First 9 Weeks Second 9 Weeks 9 Weeks Quadrilaterals & Polygons (, ) Symmetry & Transformations (G.3c, d) Fourth 9 Weeks Circles (, ) 1st Nine Weeks Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 3 days Trdtl 6 days Standard: Prerequisite for G.3, G.4 The student will use pictorial representations, including computer software, constructions, and coordinate methods, to solve problems involving symmetry and transformation. This will include investigating and using formulas for finding distance, midpoint, Notes: Key Vocabulary acute angle bisect collinear complementary coplanar horizontal intersect line obtuse plane postulate ray right [angle] segment supplementary theorem vertex vertical DOE Lesson Plans
Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 2 days Trdtl 4 days Standard: Prereq. for G.2, G.10 The student will use the relationships between angles formed by two lines cut by a transversal SOL 8.6a: Verify/describe relationships among vertical/adjacent/supplementary/ complementary angles Notes: Key Vocabulary adjacent angles alternate exterior angles alternate interior angles complementary angles consecutive (same- side) interior angles corresponding angles exterior angle interior angle intersection linear pair parallel perpendicular skew supplementary angles theorem transversal DOE Lesson Plans
Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 1 day Trdtl 2 days Standard:G.4 The student will construct and justify the constructions of a) a line segment congruent to a line segment; b) the perpendicular bisector of a line segment; e) the bisector of a given angle; f) an angle congruent to a given angle Prerequisite/Extension SOL 6.12 determine congruence of segments/angles/polygons Essential Understandings Construction techniques are used to solve problems in engineering, architectural design, and building construction. Construction techniques include using a straightedge and compass, paper folding, and dynamic geometry software. Essential Knowledge/Skills Construct and justify the constructions of a line segment congruent to a line segment; Key Vocabulary bisector centroid circumcenter circumscribed compass congruence construction incenter inscribed intersection parallel lines perpendicular bisector perpendicular lines
line or two points on line ‐ slope as rate of change; b) write equation of line given graph of line/two points on line or slope‐point on line 8.8 a) apply transformations to plane figures; b) ID applications of transformations 7.8 represent transformations (reflections, dilations, rotations, and translations) of polygons in the coordinate by graphing Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources 7 days Trdtl 14 days Standard: G.1 The student will construct and judge the validity of a logical argument consisting of a set of premises and a conclusion. This will include a) identifying the converse, inverse, and contrapositive of a conditional statement; b) translating a verbal argument into symbolic form; c) using Venn diagrams to represent set relationships; and d) using deductive reasoning. Essential Understanding Inductive reasoning, deductive reasoning, and are critical in establishing general claims. Deductive reasoning is the method that uses logic to draw conclusions based on definitions, postulates, and theorems. Inductive reasoning is the method of drawing conclusions from a limited set of observations. Proof is a justification that is logically valid and based on initial assumptions, definitions, postulates, and theorems. Logical arguments consist of a set of premises or hypotheses and a conclusion. Key Vocabulary algebraic method assumption biconditional statement conclusion conditional statement conjecture contrapositive converse coordinate method counterexample deductive reasoning hypothesis (premise) inductive reasoning inverse Law of Detachment Law of Syllogism Law of the
Recognize and use the symbols of formal logic, which include →, ↔, ~, , , and . Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 2 day 4 days Standard: G.2 The student will use the relationships between angles formed by two lines cut by a transversal to a) determine whether two are parallel; b) verify the parallelism, using algebraic and coordinate methods as well as deductive proofs; and c) solve real-world problems involving angles formed when parallel lines are cut by a transversal. Prerequisite/Extension SOL Essential Understanding Parallel lines intersected by a transversal form angles with specific relationships. Some angle relationships may be used when proving two lines intersected by a transversal are parallel. The Parallel Postulate differentiates Euclidean from non-Euclidean geometries such as spherical geometry and hyperbolic geometry. Essential Knowledge/Skills Key Vocabulary alternate exterior angles alternate interior angles consecutive (same- side) interior angles corresponding angles exterior angle interior angle parallel perpendicular skew transversal DOE Lesson Plans Textbook Resources 3.1 – 3.4
A.6 linear equations/linear inequal (2 vars) ‐ a) determine slope of line given equation of line/graph of line or two points on line ‐ slope as rate of change; b) write equation of line given graph of line/two points on line or slope‐point on line 8.6 a) verify/describe relationships among vertical/adjacent/ supplementary/complementary angles; Use algebraic and coordinate methods as well as deductive proofs to verify whether two lines are parallel. Solve problems by using the relationships between pairs of angles formed by the intersection of two parallel lines and a transversal including corresponding angles, alternate interior angles, alternate exterior angles, and same-side (consecutive) interior angles. Solve real-world problems involving intersecting and parallel lines in a plane. Resources/Lessons Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 1day Trdtl 2 days Standard:G.4 The student will construct and justify the constructions of c) a perpendicular to a given line from a point not on the line; d) a perpendicular to a given line at a given point on the line; g) a line parallel to a given line through a point not on the given line. Essential Understandings Construction techniques are used to real-world problems in engineering, architectural design, and building construction. Construction techniques include using a straightedge and compass, paper folding, and dynamic geometry software. Essential Knowledge/Skills Construct and justify the constructions of a perpendicular to a given line from a point not on the line; Key Vocabulary bisector centroid circumcenter circumscribed compass congruence construction incenter inscribed intersection parallel lines perpendicular bisector perpendicular bisector straightedge
2nd Nine Weeks Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block ½ days Trdtl 1 day Standard: G.10 The student will solve real-world problems involving angles of polygons. Essential Knowledge/Skills Solve real-world problems involving the measures of interior and exterior angles of polygons. Note: This section involves only interior and exterior angles in triangles. Key Vocabulary DOE Lesson Textbook Resources 3.5 Resources/Lessons Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 6 days Trdtl 12 days Standard: G.6 The student, given information in the form of a figure or statement, will prove two triangles are congruent, using algebraic and coordinate methods as well as deductive proofs. Essential Understanding Congruence has applications in a variety of areas, including art, architecture, and the sciences. Congruence does not depend on the position of the triangle. Concepts of logic can demonstrate congruence or similarity. Congruent figures are also similar, but similar figures are not necessarily congruent. Essential Knowledge/Skills Use definitions, postulates, and theorems to prove triangles congruent. Use coordinate methods, such as the distance formula and the slope formula, to prove two triangles are congruent. Use algebraic methods to prove two triangles are congruent. Key Vocabulary SSS (side-side-side) SAS (side-angle-side) ASA (angle-side-side) AAS (angle-angle-side) Scalene Isosceles – legs, vertex angle Equilateral Acute Obtuse Equiangular Right DOE Lesson Plans Textbook Resources Chapter 4
Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources 3 days 6 days Standard: G.5 The student, given information concerning the lengths of sides and/or measures of angles in triangles, will a) order the sides by length, given the angle measures; b) order the angles by degree measure, given the side lengths; c) determine whether a triangle exists; and d) determine the range in which the length of the third side must lie. These concepts will be considered in the context of real-world situations. Essential Understanding The longest side of a triangle is opposite the largest angle of the triangle and the shortest side is opposite the smallest angle. In a triangle, the length of two and the included angle determine the length of the side opposite the angle. In order for a triangle to exist, the length of each side must be within a range that is determined by the lengths of the other two sides. Essential Knowledge/Skills the sides of a triangle by their lengths when given the measures of the angles. Order the angles of a triangle by their measures when given the lengths of the sides. Given the lengths of three segments, determine whether a triangle could be formed. Given the lengths of two sides of a triangle, determine the range in which the length of the third side must lie. Solve real-world problems given information about the lengths of sides and/or measures of angles in triangles. Key Vocabulary altitude isosceles linear pair median scalene triangle Triangle Inequality Theorem DOE Lesson Plans Textbook Resources 5.1 – 5.4 (supplemental to G.5), 5.6, 5.7 Resources/Lessons
Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 5 days Trdtl 10 days Standard: G.7 The student, information in the form of a figure or statement, will prove two triangles are similar, using algebraic and coordinate methods as well as deductive proofs. Prerequisite/Extension SOL 6.12 determine congruence of segments/angles/polygons 7.6 determine similarity of plane figures and write proportions to express relationships between similar quads and triangles Essential Understanding Similarity has real-world applications in a variety of areas, including art, architecture, and the sciences. Similarity does not depend on the position of the triangle. Congruent figures are also similar, but similar figures are not necessarily congruent. Essential Knowledge/Skills Use definitions, postulates, and theorems to prove triangles similar. Use algebraic methods to prove that triangles are similar. Use coordinate methods, such as the distance formula, to prove two triangles are similar. Key Vocabulary area deductive proof perimeter proportion ratio scale factor similar figures similar triangles AA Similarity SSS Similarity SAS Similarity volume DOE Lesson Plans Textbook Resources 5.1 – 5.4 (supplemental to G.5), 5.6, 5.7 Resources/Lessons Section 7.1-7.5
Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 7.5days 15 days Standard: G.8 The student will solve real-world problems involving right triangles by using the Pythagorean Theorem and its converse, properties of special right triangles, and right triangle trigonometry. Prerequisite/Extension SOL 8.10 a) verify the Pythagorean Theorem; b) apply the Pythagorean Theorem A.3 express sq roots/cube roots of whole numbers/the square root of monomial alg exp (simplest radical form) A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal equation; b) justify steps used in simplifying expresessions and solving equations; c) solve quad equations (alg/graph); d) solve multistep linear equations (alg/graph) Essential Understanding The Pythagorean Theorem is essential for solving problems involving right triangles. The relationships between the sides and angles of right triangles are useful in many applied fields. Some practical problems can be solved by choosing an efficient representation of the problem. The ratios of side lengths in similar right triangles (adjacent/hypotenuse or opposite/hypotenuse) are independent of the scale factor and depend only on the angle the hypotenuse makes with the adjacent side, thus justifying the definition and calculation of trigonometric functions using the ratios of side lengths for similar right triangles. Essential Knowledge/Skills Determine whether a triangle formed with three given lengths is a right triangle. Solve for missing lengths in geometric figures, using properties of 45-45-90 triangles. Solve for missing lengths in geometric figures, using properties of 30-60-90 triangles. Solve problems involving right triangles, using sine, cosine, and tangent ratios. Solve real-world problems, using right triangle trigonometry and properties of right triangles. Explain and use the relationship between the sine and cosine of complementary angles Key Vocabulary angle of depression angle of elevation area of a triangle cosine hypotenuse Pythagorean Theorem right triangle sine tangent trigonometry 45°-45°-90° triangle 30°-60°-90° triangle DOE Lesson Textbook Resources Chapter 8: 8.1 – 8.5 Resources/Lessons
Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 1 day Trdtl 1 ½ days Standard: G.10 The student will solve real-world problems involving angles of polygons. Prerequisite/Extension SOL A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal equation; b) justify used in simplifying expresessions and solving equations; c) quad equations (alg/graph); d) solve multistep linear equations (alg/graph) 6.12 determine congruence of segments/angles/polygon Essential Understanding Two intersecting lines form angles with specific relationships. An exterior angle is formed by extending a side of a polygon. The exterior angle and the corresponding interior angle form a linear pair. The sum of the measures of the interior angles of a convex polygon may be found by dividing the interior of the polygon into nonoverlapping triangles. Essential Knowledge/Skills Find the sum of the measures of the interior and exterior angles of a convex polygon. Find the measure of each interior and exterior angle of a regular polygon. Find the number of sides of a regular polygon, given the measures of interior or exterior angles of the polygon. Solve real-world problems involving the measures of interior and exterior angles of polygons. Key Vocabulary concave convex pentagon polygon quadrilateral regular/irregular polygon diagonal dodecagon exterior angle heptagon hexagon interior isosceles n-gon nonagon octagon decagon DOE Lesson Plans Textbook Resources Chapter 6: 6.1 Resources/Lessons
Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 8 days Trdtl 12 days Standard: G.9 The student will verify characteristics of quadrilaterals and use properties of quadrilaterals to solve real-world problems. Prerequisite/Extension SOL A.4 solve multistep linear/ quad equation (2 vars) ‐ a) solve literal equation; b) justify steps used in simplifying expresessions and solving equations; c) solve quad equations (alg/graph); d) solve multistep linear equations (alg/graph) 7.7 compare/contrast quadrilaterals based on properties 6.13 ID/describe properties of quadrilaterals Essential Understanding The terms characteristics and properties can be used interchangeably to describe quadrilaterals. The term characteristics is used in elementary and middle school mathematics. Quadrilaterals have a hierarchical nature based on the relationships between their sides, angles, and diagonals. Characteristics of quadrilaterals can be used to identify the quadrilateral and to find the measures of sides and angles. Essential Knowledge/Skills Solve problems, including real-world problems, using the properties specific to parallelograms, rectangles, rhombi, squares, isosceles trapezoids, and trapezoids. Prove that quadrilaterals have specific properties, using coordinate and algebraic methods, such as the distance formula, slope, and midpoint formula. Prove the characteristics of quadrilaterals, using deductive reasoning, algebraic, and coordinate methods. properties of angles for a quadrilateral inscribed in a circle.† Key Vocabulary base base angles characteristics diagonal isosceles trapezoid kite legs median of a trapezoid parallelogram quadrilateral rectangle rhombus square trapezoid DOE Lesson Plans Textbook Resources Chapter 6: 6.2 – 6.7 Resources/Lessons Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources
4th Nine Weeks Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 10 days Trdtl 15 days Standard: G.11 The student will use angles, arcs, chords, tangents, and secants to a) investigate, verify, and properties of circles; Essential Understanding Many relationships exist between and among angles, arcs, secants, chords, and tangents of a circle. All circles are similar. Key Vocabulary Radius Diameter Chord Arc Secant Tangent Central Angle
b) solve real-world problems involving properties of circles; and c) find arc lengths and areas of sectors in circles. Prerequisite/Extension SOL 6.10 a) define π; b) solve practical problems w/circumference/area of circle; c) solve practical problems involving area and perimeter given radius/diameter; d) describe/determine volume/surface area of rectangular prism A chord is part of a secant. Real-world applications may be drawn from architecture, art, and construction. Essential Knowledge/Skills Find lengths, angle measures, and arc measures associated with two intersecting chords; two intersecting secants; an intersecting secant and tangent; two intersecting tangents; and central and inscribed angles. Calculate the area of a sector and the length of an arc of a circle, using proportions. Solve real-world problems associated with circles, using properties of angles, lines, and arcs. Verify properties of circles, using deductive reasoning, algebraic, and coordinate methods. Inscribed Angle Sector DOE Lesson Plans Textbook Resources Chapter 12 & additional resources Resources/Lessons Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 2 days Trdtl 4 days Standard: G.12 The student, given the coordinates of the center of a circle and a point on Essential Understanding Key Vocabulary Standard Form for the Equation of a Circle
the circle, will write the equation of the circle. Prerequisite/Extension SOL 6.10 a) define π; b) practical problems w/circumference/area of circle; c) solve practical problems involving area and perimeter given radius/diameter; d) describe/determine volume/surface area of rectangular prism A circle is a locus of points equidistant from a given point, the center. Standard form for the equation of a circle is , where the coordinates of the center of the circle are and is the length of the radius. The circle is a conic section. Essential Knowledge/Skills Identify the center, radius, and diameter of a circle from a given standard equation. Use the distance formula to find the radius of a circle. Given the coordinates of the center and radius of the circle, identify a point on the circle. Given the equation of a circle in standard form, identify the coordinates of the center and find the radius of the circle. Given the coordinates of the endpoints of a diameter, find the equation of the circle. Given the coordinates of the center and a point on the circle, find the equation of the circle. Recognize that the equation of a circle of center and radius is derived using the Pythagorean Theorem.† DOE Lesson Plans Textbook Resources Chapter 12 & additional resources Resources/Lessons Circle Match Game Dates Standards/Essential Questions Essential Knowledge/Skills/Understanding Resources Block 4 days 8 days Standard: G.13 The student will use formulas for surface area and volume of objects to solve problems. Essential Understanding The surface area of a three-dimensional object is the sum of the areas of all its faces. Key Vocabulary DOE Lesson Plans