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Bensalem Township School District Geometry Curriculum Based upon the Pennsylvania State Standards, Assessment Anchors and Eligible Content For the Keystone Exams: Geometry Pennsylvania Department of Education www.education.state.pa.us 2011 Last Updated April 2011 Unit 1 - Basic Terms and Definitions Standards Link: 2.5.G.B.Use symbols, mathematical terminology, standard notation, mathematical rules, graphing, and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas, and results. 2.8.G.B. Use algebraic representations to solve problems using coordinate geometry Big Idea: Points, lines, angles, and planes are the foundation blocks for the study of geometry. Enduring Understanding(s): The foundation of Euclidean geometry is the undefined terms point, line, angle and plane. Parallel lines with transversals create angle pairs with congruent or supplementary relationships. Relationship between parts and wholes of segments and angels. Essential Question(s): What are the foundation blocks for the structure of geometry? What are the techniques used to measure segments and angles? How are angle pairs related? Knowledge: Skill(s): There are 3 undefined terms that play an important role in Geometry. (2.5.G.B) Identify points as collinear and/or coplanar (2.5.G.B) There are different characteristics of a line, ray and line segment. (2.5.G.B) The lengths of a segment can be used to determine congruency. (2.8.G.B) The segment addition postulate can be used to find missing lengths of segments. (2.8.G.B) Distinguish the difference between parallel and skew lines. (2.5.G.B) Given a picture, identify and name the following objects: point, line, segment, plane, ray, angle, parallel, perpendicular, and skew lines (2.5.G.B) Find the length of a segment (segment addition postulate) (2.8.G.B) Angles can be classified and acute, right, obtuse or straight, based on its measure. (2.5.G.B) The measures of angles can be used to determine congruency. (2.8.G.B) The angle addition postulate can be used to find missing measures of angles. (2.8.G.B) Two angles may be classified as complementary or supplementary if the sum of their measures is 90 or 180. (2.5.G.B) The distance formula can be used to find the length of a segment. (2.8.G.B) Find the measure of angle (angle addition postulate) Problems should require the use of algebraic equations to solve. 2.8.G.B Using a protractor, draw various angles of given measurements. (2.5.G.B) Find the measure of an angle and/or variable knowing that two angles are complementary or supplementary. (2.8.G.B) Calculate the distance of a segment using the distance formula. (2.8.G.B) Determine the midpoint of a segment using the midpoint formula. (2.8.G.B) Find the missing endpoint of a segment when given one endpoint and its midpoint. (2.8.G.B) The midpoint formula can be used to find the midpoint and the endpoint of a segment. (2.8.G.B) Assessment/Evidence of Learning: Chapter 1 Common Assessment – found on staff drive Learning Activities: Mrs. Fischer had to go to two dinners, one right after the other. Her parents live at (7, -11) and her brother lives at (4, -5). Her brother’s house is located at the midpoint between Mrs. Fischer and her parents. a. Find the coordinates of Mrs. Fischer’s house. How far she will travel to eat at her parents’ house? Unit 2 - Reasoning and Proof Standards Link: 2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of counter-examples, truth tables, etc.) to validate conjectures or arguments. 2.4.G.B. Use statements, converses, inverses, and contrapositives to construct valid arguments or to validate arguments relating to geometric theorems. Big Idea: Enduring Understanding(s): Reasoning and Proof There are direct and indirect ways of coming to a conclusion or proving something. Essential Question(s): How do proofs support geometric reasoning? How do you prove theorems and how do you demonstrate conjectures? Knowledge: Skill(s): Conditional statements are the foundation for theorems and postulates in Geometry. (2.4.G.B) Proofs are used to support geometric reasoning through a logical sequence of statements and justifications. (2.4.G.A) Proofs can be written in a 2-column, paragraph or flow format. (2.4.G.A) Inductive reasoning is drawing conclusions based on a pattern. (2.4.G.A) Identify the different forms of a conditional statement. (2.4.G.B) Write a conditional statement in the different forms. (2.4.G.B) Recognize proper definitions using biconditionals. (2.4.G.B) Recognize conditional statements and be able to write the converse, inverse, biconditional and contrapositive. (2.4.G.B) Deductive reasoning is drawing Identify the truth value of each statement. conclusions based on logical progression (2.4.G.A) of truths. (2.4.G.A) Draw conclusions using deductive reasoning, the law of syllogism and the law of detachment. (2.4.G.B) Use equality, transitive, symmetric, reflexive, substitution and distributive properties to write formal algebraic proofs (in 2-column form). (2.4.G.A ) Justify statements about congruent segments and angles using properties above (including proof of the vertical angle theorem). (2.4.G.A) **Honors** Prove all theorems. (2.4.G.A) Assessment/Evidence of Learning: Ad Project – Found in Prentice Hall Resources Chapter 2 Common Assessment – found on staff drive Unit 3 - Parallel and Perpendicular Lines Standards Link: 2.3.G.C. Use properties of geometric figures and measurement formulas to solve for a missing quantity (e.g., the measure of a specific angle created by parallel lines and is a transversal). 2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of counter-examples, truth tables, etc.) to validate conjectures or arguments. 2.9.G.A. Identify and use properties and relations of geometric figures; create justifications for arguments related to geometric relations. 2.9.G.C. Use techniques from coordinate geometry to establish properties of lines, 2-dimensional shapes. Big Idea: Enduring Understanding(s): Properties of angles formed by Parallel and Perpendicular Lines Two intersecting lines form angles with specific relationships. Parallel lines cut by a transversal form angles with specific relationships. Essential Question(s): How can we use the properties of special pairs of angles to solve algebraic and geometric problems? How do we identify and name the angles formed by a transversal crossing two or more lines? How is slope used to determine the relationships between pairs of lines? Knowledge: Skill(s): When two or more lines are intersected by a transversal, there are special angle relationships that exist. These include alternate interior angles, alternate exterior angles, same-side interior angles, same side exterior angles and corresponding angles. (2.3.G.C) When two parallel lines are interested by a transversal: a. Corresponding angles are congruent b. Alternate interior angles are congruent c. Alternate exterior angles are congruent d. Same-side interior angles are supplementary e. Same-side exterior angles are supplementary (2.3.G.C) Lines can be proven parallel when specific information is known about angle relationships. (2.9.G.A) Identify the different angle pair relationships given 2 or more lines cut by a transversal: same-side interior, same-side exterior, alternate exterior, alternate interior, corresponding, vertical, and linear pairs. (2.3.G.C) Identify the relationships between the angles when the lines are parallel. (2.9.G.A) Determine what you need to know to prove two lines are parallel. (2.9.G.A) Know what relationships exist when the two lines are perpendicular to the transversal. (2.3.G.C) Write a two-column proof using the postulates and theorems. Include fill in the blank. Honors will write a 2-column proof in its entirety. (2.4.G.A) Use slope to determine if two lines are parallel. (2.9.G.A) Two lines are parallel if and only if they have the same slope. (2.9.G.A, Use the slope-intercept form and/or point2.9.G.C.) slope form of an equation to write equations for parallel and perpendicular lines. (2.9.G.A, If two lines are perpendicular, the 2.9.G.C) product of their slopes is -1. (2.9.G.A, 2.9.G.C) Given the equation of a line and a point not on the line, you can write the equation of a line that is either parallel or perpendicular to the given line. (2.9.G.A, 2.9.G.C.) Assessment/Evidence of Learning: Chapter 3 Common Assessment – found on staff drive Parallel and Perpendicular Lines Performance Task – found on staff drive Unit 4 - Congruent Triangles Standards Link: 2.1.G.C. Use ratio and proportion to model relationships between quantities. 2.4.G.A. Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of counter-examples, truth tables, etc.) to validate conjectures or arguments. 2.5.G.B. Use symbols, mathematical terminology, standard notation, mathematical rules, graphing, and other types of mathematical representations to communicate observations, predictions, concepts, procedures, generalizations, ideas, and results. 2.9.G.A. Identify and use properties and relations of geometric figures; create justifications for arguments related to geometric relations. Big Idea: Enduring Understanding(s): Similar Polygons and Congruent Triangles Proof is a justification that is logically valid and based on definitions, postulates, and theorems. Relationships that exist between the angles and sides of geometric figures can be proved. Essential Question(s): What are Sums of Interior and Exterior angles of regular polygons? How can geometric properties be used to prove relationships between the angles and sides of geometric figures? Why is it important to be able to identify congruent triangles? Knowledge: Skill(s): Triangles can be classified as equilateral, isosceles, scalene, acute, equiangular, obtuse and right. (2.9.G.A) Classify triangles as equilateral, isosceles, scalene, acute, equiangular, obtuse or right. (2.9.G.A) There are specific names for the parts of a triangle. (2.5.G.B) Identify the vertex, legs, base angles and base of a triangle. (2.5.G.B) Congruent figures are figures whose corresponding parts (sides and angles) are congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Name corresponding parts of congruent figures. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent by side-angle-side (SAS). (2.4.G.A) IF two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent by angle-side-angle (ASA). (2.4.G.A) If two angles and a nonincluded side of one triangle are congruent to two angles and the corresponding nonincluded side of another triangle, then the triangles are congruent by angle-side-angle (AAS). (2.4.G.A) Corresponding parts of congruent triangles are congruent (CPCTC). (2.4.G.A) If the hypotenuse and a leg of one right triangle are congruent to the hypotenuse and a leg of another right triangle, then the triangles are congruent by hypotenuse-leg (HL). (2.4.G.A) Determine if two figures are congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Write a formal proof to demonstrate the process of determining that two figures are congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Determine if two triangles are congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Identify the theorem (SSS, SAS, ASA, AAS, HL) that allows us to conclude the two triangles are congruent. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Determine what information is necessary to prove the two triangles are congruent by either SSS, SAS, ASA, AAS or HL. (2.1.G.C, 2.4.G.A, 2.5.G.B, 2.9.G.A) Write a formal proof to demonstrate the process of determining that two triangles are congruent. (2.4.G.A) Write a formal proof to demonstrate the process of determining congruence among corresponding parts of congruent triangles. (2.4.G.A) Assessment/Evidence of Learning: Chapter 4 Common Assessment – found on staff drive Learning Activities: Company Logo Project – found on staff drive Unit 5 - Properties of Triangles Standards Link: 2.9.G.A Identify and use properties and relations of geometric figures; create justifications for arguments related to geometric relations. 2.5.G.A Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts. 2.8.G.B Use algebraic representations to solve problems using coordinate geometry. 2.9.G.C Use techniques from coordinate geometry to establish properties of lines, 2dimensional shapes. Big Idea: Enduring Understanding(s): The classification and properties of triangles can be determined by their distinct characteristics. Knowing properties of triangles helps us solve various problems. Essential Question(s): What are the special properties of triangles? How can we compare angle measures and side lengths in triangles? Knowledge: Skill(s): The midsegent of a triangle joins the midpoints of two sides of a triangle.(2.9.G.A, 2.5.G.A) The midsegment is parallel to the base of the triangle(2.9.G.A, 2.5.G.A) The midsegment is half the length of the base of the triangle. (2.9.G.A, 2.5.G.A) The line perpendicular to the base of a triangle at its midpoint is the perpendicular bisector(2.9.G.A, 2.5.G.A) A point equidistant from the endpoints of a segment lies on the perpendicular bisector of the segment. (2.9.G.A, 2.5.G.A) An angle bisector is a ray that cuts an angle into two equal parts. (2.9.G.A, 2.5.G.A) A point on the angle bisector is equidistant from the sides of the angle. (2.9.G.A, 2.5.G.A) A interior point equidistant from the sides of an angle lies on the angle bisector. (2.9.G.A, 2.5.G.A) Concurrent lines are three or more lines that intersect in one point. (2.9.G.A, 2.5.G.A) The circumcenter is the center of the circle that circumscribes a triangle. (2.9.G.A, 2.5.G.A) The point at which concurrent lines intersect is the point of concurrency. (2.9.G.A, 2.5.G.A) The median is a segment whose endpoints are a vertex of the triangle and the midpoint of the opposite side. Identify and/or construct the midsegment of a triangle. (2.9.G.A, 2.5.G.A) Determine the length of the side parallel to the midsegment of a triangle(2.9.G.A, 2.5.G.A) Use the properties of midsegments to find missing lengths(2.9.G.A, 2.5.G.A) Identify and/or construct a perpendicular bisector(2.9.G.A, 2.5.G.A) Determine the length of segments using the properties of perpendicular bisectors(2.9.G.A, 2.5.G.A) Identify and/or construct an angle bisector. (2.9.G.A, 2.5.G.A) Determine the length of segments and measures of angles using the properties of angle bisectors. (2.9.G.A, 2.5.G.A) Identify and/or construct concurrent lines (perpendicular bisectors, angle bisectors, medians, altitudes) within a triangle. (2.9.G.A, 2.5.G.A) Identify and/or construct points of concurrency (circumcenter, incenter, centroid, orthocenter) and match it with the corresponding segments. (2.9.G.A, 2.5.G.A) Use coordinate geometry to determine the coordinates of the points of concurrency. .(2.8.G.B, 2.9.G.A, 2.5.G.A, 2.9.G.C) Use coordinate geometry to find the center of a circle that circumscribes a right triangle. (2.9.G.A, 2.5.G.A, 2.8.G.B, 2.9.G.C) Use the properties of the median to find lengths of segments within a triangle. (2.9.G.A, 2.5.G.A) Order the sides/angles of the triangle from least to greatest. (2.9.G.A, 2.5.G.A) Determine if the given segment lengths could (2.9.G.A, 2.5.G.A) The medians of a triangle are concurrent at a point that is two thirds the distance from each vertex to the midpoint of the opposite side. (2.9.G.A, 2.5.G.A) The altitude of a triangle is the perpendicular segment from the vertex to the line containing the opposite side. (2.9.G.A, 2.5.G.A) form a triangle. (2.9.G.A, 2.5.G.A) The largest side of the triangle lies opposite the largest angle. (2.9.G.A, 2.5.G.A) The sum of the lengths of any two sides of a triangle is greater than the length of the third side. (2.9.G.A, 2.5.G.A) Assessment/Evidence of Learning: Chapter 5 Common Assessment – found on staff drive Learning Activities: Create a Flag Project – found on staff drive Unit 6 - Quadrilaterals Standards Link: 2.9.G.C Use techniques from coordinate geometry to establish properties of lines, 2dimensional shapes. 2.8.G.B Use algebraic representations to solve problems using coordinate geometry. 2.11.G.A Find the measures of the sides of a polygon with a given perimeter that will maximize the area of the polygon. Big Idea: Enduring Understanding(s): Quadrilaterals Quadrilaterals can be classified by their properties. Polygons and their angles are useful for solving problems in architecture, construction, plumbing, engineering, landscaping, etc. Essential Question(s): What types of quadrilaterals exist and what properties are unique to them? Knowledge: Skill(s): The slope formula can be used to find the slope of a line.(2.9.G.C) Identify, name and describe polygons based on the number of sides (3-12) through n-gons. (2.9.G.A) The slope of a line determines whether the two lines are parallel, perpendicular or neither. (2.9.G.C) Classify polygons as regular or irregular.(2.11.G.A) The distance formula can be used to determine the length of a line. (2.9.G.C) Classify polygons as convex or concave. (2.9.G.A) In a parallelogram, the opposite sides are congruent and parallel. (2.9.G.A) A parallelogram is a rhombus if all four sides are congruent. (2.9.G.A) Classify quadrilaterals as parallelograms, trapezoids and kites based on the number of parallel and congruent sides using coordinate geometry. (2.9.G.C, 2.8.G.B) A parallelogram is a rectangle if all four angles are congruent. (2.9.G.A) Identify the properties of special parallelograms. Use those properties to solve problems. (2.9.G.A) A parallelogram is a square if all four sides and four angles are congruent. (2.9.G.A) A quadrilateral is a trapezoid if it has one pair of parallel sides. (2.9.G.A) Use the properties of a midsegment of a trapezoid to find lengths of trapezoids. (Include problems that require multiplying binomials and factoring in Advanced and Honors) (2.9.G.A) A trapezoid is isosceles if the two non parallel sides are congruent. (2.9.G.A) Calculate the measure of the interior angles of a polygon. (2.9.G.A) A kite is a quadrilateral with 2 congruent adjacent sides and no two pair of opposite sides congruent. (2.9.G.A) Determine the number of sides a polygon has based on the sum of the angles. (2.9.G.A) Assessment/Evidence of Learning: Chapter 6 Common Assessment – found on staff drive Name that Quadrilateral Open ended question – found on staff drive Unit 7 - Similarity Standards Link: 2.9.G.B Use arguments based on transformations to establish congruence or similarity of 2-dimensional shapes. 2.4.G.A Write formal proofs (direct proofs, indirect proofs/proofs by contradiction, use of counter-examples, truth tables, etc.) to validate conjectures or arguments. 2.1.G.C Use ratio and proportion to model relationships between quantities. Big Idea: Enduring Understanding(s): Proportionality and scale factors can be used to solve practical problems. We can use congruence and similarity to help us measure large objects or distances indirectly. Objects are similar if they have a common scale factor for their sides. We can compare figures that are congruent or similar to help solve problems. Essential Question(s): What is the difference between a ratio and a proportion? What are the requirements for two shapes to be called similar? How do you use properties of similar figures to solve practical problems? How can you use ratios and proportions to connect mathematical ideas? How do we measure large objects or distances since we cannot measure them directly? Knowledge: Skill(s): A ratio is a comparison of two quantities.(2.1.G.C) Define and write ratios.(2.1.G.C) Identify and solve proportions. (2.1.G.C) A proportion is a statement that two ratios are equal. (2.1.G.C) There exist many properties of proportions, including the cross-product property. (2.1.G.C) The scale compares each length in a scale drawing to the actual length.(2.9.G.B) In order for two figures to be similar, corresponding sides must be proportional and corresponding angles must be congruent.(2.9.G.B) The similarity ratio is the ratio of corresponding lengths of two similar figures.(2.1.G.C) If two angles of one triangle are congruent to two angles of another triangle, then the triangles are similar by angle-angle (AA).(2.9.G.B) If an angle of one triangle is congruent to an angle of a second triangle, and the sides including the two angles are proportional, then the triangles are similar by side-angle-side (SAS).(2.9.G.B) If the corresponding sides of two triangles are proportional, then the triangles are similar by side-side-side (SSS).(2.9.G.B) The altitude to the hypotenuse of a right triangle divides the triangle into two triangles that are similar to the original triangle and to each other.(2.1.G.C, 2.9.G.B) The side-splitter theorem states that if a line is parallel to one side of a triangle Complete the proportion to make the statement true. (2.1.G.C) Identify and use properties of similar polygons to find missing lengths. (2.1.G.C) Determine the similarity ratio of similar figures.(2.1.G.C) Use the similarity ratio to find missing lengths of similar figures.(2.1.G.C) Apply similar figures to practical problems.(2.1.G.C) Use the definition of similarity to write a formal proof showing two figures are similar.(2.9.G.B) Identify whether the triangles are similar by AA, SAS, and/or SSS.(2.9.G.B) Write a formal proof of similar triangles by AA, SAS, and/or SSS.(2.9.G.B, 2.4.G.A) Find and use relationships in similar right triangles to determine missing lengths and angles.(2.1.G.C) Use the side splitter theorem to write formal proofs and find missing sides and angles of a triangle.(2.4.G.A) and intersects the other two sides, then it divides those sides proportionally.(2.1.G.C, 2.9.G.B) Assessment/Evidence of Learning: Chapter 7 Common Assessment – found on staff drive Learning Activities: Unit 8 - Right Triangles and Trigonometry Standards Link: 2.10.G.A Identify, create, and solve practical problems involving right triangles using the trigonometric ratios and the Pythagorean Theorem. 2.1.G.C Use ratio and proportion to model relationships between quantities. Big Idea: Enduring Understanding(s): Mathematical principals involving right triangles are the cornerstone for architectural design. There are many applications of the Pythagorean Theorem. Angles of elevation and depression are needed to help solve indirect measurement problems. Essential Question(s): What are the properties of right triangles and how are they used? What are the different methods for finding the missing sides/angles of right triangles? How can we apply these methods to practical situations? How does the Pythagorean Theorem help us solve authentic applications? Knowledge: Skill(s): The Pythagorean theorem states that the sum of squares of the lengths of the legs in a right triangle is equal to the square of the length of the hypotenuse.(2.10.G.A) Use the Pythagorean Theorem to find missing sides of a right triangle.(2.10.G.A) The converse of a the Pythagorean Theorem states that if the square of one length of a triangle is equal to the sum of the squares of the lengths of the other two sides, then the triangle is a right triangle.(2.10.G.A) Use the converse of the Pythagorean theorem to classify a triangle as acute, right, or obtuse.(2.10.G.A) Apply the Pythagorean Theorem to practical situations.(2.10.G.A) Use the properties of special right triangles to find missing sides.(2.1.G.C) Apply properties of special right triangles to practical situations.(2.1.G.C) Special right triangles are triangles whose angles are 30-60-90 and 45-45- Write the ratios for the tangent, sine, and cosine or an angle within a right 90.(2.1.G.C) triangle.(2.10.G.A) The lengths of the sides of special right triangles have special properties.(2.1.G.C) Use trigonometric ratios to write and solve for missing sides of a right triangle.(2.10.G.A) Trigonometric ratios can be used to find missing sides of right triangles.(2.10.G.A) Inverse trig can be used to find the missing angles of right triangles.(2.10.G.A) Use inverse trig to find the missing angles of a right triangle.(2.10.G.A) Identify angles of elevation and depression.(2.10.G.A) Complete application problems using angles of elevation and depression.(2.10.G.A) The angle of elevation is the angle formed by a horizontal line and the line of sight above the horizontal line.(2.10.G.A) The angle of depression is the angle formed by a horizontal line and the line of sight below the horizontal line.(2.10.G.A) Angles of elevation and depression can be used to find heights.(2.10.G.A) Assessment/Evidence of Learning: Chapter 8 Common Assessment – found on staff drive Learning Activities: Unit 9 - Measurement in Two Dimensions Standards Link: 2.3.G.E Describe how a change in the value of one variable in area and volume formulas affect the value of the measurement. 2.10.G.A. Identify, create, and solve practical problems involving right triangles using the trigonometric ratios and the Pythagorean theorem. 2.11.G.A. Find the measures of the sides of a polygon with a given perimeter that will maximize the area of the polygon 2.11.G.C. Use sums of areas of standard shapes to estimate the areas of complex shapes. 2.7.G.A. Use geometric figures and the concept of area to calculate probability. Big Idea: Enduring Understanding(s): Measurement in two dimensions The use of 2-dimensional measurements is critical for understanding our world. Essential Question(s): What do changes in dimensions do to area and volumes of shapes? What life situations might require us to calculate perimeter/circumference or area? Knowledge: Skill(s): When all of the dimensions of a shape are changed proportionally, the value of the perimeter and area are affected by the changes in all dimensions. (2.3.G.E) Compare the area and perimeter when a change in dimension has occurred.(2.3.G.E) Formulas for two dimensional figures can be used to calculate perimeter/circumference and area. (2.11.G.A.) Given perimeter/circumference or area one missing dimension can be solved for. (2.11.G.A.) Relate the ratios of area of two specific Apply the formulas for perimeter and area of triangles, quadrilaterals, and circles (circumference) (2.11.G.A.) Manipulate the formulas to find missing lengths. (2.11.G.A.) Using the correct formula for area for given shapes, find the probability of getting a specific outcome. (2.7.G.A.) shapes to probability. (2.7.G.A.) Assessment/Evidence of Learning: Chapter 9 Common Assessment – found on staff drive Learning Activities: Unit 10 - Measurements in Three Dimensions Standards Link: 2.3.G.E Describe how a change in the value of one variable in area and volume formulas affect the value of the measurement. 2.10.G.A. Identify, create, and solve practical problems involving right triangles using the trigonometric ratios and the Pythagorean theorem. 2.11.G.A. Find the measures of the sides of a polygon with a given perimeter that will maximize the area of the polygon 2.11.G.C. Use sums of areas of standard shapes to estimate the areas of complex shapes. 2.5.G.A Develop a plan to analyze a problem, identify the information needed to solve the problem, carry out the plan, check whether an answer makes sense, and explain how the problem was solved in grade appropriate contexts. Big Idea: Enduring Understanding(s): Measurements in three dimensions Changing the parameters of a figure affects the surface area and volume. Essential Question(s): What life situations might require us to calculate surface area or volume? How can we break a 3-dimensional object into something we can measure with 2dimensional tools? What effect does a change in the dimensions of an object have on the surface area and volume? Knowledge: Skill(s): Formulas for three dimensional figures can be used to calculate surface area and volume.(2.5.G.A) Apply the correct formula for surface area and volume. (2.5.G.A) When some of the dimensions of a shape are changed proportionally, the value of the surface area and volume are affected by the changes in the dimensions. (2.3.G.E) Compare the surface area and volume when a change in dimension has occurred.(2.3.G.E) Assessment/Evidence of Learning: Chapter 10 common assessment – found on staff drive Open ended question: Given two equilateral triangles, find a third one whose area is the sum of the other two. What percent of the area of a circle is enclosed by an isosceles triangle one of whose sides is the diameter of the circle? Unit 11 - Circles Standards Link: 2.9.G.A Identify and use properties and relations of geometric figures; create justifications for arguments related to geometric relations. Big Idea: Enduring Understanding(s): Circles Basic properties of a circle can be used to determine measurements in real life situations. That pi is the relationship between the circumference and diameter of a circle. Theorems about circles were developed to help early astronomers study spatial bodies. Essential Question(s): How do you apply the relationships among circles, lines, segments, and the angles they form? Knowledge: Skill(s): Theorems, postulates and definitions for arcs, chords, sectors, segments, secants, tangents, and angles can be used to find missing information. ( 2.9.G.A) Distinguish between the basic components of a circle: arc, chord, sector, segment, tangent, secant, central angle, semi-circle, minor and major arcs, inscribed angles. (2.9.G.A) Use properties of tangents, chords, arcs, and central angles to find missing lengths and angles (2.9.G.A) Compare arc length and sectors Find the arc length of a circle. Use properties of inscribed angles to find missing angles Use properties of secants to find missing lengths and angles Assessment/Evidence of Learning: Unit 12 - Transformations Standards Link: 2.9.G.B Use arguments based on transformations to establish congruence or similarity of two dimensional shapes. Big Idea: Enduring Understanding(s): Transformations of figures in a plane can preserve distance and/or shape. Transformations are changes in geometric figures’ positions, shapes, and/or sizes. Essential Question(s): How do you apply the basic properties of transformations? What are the results of those transformations? Knowledge: Skill(s): A transformation is a change in a geometric figure’s position, shape, or size. (2.9.G.B) Identify whether or not the transformation was an isometry. (2.9.G.B) The original figure is known as the preimage. (2.9.G.B) Name images and corresponding parts. (2.9.G.B) The resulting figure is known as the image. (2.9.G.B) Given an image and preimage, identify the transformation that has taken place as a reflection, rotation, or translation. (2.9.G.B) An isometry is a transformation in which the preimage and image are congruent. (2.9.G.B) Contruct a geometry preimage and image using reflections, translations, and rotations. (2.9.G.B) A translation maps all points of a figure the same distance in the same direction (slide). (2.9.G.B) Determine whether a figure has point, line, rotational symmetry or neither. (2.9.G.B) A reflection is a transformation in which a figure and its image have opposite orientations (flip). A rotation is a transformation that turns a figure a specified number of degrees about a point. Some transformation result in point, line, or rotational symmetry. Assessment/Evidence of Learning: Learning Activities: Resources: