Number and Number Systems - Perry Localperrylocal.org/curriculum/files/2010/12/geometry-Level-8.doc · Web viewG. Estimate, compute and solve problems involving real numbers, including

Geometry Level 8 Course of Study

2011-2012

Course Description

This course builds on the mathematics learned in Algebra and incorporates current 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 COS level 2011 - 1 -


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 BenchmarksBy the end of the 8-10 program:

A. 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 8Course 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.



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.


Real

Rational # Irrational

Integers

Natural


2011-1012


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

Not 43 ∙ 43 or (42)3



2011-1012


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 equationsWord problemsPythagorean 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.



2011-2012


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



2011-2012


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.



2011-2012


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



2011-1012


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.



2011-1012


Vocabulary

ClosureMagnitude of Quantities

* MEPCV

* MEPCV – Maintain and Enhance Previous Content Vocabulary – Previous Content Vocabulary is now enhanced to the current grade appropriate indicators.



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.


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.



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 BenchmarksBy 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.



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.



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.



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.



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


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.



2011-2012


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.



2011-1012


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.



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.



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.



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.


2011-2012



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.



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.



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.



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).

Performance skills: Examples: Find the height of a tree using the shadow of the tree and the height and shadow of a person.

E1D: Use the definition of similarity to establish the congruence of angles, proportionality of sides, and scale factor of two similar polygons.

E1F: Apply relationships between perimeters of similar figures, areas of similar figures,a dn volumes of similar figures, in terms of scale factor, to solve mathematical and real-world problems.

E1G: Determine the geometric mean between two numbers and use it to solve problems (e.g., find the lengths of segments in right triangles).

Performance Skills: Example: find the lengths of segments in right triangles

E1H: Identify and give properties of congruent or similar solids.



2011-2012

ACT: Using length, area, perimeter, and volume to find quantities and solve problems-area and perimeter

F1A: Find the perimeter and area of common plane figures, including triangles, quadrilaterals, regular polygons, and irregular figures, from given information using appropriate units of measurement.

F1B: Manipulate perimeter and area formulas to solve problems (e.g., finding missing lengths).

Performance skills: Example: Find the missing lengths when given perimeter or area.

F1C: Use area to solve problems involving geometric probability.

F1D: Find arc lengths and circumferences of circles from given information (e.g., radius, diameter, coordinates).

Performance skills: Example: Find arc lengths and circumference when given a radius, diameter or coordinates.

F1E: Find the area of a circle and the area of a sector of a circle from given information (e.g., radius, diameter, coordinates).

Performance skills: Example: Find area when given a radius, diameter, or coordinates.



2011-2012

ACT: Using length, area, perimeter, and volume to find quantities and solve problems-lateral area, surface area, and volume

F2A: Find the lateral area, surface area, and volume of prisms, cylinders, cones, and pyramids in mathematical and real-world settings.

Measurement

Indicator 4 Benchmarks 8-10 Use Measurement Techniques and Tools Systems

(organizer)

4. Derive formulas for surface area and volume and justify them using geometric models and common materials. For example, find:

a. the surface area of a cylinder as a function of its height and radius;b. that the volume of a pyramid (or cone) is one-third of the volume of a prism (or

cylinder) with the same base area and height.

Indicator 5 Benchmarks 8-10 Use Measurement Techniques and Tools Systems

(organizer)

5. Determine surface area for pyramids by analyzing their parts.


2011-2012



2011-1012


F2B: Use cross sections of prisms, cylinders, pyramids, and cones to solve volume problems.



2011-2012


F2C: Find the surface area and volume of a sphere in mathematical and real-world settings.

Measurement

Indicator 9 Benchmarks 8-10 Use Measurement Techniques and Tools (organizer)

9. Demonstrate understanding of the concepts of perimeter, circumference and area by using established formulas for triangles, quadrilaterals, and circles to determine the surface area and volume of prisms, pyramids, cylinders, spheres, and cones. (Note: only volume should be calculated for spheres and cones.)

Indicator 10 Benchmarks 8-10 Use Measurement Techniques and Tools (organizer)

10. Use conventional formulas to find the surface area and volume of prisms, pyramids and cylinders and the volume of spheres and cones to a specified level of precision.



2011-1012

Common Core

G-GMD #1 Give an informal argument for the formulas for the circumference of a circle, area of a circle, and volume of a cylinder, pyramid and cone.

Performance skill: Use dissection arguments Cavalieri’s principle and informal limit arguments.

G-GMD #2 Give an informal argument using Cavalieri’s principle for the formulas for the volume of a sphere and other sold figures.

G-MG #3 Apply geometric methods to solve design problems.

Performance skills: Design an object or structure to satisfy physical constraints or minimize cost.

Work with typographic grid systems based on ratios.



2011-1012

ACT: Relating geometric ideas to the coordinate plane-coordinate geometry

G1A: Use slope to distinguish between and write equations for parallel and perpendicular lines.

G1B: apply the midpoint and distance formulas to points and segments to find midpoints, distances, and missing information.

G1C: Use coordinate geometry to solve problems about geometric figures (e.g., segments, triangles, quadrilaterals).

Performance skills: Solve problems about segments, triangles, and quadrilaterals.

G1D: Write equations for circles in standard form and solve problems using equations and graphs.



2011-2012

ACT: Relating geometric ideas to the coordinate plane-coordinate geometry

G1E: Determine the effect of reflections, rotations, translations, and dilations and their compositions on the coordinate plane.

Geometry and Spatial Standard

Indicator 5 Benchmarks 8-10 Transformation and Symmetry (organizer)

5. Draw the results of translations, reflections, rotations and dilation of objects in the coordinate plane, and determine properties that remain fixed; e.g., lengths of sides remain the same under translations.



2011-2012

ACT: Investigating and applying basic ideas of trigonometry-introduction to trigonometry

H1A: Apply properties of 45◦ - 45◦ - 90◦ and 30◦ - 60◦ - 90◦ triangles to determine lengths of sides of triangle.

H1B: Find the sine, cosine, and tangent ratios of acute angles given the side lengths of right triangles.

H1C: Use trigonometric rations to find the sides or angles of right triangles and to solve real-world problems (e.g., use angles of elevation and depression to find missing measures).



2011-2012

Common Core Standards

G-SRT #7 Explain and use the relationship between the sine and cosine of complementary angles.

G-SRT #9 Derive the formula A = ½ ab sin (c) for the area of a triangle by drawing an auxiliary line from a vertex perpendicular to the opposite side

G-SRT #10 Prove the Law of Sines and Cosines and use them to solve problems.

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.

Examples: surveying problems, resultant forces

Performance skills: Example: Use angles of elevation and depression to find missing measures.

Geometry Level 8


Course of Study2011-1012

Measurement

Measurement Units

1. Compare and order the relative size of common U.S. customary units and metric units; e.g., mile and kilometer, gallon and liter, pound and kilogram.

2. Use proportional relationships and formulas to convert units from one measurement system to another; e.g., degrees Fahrenheit to degrees Celsius.

Use Measurement Techniques and Tools

3. Use appropriate levels of precision when calculating with measurements

4. Derive formulas for surface area and volume and justify them using geometric models and common materials. For example, find:

a. the surface area of a cylinder as a function of its height and radius;b. that the volume of a pyramid (or cone) is one-third of the volume of a prism

(or a cylinder) with the same base area and height.

5. Determine surface area for pyramids by analyzing their parts.

6. Solve and determine the reasonableness of the results for problems involving rates and derived measurements, such as velocity and density, using formulas, models and graphs.

7.Apply proportional reasoning to solve problems involving indirect measurements or rates.

10. Use conventional forumlas to find the surface area and volume of prisms, pyramids and cylinders and volume of spheres and cones to a specified level of precision.



2011-2012

Measurement (Continued)

Vocabulary

Scale DrawingsRight Triangle Trigonometry

* MEPCV




2011-1012

Geometry and Spatial Sense

Characteristics and Properties (organizer)1. Make and test conjectures about characteristics and properties (e.g., sides, angles, symmetry) of two-dimensional figures and three-dimensional objects.

2. Recognize the angles formed and the relationship between the angles when two lines intersect and when parallel lines are cut by a transversal.

3. Use proportions in several forms to solve problems involving similar figures (part-to-part, part-to-whole, corresponding sides between figures).

Special relationships (organizer)4. Represent and analyze shapes using coordinate geometry; e.g., given three

vertices and the type of quadrilateral, find the coordinates of the fourth vertex

Visualization and geometric models (organizer)6. Draw nets for a variety of prisms, pyramids, cylinders and cones.



2011-1012

Geometry and Spatial Sense (Continued)

Vocabulary

Slope FormulasDistance FormulasTrigonometric RatiosSineCosineTangent

* MEPCV

MEPCV – Maintain and Enhance Previous Content Vocabulary – Previous Content Vocabulary is now enhanced to the current grade appropriate indicators.



2011-2012

Mathematics Course of StudyPatterns, Functions, and Algebra

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 BenchmarksBy the end of 8-10 program:

A. Generalize and explain patterns and sequences in order to find the next term and the nth term.

B. Identify and classify functions as linear or nonlinear, and contrast their properties using tables, graphs or equations.

C. Translate information from one representation (words, table, graph or equation) to another representation of a relation or function.

D. Use algebraic representations, such as tables, graphs, expressions, functions and inequalities, to model and solve problem situations.

E. Analyze and compare functions and their graphs using attributes, such as rates of change, intercepts and zeros.

F. Solve and graph linear equations and inequalities.

G. Solve quadratic equations with real roots by graphing, formula and factoring.

H. Solve systems of linear equations involving two variables graphically and symbolically. I. Model and solve problem situations involving direct and inverse variation. J. Describe and interpret rates of change from graphical and numerical data.




2011-2012

Patterns, Functions, and Algebra

Use patterns, relations, and functions (organizer)1. Relate the various representations of a relationship; i.e., relate a table to graph,

description and symbolic form.

2. Generalize patterns and sequences by describing how to find the nth term.

3. Identify functions as linear or nonlinear based on information given in a table, graph or equation.

Use algebraic representations (organizer)4. Extend the uses of variables to include covariants where y depends on x.

5. Use physical models to add and subtract monomials and polynomials, and to multiply a polynomial by a monomial.

6. Describe the relationship between the graph of a line and its equation, including being able to explain the meaning of slope as a constant rate of change and y-intercept in real-world problems.

7. Use symbolic algebra (equations and inequalities), graphs and tables to represent situations and solve problems.

8. Write, simplify and evaluate algebraic expressions (including formulas) to generalize situations and solve problems.

9. Solve linear equations and inequalities graphically, symbolically and using technology.

10. Solve 2 by 2 systems of linear equations graphically and by simple substitution.

11. Interpret the meaning of the solution of a 2 by 2 system of equations; i.e., point, line, no solution.

12. Solve simple quadratic equations graphically; e.g., y = x2 – 16.

13. Compute and interpret slope, midpoint and distance given a set of ordered pairs.



2011-1012

Analyze Change (organizer)14. Differentiate and explain types of changes in mathematical relationships, such as linear

vs. nonlinear, continuous vs. noncontinuous, direct variation vs. inverse variation.

15.Describe and compare how changes in an equation affects the related graphs; e.g., for a linear equation changing the coefficient of x affects the slope and changing the constant affects the intercepts.

16. Use graphing calculators or computers to analyze change; e.g., interest compounded over time as a nonlinear growth pattern.



2011-2012

Patterns, Functions, and Algebra (Continued)

Vocabulary

Functions/Relationships-LinearQuadraticExponential

Freely TranslateDomainRangeFunctionTabularExponential Growth and DecayQuadratic Formula

* MEPCV




2011-2012

Mathematics Course of StudyData Analysis and Probability

Standard

Students pose questions and collect, organize, represent, interpret and analyze data to answer those questions. Students develop and evaluate inferences, predictions and arguments.


A. Create, interpret and use graphical displays and statistical measures to describe data; e.g., box-and-whisker plots, histograms, scatterplots, measures of center and variability.

B. Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose.

C. Compare the characteristics of the mean, median and mode for a given set of data, and explain which measure of center best represents the data.

D. Find, use and interpret measures of center and spread, such as mean and quartiles, and use those measures to compare and draw conclusions about sets of data.

E. Evaluate the validity of claims and predictions that are based on data by examining the appropriateness of the data collection and analysis.

F. Construct convincing arguments based on analysis of data and interpretation of graphs.

G. Describe sampling methods and analyze the effects of method chosen on how well the resulting sample represents the population.

H. Use counting techniques, such as permutations and combinations, to determine the total number of options and possible outcomes.

I. Design an experiment to test a theoretical probability, and record and explain results.

J. Compute probabilities of compound events, independent events, and simple dependent events.


K. Make predictions based on theoretical probabilities and experimental results.


2011-2012

Data Analysis and ProbabilityData Collection (organizer)

1. Use, create and interpret scatterplots and other types of graphs as appropriate.

2. Evaluate different graphical representations of the same data to determine which is the most appropriate representation for an identified purpose; e.g., line graph for change over time, circle graph for part-to-whole comparison, scatterplot for relationship between two variants.

3. Differentiate between discrete and continuous data and appropriate ways to represent each.

Statistical Methods (organizer)4. Compare two sets of data using measures of center (mean, mode, median) and measures

of spread (range, quartiles, interquartile range, percentiles).

5. Explain the mean's sensitivity to extremes and its use in comparison with the median and mode.

6. Make conjectures about possible relationship in a scatterplot and approximate line of best fit.

7. Identify different ways of selecting samples, such as survey response, random sample, representative sample and convenience sample.

8. Describe how the relative size of a sample compared to the target population affects the validity of predictions.

9. Construct convincing arguments based on analysis of data and interpretation of graphs.

Probability (organizer)

10. Calculate the number of possible outcomes for a situation, recognizing and accounting for when items may occur more than once or when order is important.

11. Demonstrate an understanding that the probability of either of two disjoint events occurring can be found by adding the probabilities for each and that the probability of one independent event following another can be found by multiplying the probabilities.



2011-2012

Data Analysis and Probability (Continued)

Vocabulary

Quantitative/QualitativeSkewnessSampling MethodsFundamental Counting PrincipleSample SpaceSimple Dependent Events

* MEPCV




2011-2012

Mathematics Course of StudyMathematical Processes

Standard

Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques, and communicate mathematical ideas. The benchmarks for mathematical processes articulate what students should demonstrate in problem solving, representation, communication, reasoning, and connections at key points in their mathematics program.

K-12 Mathematics BenchmarksBy the end of the 8-10 program:

A. Formulate a problem or mathematical model in response to a specific need or situation, determine information required to solve the problem, choose method for obtaining this information, and set limits for acceptable solution.

B. Apply mathematical knowledge and skills routinely in other content areas and practical situations.

C Recognize and use connections between equivalent representations and related procedures for a mathematical concept; e.g., zero of a function and the x-intercept of the graph of the function, apply proportional thinking when measuring, describing functions, and comparing probabilities.

D. Apply reasoning processes and skills to construct logical verifications or counter-examples to test conjectures and to justify and defend algorithms and solutions.

E. Use a variety of mathematical representations flexibly and appropriately to organize, record and communicate mathematical ideas.

F. Use precise mathematical language and notations to represent problem situations and mathematical ideas.

G. Write clearly and coherently about mathematical thinking and ideas.

H. Locate and interpret mathematical information accurately, and communicate ideas, processes and solutions in a complete and easily understood manner.



2011-2012

Mathematical Processes (Continued)Grade Level Indicators

Students use mathematical processes and knowledge to solve problems. Students apply problem-solving and decision-making techniques and communicate mathematical ideas. The benchmarks for mathe-matical processes articulate what students should demonstrate in problem solving, representation, communication, reasoning and connections at key points in their mathematics program.

Vocabulary/Strategies

Compare: to determine how two things are alike and/or different; the common/critical attributes must be identified.

Compare is involved in ALL of the following:

Analyze: to investigate by breaking it down so as to more clearly understand the impact to the situation.

Describe: to analyze into its parts but less detailed than explain.

Determine: to reach a decision after a thorough investigation; to find the cause of and then to solve or set limits to a situation.

Identify: to show or prove the sameness of.

Interpret: a student must 1st analyze and then make an inference; this is more subjective than an evaluation.

Predict: to state what one believes will happen (based on data).

Recognize: to examine closely and identify the common/critical attributes.

Other Stated Verbs in the Indicators:Justify ApplyOrder GeneralizeExplain FindDemonstrate SimplifyEstimate ModelConvert ClassifyUse CreateSolve Define


Documents

Number and Number Systems - Perry Localperrylocal.org/curriculum/files/2010/12/geometry-Level-8.doc · Web viewG. Estimate, compute and solve problems involving real numbers, including