Topic/Unit: Points, Lines, and Planes Big Ideas/Enduring Understandings: Describing geometry figures Measuring geometric figures Understanding equality and congruence Use inductive and deductive reasoning 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 2-column, indirect 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
Mathematical Practices: 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. 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 student will be able to: Identify and classify geometric figures. Measure segments in the coordinate plane. Classify angles and find their measures. 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. 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
The uses of inductive and deductive reasoning. Proofs use postulates, theorems and properties to support them. Lines are parallel and perpendicular or neither. Use perimeter and area formulas. 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. Construct a proof using either the two column proof or paragraph form. Finding real life examples of geometric figures in relevant real world places. 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. Math journal on what the students understand and still need to know. 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. Proof based assessment on postulates and theorems. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Challenge based learning problems. Page 3
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 the orthocenter, centroid, incenter, and circumcenter relate to the segments of triangles? How do side lengths of a triangle compare to the angle measures of a triangle? How do you use the ideas of 2-column, 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
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, Dilation, Reduction, Enlargement Concepts: Competencies: Instructional Practices: Assessments: The student will know: The types of information about certain angle and side relationships lead to short cuts in proving triangles congruent and/or similar. How coordinate geometry is used to prove triangles are congruent. How 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. 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. AA, SSS, and SAS Page 5
Difference between writing a coordinate proof and an indirect proof. The points of concurrency in triangles. Basic proportions in geometric shapes. Recall and use parallel line properties to find proportional polygons. Solve inequalities of angles and sides in triangles. Perform similarity transformations. Apply proportions to 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. Test involving writing proof of congruence postulates. Students will present Pythagorean theorem sections. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Challenge based learning problems. Page 6
Topic/Unit: Figures in the Plane Big Ideas/Enduring Understandings: Using the Pythagorean theorem and its converse Using special relationships in right triangles Using trigonometric rations to solve right triangles Using angle relationships in polygons Using properties of parallelograms Classifying quadrilaterals by their properties Performing congruence and similarity transformations Making real-world connections to symmetry and tessellations Applying matrices and vectors in geometry Essential Questions: If you know the lengths of two sides of a right triangle, how do you find the length of the third side? How can you use the sides of a triangle to determine if it is right? What is the importance of the Pythagorean theorem and its converse? How do you find the lengths of the sides of a 30-60-90 triangle and a 45-45-90 triangle? How can you find the side length of a triangle if you only know one side and one angle in a right triangle? How can you find the angle measure of one angle in a triangle if you only know two side lengths in a right 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? How do you translate, reflect, rotate, and identify symmetry in a figure? 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 Page 7
Mathematical Practices: 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. 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. Trigonometric ratios in right triangles. How to classify quadrilaterals by their sides, angles and diagonals. Formulas for the angle sums in any polygon. 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 flipped classroom through my wiki (ruiz wiki: https://wiki.stsd.org/groups/r 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. 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. Page 8
Relationship of triangles to the sum formulas. Properties of parallelograms. That transformations of geometric figures result in images congruent to or similar to the original figures are called reflections, rotations and dilations. Use converse of Pythagorean theorem to classify triangle by its side lengths. Use 30-60-90 and 45-45-90 rules as a shortcut to solving special right triangles. Find the missing interior or exterior angle in a convex polygon. Recall parallelogram rules to determine properties of quadrilaterals. Explain the similarities and differences that exist among the basic geometric transformations. 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. Use teachertube.com and youtube.com to further explain information. Guided teacher instruction. Proof writing for properties of parallelograms. Challenge based learning problems. Quizzes/tests on classifying quadrilaterals and trigonometric ratios and other content. Students perform skit/rap/poem to show their understanding of the Pythagorean theorem and right triangles. Rotate, reflect, translate, dilate any geometric shape in the coordinate plane. Page 9
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? How do you find the measure of an arc of a circle? How can you tell if two chords in a circle are congruent? How do you find the measure of angles in a circle? How do you find the measure of the angles of two chords that intersect on, inside, or outside the circle? What are some of the properties of chords, secants and tangents to a circle? What do you need to know to write the standard equation of a circle? How do you find the length of the arc of a circle? How do you find the area of a sector of a circle? What is the relationship between radian measure and degree measure? 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.C.8, CC.2.2.HS.D.9 Page 10
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: 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, radians 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 segments of chords, secants and tangents in a circle. Find the measure of arcs of a circle and angles of a circle. Determine the relationship between Pythagorean theorem, the distance formula, and the equation of a circle. 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. 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. Page 11
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. Graph circles on the coordinate plane given the equation of the circle. Use distance and midpoint formulas to aid in finding the equations of circles. Convert between radian measures and degree measures in the unit circle. 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. Challenge based learning problems. Peer assessments/ critical friend activity where students check their work against other students and find errors and make corrections. Students will write proofs for relationships of chords within a circle. Page 12
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 use coordinates and algebraic techniques to represent, interpret, and verify geometric relationships? 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? 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? 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 13
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: 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. How to find geometric probability. Ratios are used to find areas of similar polygons and can be applied to probability problems. Formulas to solve for surface area and volume. 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 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. Exit card to assess understanding of ratios. Homework assignments and corrections. Quizzes/tests on measurement. Group presentations presenting the material using imovie or similar video based instruction. Students will use small whiteboards to display their answers and understanding for various problems. 14
The connection between similarity and solids (3- dimensional figures). Use area and volume to determine probability in figures. Solve real world problems using surface area and volume of solids. Math journal on what the students understand and still need to know. Word splash of formulas. Use teachertube.com and youtube.com to further explain information. Peer assessments/ critical friend activity where students check their work against other students and find errors and make corrections. Guided teacher instruction. Challenge based learning problems. Use calculators to solve formulas for surface area and volume. Smartboard Kooshball Jeopardy Review 15