle using the given ratio as the scale factor and find the intersection between the dilation and the equation of the line passing through the given points, Justify and explain the reasons for each step in the process of finding a point that partitions a segment in a given ratio. Directed line segment Partitions Students know: Techniques for finding the distance between two points and the equation of a line passing through two points, Forms for writing the equation of a circle dependent on the information given to find the equation of the dilation of a circle, Techniques to find the intersection between a line and a circle. Students understand/are able to: Accurately find the distance between two points and the equation of a line passing through two points, Accurately find the equation of a dilation of a circle, Find the intersection point(s) of a line and a circle. A radius of a circle may be used to show the distance between two points, A dilation of a circle may be used to partition a line segment by making it the radius, in a given ratio. EEG-GPE.5-6. N/A (See EEG.CO.1)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given circles with two points on the circle, Compare the measures of the angles (with and without technology) formed by creating radii to the given points, creating chords from a third point on the circle to the given points, and creating tangents from a third point outside the circle to the given points, and conjecture about possible relationships among the angles, Use logical reasoning to justify (or deny) the conjectures (in particular justify that an inscribed angle is one half the central angle cutting off the same arc, and the circumscribed angle cutting off that arc is supplementary to the central angle relating all three). Given circles with chords from a point on the circle to the endpoints of a diameter, Find the measure of the Central angles Inscribed angles Circumscribed angles Students know: Definitions and characteristics of central, inscribed, and circumscribed angles in a circle, Techniques to find measures of angles including using technology (dynamic geometry software). Students understand/are able to: Explain and justify possible relationships among central, inscribed, and circumscribed angles sharing intersection points on the circle, Accurately find measures of angles (including using technology (dynamic geometry software)) formed from inscribed angles, radii, chords, central angles, circumscribed angles, and tangents Relationships that exist among inscribed angles, radii, and chords may be used to find the measures of other angles when appropriate conditions are given, Identifying and justifying relationships exist in geometric figures. EEG-C.1-3. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a triangle, Use tools (e.g., compass, straight edge, geometry software) to construct inscribed and circumscribed circles, Explain and justify the sequence of steps taken to complete the construction. Given a quadrilateral inscribed in a circle, Conjecture possible relationships among the angles through the use of inscribed angles use logical reasoning to justify (or deny) the conjectures (in particular, justify that diagonally opposite angles of a quadrilateral are supplementary). Construct Inscribed and circumscribed circles of a triangle Quadrilateral inscribed in a circle Students know: Techniques to find circumcenter and in center of a triangle, and to use this in inscribing or circumscribing the triangle, Properties of inscribed angles. Students understand/are able to: Use appropriate tools to accurately construct inscribed and circumscribed circles of a triangle, Explain and justify the steps that are used when creating the construction, Apply the properties of inscribed angles to reach a conclusion. Every triangle has a point which is equidistant from each vertex of the triangle and a point which is equidistant from each side of the triangle, Opposite angles of a quadrilateral inscribed in a circle are supplementary. EEG-C.1-3. N/A
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given coordinates and geometric theorems and statements defined on a coordinate system, Use the coordinate system and logical reasoning to justify (or deny) the statement or theorem, and to critique arguments presented by others. Simple geometric theorems Students know: Relationships (e.g. distance, slope of line) between sets of points, Properties of geometric shapes, Coordinate graphing rules and techniques, Techniques for presenting a proof of geometric theorems. Students understand/are able to: Accurately determine what information is needed to prove or disprove a statement or theorem, Accurately find the needed information and explain and justify conclusions, Communicate logical reasoning in a systematic way to present a mathematical proof of geometric theorems. EEG-GPE.4. N/A (See EEG-GPE)
Grade Level/ Course (HS): Geometry Unit 5 Standard: G.C. 5 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. Domain: Circles Cluster: Find arc lengths and areas of sectors of circles. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Recall how to find the area and circumference of a circle. Explain that 1° = Π/180 radians Recall from G.C.1, that all circles are similar. Determine the constant of proportionality (scale factor). Justify the radii of any two circles (r1 and r2) and the arc lengths (s1 and s2) determined by congruent central angles are proportional, such that r1 /s1 = r2/s2 Verify that the constant of a proportion is the same as the radian measure, Θ, of the given central angle. Conclude s = r Θ From Appendix A: Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular trigonometry in this course. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a real-world object, Select an appropriate geometric shape to model the object, Provide a description of the object through the measures and properties of the geometric shape which is modeling the object, Explain and justify the model which was selected. Students know: Techniques to find measures of geometric shapes, Properties of geometric shapes. Students understand/are able to: Model a real-world object through the use of a geometric shape, Justify the model by connecting its measures and properties to the object. Geometric shapes may be used to model real-world objects, Attributes of geometric figures help us identify the figures and find their measures, therefore matching these figures to real world objects allows the application of geometric techniques to real world problems. Level IV Students will: EEG-MG.1-3. Apply geometric methods to solve design problems. Ex. Identify the two-dimensional shapes that create the three-dimensional figure (e.g., I can see four triangles in a pyramid; I can see six squares on the outside of a cube). Ex. Determine the least number of tiles needed to cover the outside of a cubed figure- glue tiles onto a box. Ex. How many cups of water will this cylinder hold? Ex. Determine the amount of materials needed to wrap a present. Level III Students will: EEG-MG.1-3. Use properties of geometric shapes to describe real-life objects. Ex. Name everyday objects in terms of geometric shapes (can of soda is a cylinder, box of cereal is a rectangular prism). Ex. Describe the sides of a box of tissues (ends are squares, sides are rectangles). Ex. How many small square boxes (cubes)can I fit into a large cube? Ex. How many boxes (cubes) will fit on this shelf? Ex. Determine the dimensions of a classroom (length, width, and height.) Level II Students will: EEG-MG.1-3. Identify geometric shapes. Ex. Find real-life objects that have similar characteristics to a sphere. Ex. Given a cube, determine what real-life object has similar characteristics to a cube. Level I Students will: EEG-MG.1-3. Compare the capacity of three-dimensional objects. Ex. Which has the greatest capacity, a house or a school? Ex. Determine, which holds more, a cup or a barrel?
Grade Level/ Course: Geometry Unit 6 Standard: S.CP 2 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. Statistics and Probability is a Modeling Conceptual Category. Domain: Conditional Probability and the Rules of Probability Cluster: Understand independence and conditional probability and use them to interpret data. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Categorize events as independent or not using the characterization that two events A and B are independent when the probability of A and B occurring together is the product of their probabilities. From Appendix A: Build on work from 2-way tables from Algebra 1 Unit 3 (S.ID.5) to develop understanding of conditional probability and independence. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Grade Level/ Course: Geometry Unit 6 Standard: S.CP.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. Statistics and Probability is a Modeling Conceptual Category. Domain: Conditional Probability and Rules of Probability Cluster: Use rules of probability to compute probabilities of compound events in a uniform probability model. Quarter 1: Quarter 2: Quarter 3: Quarter 4: Find the conditional probability of A given B as the fraction of B’s outcomes that also belong to A. Interpret the answer in terms of the model. Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others. Model with mathematics. Use appropriate tools strategically. Attend to precision. Look for and make use of structure. Look for and express regularity in repeated reasoning.
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation consisting of two events, Determine the simple probability of each event, Determine the P(A or B) and P(A and B), Interpret the Addition Rule by counting outcomes in the four events A, B, A and B, A or B and showing the relationship to P(A or B) = P(A) + P(B) - P(A and B), Interpret the Addition Rule in the case that the P(A and B) = 0. Addition Rule Students know: Techniques for finding probabilities of simple and compound events. Students understand/are able to: Accurately determine the probability of simple and compound events. Formulas are useful to generalize regularities, but must be justified, The Addition Rule may be used for finding compound probability. EES-CP.6-7. N/A (See EES-IC.1-2)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation consisting of two events A and B, Use the definition of conditional probability P(B|A) = P(A and B)/P(A) to determine the probability of the compound event (A and B) when the P(A|B) and the P(A) are known or may be determined. Interpret the probability as it relates to the context. Uniform probability model General Multiplication Rule Students know: Techniques for finding probabilities of simple and conditional events. Students understand/are able to: Determine the probability of a single event. Determine the probability of a conditional event. The general Multiplication Rule for probability is a manipulation of the formula for conditional probability. The formula P(A and B) = P(A)P(B|A) will always apply regardless of whether the events are independent or dependent. EES-CP.8-9. N/A (+)
Evidence of Student Attainment/Assessment Vocabulary Knowledge Skills Instructional Achievement Level Descriptors Students: Given a contextual situation, Choose the appropriate counting technique (permutation or combination), find the number of ways an event(s) can occur, and use these counts to determine probabilities of the event, including compound events. Permutations Combinations Compound events Students know: Order is the determining factor in whether an event requires a permutation or a combination to count the number of possible outcomes of the event. Techniques for finding probabilities of simple and compound events. Techniques for finding the number of permutations or combinations of an event. Students understand/are able to: Evaluate factorial expressions. Apply the multiplication and addition rules to determine probabilities. Interpret and apply the different notations for combinations and permutations. Perform procedures to evaluate expressions involving the number of combinations and permutations of n things taken r at a time. There are contextual situations that can be interpreted through the use of combinations and permutations. The contextual situation determines whether combinations or permutations must be utilized. Mathematics is a coherent EES-CP.8-9. N/A (+)