Geometry Level 8
Course of Study
2011-2012
Course Description
This course builds on the mathematics learned in Algebraand incorporatescurrent state standards and more advanced topics from the state adopted common core standards and ACT quality core strands:congruence; similarity, right triangles and trigonometry; circles; expressing geometric properties with equations;geometric measurement and dimension; and modeling with geometric.
Examples of topics studied may include experimenting with transformations in a plane, understanding congruence in terms of rigid motion, proving geometric theorems, making geometric constructions, understanding similarity in terms of similarity transformations, proving theorems involving similarity, defining trigonometric ratios and solving problems involving right triangles, understanding and applying theorems about circles, finding arc lengths and areas of sectors of circles, using coordinates to prove simple geometric theorems algebraically, explaining volume formulas and using them to solve problems, and applying geometric concepts in modeling situations.
Mods:2
Prerequisite:Teacher Recommendation
Credit:1
Geometry Level 8
Course of Study
2011-2012
Number, Number Sense and Operations
Standard
Students demonstrate number sense, including an understanding of number systems and operations and how they relate to one another. Students compute fluently and make reasonable estimates using paper and pencil, technology-supported and mental methods.
Mathematics Benchmarks
By the end of the 8-10 program:
- Use scientific notation to express large numbers and numbers less than one.
B.Identify subsets of the real number system.
C.Apply properties of operations and the real number system, and justify when they hold for a set of numbers.
D.Connect physical, verbal and symbolic representations of integers, rational numbers and irrational numbers.
E.Compare, order and determine equivalent forms of real numbers.
F.Explain the effects of operations on the magnitude of quantities.
G. Estimate, compute and solve problems involving real numbers, including ratio, proportion and percent, and explain solutions.
H. Find the square root of perfect squares, and approximate the square root of non-perfect squares.
I. Estimate, compute and solve problems involving scientific notation, square roots and numbers with integer exponents.
Geometry Level 8
Course of Study
2011-1012
Number, Number Sense and Operations
Indicator 1 Benchmarks 8-10 Benchmark A
- Number and Number Systems (organizer)
1.Use scientific notation to express large numbers and small numbers between 0 and 1.
Performance Skills:
- Convert a large number in standard form to scientific notation.
- Convert a small number between zero and 1 to scientific notation.
- Convert from scientific notation using positive and negative exponents to standard form.
Geometry Level 8
Course of Study
2011-2012
Number, Number Sense and Operations (Continued)
Indicator 2 Benchmarks 8-10 Benchmark B, I
- Number and Number Systems (organizer)
2.Recognize that natural numbers, whole numbers, integers, rational numbers and irrational numbers are subsets of the real number system.
Performance Skills:
- Classify natural numbers as a subset of whole numbers.
- Classify whole numbers as a subset of the integers.
- Classify the integers as a subset of the rational numbers.
- Classify rational and irrational numbers as a subset of the real numbers.
Geometry Level 8
Course of Study
2011-1012
Number, Number Sense and Operations (Continued)
Indicator 3 Benchmarks 8-10 Benchmark B, I
- Meaning of Operations (organizer)
3. Apply order of operations to simplify expressions and perform computations involving integer exponents and radicals.
Performance Skills:
- Apply order of operations to simplify expressions (including integer exponents and radicals).
Ex.22 ∻√121 ∙ (4 – 7)
Ex.(4 + 3)2∙3 – 1
Not43 ∙ 43 or (42)3
Geometry Level 8
Course of Study
2011-1012
Number, Number Sense and Operations (Continued)
Indicator 4 Benchmarks 8-10 Benchmark C
- Meaning of Operations (organizer)
4.Explain and use the inverse and identity properties and use inverse relationships (addition/subtraction, multiplication/division, squaring/square roots) in problem solving situations.
Performance Skills:
- Explain inverse and identity properties.
- Apply inverse and identity relationships in problem solving situations.
Eg.
Solving equations
Word problems
Pythagorean Theorem
ACT: Prerequisites-skills acquired by students in a previous course and refined in this course.
A1A: Apply algebraic properties (e.g., commutative, associative, distributive, identity, inverse, substitution) to simplify algebraic expressions.
Geometry Level 8
Course of Study
2011-2012
Number, Number Sense and Operations (Continued)
Indicator 5 Benchmarks 8-10 Benchmark F, G
- Computation and Estimation (organizer)
5.Determine when an estimate is sufficient and when an exact answer is needed in problem situations, and evaluate estimates in relation to actual answers; e.g., very close, less than, greater than.
Performance Skills:
- Identify key words that imply an exact answer is not needed:
-About
-Estimate
-Approximately
- Differentiate when an exact answer is required and when an estimate is acceptable using real-life examples.
-Bank Account
-Measurement
-Large Purchases
-Time
Geometry Level 8
Course of Study
2011-2012
Number, Number Sense and Operations (Continued)
Indicator 6 Benchmarks 8-10 Benchmark G
- Computation and Estimation (organizer)
6.Estimate, compute and solve problems involving rational numbers, including ratio, proportion and percent, and judge the reasonableness of solutions.
Performance Skills:
- Estimate, compute, and solve problems involving rational numbers, including ratio, proportion, and percent.
- Determine the reasonableness of solutions to problems involving rational numbers.
Geometry Level 8
Course of Study
2011-2012
Number, Number Sense and Operations (Continued)
Indicator 7 Benchmarks 8-10 Benchmark H
- Computation and Estimation (organizer)
7. Find the square root of perfect squares, and approximate the square root of non-perfect squares as consecutive integers between which the root lies; e.g., the square root of 130 is between 11 and 12.
Performance Skills:
- Find the square root of perfect squares.
Eg.√81 = 9
- Approximate the square root of non-perfect squares as consecutive integers between which the root lies.
Eg.√130 is between 11 and 12
Geometry Level 8
Course of Study
2011-1012
Number, Number Sense and Operations (Continued)
Indicator 8 Benchmarks 8-10 Benchmark I
- Computation and Estimation (organizer)
8.Add, subtract, multiply, divide and compare numbers written in scientific notation.
Performance Skills:
- Add, subtract, multiply, and divide numbers written in scientific notation.
- Compare and order numbers written in scientific notation.
Geometry Level 8
Course of Study
2011-1012
Number, Number Sense and Operations (Continued)
Vocabulary
Closure
Magnitude of Quantities
*MEPCV
*MEPCV – Maintain and Enhance Previous Content Vocabulary – Previous Content Vocabulary is now enhanced to the current grade appropriate indicators.
Geometry Level 8
Course of Study
2011-2012
Measurement
Standard
Students estimate and measure to a required degree of accuracy and precision by selecting and using appropriate units, tools, and technologies.
Mathematics Benchmarks
By the end of the 8-10 program:
A.Solve increasingly complex non-routine measurement problems and check for reasonableness of results.
B.Use formulas to find surface area and volume for specified three-dimensional objects accurate to a specified level of precision.
C.Apply indirect measurement techniques, tools and formulas, as appropriate, to find perimeter, circumference and area of circles, triangles, quadrilaterals and composite shapes, and to find volume of prisms, cylinders, and pyramids.
D. Use proportional reasoning and apply indirect measurement techniques, including right triangle trigonometry and properties of similar triangles, to solve problems involving measurements and rates.
E. Estimate and compute various attributes, including length, angle measure, area, surface area and volume, to a specified level of precision.
F. Write and solve real-world, multi-step problems involving money, elapsed time and temperature, and verify reasonableness of solutions.
Geometry Level 8
Course of Study
2011-2012
Geometry and Spatial Sense
Standard
Students identify, classify, compare and analyze characteristics, properties and relationships of one-, two-, and three-dimensional geometric figures and objects. Students use spatial reasoning, properties of geometric objects, and transformations to analyze mathematical situations and solve problems.
Mathematics Benchmarks
By the end of 8-10 program:
A. Formally define geometric figures.
B.Describe and apply the properties of similar and congruent figures; and justify conjectures involving similarity and congruence.
C. Recognize and apply angle relationships in situations involving intersecting lines, perpendicular lines and parallel lines.
D. Use coordinate geometry to represent and examine the properties of geometric figures.
E. Draw and construct representations of two- and three-dimensional geometric objects using a variety of tools, such as straightedge, compass and technology.
F. Represent and model transformations in a coordinate plane and describe the results.
G. Prove or disprove conjectures and solve problems involving two- and three-dimensional objects represented within a coordinate system.
H.Establish the validity of conjectures about geometric objects, their properties and relationships by counter-example, inductive and deductive reasoning, and critiquing arguments made by others.
I. Use right triangle trigonometric relationships to determine lengths and angle measures.
Geometry Level 8
Course of Study
2011-1012
ACT: Prerequisites-skills acquired by students in a previous course and refined in this course.
A1B: Solve single-step and multistep equations and inequalities in one variable.
A1C: Write linear equations in standard form and slope-intercept form when given two points, a point and a slope, or the graph of the equation.
A1D: Recognize the concept of slope as a rate of change and determine the slope when given the equation of a line in standard form or slope-intercept form, the graph of a line, two points, or a verbal description.
A1E: Graph a linear equation using a table of values x- and y-intercepts, or slope-intercept form.
Geometry Level 8
Course of Study
2011-2012
ACT: Exploring the skills and strategies underlying mathematics-process objectives learned in the context of increasingly complex mathematical and real-world problems.
B1A: Apply problem-solving skills (e.g., identifying irrelevant or missing information, making conjectures, extracting mathematical meaning, recognizing and performing multiple steps when needed, verifying results in the context of the problem) to the solution of real-world problems.
Performance skills: Identify irrelevant or missing information, making conjectures, extracting mathematical meaning, recognizing and performing multiple steps when needed, verifying results in the context of the problem
B1B: Use a variety of strategies to set up and solve increasingly complex problems.
B1C: Represent data, real-world situations, and solutions in increasingly complex contexts (e.g., expressions, formulas, tables, charts, graphs, relations, functions) and understand the relationships.
Performance skills: Solve real-world problems and represent solutions as expressions, formulas, tables, charts, graphs, relations and functions.
B1D: Use the language of mathematics to communicate increasingly complex ideas orally and in writing, using symbols and notations correctly.
B1E: Make appropriate use of estimation and mental mathematics in computations and to determine the reasonableness of solutions to increasingly complex problems.
B1F: Make mathematical connections among concepts, across disciplines, and in everyday experiences.
Geometry Level 8
Course of Study
2011-1012
B1G: Demonstrate the appropriate role of technology (e.g., calculators, software programs) in mathematics (e.g., organize data, develop concepts, explore relationships, decrease time spent on computations after a skill has been established).
Performance Skills: Use technology to organize data, develop concepts, explore relationships, and decrease time spent on computations.
B1H: Apply previously learned algebraic concepts in geometric contexts.
Geometry Level 8
Course of Study
2011-1012
ACT: Using logic and proof to reason mathematically-logic and proof
C1A: Use definitions, basic postulates, and theorems about points, segments, lines, angles, and planes to write proofs and to solve problems.
C1B: Use inductive reasoning to make conjectures and deductive reasoning to arrive at valid conclusions.
C1C: Identify and write conditional and biconditional statements along with the converse, inverse, and contrapositive of a conditional statement; use these statements to form conclusions
C1D: Use various methods to prove that two lines are parallel or perpendicular (e.g., using coordinates, angle measures).
Performance Skills: Use coordinates
Use angle measure
C1E: Read and write different types and formats of proofs including two-column, flowchart, paragraph, and indirect proofs.
C1F: Prove that two triangles are congruent by applying the SSS, SAS, ASA, AAS, and HL congruence statements.
C1G: Use the principle that corresponding parts of congruent triangles are congruent to solve problems.
C1H: Use several methods, including AA, SAS, and SSS, to prove that two triangles are similar, corresponding sides are proportional, and corresponding angles are congruent.
C1I: Use properties of special quadrilaterals in a proof.
Performance Skills: Use properties of parallelograms, rectangles, rhombi, squares, trapezoids, kites
Geometry Level 8
Course of Study
2011-2012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-point, lines, planes, and space
D1A: Identify and model plane figures, including collinear and noncollinear point, lines, segments, rays, and angles using appropriate mathematical symbols.
Geometry Level 8
Course of Study
2011-2012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-point, lines, planes, and space
D1B: Identify vertical, adjacent, complementary, and supplementary angle pairs and use them to solve problems (e.g., solve equations, use in proofs).
Performance Skills: Solve equations
Use in proofs
Measurement
Indicator 8 Benchmarks 8-10
- Use measurement Techniques and tools (organizer)
8. Find the sum of the interior and exterior regular convex polygons with and without measuring the angles with a protractor.
Geometry Level 8
Course of Study
2011-1012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-point, lines, planes, and space
D1C: Identify corresponding, same-side interior, same-side exterior, alternate interior, and alternate exterior angle pairs formed by a pair of parallel lines and a transversal and use the special angle pairs to solve problems (e.g., solve equations, use in proofs).
Performance Skills: Solve equations
Use in proofs
D1D: Use construction techniques, including straightedge and compass, to bisect and trisect segments and to create parallel and perpendicular lines, perpendicular bisectors, and angle bisectors.
D1E: Locate, describe, and draw a locus in a plane or space.
D1F: Apply properties and theorems of parallel and perpendicular lines to solve problems.
Geometry Level 8
Course of Study
2011-2012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-polygons
D2A: Identify and classify triangles by their sides and angles.
D2B: Identify medians, altitudes, perpendicular bisectors, and angle bisectors of triangles and use their properties to solve problems (e.g., find points of concurrency, segment lengths, or angle measures).
Performance Skills: Find points of concurrency
Find segment lengths
Find angle measures
D2C: Apply the Triangle Inequality Theorem to determine if a triangle exists and the order of sides and angles.
D2D: Solve problems involving the relationships formed when the altitude to the hypotenuse of a right triangle is drawn.
D2E: Apply the Pythagorean theorem in right triangles to find lengths of the unknown side.
Performance skills: Examples: shadow problem, ladder problems
D2F: Identify and use Pythagorean triples in right triangles to find lengths of the unknown side.
D2G: Identify and classify quadrilaterals, including parallelograms, rectangles, rhombi, squares, kits, trapezoids and isosceles trapezoids, using their properties.
D2H: Identify and classify regular and nonregular polygons (e.g., pentagons, hexagons heptagons, octagons, nonagons, decagons, dodecagons) based on the number of sides, the angle measures, and the side lengths.
Geometry Level 8
Course of Study
2011-1012
D2I: Apply the Angle Sum Theorem for triangles and polygons to find interior and exterior angle measures given the number of sides, to find the number of sides given angle measures, and to solve real-world problems.
D2J: Apply the Isosceles Triangle Theorem and its converse to triangles to solve mathematical and real-world problems.
Geometry Level 8
Course of Study
2011-2012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-circles
D3A: Identify and define line segments associated with circles (e.g., radii, diameters, chords, secants, tangents).
Performance skills: Identify and define radius, diameter, chords, secants, tangents.
D3B: Determine the measure of central and inscribed angles and their intercepted arcs.
D3C: Find segment lengths, angle measures, and intercepted arc measures formed by chords, secants, and tangents intersecting inside and outside circles.
D3D: Solve problems using inscribed and circumscribed polygons.
Geometry Level 8
Course of Study
2011-2012
Geometry Level 8
Course of Study
2011-1012
Common Core Standards
G-C #1 Prove that all circles are similar
G-C #3 Construct the inscribed and circumscribed circles of a triangle and prove properties of angles for a quadrilateral inscribed in a circle.
G-C #4 Consttruct a tangent line from a point outside a given circle to the circle.
Geometry Level 8
Course of Study
2011-2012
ACT: Identifying, classifying, and applying the properties of geometric figures in space-solids
D4A: Identify and classify prisms, pyramids, cylinders, cones, and spheres and use their properties to solve problems.
D4B: Describe and draw cross sections of prisms, cylinders, pyramids and cones.
Geometry Level 8
Course of Study
2011-2012
Common Core
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.
Geometry Level 8
Course of Study
2011-2012
ACT: Comparing congruent and similar geometric figures-similarity and congruence
E1A: Determine points or lines of symmetry and apply the properties of symmetry to figures.
E1B: Identify congruent figures and their corresponding parts.
E1C: Identify similar figures and use ratios and proportions to solve mathematical and real-world problems (e.g., finding the height of a tree using the shadow of the tree and the height and shadow of a person).