Unit 1: Congruence, Proof, and Constructions In previous grades, students were asked to draw triangles based on given measurements. They also have prior experience with rigid motions: translations, reflections, and rotations and have used these to develop notions about what it means for two objects to be congruent. In this unit, students establish triangle congruence criteria, based on analyses of rigid motions and formal constructions. They use triangle congruence as a familiar foundation for the development of formal proof. Students prove theorems—using a variety of formats—and solve problems about triangles, quadrilaterals, and other polygons. They apply reasoning to complete geometric constructions and explain why they work. Unit 1: Congruence, Proof, and Constructions Cluster Standard Practices Experiment with transformations in the plane. Cluster Note: Build on student experience with rigid motions from earlier grades. Point out the basis of rigid motions in geometric concepts, e.g., translations move points a specified distance along a line parallel to a specified line; rotations move objects along a circular arc with a specified center through a specified angle. G.CO.1 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. Essential Skills and Knowledge Ability to fluently use mathematical vocabulary G.CO.2 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) Essential Skills and Knowledge Ability to make connections between function transformations (F.BF.3) and geometric transformations Knowledge that rigid transformations preserve the shape of a figure 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.
triangles are equal and that these relationships lead to the definition of the six trigonometric ratios G.SRT.7 Explain and use the relationship between the sine and cosine of complementary angles. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. G.SRT.8 Use trigonometric ratios and the Pythagorean Theorem to solve right triangles in applied problems.★ Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. 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. Apply geometric concepts in modeling situations. Cluster Note: Focus on situations well modeled by trigonometric ratios for acute angles Note: These are overarching standards that have applications in multiple units Make sense of problems and persevere in solving them.
a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side. Essential Skills and Knowledge Ability to make connections between the formulaand right triangle trigonometry G.SRT.10 (+) Prove the Laws of Sines and Cosines and use them to solve problems. Essential Skills and Knowledge Ability to recognize when it is appropriate to use the Law of Sines and the Law of Cosines G.SRT.11 (+) 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). Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. 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. Unit 3: Extending to Three Dimensions Students’ experience with two-dimensional and three-dimensional objects is extended to include informal explanations of circumference, area and volume formulas. Additionally, students apply their knowledge of two-dimensional shapes to consider the shapes of cross-sections and the result of rotating a two-dimensional object about a line. Unit 3: Extending to Three Dimensions
Cluster Standard Practices Explain volume formulas and use them to solve problems. Cluster Note: Informal arguments for area and volume formulas can make use of the way in which area and volume scale under similarity transformations: when one figure in the plane results from another by applying a similarity transformation with scale factor k, its area is times the area of the first. Similarly, volumes of solid figures scale by under a similarity transformation with scale factor k. G.GMD.1 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. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. G.GMD.3 Use volume formulas for cylinders, pyramids, cones, and spheres to solve problems.★ Note: This is an overarching standard that has applications in multiple units Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. 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. Visualize the relation between two and objects. G.GMD.4 Identify the shapes of two-dimensional cross-sections of three dimensional objects, and identify three-dimensional objects generated by rotations of two-dimensional objects. Essential Skills and Knowledge Ability to make connections between Make sense of problems and persevere in solving them. Reason abstractly dimensional
figures such as rectangles, squares, circles, and triangles and three-dimensional figures such as cylinders, spheres, pyramids and cones 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. Apply geometric concepts in modeling situations. 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). ★ Note: Focus on situations that require relating and objects, determining and using volume, and the Make sense of problems and persevere in solving them. Reason abstractly
See the skills and knowledge that are stated in the Standard. G.C.2 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. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. G.C.3 Construct the inscribed and circumscribed circles of a triangle, and prove properties of angles for a quadrilateral inscribed in a circle. Essential Skills and Knowledge Ability to use concurrence of perpendicular bisectors and angle bisectors for the basis of the construction G.C.4 (+) Construct a tangent line from a point outside a given circle to the circle. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. 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. Find arc lengths and areas of sectors of circles. G.C.5 Derive using similarity the fact that the length of the arc intercepted by an angle is proportional to the radius, and define the radian measure of the angle as the constant of proportionality; derive the formula for the area of a sector. Note: Emphasize the similarity of all circles. Note that by similarity of sectors with the same central angle, arc lengths are proportional to the radius. Use this as a basis for introducing radian as a unit of measure. It is not intended that it be applied to the development of circular Trigonometry in this course. Essential Skills and Knowledge See the skills and knowledge that are stated in the Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others.
Standard. 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. Translate between the geometric description and the equation for a conic section. G.GPE.1 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. Essential Skills and Knowledge See the skills and knowledge that are stated in the Standard. 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. Apply geometric concepts in modeling situations. 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). ★ Note: Focus on situations in which the analysis of circles is required. Note: This is an overarching standard that has applications in multiple units Essential Skills and Knowledge Ability to connect experiences from Unit 2 and Unit 3 with two dimensional and three shapes Make sense of problems and persevere in solving them. Reason abstractly and quantitatively. Construct viable arguments and critique the reasoning of others.