# Florida Geometry EOC Assessment Study Guide

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1 Florida Geometry EOC Assessment Study Guide The Florida Geometry End of Course Assessment is computer-based. During testing students will have access to the Algebra I/Geometry EOC Assessments Reference Sheet (at the end of this document) and a scientific calculator. View the epat Geometry Practice Test for additional information. The exam will be given in two 80 minute sessions for a total duration of 160 minutes. It will contain multiple choice items and gridded-response items % of the test questions will be low complexity items, 60-80% will be moderate complexity items, and 10-20% will be high complexity items. In addition, 65% of the exam questions will come from Two-Dimensional Geometry, 20% from Three-Dimensional Geometry, and 15% from Trigonometry and Discrete Mathematics. Students in Geometry and Geometry Honors will be required to take the Geometry End of Course Assessment which is based on the regular Geometry Course (Course Code: ). Additional Resources: Geometry Course Description: Florida Algebra EOC Item Specifications: Test Item Specifications, Computer-Based Practice Tests (epats) for EOC Assessments: / Two-Dimensional Geometry (65%) MA.912.G.1.1: Find the lengths and midpoints of line segments in two-dimensional coordinate systems. Benchmark Clarifications: Students will find the length or midpoint or one of the end points of a segment. Students will justify lengths of segments. Content Limits Items may require multiple steps. Items may include both distance and midpoint. Stimulus Attributes: Graphics should be used for most of these items, as appropriate. Items may be set in either real-world or mathematical contexts. Response Attributes: Fill-in response items may require that students provide the length of a segment or the x-coordinate (or y-coordinate) of a point of interest. Fill-in response items may have a negative answer. MC,FR Page 1

2 MA.912.G.1.3: Identify and use the relationships between special pairs of angles formed by parallel lines and transversals. Benchmark Clarification: Students will recognize, represent, apply, and/or explain properties of angles formed by parallel lines and transversals. Content Limits: Items may have multiple sets of parallel lines. Items will not include more than six lines in the graphic. Stimulus Attributes: Items may be set in either real-world or mathematical contexts. Graphics should be used in these items, as appropriate. Response Attribute: Fill-in response items may require that students provide an angle measure. Item Type: MC, FR. Page 2

3 Examples from Standards Website: Example 1: In the diagram, the lines k and l are parallel. Find the value of x. Find all angle values in the diagram. Explain your answer. Example 2: In the diagram, the lines m and n are parallel. Find the value of x. Explain your answer. MA.912.G.2.2: Determine the measures of interior and exterior angles of polygons, justifying the method used. Benchmark Clarification: Students will determine the measures of interior and exterior angles of polygons. Content Limit: All angle measurements will be in degrees. Stimulus Attributes: Items may be set in either real-world or mathematical contexts. Graphics should be used in these items, as appropriate. Item Type: MC, FR Page 3

4 Examples from Standards Website: Example 1: Calculate the measure of one interior angle and one exterior of a regular octagon. Explain your method. Example 2: Suppose that you will make a picture frame like the one shown below. To make the regular hexagonal frame, you will use identical trapezoidal pieces. What are the measures of the angles of the trapezoids? Explain your answer. MA.912.G.2.3: Use properties of congruent and similar polygons to solve mathematical or real-world problems. Benchmark Clarification: Students will use properties of congruent and/or similar polygons to solve problems. Content Limits: All angle measurements will be in degrees. Items may require statements and/or justifications to complete formal and informal proofs. Stimulus Attribute: Graphics should be used in these items, as appropriate. High Item Type: MC, FR Page 4

5 Example from Standards Website: Suppose a building is in the shape of a regular hexagon. The architect wants to put walkways as indicated. Show that the triangles formed are equal in size and shape. Page 5

6 Items assessing MA.912.G.2.3 also assess: MA.912.G.2.1: Identify and describe convex, concave, regular, and irregular polygons. High Example from the Standards Website: Example 1: Draw a hexagon. Is it convex or concave? Is it regular or irregular? Explain your answers. Example 2: Define the terms convex, concave, regular and irregular polygon and draw a picture of the tern next to the definition. MA.912.G.4.1: Classify, construct, and describe triangles that are right, acute, obtuse, scalene, isosceles, equilateral, and equiangular. Triangle Classification Game This game explores triangle geometry in a discrete environment. MA.912.G.4.2: Define, identify, and construct altitudes, medians, angle bisectors, perpendicular bisectors, orthocenter, centroid, incenter, and circumcenter. Example from the Standards Website: Draw several triangles. Construct their angle bisectors. What do you observe from your drawings? MA.912.G.4.4: Use properties of congruent and similar triangles to solve problems involving lengths and areas. Example from the Standards Website: Of two similar triangles, the second has sides half the length of the first. The area of the first triangle is 20, What is the area of the second triangle? MA.912.G.4.5: Apply theorems involving segments divided proportionally. Example from the Standards Website: In triangle ABC shown below, is parallel to. What is the length of? MA.912.G.2.4: Apply transformations (translations, reflections, rotations, dilations, and scale factors) to polygons. to determine congruence, similarity, and symmetry. Know that images formed by translations, reflections, and rotations are congruent to the original shape. Create and verify tessellations of the plane using polygons. Remarks: Physical objects, drawings, and dynamic geometry software might help students explore this benchmark. Students' early work in elementary and middle school should form a base for teaching this benchmark (see MA.3.G.3.3, MA.4.G.5.2, and MA.7.G.4.2). Students should explore different types of transformations and observe that some transformations (translations, reflections, and rotations) result in congruent shapes. Benchmark Clarification: Students will apply transformations to polygons to determine congruence, similarity, and symmetry. Content Limits: Items may include using coordinate geometry to perform transformations in the plane. Items may require statements and/or justifications to determine congruence, similarity, and symmetry. Stimulus Attributes: Items may assess transformations, including translations, reflections, rotations, dilations, and scale factors. Graphics should be used for most of these items, as appropriate. Items may be set in either real-world or mathematical contexts. Response Attributes: Fill-in response items may require that students provide the length of a segment or the x-coordinate (or y-coordinate) of a point of interest. Fill-in response items may have a negative answer. High MC, FR Page 6

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8 Example from Standards Website: Explore regular polygons through manipulatives and/or drawing programs. Describe which of the polygons would be best for tiling a rectangular floor. Explain your reasoning. MA.912.G.2.5: Explain the derivation and apply formulas for perimeter and area of polygons (triangles, quadrilaterals, pentagons, etc.). Benchmark Clarification: Students will solve problems by using and/or deriving formulas for perimeter and/or area of polygons. Content Limits: Items requiring students to calculate area may require the use of the apothem. Composite figures may include circles. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in most of these items, as appropriate. MC, FR Page 8

9 Examples from the Standards Website: Example 1: A rectangle of area 360 square yards is ten times as long as it is wide. Find its length and width. Example 2: Explain the derivation of the formula for the area of a triangle. Example 3: The design below is called the Ohio Star. Assuming that it measures 9 inches by 9 inches, calculate the total area of all the orange patches, the total area of all the yellow patches, and the total area of all the green patches. How much fabric of each color will you need to cover an area that measures 72 inches by 90 inches? Items Assessing MA.912.G.2.5 also assess: MA.912.G.2.7: Determine how changes in dimensions affect the perimeter and area of common geometric figures. Example from the Standards Website: If the lengths of each side of a trapezoid are tripled, determine the change in its area, and justify your answer. MA.912.G.3.3: Use coordinate geometry to prove properties of congruent, regular, and similar quadrilaterals. Benchmark Clarification: Students will use coordinate geometry and geometric properties to justify measures and characteristics of congruent, regular, and similar quadrilaterals. Content Limits: Items may include statements and/or justifications to complete formal and informal proofs. Items may include the use of coordinate planes. Stimulus Attributes: Graphics should be used for most of these items, as appropriate. Items may be set in either real-world or mathematical contexts. High MC Examples from the Standards Website: Example: Given a quadrilateral with vertices (0, 0), (5/2, 5 3/2), (5, 0), (7, 7 3/3), prove that the diagonals of this quadrilateral are perpendicular. Example: Is rectangle ABCD with vertices at A(0, 0), B(4, 0), C(4, 2), D(0, 2) congruent to rectangle PQRS with vertices at P(-2, -1), Q(2, -1), R(2, 1), S(-2, 1)? Justify your answer. Page 9

10 MA.912.G.3.4: Prove theorems involving quadrilaterals. Benchmark Clarification: Students will use geometric properties to justify measures and characteristics of quadrilaterals. Content Limit: Items may require statements and/or justifications to complete formal and informal proofs. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in these items, as appropriate. High MC, FR Example from the Standards Website: Prove that the diagonals of a rectangle are congruent. Page 10

11 Items Assessing MA.912.G.3.4 also assess: MA.912.D.6.4: Use methods of direct and indirect proof and determine whether a short proof is logically valid. Example from the Standards Website: If somebody argues, If it s Thursday, it is raining. along with It is raining implies that "it is Thursday.", is this a valid or invalid argument? Explain your answer. MA.912.G.3.1: Describe, classify, and compare relationships among quadrilaterals including the square, rectangle, rhombus, parallelogram, trapezoid, and kite. Remark: This benchmark examines properties of quadrilaterals one at a time. Example from the Standards Website: Explore a trapezoid through manipulatives, drawings and/or technology. Draw the diagonals and determine whether they are perpendicular. Give a convincing argument that your judgment is correct. MA.912.G.3.2: Compare and contrast special quadrilaterals on the basis of their properties. Remark: This benchmark examines properties of quadrilaterals one at a time. Example from the Standards Website: Explore a trapezoid through manipulatives, drawings and/or technology. Draw the diagonals and determine whether they are perpendicular. Give a convincing argument that your judgment is correct. MA.912.G.8.5: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs. High Examples from the Standards Website: Example 1: Prove that the sum of the measures of the interior angles of a triangle is 180. Example 2: Prove that the perpendicular bisector of line segment AB is the set of all points equidistant from the endpoints A and B. Example 3: Prove that two lines are parallel if and only if the alternate interior angles the lines make with a transversal are equal. MA.912.G.4.6: Prove that triangles are congruent or similar and use the concept of corresponding parts of congruent triangles. Benchmark Clarification: Students will use geometric properties to justify measures and characteristics of triangles. Content Limit: Items may require statements and/or justifications to complete formal and informal proofs. Stimulus Attributes: Items may be set in either real-world or mathematical contexts. Graphics should be used in these items, as appropriate. High MC Page 11

12 Example from the Standards Website: Prove that triangles ABC and APQ are similar. Items Assessing MA.912.G.4.6 also assess: MA.912.D.6.4: Use methods of direct and indirect proof and determine whether a short proof is logically valid. Example from the Standards Website: If somebody argues, If it s Thursday, it is raining. along with It is raining implies that "it is Thursday.", is this a valid or invalid argument? Explain your answer. MA.912.G.8.5: Write geometric proofs, including proofs by contradiction and proofs involving coordinate geometry. Use and compare a variety of ways to present deductive proofs, such as flow charts, paragraphs, two-column, and indirect proofs. High Examples from the Standards Website: Example 1: Prove that the sum of the measures of the interior angles of a triangle is 180. Example 2: Prove that the perpendicular bisector of line segment AB is the set of all points equidistant from the endpoints A and B. Example 3: Prove that two lines are parallel if and only if the alternate interior angles the lines make with a transversal are equal. Page 12

13 MA.912.G.4.7: Apply the inequality theorems: triangle inequality, inequality in one triangle, and the Hinge theorem. Benchmark Clarification: Students will apply the inequality theorems to determine relationships about sides and angles within a triangle and between triangles. Content Limit: Items may assess methods of proving triangles congruent. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in these items, as appropriate. MC Example from the Standards Website: Can you draw a triangle with sides of length 7 cm, 4 cm, and 15 cm? Explain your answer. Page 13

14 MA.912.G.5.4: Solve real-world problem s involving right triangles. Content Limits: Items may require students to apply the Pythagorean theorem, special right triangle relationships, and/or characteristics of triangles resulting from the altitude of a right triangle drawn from the right angle to the hypotenuse. Items may include the application of the geometric mean. Stimulus Attributes: Items assessing MA.912.G.5.2 may be set in either mathematical or real-world contexts. All other items must be set in real-world context. Any radical expressions in the item stem must be in simplified or rationalized form. Graphics should be used in most of these items, as appropriate. Response Attributes: Any radical expressions in multiple-choice options will be provided in simplified or rationalized form. High MC, FR Example from the Standards Website: The distance of the base of a ladder from the wall it leans against should be at least 1/3 of the ladder's total length. Suppose a 12-ft ladder is placed according to these guidelines. Give the minimum distance of the base of the ladder from the wall. How far up the wall will the ladder reach? Explain and include a sketch in your explanation. Page 14

15 Items assessing MA.912.G.5.4 also assess: MA.912.G.5.1: Prove and apply the Pythagorean theorem and its converse. High Example: Determine if the triangle with side lengths of 10, 12, and 18 is a right triangle. Justify your reasoning. MA.912.G.5.2: State and apply the relationships that exist when the altitude is drawn to the hypotenuse of a right triangle. Example from the Standards Website: Find the value of x in the right triangle shown here. MA.912.G.5.3: Use special right triangles ( and ) to solve problems. Example: An isosceles right triangle has one leg 6 cm long. Find the lengths of the other two sides. MA.912.G.6.5: Solve real-world problems using measures of circumference, arc length, and areas of circles and sectors. Benchmark Clarification: Students will solve problems related to circles. Content Limits: All angle measurements will be in degrees. Items may require statements and/or justifications to complete formal and informal proofs. Stimulus Attributes: Items may be set in either real-world or mathematical contexts. Graphics should be used in most of these items, as appropriate. High MC, FR Page 15

16 Example from the Standards Website: Which will give you more: three 6-inch pizzas or two 8-inch pizzas? Explain your answer. Items assessing MA.912.G.6.5 also assess: MA.912.G.6.2: Define and identify: circumference, radius, diameter, arc, arc length, chord, secant, tangent and concentric circles. Low Example from the Standards Website: What is the angle between a tangent to a circle and the radius at the point where the tangent meets the circle? MA.912.G.6.4: Determine and use measures of arcs and related angles (central, inscribed, and intersections of secants and tangents). Example from the Standards Website: Find the measure of angle ABC in the diagram shown to the right. MA.912.G.6.6: Given the center and the radius, find the equation of a circle in the coordinate plane or given the equation of a circle in center-radius form, state the center and the radius of the circle. Benchmark Clarification: Students will identify the center, radius, and/or graph of a circle given the equation of a circle, or write the equation of a circle given the center, radius, and/or graph. Content Limit: Equations of circles must be presented in center-radius form, where h and k are rational and r may be irrational. Items will not require students to manipulate equations to or from standard form. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in most of these items, as appropriate. MC Page 16

17 Example from the Standards Website: Find the equation of the circle with radius 10 and center (6, -3). Items assessing MA.912.G.6.6 also assess: MA.912.G.6.7: Given the equation of a circle in center-radius form or given the center and the radius of a circle, sketch the graph of the circle. Example: Sketch the graph of the circle whose equation is MA.912.G.8.4: Make conjectures with justifications about geometric ideas. Distinguish between information that supports a conjecture and the proof of a conjecture. Benchmark Clarification: Students will provide statements and/or reasons in a formal or informal proof or distinguish between mere examples of a geometric idea and proof of that idea. Content Limits: Items must adhere to the content limits stated in other benchmarks. Items may include proofs about congruent/similar triangles and parallel lines. High MC Example from the Standards Website: Calculate the ratios of side lengths in several different-sized triangles with angles of 90, 50, and 40. What do you notice about the ratios? How might you prove that your observation is true (or show that it is false)? Page 17

18 Three-Dimensional Geometry (20%) MA.912.G.7.1: Describe and make regular, non-regular, and oblique polyhedra, and sketch the net for a given polyhedron and vice versa. Benchmark Clarifications: Students will identify the net for a given polyhedron and vice versa. Students will identify and determine the types of faces and/or the numbers of edges, faces, and vertices of a given polyhedron or a given net. Content Limits: Items will only include: The five Platonic solids (tetrahedron, hexahedron or cube, octahedron, dodecahedron, and icosahedron); Right or oblique prisms or pyramids with up to 12 edges on the base or composites; Composites of the right or oblique prisms or pyramids; and Other solids with fewer than 15 faces. Items must not require use of formulas relating faces, edges, and vertices. Items may not include cones, spheres, or cylinders. Stimulus Attributes: Graphics should be used for most of these items, as appropriate. Items may be set in either real-world or mathematical contexts. Response Attribute: Fill-in response items may require that students provide the number of edges, faces, or vertices of a given polyhedron. MC, FR Example from the Standards Website: Make a net for a tetrahedron out of poster board and fold it up to make the tetrahedron. Is this a regular polyhedron? Explain why or why not. Items assessing MA.912.G.7.1 also assess: MA.912.G.7.2: Describe the relationships between the faces, edges, and vertices of polyhedra. Example from Standards Website: Use manipulatives to investigate the relationships between faces, edges, and vertices of polyhedra i.e., Euler's Theorem. Page 18

19 MA.912.G.7.5: Explain and use formulas for lateral area, surface area, and volume of solids. Benchmark Clarification: Students will explain and/or apply formulas to determine surface area, lateral area, and volume of solids. Content Limits: Solids will be limited to right prisms, right-circular cylinders, spheres, right pyramids, right-circular cones, and/or composites of these solids. Items may not include oblique figures. Items may ask students to apply knowledge of congruent and similar solids. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in most of these items, as appropriate. MC, FR Page 19

20 Example from the Standards Website: A gold class ring is dropped into a glass that is a right cylinder with a 6 cm diameter. The water level rises 1 mm. What is the volume of the ring? Example: Given the composite solid consisting of a hemisphere and a cone, calculate the surface area and the volume. Items assessing MA.912.G.7.5 also assess: MA.912.G.7.4: Identify chords, tangents, radii, and great circles of spheres Low Example from Standards Website: On Earth, is the equator a great circle? Explain your answer. MA.912.G.7.6: Identify and use properties of congruent and similar solids. Example from Standards Website: Explain how the surface area and volume of similar cylinders are related. MA.912.G.7.7: Determine how changes in dimensions affect the surface area and volume of common geometric solids. Benchmark Clarifications: Students will determine how changes in parameter(s) affect perimeter, area, surface area, or volume, or vice-versa. Students will determine how changes to one parameter will change other parameters when the perimeter, area, surface area, or volume is held constant. Content Limits: One or two parameters may be changed, resulting in the change of another parameter. Three parameters may be changed in one item only if all three are changed by a constant factor. Solids will be limited to right prisms, right circular cylinders, spheres, right pyramids, right circular cones, and/or composites of these solids. Items may not include oblique figures. Items may involve, explicitly and/or implicitly, no more than four parameters. Changes in dimension may or may not result in similar figures. Stimulus Attributes: Items may be set in either mathematical or real-world contexts. Graphics should be used in most of these items, as appropriate. MC, FR Example from the Standards Website: Explain how changing the radius or height of a cylinder affects its surface area and volume. Page 20

21 Items assessing MA.912.G.7.7 also assess: MA.912.G.2.7: Determine how changes in dimensions affect the perimeter and area of common geometric figures. Example from the Standards Website: If the lengths of each side of a trapezoid are tripled, determine the change in its area, and justify your answer. Trigonometry and Discrete Mathematics (15%) MA.912.T.2.1: Define and use the trigonometric ratios (sine, cosine, tangent, cotangent, secant, cosecant) in terms of angles of right triangles. Benchmark Clarification: Students will solve real-world problems involving righttriangle trigonometry. Content Limits: Items should not include special right triangles ( and ) or the Pythagorean theorem. Angle measures will be in degree measure. Items will assess only sine, cosine, and tangent to determine the length of a side or an angle measure. Stimulus Attributes: Items may be set in either real-world or mathematical contexts. Graphics should be used in all items. Response Attributes: Fill-in response items may require the student to provide an angle measure or a length. Radian equivalents of correct answers will not be used as distractors. Fill-in response items will specify the nature of the response, if the response is not an integer. MC, FR Page 21

22 Example: In triangle ABC, tan A = 1/5. Find sin A and cot A. Example: Show that the slope of a line at 135º to the x-axis is the same as the tangent of 135º. MA.912.D.6.2: Find the converse, inverse, and contrapositive of a statement. Benchmark Clarification: Students will identify the converse, inverse, or contrapositive of a given statement. Content Limits: Truth tables or validity of a given statement will not be assessed. Items must present propositions as a sentence, and not by using symbols, e.g., p q or 3x + 1 = 7 x = 2. Stimulus Attribute: Items may be set in either real-world or mathematical contexts. MC Example from the Standards Website: Determine the inverse, converse and contrapositive of the statement, If it is Thursday, there will be rain. Items assessing MA.912.D.6.2 also assess: MA.912.D.6.3: Determine whether two propositions are logically equivalent. Example from the Standards Website: Determine whether the propositions and are logically equivalent. Page 22

23 Benchmarks that are not assessed by the End-of-Course Exam MA.912.G.1.2: Construct congruent segments and angles, angle bisectors, and parallel and perpendicular lines using a straight edge and compass or a drawing program, explaining and justifying the process used. Not assessed MA.912.G.4.3: Construct triangles congruent to given triangles. High Not assessed MA.912.G.8.1: Analyze the structure of Euclidean geometry as an axiomatic system. Distinguish between undefined terms, definitions, postulates, and theorems. MA.912.G.8.2: Use a variety of problem-solving strategies, such as drawing a diagram, making a chart, guess-and-check, solving a simpler problem, writing an equation, and working backwards. MA.912.G.8.3: Determine whether a solution is reasonable in the context of the original situation. MA.912.G.8.6: Perform basic constructions using straightedge and compass, and/or drawing programs describing and justifying the procedures used. Distinguish between sketching, constructing, and drawing geometric figures. High High Not assessed Not assessed Not assessed Not assessed The next two pages contain a copy of the Algebra I / Geometry EOC Assessment Reference Sheet which may be used on this exam. Not all references found on the sheet will be used on this exam. A link to this reference sheet is: %20EOC%20Ref%20Sheet%20.pdf Page 23

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