Salisbury Township School District Planned Course of Study Geometry Salisbury Inspire, Think, Learn, Grow Together!

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1 Topic/Unit: Points, Lines, and Planes Big Ideas/Enduring Understandings: Geometric figures can be described and measured There is a relationship between equality and congruence Inductive and deductive reasoning can be used to prove conjectures Understanding geometric relationships in diagrams Writing proofs of geometric relationships Using properties of parallel and perpendicular Lines Proving relationships using angle measures Making connections to lines in algebra Essential Questions: How do you name geometric figures? What are congruent segments? How do you find the distance and the midpoint between two points in the coordinate plane? How do you classify angles? How do you identify complementary and supplementary angles? How does classifying angles relate to polygons? How do you use inductive reasoning in mathematics? How do you write a geometric proof? How do you construct a logical argument and write proofs (ie direct proof and counter-examples)? What are the relationships of the angles formed when two lines are cut by a transversal? How can you explain the relationship between congruence and similarity in both 2- and 3-dimensional figures? How can a change in one measurement of a 2- or 3-dimensional figure effect other measurements such as perimeter, area, surface area or volume of that figure? PA Academic Standards (PA Common Core): C.C.2.3.HS.A.1, C.C.2.3.HS.A.2, C.C.2.3.HS.A.3, C.C.2.3.HS.A.4, C.C.2.3.HS.A.5, C.C.2.3.HS.A.6, C.C.2.3.HS.A.7, C.C.2.3.HS.A.11, C.C.2.3.HS.A.12, C.C.2.3.HS.A.13, C.C.2.3.HS.A.14, CC.2.2.HS.D.9 Page 1

2 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: Point, Line, Plane, Collinear Points, Coplanar Points, Line segments, Endpoints, Ray, Intersection, Postulate, Coordinate, Distance, Congruent Segments, Midpoint, Segment bisector, Angle, Acute, Right, Obtuse, Straight Angle, Congruent Angles, Angle Bisector, Construction, Complementary Angles, Supplementary Angles, Adjacent Angles, Linear Pair, Vertical Angles, Polygon, Convex, Concave, Equilateral, Equiangular, Regular, Conjecture, Inductive Reasoning, Counterexample, Converse, Inverse, Contrapositive, Hypothesis, Conclusion, Negation, Equivalent Statements, Perpendicular Lines, Biconditional Statement, Deductive Reasoning, Proof, Two Column Proof, Theorem, Parallel Lines, Skew Lines, Parallel Planes, Transversal, Corresponding Angles, Alternate Interior Angles, Alternate Exterior Angles, Consecutive Interior Angles, Paragraph Proof, Slope, Slope-intercept Form Concepts: Competencies: Instructional Practices: Assessments: The student will know: The basic elements in geometry such as points, lines, angles, and polygons. The definition and relationship among perimeter and area of simple geometric figures. The uses of inductive and deductive reasoning The student will be able to: Identify and classify geometric figures. Measure segments in the coordinate plane. Classify angles and find their measures. Use perimeter and area formulas. Group assignment using the sun position in the sky and angles formed to determine basic directions. Using geometer s sketchpad, Grapher to make conjectures on classifications and proofs. Finding real life examples of geometric figures in relevant real world places. Quizzes/tests covering content of concepts and vocabulary. Finding perimeter and area at home/school and writing a budget plan for costs associated with carpeting or painting a room. Exit card to assess understanding of differences and similarities between geometric reasoning. Page 2

3 Proofs use postulates, theorems and properties to support them. The various methods to prove lines are parallel and perpendicular. Describe and identify congruent segments and angle measures. Explain the differences between inductive and deductive reasoning. Write a logical argument to show a geometric relationship is true. Videos and graphical representations of the points, lines, angles, and polygons and their relationships. Smartboard Kooshball Jeopardy Review. Class discussions and debate for inductive and deductive reasoning. Homework assignments and corrections. Snowball fight where students pick up thrown paper and solve problem written on it. Use small whiteboards as periodic formative assessment. Construct a proof using either the two column proof or paragraph form. Math journal on what the students understand and still need to know. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Page 3

4 Topic/Unit: Triangles Big Ideas/Enduring Understandings: Classifying triangles by sides and angles Proving that triangles are congruent Using coordinate geometry to investigate triangle relationships Using properties of special segments in triangles Using triangle inequalities to determine what triangles are possible Extending methods for justifying and proving relationships Using ratios and proportions to solve geometry problems Showing that triangles are similar Using indirect measurement and similarity Essential Questions: What are congruent geometric figures? How do you prove triangles congruent and prove their angles and sides are congruent? How do you prove triangles are similar? How do you find the orthocenter, centroid, incenter, and circumcenter and how they relate to the segments of triangles? How do the lengths of a triangle and the measure of its angles compare? How do you use the ideas of direct and indirect proof, and counter-examples to verify valid conjectures and refute invalid conjectures? How can you explain the relationship between congruence and similarity in both 2- and 3-dimensional figures? How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships? How can a change in one measurement of a 2- or 3-dimensional figure effect other measurements such as perimeter, area, surface area or volume of that figure? PA Academic Standards (PA Common Core): C.C.2.3.HS.A.1, C.C.2.3.HS.A.2, C.C.2.3.HS.A.3, C.C.2.3.HS.A.4, C.C.2.3.HS.A.5, C.C.2.3.HS.A.6, C.C.2.3.HS.A.7, C.C.2.3.HS.A.11, C.C.2.3.HS.A.12, C.C.2.3.HS.A.13, C.C.2.3.HS.A.14, CC.2.2.HS.D.9 Page 4

5 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Tier 3 Vocabulary: Triangles, Scalene Isosceles, Equilateral, Acute, Right, Obtuse, Equiangular, Interior Angles, Exterior Angles, Congruent figures, Corresponding parts, Right Triangle, Hypotenuse, Flow proof, Transformation, Midsegment, Coordinate proof, Perpendicular bisector, Equidistant, Point of concurrency, Circumcenter, Incenter, Median, Centroid, Altitude, Orthocenter, Indirect Proof, Ratio, Proportion, Geometric Mean, Scale, Similar Polygons, Scale Factor Concepts: Competencies: Instructional Practices: Assessments: The student will know: The types of information known about angle and side relationships in a triangle lead to short cuts in proving triangles congruent and/or similar. Coordinate geometry is an aid in proving triangles are congruent. Inequalities are used to make comparisons between side lengths and angles. The student will be able to: Write proofs for triangle congruency using SAS, SSS, ASA, HL, and AAS. Describe basic transformations in coordinate geometry. Determine the points of concurrency. Write proofs for similar polygons using. AA, SSS, and SAS Group assignment to discuss how and why the theorems of SSS, SAS, ASA, AAS, and HL are true and why SSA is not. Using khanacademy.com for further understanding/support for proving triangles congruent. Finding real life examples of scale factor and how it can be used for scale drawings. Quizzes/tests covering content in unit. Exit card to assess understanding of basic proof writing. Homework assignments and corrections. Geometer sketchpad assignment involving points of concurrency in triangles. Use small whiteboards as periodic formative assessment. Page 5

6 Difference between writing a coordinate proof and a 2- column proof. The points of concurrency in triangles. Basic proportional relationships in geometric shapes. Recall and use parallel line properties to find proportional polygons. Solve inequalities of angles and sides in triangles. Perform similarity transformations. Classify the angles of triangles and find their measures. Apply proportions to triangles and other polygons. Movement activities to determine where point of concurrency is on various types of triangles. Use calculators to solve proportions in triangles. Smartboard Kooshball Jeopardy Review Math journal on what the students understand and still need to know. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Page 6

7 Topic/Unit: Figures in the Plane Big Ideas/Enduring Understandings: Use the Pythagorean theorem and its converse Use special relationships in right triangles Use trigonometric ratios to solve right triangles Use angle relationships in polygons Use properties of parallelograms Classify quadrilaterals by their properties Essential Questions: What is the importance of the Pythagorean theorem and its converse? How do you find the lengths of the sides of a triangle and a triangle? How are trigonometric ratios used to solve a triangle? How do you find a missing angle measurement in a convex polygon? How can you identify special quadrilaterals? How can you explain the relationship between congruence and similarity in both 2- and 3-dimensional figures? How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships PA Academic Standards (PA Common Core): C.C.2.3.HS.A.3, C.C.2.3.HS.A.4, C.C.2.3.HS.A.6, C.C.2.3.HS.A.7, C.C.2.3.HS.A.11, C.C.2.3.HS.A.12, C.C.2.3.HS.A.13, C.C.2.3.HS.A.14, C.C.2.2HS.C.9, CC.2.2.HS.D.9 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Page 7

8 Tier 3 Vocabulary: Pythagorean triple, trigonometric ratio, tangent, sine, consine, angle of elevation, angle of depression, solve a right triangle, inverse tangent, inverse sine, inverse cosine, diagonal, parallelogram, rhombus, rectangle, square, trapezoid, midsegment of a trapezoid, isosceles trapezoid, kite, image, preimage, isometry, vector, component form, matrix, element, dimensions, line of reflection, center of rotation, angle of rotation, glide reflection, composition of transformations, line symmetry, rotational symmetry, scalar multiplication Concepts: Competencies: Instructional Practices: Assessments: The student will know: The Pythagorean theorem and its converse. The trigonometric ratios. How to classify quadrilaterals by their sides, angles and diagonals. The formula for angle sums in any polygon and The relationship to triangles and the sum formula. Properties of parallelograms. The student will be able to: Solve right triangles using trigonometric ratios. Solve for a missing side length using the Pythagorean theorem. Determine the differences and similarities among quadrilaterals. Use the converse of the Pythagorean theorem to classify triangle by its side lengths. Use and rules as a shortcut to solving special right triangles. Use the flipped classroom through my wiki (ruiz wiki: uizwikiwikiwiki/) as a way to reintroduce Pythagorean theorem. Use calculator to solve geometric ratios and other applicable calculations. Students will use the quadrilateral dance as a way to remember their properties. Find and explain real world examples using Pythagorean theorem as well as trigonometric ratios. Math journal on what the students understand and still need to know. Group presentation of a skit/play on quadrilateral differences and similarities. Exit card to assess understanding of trigonometric ratios. Homework assignments and corrections. Students will use geometer s sketchpad to show their findings in parallel and perpendicular lines. Quizzes/tests on classifying quadrilaterals and trigonometric ratios and other content. Students will turn in a skit/rap/poem to show their understanding of the Pythagorean theorem and right triangles. Page 8

9 Find the missing interior or exterior angle in a convex polygon. recall parallelogram rules to determine properties of quadrilaterals. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Page 9

10 Topic/Unit: Circles Big Ideas/Enduring Understandings: Use properties of segments that intersect circles Apply angle relationships in circles Use circles in the coordinate plane Compare measures for parts of circles and the whole circle Essential Questions: How can you verify that segment is tangent to a circle? What are the relationships between measures of the arcs of a circle and the measures of the angles in a circle? How can you tell if two chords in a circle are congruent? How do you find the measure of the angles of two chords that intersect on, inside, or outside the circle? What are the properties of chords, secants and tangents to a circle? What do you need to know to write the standard equation of a circle? What is the relationship between the length of an arc and the circumference of a circle? What is the relationship between the area of sectors and the area of a circle? How are radian measures related to degree measures? PA Academic Standards (PA Common Core): C.C.2.3.HS.A.3, C.C.2.3.HS.A.5, C.C.2.3.HS.A.6, C.C.2.3.HS.A.7, C.C.2.3.HS.A.8, C.C.2.3.HS.A.9, C.C.2.3.HS.A.10,C.C.2.3.HS.A.11, C.C.2.3.HS.A.12, C.C.2.3.HS.A.13, C.C.2.3.HS.A.14, CC.2.2.HS.D.9 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. Page 10

11 Tier 3 Vocabulary: Circle, center, radius, diameter, chord, secant, tangent, central angle, minor arc, major arc, semicircle, congruent circles, congruent arcs, inscribed angle, intercepted arc, standard equation of a circle, circumference, arc length, sector of a circle, center of a polygon, radius of a polygon, central angle of a regular polygon, probability, geometric probability, radian Concepts: Competencies: Instructional Practices: Assessments: Students will know: Relationships among lengths, arc measures, and angles formed when two lines intersect circles in one or two points. Circles in a plane are described using equations. The relationship of circumference and area to arc length and area of a sector. The conversion factor between radians and degrees. Students will be able to: Find the length of the segments of chords, secants and tangents in a circle. Find the measure of arcs and angles in a circle. Determine the relationship between Pythagorean theorem, the distance formula, and the equation of a circle. Use the area and circumference formula and apply those to arc length and sector. Use the properties of chords in a circle to find angle and arc measures. Find the equation of a circle given information about its radius and center. Use calculator to solve formulas and other content. Students will use the smartboard to label parts of a circle. Group assignment where students will explain the formula for a sector and the formula for arc length. Use khanacademy.org for reinforcement of arc length and area of a sector. Math journal on what the students understand and still need to know. Word splash of vocabulary. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Exit card to assess understanding of ratios. Homework assignments and corrections. Quizzes/tests on circles and measurement. Group presentations of concepts and formulas of circles. Students will use small whiteboards to display their answers and understanding for various problems. Peer assessments/ critical friend activity where students check their work against other students and find errors and make corrections. Page 11

12 Explain how and use distance and midpoint formulas to aid in finding the equations of circles. Convert between radian and degree measures in the unit circle. Page 12

13 Topic/Unit: Measurement Big Ideas/Enduring Understandings: Exploring solids and their properties Solving problems using surface area and volume Connecting similarity to solids Using area formulas for polygons Relating length, perimeter, and area ratios in similar polygons Apply probability to geometric figures. Essential Questions: How can you use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships? How can you represent the probability of an event using geometric properties of length or area? How can you explain the relationship between congruence and similarity in both 2- and 3-dimensional figures? How can a change in one measurement of a 2- or 3-dimensional figure effect other measurements such as perimeter, area, surface area or volume of that figure? How do you find the area and perimeter of a parallelogram, triangle, rhombus, kite, trapezoid, and regular polygon? How can you find the surface area and volume of 3-dimensional figures? How can we represent the probability of an event using geometric properties of length, area, and volume? PA Academic Standards (PA Common Core): C.C.2.3.HS.A.3, C.C.2.3.HS.A.4, C.C.2.3.HS.A.6, C.C.2.3.HS.A.7, C.C.2.3.HS.A.11, C.C.2.3.HS.A.12, C.C.2.3.HS.A.13, C.C.2.3.HS.A.14, C.C.2.4.HS.B.5, C.C.2.4.HS.B.6, CC.2.2.HS.D.9 Mathematical Practices: 1. Make sense of problems and persevere in solving them. 2. Reason abstractly and quantitatively. 3. Construct viable arguments and critique the reasoning of others. 4. Model with Mathematics. 5. Use appropriate tools strategically. 6. Attend to precision. 7. Look for and make use of structure. 8. Look for and express regularity in repeated reasoning. 13

14 Tier 3 Vocabulary: Face, Edge, Vertex, Base, Polyhedron, Platonic Solids, Tetrahedron, Cube, Cross Section, Prism, Surface Area, Lateral Area, Net, Right Prism, Oblique Prism, Cylinder, Pyramid, Slant height, Cone, Lateral Surface, Volume, Sphere, Hemisphere, Similar Solids Concepts: Competencies: Instructional Practices: Assessments: Students will know: Formulas for areas of quadrilaterals, regular polygons, and sectors of circles. Relationship of probability to finding geometric probabilities. Ratios are used to find areas of similar polygons and can be applied to probability problems. Formulas to solve for surface area and volume. The connection between similarity and area and volume of solids (3- dimensional figures). Students will be able to: Use geometric probability from solving area equations. Recognize and use the formulas for areas of quadrilaterals. Use formulas to solve for variables in surface area and volume equations. Identify formula and its application to its figure. Use area and volume to determine probability in figures. Solve real world problems using surface area and volume of solids. Use calculator to solve formulas and other content. Students will use the smartboard to label parts of a circle. Group assignment where students will explain the formulas found for surface area and volume. Real life problems showing volume and surface area. Use khanacademy.org for reinforcement of arc length and area of a sector. Math journal on what the students understand and still need to know. Word splash of formulas. Exit card to assess understanding of ratios. Homework assignments and corrections. Quizzes/tests on measurement. Group presentations of concepts and formulas. Students will use small whiteboards to display their answers and understanding for various problems. Peer assessments/ critical friend activity where students check their work against other students and find errors and make corrections. Use teachertube.com and youtube.com to further explain information. 14

15 Guided teacher instruction. Use calculators to solve formulas for surface area and volume. Smartboard Kooshball Jeopardy Review 15