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Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #1 Geometric Tools Anticipated timeframe: Day 1-9 Standards addressed: G.CO.1 Experiment with transformations in the plane. 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. G.CO.12 Make geometric constructions. 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.G.MG.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) 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) N.Q.1 Reason quantitatively and use units to solve problems. Transfer Goals: Visualizations help connect properties to real objects. Definitions establish meaning. There are many items that must be measured and there are various tools to do so. Conclusions come from basic facts and clear information. Enduring Understandings: Essential Questions: 3 dimensional objects can be represented with 2 How can one represent a 3-dimensional figure in 2 dimensional figures. dimensions? Geometry is a math system built on accepted facts, terms, What are the foundations of geometry? and definitions. How does one describe the parts of an angle? Number operations can be used to find and compare measures of segments and angles Special angle pairs can identify relationships and find angle measures Midpoints and lengths of any segment can be found Perimeter and area are two measures of figures. Learners will know: Learners will be able to: Figures have attributes Make nets for solid figures Terms appropriate to the content Define basic geometric figures Postulates are the foundations of proofs Measure line segments Key terms: pg. 70 chapter vocabulary (text); pg. 2, ELL Use distance and midpoint formulas Use a protractor to measure angles Performance Tasks: Mid-unit quiz Unit test Group station products Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Authentic Assessment: “pull it all together”, pg. 69, choose 1 of 2 Anticipated daily sequence of activities: Nets and drawings (textbook 1-1) Points, lines, planes (1-2) Measuring segments (1-3) Measuring angles (1-4) Angle pairs (1-5) Basic constructions (1-6) Midpoint and distance in coordinate plane (1-7) Perimeter circumference, area (1-8) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #2 Logic and Reasoning Anticipated timeframe: Day 9-14 Standards addressed: G.CO.9 Prove geometric theorems. 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. G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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. G.CO.11 Prove geometric theorems. 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. Transfer Goals: It is often possible to verify complex truths by reasoning from simpler ones using deductive reasoning. Enduring Understandings: Essential Questions: Patterns can be used to discover relationships. How can you make a conjecture and prove it is true? Some relationships can be described using If-Then statements. A definition is good if it can be written as a biconditional Algebraic properties of equality are used in geometry to solve problems and support reasoning. True information and information that can be demonstrated as true can be used to prove bigger ideas. Learners will know: Learners will be able to: Deductive and inductive reasoning Determine the truth of statements Inverse, converse, and contrapositive of statements Use true statements to prove other statements or counterexamples to disprove them Terms appropriate to the content Create if-then statements What makes a good definition The Law of Detachment The Law of Syllogism Key terms: pg. 129 chapter vocabulary (text); pg. 80, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Authentic Assessment: “pull it all together”, pg. 128, choose 1 of 2 Anticipated daily sequence of activities: Patterns and inductive reasoning (text 2-1) Conditional statements (2-2) Biconditionals and definitions (2-3) Deductive reasoning (2-4) Reasoning in algebra and geometry (2-5) Proving angles equal (2-6) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #3 Lines Anticipated timeframe: Day 15-24 Standards addressed: G.CO.1 Experiment with transformations in the plane. 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. G.CO.9 Prove geometric theorems. 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. G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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. G.CO.12 Make geometric constructions. 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. G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.GPE.5 Use coordinates to prove simple geometric theorems algebraically. 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). G.MG.3 Apply geometric concepts in modeling situations. 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). Transfer Goals: Everything can be located on a grid of some nature. Items in nature are related in many ways, can be used together to make new information Enduring Understandings: Essential Questions: Not all lines and not all planes intersect. How do you prove two lines are parallel? Intersections create special angle pairs How do you prove two lines are perpendicular? The sum of the angle measures of a triangle are always the What is the sum of the measures of a triangle? same. How do you write an equation of a line in a coordinate A line can be graphed and its equation written when specific plane? facts are known. Comparing slopes of lines can determine whether they are parallel or perpendicular, or not. Learners will know: Learners will be able to: Properties of the coordinate plane Find the measure of an angle indirectly Terms appropriate to the content Determine whether lines are parallel, perpendicular, or neither. Characteristics of parallel and perpendicular lines Graph lines on a coordinate plane given the equation or the Terms involved with lines, intersections, and angles slope and y-intercept. Key terms: pg. 206 chapter vocabulary (text); pg. 138, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Authentic Assessment: “pull it all together”, pg. 205, choose #1 Anticipated daily sequence of activities: Lines and angles (text 3-1) Parallel lines (3-2) Proving lines parallel (3-3) Perpendicular lines (3-4) Parallel lines and triangles (3-5) Constructing lines (3-6) Linear equations (3-7) Slopes vs. parallel/perpendicular (3-8) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #4 Triangles Anticipated timeframe: Day 25-32 Standards addressed: G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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. G.CO.12 Make geometric constructions. 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. G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.SRT.5 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Transfer Goals: Identification of relationships between items can lead to an understanding of additional relationships. Larger ideas can be proven true by building upon known truths. Enduring Understandings: Essential Questions: Comparing the corresponding parts of two figures can show How do you identify corresponding parts of congruent whether the figures are congruent triangles? If two figures are congruent, then every corresponding part How do you show two triangles are congruent? of them is identical How can you tell when a triangle is isosceles or equilateral? Isosceles and equilateral triangles have special relationships Learners will know: Learners will be able to: Triangle congruence theorems and postulates Write congruence statements for triangle pairs Terms appropriate to the content Demonstrate triangles are congruent through proofs Methods for identifying corresponding parts of figures Identify overlapping congruent triangles Steps in writing a proof Key terms: pg. 273 chapter vocabulary (text); pg. 216, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 272, choose #2 Anticipated daily sequence of activities: Congruent figures (text 4-1) Triangle congruence (4-2; 4-3) Using corresponding parts of triangles (4-4) Special triangles (4-5) Right triangles (4-6) Overlapping triangles (4-7) Anticipated resources: Pearson/Prentice Hall Geometry text Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #5 Parts of Triangles Anticipated timeframe: Day 33-39 Standards addressed: A.CED.1 Create equations that describe numbers or relationship. 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.* G.C.3 Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.CO.9 Prove geometric theorems. 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. G.CO.10 Prove geometric theorems. Prove theorems about triangles. Theorems include: measures of interior angles of a triangle sum to 180 degrees; 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. G.CO.12 Make geometric constructions. 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. G.SRT.5 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Transfer Goals: Breaking items into its parts can sometimes make it easier to learn more about it. There are many ways to arrive at a solution. Enduring Understandings: Essential Questions: The midsegment of a triangle can be used to find How is coordinate geometry used to find relationships relationships within triangles? Triangles are key to understanding bisectors How do you solve problems involving the measurements of triangles? Bisectors can be used to help with triangle measures The measures of angles of a triangle are related to the length How do you write indirect proofs of the opposite side The length of medians and altitudes can be found from other segments Indirect proofs provide a different way to a determined result In indirect reasoning, when all possibilities are considered and then all but one are proved false, the remaining possibility must be true. Learners will know: Learners will be able to: How indirect proofs are formed Find the circumcenter of a triangle The relationships between parts of a triangle Find the orthocenter of a triangle Terms appropriate to the content Determine comparative measures of parts of triangles Key terms: pg. 341chapter vocabulary (text); pg. 282, ELL Find the exact measures of part of triangles Create indirect proofs Performance Tasks: Mid-unit quiz Unit test Group station products Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Authentic Assessment: “pull it all together”, pg. 340, choose #1 Anticipated daily sequence of activities: Midsegments of triangles (text 5-1) Bisectors (5-2; 5-3) Medians and altitudes (5-4) Indirect proof (5-5) Inequalities in triangles (5-6; 5-7) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #6 Multi-sided shapes Anticipated timeframe: Day 40-49 Standards addressed: G.CO.11 Prove geometric theorems. 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. G.CO.12 Make geometric constructions. 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. G.GPE.4 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). G.GPE.7 Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.SRT.5 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Transfer Goals: Items with specific properties often have special names Relationships between and within figures can be identified, isolated, and proven Enduring Understandings: Essential Questions: The sum of the measures of a polygon is related to the How can the sum of the measures of the angles in a polygon number of sides be determined? Certain quadrilaterals have specific properties and names How are quadrilaterals classified? Variables can be used to name coordinates in a figure to help How can coordinate geometry be used to prove general identify relationships between parts of the figure relationships? Learners will know: Learners will be able to: The formula for deriving angle measures of a polygon Compute the sum of the angle measures for any polygon Properties of parallel and perpendicular lines to classify Graph a quadrilateral given coordinate quadrilaterals Name all special quadrilaterals Ordered pairs in a coordinate plane Find midpoints and slopes of segments in a figure Terms appropriate to the content Demonstrate congruence of figures The names for special quadrilaterals The distance formula Key terms: pg. 420 chapter vocabulary (text); pg. 350, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 419, choose #3 Anticipated daily sequence of activities: Polygon-angle sum theorems (text 6-1) Properties of parallelograms (6-2) Proving parallelograms (6-3) Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Properties of special quadrilaterals (6-4) Conditions of special quadrilaterals (6-5) Trapezoids and kites (6-6) Polygons in coordinate plane (6-7) Applying coordinate geometry (6-8) Proofs using coordinate geometry (6-9) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #7 Similar Figures Anticipated timeframe: Day 50-55 Standards addressed: A.CED.1 Create equations that describe numbers or relationship. 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. G.GPE.5 Use coordinates to prove simple geometric theorems algebraically. 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). G.SRT.4 Prove theorems involving similarity. 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. G.SRT.5 Prove theorems involving similarity. Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures. Transfer Goals: Many things can be similar if enough of their components are the same Similar items can be compared for relative size Enduring Understandings: Essential Questions: Equations can be written with equal ratios and, with a How are proportions used to find the side lengths of variable, can be solved to find the value of the variable. polygons? Similarity can be determined from using ratios and How can two triangles be shown to be similar? proportions How are corresponding parts of similar triangles identified? Lines intersecting parallel lines form proportional segments Triangles can be shown to be similar based on the relationships of their parts Learners will know: Learners will be able to: Ratios Solve algebraic proportions Proportions Find the length of a side in a figure given sufficient information Terms appropriate to the content Determine if two triangles are similar Similarity Key terms: pg. 480 chapter vocabulary (text); pg. 430, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 479, choose #2 Anticipated daily sequence of activities: Ratios and proportions (text 7-1) Similar polygons (7-2) Proving triangles similar (7-3) Similarity with right triangles (7-4) Proportions in triangles (7-5) Anticipated resources: Pearson/Prentice Hall Geometry text Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #8 Intro to Trigonometry Anticipated timeframe: Day 56-62 Standards addressed: G.MG.1 Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.SRT.4 Prove theorems involving similarity. 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. G.SRT.7 Define trigonometric ratios and solve problems involving right triangles. Explain and use the relationship between the sine and cosine of complementary angles. G.SRT.8 Define trigonometric ratios and solve problems involving right triangles. Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems. G.SRT.10 Apply trigonometry to general triangles. Prove the Laws of Sines and Cosines and use them to solve problems. G.SRT.11 Apply trigonometry to general triangles. 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). Transfer Goals: Values usually measured by hand can be determined through mathematical functions. What is done, can usually be undone by an inverse operation. Enduring Understandings: Essential Questions: If the lengths of two sides of a triangle are known, the third How are side lengths found in a right triangle? side can be determined How do trigonometric ratios relate to similar right triangles? Some right triangles have specific properties What is a vector? Ratios can be used to find measures of parts of a triangle if certain other measures of the triangle are known Vectors can be used to model motion and direction Learners will know: Learners will be able to: The Pythagorean Theorem Find the length of a side or measure of an angle of a right triangle given sufficient other measures Concepts of 30-60-90 and 45-45-90 triangles Determine triangles that are acute, obtuse, or right triangles Terms appropriate to the content Apply the definition of sine, cosine, and tangent. Trigonometric functions and their definitions Find inverse trigonometric function values Inverse trigonometric functions Determine the magnitude and direction of vectors Angle of depression or elevation Magnitude and direction for vectors Vector addition Key terms: pg. 534 chapter vocabulary (text); pg. 488, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 533, choose #2 Anticipated daily sequence of activities: Pythagorean Theorem (text 8-1) Converse of Pythagorean Theorem (8-1) Special right triangles (8-2) Trigonometric functions (8-3) Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Angles of elevation and depression (8-4) Vectors (internet or other resources) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #9 Transformations Anticipated timeframe: Day 63-69 Standards addressed: G.CO.2 Experiment with transformations in the plane. 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). G.CO.3 Experiment with transformations in the plane. Given a rectangle, parallelogram, trapezoid, or regular polygon, describe the rotations and reflections that carry it onto itself. G.CO.4 Experiment with transformations in the plane. Develop definitions of rotations, reflections, and translations in terms of angles, circles, perpendicular lines, parallel lines, and line segments. G.CO.5 Experiment with transformations in the plane. 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. G.CO.6 Understand congruence in terms of rigid motions. 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. G.CO.7 Understand congruence in terms of rigid motions. 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. G.CO.8 Understand congruence in terms of rigid motions. Explain how the criteria for triangle congruence (ASA, SAS, and SSS) follow from the definition of congruence in terms of rigid motions. G.SRT.1a Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: 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. G.SRT.1b Understand similarity in terms of similarity transformations. Verify experimentally the properties of dilations given by a center and a scale factor: The dilation of a line segment is longer or shorter in the ratio given by the scale factor. G.SRT.2 Understand similarity in terms of similarity transformations. 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. G.SRT.3 Understand similarity in terms of similarity transformations. Use the properties of similarity transformations to establish the AA criterion for two triangles to be similar. Transfer Goals: Scale factors are used in many situations and more many purposes Many events are much alike although larger or smaller in scale or different in environment or relationship Enduring Understandings: Essential Questions: Figures can be moved without changing their size or shape How can a figure’s position change without changing its size and shape? A scale factor is used to change the size of a figure to a similar figure How can a figure’s size change without changing its shape? Identical figures can be mapped onto each other using How is a transformation represented in a coordinate plane? reflections, translations and/or rotations How can symmetry in a figure be recognized Some shapes fit together to make a repeating pattern Some shapes appear unchanged after a reflection or rotation Learners will know: Learners will be able to: Translations Find coordinates of a figure following a translation, reflection, rotation, or dilation. Reflections Identify the transformation type from given information Rotations Identify the type of symmetry between two figures Dilations Determine the ability of a figure to tessellate Isometry Scale factor Line of symmetry Rotational symmetry Tessellation Key terms: pg. 602 chapter vocabulary (text); pg. 542, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 601, #2 Anticipated daily sequence of activities: Translations (text 9-1) Reflections (9-2) Rotations (9-3) Symmetry (9-3) Dilations (9-6) Composition of reflections (9-4) Tessellations (internet or other resources) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Identify the isometry between two congruent figures Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #10 Inside the box Anticipated timeframe: Day 70-76 Standards addressed: G.C.1 Understand and apply theorems about circles. Prove that all circles are similar. G.C.5 Find arc lengths and areas of sectors of circles. 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. G.CO.1 Experiment with transformations in the plane. 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. G.CO.13 Make geometric constructions. Construct an equilateral triangle, a square, and a regular hexagon inscribed in a circle. G.GMD.1 Explain volume formulas and use them to solve problems. 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. G.GMD.3 Explain volume formulas and use them to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GPE.7 Use coordinates to prove simple geometric theorems algebraically. Use coordinates to compute perimeters of polygons and areas of triangles and rectangles, e.g., using the distance formula. G.MG.1 Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.SRT.9 Apply trigonometry to general triangles. 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. S.CP.1 Describe events and subsets of a sample space (the set of outcomes) using characteristics (or categories) of the outcomes, or as unions, intersections, or compliments of other events (“or”, “and”,”not”). Transfer Goals: The value of new items can be found by using the familiar information Many things can be classified to help enable us to deal with them in the same way Enduring Understandings: Essential Questions: The area of a quadrilateral with two parallel sides can be How are the area and perimeter of a polygon found? found by multiplying its height by the average of its bases. How are the area and circumference of a circle found? The area of a regular polygon is a function of the distance How do perimeters and areas of similar figures compare? from its center to a side and its perimeter The length of an arc in a circle is related to the angle that forms it. The area of portions of a circle formed by arcs and radii can be found when its radius is known Learners will know: Learners will be able to: Formulas for finding areas and perimeters of various Find the area and perimeter of various polygons polygons Find the area and circumference for circles Formulas for finding circumference and area of circles Find the ratio of the area between two figures Use of trigonometry to find area Find the measure of arc angles Terms appropriate to the content Find the length of arcs Key terms: pg. 676 chapter vocabulary (text); pg. 612, ELL Find the area of portions of a figure Performance Tasks: Mid-unit quiz Unit test Group station products Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Authentic Assessment: “pull it all together”, pg. 675, choose #1 Anticipated daily sequence of activities: Trigonometric functions and area (text 10-5) Circles and arcs (10-6) Areas of circles and sectors (10-7) Geometric probability (10-8) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #11 Outside & Inside the Figure Anticipated timeframe: Day 77-84 Standards addressed: A.CED.4 Create equations that describe numbers or relationship. 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. G.GMD.1 Explain volume formulas and use them to solve problems. 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. G.GMD.2 Explain volume formulas and use them to solve problems. Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other solid figures. G.GMD.3 Explain volume formulas and use them to solve problems. Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems. G.GMD.4 Visualize relationships between two-dimensional and three-dimensional objects. Identify the shapes of twodimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. G.MG.1 Apply geometric concepts in modeling situations. Use geometric shapes, their measures, and their properties to describe objects (e.g., modeling a tree trunk or a human torso as a cylinder). G.MG.2 Apply geometric concepts in modeling situations. Apply concepts of density based on area and volume in modeling situations (e.g., persons per square mile, BTUs per cubic foot). Transfer Goals: Events and objects are the sum of their parts Events and objects can often be better understood by examining their details in a way that is not easily seen Enduring Understandings: Essential Questions: 3 dimensional figures can be analyzed through the How are the intersection of a plane and a solid found? relationships among their vertices, edges, and faces How is the surface area of a solid found? Surfaces areas of portions of a figure can be added to make How is the volume of a solid found? the whole How do the volume and surface area of similar solids The volume and surface area of a figure can be found from compare? its key dimensions Ratios can be used to compare the areas and volumes of similar solids Learners will know: Learners will be able to: How cross sections are formed Compute the volume and surface area of cylinders, prisms, spheres, pyramids, and cones. Formulas for surface area and volume for cylinders, prisms, spheres, pyramids, and cones Describe the figure formed by cross sections Terms appropriate to the content Count the number of edges and faces of various figures The role of ratio to similar solids Euler’s Formula Key terms: pg. 751 chapter vocabulary (text); pg. 686, ELL Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 750, choose #2 Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Anticipated daily sequence of activities: 3 dimensional figures and cross sections (text 11-1) Prisms and cylinders (11-2) Pyramids and cones (11-3) Spheres (11-6) Areas and volumes of similar solids (11-7) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS Asbury Park High School Unit Plan Department: Mathematics Course: Geometry Unit designation: #12 Around and Through Anticipated timeframe: Day 85-90 Standards addressed: G.C.2 Understand and apply theorems about circles. 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. G.C.3 Understand and apply theorems about circles. Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. G.C.4 Understand and apply theorems about circles. Construct a tangent line from a point outside a given circle to the circle. G.GMD.4 Visualize relationships between two-dimensional and three-dimensional objects. Identify the shapes of twodimensional cross-sections of three-dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. G.GPE.1 Translate between the geometric description and the equation for a conic section. 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. Transfer Goals: A locus can often help us see a pattern in a figure or events A circle is defined in a small number of ways, but its parts can be defined in many ways Enduring Understandings: Essential Questions: A radius of a circle and the tangent that intersects the How can relationships between arcs and angles of a circle be endpoint of the radius have a special relationship proven? Information about parts of a circle can be used to learn more How do you find the measures of resulting angles, arcs, and about other parts of the circle segments when lines intersect in a circle or within a circle? The description of a locus can be used to sketch a geometric How do you find the equation of a line in a coordinate relationship plane? Angles formed by intersecting lines have a special relationship to the arcs those lines intercept. The information contained in the equation of a circle enables the circle to be graphed on a coordinate plane The equation of a circle can be written if its radius and center point are known Learners will know: Learners will be able to: Relationships between arcs and angles Find the lengths of various segments given information about secants, chords, and radii of circles Properties of lines tangent to a circle Find the arc of a circle given information about a secant and Secants and tangents infer angles radius A center and radius of a circle is defined by its equation and Graph a circle given its equation. visa-versa. Derive a circle’s center and radius from an equation Key terms: pg. 813 chapter vocabulary (text); pg. 760, ELL Graph a locus of points from its definition. Performance Tasks: Mid-unit quiz Unit test Group station products Authentic Assessment: “pull it all together”, pg. 812, choose #2 Other Evidence: Daily class work Class participation Group collaborations Do Now quizzes Anticipated daily sequence of activities: Tangent lines (text 12-1) Chords and arcs (12-2) Inscribed angles (12-3) Angle measures and segment lengths (12-4) Circles in the coordinate plane (12-5) Locus (12-6) Anticipated resources: Pearson/Prentice Hall Geometry text Student Skills Handbook Student Visual Glossary Spanish version of materials Pearson MathXL system and resources Pearson test bank generating software ALEKS