Number and Number Systems - Perry Local



Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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:

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.

No Grade Level Indicators for Benchmarks D, E

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Number, Number Sense and Operations (Continued)

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.

8.EE3 Use numbers expressed in the form of a single digit times an integer power of 10 to estimate very large or very small quantitites, and to express how many times as much one is than the other.

8.EE4 Perform operations with numbers expressed in scientific notation, including problems where both decimal and scientific notation are used.

Use scientific notation and choose units of appropriate size for measurements of very large or very small quantities.

Interpret scientific notation that has been generated by technology.

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.

▪ Convert a small number in standard form to scientific notation

▪ Estimate the population of the United States as 3 x 108 .

▪

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Number, Number Sense and Operations (Continued)

Indicator 2 Benchmarks 5-7

▪ Number and Number Systems (organizer)

2. Explain the meaning of exponents that are negative or 0.

Performance Skills:

▪ Explain that a power with a negative exponent (3-2) simplifies to one over that power with the exponent changed to a positive exponent (1/32).

Example: 3-2 = 1/32, which can simplify to 1/9.

▪ Identify that any power with an exponent of zero is equal to one.

Example: 50 = 1, 90 = 1, 1000 = 1, (-5)0 = 1

3. Simplify expressions with integer exponents.

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

Ex. 43 ∙ 43 or (42)3

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Number, Number Sense and Operations

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.

8.NS1 Know that numbers that are not rational are called irrational; understand informally that every number has a decimal expansion; for rational numbers show that the decimal expansion repeats eventually, and convert a decimal expansion which repeats eventually into a rational number.

8.NS2 Use rational approximations of irrational numbers to compare the size of irrational numbers, locate them approximately on a number line diagram, and estimate the value of expressions.

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.

▪ Truncate the decimal expansion of to show that is between 1 and 2, then between 1.4 and 1.5, and explain how to continue on to get better approximations.

Imaginary

N numbers

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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

Not 43 ∙ 43 or (42)3

▪ Simplify a numerical expression involving integers, fractions, and decimals following order of operations.

▪ Simplify numerical expression involving integers, fractions, and decimals using properties

including:

|Example: Addition Property | Multiplication Property |

|Commutative a + b = b + a |ab = ba |

|Associative (a + b) + c = a (b + c) |(a x b) x c = a x (b x c) |

|Identity a + 0 = a |a x 1 = a |

|Inverse a + (-a) = 0 |a x 1/a = 1 |

|Distributive Property | |

| | |

|a (b + c) = ab + ac | |

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Number, Number Sense and Operations (Continued)

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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

▪ Explain through the use of examples, through the use of number line, or counters the effect of computing with integers.

a. Addition and multiplication of two integers can result in a lesser value.

Example: 4 + (-3) = 1 5 · (-2) = (-10)

b. Subtraction and division of two integers can result in a greater value.

Example: -4 – (-3) = (-1) (-8) ∻ (-4) = 2

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Number, Number Sense and Operations (Continued)

Indicator 6 Benchmarks 5-7 Benchmark I

▪ Computation and Estimation (organizer)

5. Simplify numerical expressions involving integers and use integers to solve real-life problems.

Performance Skills:

▪ Simplify numerical expressions involving integers.

▪ Use integers to solve real-life problems.

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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

▪ Use appropriate vocabulary to compare estimates with actual answer.

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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.

▪ Convert between fraction, decimal, or percent.

▪ Solve problems using the appropriate form of a rational number (fraction, decimal, percent).

▪ **For example: If a woman making $25 an hour gets a 10% raise, she will make an additional 1/10 of her salary an hour, or $2.50, for a new salary of $27.50. If you want to place a towel bar 9 3/4 inches long in the center of a door that is 27 1/2 inches wide, you will need to place the bar about 9 inches from each edge; this estimate can be used as a check on the exact computation.

▪

Pre-Algebra 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.

8.EE Use square root and cube root symbols to represent solutions to equations of the form x2=p and x3=p, where p is a positive rational number.

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

▪ Approximate the square root of a non-perfect square as consecutive integers between which the root lies.

▪ Evaluate square roots of small perfect squares and cube roots of small perfect cubes.

▪ Know that √2 is irrational

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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.

Pre-Algebra Sixth-Seven

2011-2012

Number, Number Sense and Operations (Continued)

Vocabulary

Properties

Integers

Absolute Value

Square Roots

Appropriate Form of a Rational Number

Scientific Notation

Rational

Irrational

Terminating

Repeating

Real Numbers

Numerical Expressions

Inverse

Algorithm

Identity Property

Part (Percentage)

Rate

Base

Exponential Form

Standard Notation

Radical

Perfect Square

Non-Perfect Square

* MEPCV

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

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

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.

Pre-Algebra Sixth-Seven

2011-2012

Measurement (Continued)

Indicator 1 Benchmarks 8-10 Benchmark D

▪ Measurement Units (organizer)

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.

a. Compute unit rates associated with ratios of fractions, including ratios of lengths, areas and other quantities measured in like or different units.

Performance Skills:

▪ Select appropriate units for measuring derived measurements in a problem situation.

▪ Label answer with correct derived measurement (e.g., mph, rpm, and feet per second).

▪ Compare and order the relative size of common U.S. customary units and metric units.

Eg.

Mile greater than kilometer.

Gallon greater than liter.

Pound less than kilogram.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 2 Benchmarks 8-10 Benchmark D

▪ Measurement Units (organizer)

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

Performance Skill:

▪ Convert/change units from one measurement system to another through various methods.

▪ Explain the difference between using scale factor and cross products when solving a proportional reasoning.

▪ Differentiate when best to use scale factor or cross products in proportional relationships and scale factors.

Examples

Proportional relationship

Formulas

Conversion charts

Pre-Algebra Sixth-Seven

2011-2012

Measurement (Continued)

Indicator 3 Benchmarks 8-10 Benchmark B, E

▪ Use Measurement Techniques and Tools (organizer)

3. Use appropriate levels of precision when calculating with measurements.

Performance Skills:

▪ Find least precise measurement in problem calculation.

▪ Apply appropriate level of precision when calculating with measurement.

Eg. 6.31m + 5.447m + 2.8m = 14.557

14.6

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 4 Benchmarks 8-10 Benchmark B

▪ Use Measurement Techniques and Tools (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.

Performance Skills:

▪ Derive and justify the formula for surface area of a cylinder.

Eg. Find surface area of a cylinder by finding areas of bases and wrapping cylinder with paper which produces a rectangle.

▪ Derive and justify the formula for a volume of a pyramid or cone by using geometric models or common materials.

Eg. Use cylinder and cone with same base and height and fill each with substance and compare quantities.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 5 Benchmarks 8-10 Benchmark A, C

▪ Use Measurement Techniques and Tools (organizer)

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

Performance Skills:

▪ Identify the shape of each face of the pyramid.

▪ Identify the relationship of the measurements.

Eg. Triangle base is equal to length of square.

▪ Determine the area of the faces of a pyramid and add to determine surface area.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 5 Benchmarks 8-10 Benchmark A, C

▪ Use Measurement Techniques and Tools (organizer)

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

Performance Skills:

▪ Identify the shape of each face of the pyramid.

▪ Identify the relationship of the measurements.

Eg. Triangle base is equal to length of square.

▪ Determine the area of the faces of a pyramid and add to determine surface area.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 6 Benchmarks 8-10 Benchmark A, F

▪ Use Measurement Techniques and Tools (organizer)

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.

Performance Skills:

▪ Explain labeling of a rate eg per means division.

▪ Solve problems involving rates and derived measurements using formulas, models, and graphs.

Eg.

Velocity

Density

Slope of a line is average rate of change

Roller coaster

Car used to find rate

▪ Determine reasonableness of results from problem.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 7 Benchmarks 8-10 Benchmark D

▪ Use Measurement Techniques and Tools (organizer)

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

Performance Skill:

▪ Apply proportional reasoning to solve problems involving indirect measurement of rates.

Eg.

6 Ft.

X

22 4 Ft.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 8 Benchmarks 8-10 Benchmark E

▪ Use Measurement Techniques and Tools (organizer)

8. Find the sum of the interior and exterior angles of regular convex polygons with and without measuring the angles with a protractor.

Performance Skills:

▪ Find the sum of the interior and exterior angles of a regular convex polygon with a protractor.

▪ Find the sum of the interior and exterior angles of a regular convex polygon with formulas.

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 9 Benchmarks 8-10 Benchmark C

▪ 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.)

Performance Skills:

▪ Find the area of each face for prisms, pyramids, cylinders, spheres, and cones.

▪ To find surface area, add each face together.

Find volume of prisms and pyramids by multiplying area of the base by height

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Indicator 10 Benchmarks 8-10 Benchmark E

▪ Use Measurement Techniques and Tools (organizer)

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

8.G9 Know the formulas for the volumes of cones, cylinders, and spheres and use them to solve real-world and mathematical problems.

Performance Skill:

▪ Use formulas to solve problems with surface area and volume.

Eg.

Prisms, pyramids and cylinders (surface area)

Spheres, cones, prisms, pyramids and cylinders (volume)

Pre-Algebra Level 8

2011-2012

Course of Study

Measurement (Continued)

Vocabulary

Proportional Reasoning

Reference Table

Square Feet

Square Yards

Cubic Meters

Cubic Centimeters

Area of Trapezoids

Volume of Cylinders/Prisms

Composite Shapes

Area of Triangles/Parallelograms

Proportional Relationships

Scale Factors

Unit Conversions

Rate

Unit Rate

Cross Products

Prisms

Sector

Net

Alterations

Faces

Vertex

Edge

Velocity

Density

Quadrilaterals

Spheres

Interior/exterior angle of convex polygons

Mile/kilometer

Liter/gallon

Appropriate levels of precision

* MEPCV

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

Pre-Algebra Level 8

2011-2012

Course of Study

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 the 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.

No Grade Level Indicators for Benchmarks A, G, I

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 1 Benchmarks 8-10 Benchmarks B, D, H

▪ 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.

Performance Skill:

▪ Provide examples and/or counter-examples for grouping two dimentional figures and three dimensional objects based on their characteristics.

Eg.

Quadrilaterals, triangles, cubes

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 2 Benchmarks 8-10 Benchmark C

▪ Characteristics and Properties (organizer)

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

8.G5 Use informal arguments to establish facts about the angle sum and exterior angle of triangles, about the angles created when parallel lines are cut by a transversal, and the angle-angle criterion for similarity of triangles.

Performance Skills:

▪ Identify supplementary, complementary, adjacent, vertical angles and their relationship when lines intersect.

▪ Define transversal.

▪ Identify alternate interior angles, alternate exterior angles, consecutive interior angles, corresponding angles and their relationship when two parallel lines are crossed by a transversal.

▪ Arrange three copies of the same triangle so that the sum of the three angles appears to form a line, and give an argument in terms of transversals why this is so.

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 3 Benchmarks 8-10 Benchmark B

▪ Characteristics and Properties (organizer)

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

Performance Skill:

▪ Use proportions to solve problems involving similar figures.

Eg.

Part to part

Part to whole

Corresponding sides between figures

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 4 Benchmarks 8-10 Benchmark D

▪ Spatial 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.

.

Performance Skills:

▪ Use coordinate geometry to represent shapes.

Eg.

Given one point, draw a square on coordinate plane.

▪ Analyze shapes using coordinate geometry.

Eg.

Given

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 5 Benchmarks 8-10 Benchmark F

▪ Transformations and Symmetry (organizer)

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

8.G1 Verify experimentally the properties or rotations, reflections, and translations:

a.Lines are taken to lines, and line segments to line segments of the same length.

b.Angles are taken to angles of the same measure.

c. Parallel lines are taken to parallel lines.

8.G2 Understand that a two-dimensional figure is congruent to another if the second can be obtained from the first by a sequence of rotations, reflections, and translations; given two congruent figures, describe a sequence that exhibitis the congruence between them.

8.G3 Describe the effect of dilations, translations, rotations, and reflections on two-dimensional figures using coordinates.

8. G4 Understand that a two-dimensional figure is similar to another if the second can be obtained from the first by a sequence of rotations, reflections, translations, and dilations; given two similar two dimensional figures, describe a sequence that exhibits the similariy between them.

Performance Skills:

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Indicator 6 Benchmarks 8-10 Benchmarks E

▪ Visualization and Geometric Models (organizer)

6. Draw nets for a variety of prisms, pyramids, cylinders and cones.

Performance Skills:

▪ Predict the three dimensional object when given a net (two dimensional).

▪ Confirm prediction with folding of the net.

▪ Draw nets for a variety of prisms, pyramids, cylinders, and cones.

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

8.SP 6

▪ Understand and apply Pythagorean Theorem.

6. Explain a proof of the Pythagorean Theorem and its converse.

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

8.SP 7

▪ Understand and apply Pythagorean Theorem.

7. Apply the Pythagorean Theorem to determine unknown side lenghts in right triangles in real-world and mathematical problems in two and three dimensions.

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

8.SP 8

▪ Understand and apply Pythagorean Theorem.

8. Apply the Pythagorean Theorem to find the distance between two points in a coordinate system.

Pre-Algebra Level 8

2011-2012

Course of Study

Geometry and Spatial Sense (Continued)

Vocabulary

Conditions

Rotation Symmetries

Pythagorean Theorem

Triangle Angle Sum Relationships

Corresponding

Congruent

Attributes

Supplementary

Complementary

Rotational Symmetry

Line Symmetry

Transformations

- TranslationsReflections

- Rotations

- Dilations

Transversal

Coordinate Geometry

* MEPCV

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

Pre-Algebra Level 8

2011-2012

Mathematics Course of Study

Patterns, 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 Benchmarks

By the end of the 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.

Pre-Algebra level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 1 Benchmarks 8-10 Benchmarks C

▪ Use Patterns, Relations and Function (organizer)

1. Relate the various representations of a relationship; i.e., relate a table to graph, description and symbolic form.

Performance Skills:

▪ Demonstrate the ability to plot points from a table to a graph and draw the corresponding line.

▪ Explain the meaning of points/line on a graph for real world situations.

Eg.

Linear

Scatterplots

Quadratic

Pre-Algebra level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 2 Benchmarks 8-10 Benchmark A

▪ Use Patterns, Relations and Function (organizer)

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

Analyze a pattern to find the nth term

Performance Skills

▪ Recognize a continuous relationship as a pattern.

▪ Given a pattern, create a rule and/or variable expression.

▪ Given a rule, create a pattern and/or variable expression.

▪ Given a function, create a table, rule (words), and/or graph.

▪ Create an equation to find the nth term using patterns and sequences.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 3 Benchmarks 8-10 Benchmarks B

▪ Use Patterns, Relations and Function (organizer)

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

8.EE7 Solve linear equations in one variable.

a. Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions.

b. Solve linear equations with rational number coefficients, including equations whose solutions require expanding expressions using the distributive property and collecting like terms.

8.EE8 Analyze and solve pairs of simultaneous linear equations.

a. Understand that solutions to a system of two linear equaitons in two variables correspond to points of intersectin of their graphs, because points of intersectin satisfy both equations.

b. Solve systems of two linear equations in two variable algebraically, and estimate solutions by graphing the equations.

c. Solve real-world and mathematical problems leading to two linear equations in two variables.

Performance Skills:

▪ Identify functions as linear or non-linear based on information.

- Analyze a table and determine if it represents a linear/non-linear function based on constant change in x and y.

- Analyze a graph and determine if it represents a linear/non-linear function based on appearance.

- Analyze an equation and determine if it represents a linear/non-linear function based on form.

▪ Represent linear and nonlinear equations by plotting points in the coordinate plane.

▪ Plot points on the coordinate plane given a table of values or an equation.

▪ 3x + 2y =5 and 3x + 2y =6 have no solutions because 3x + 2y cannot simultaneously be 5 and 6.

▪ Given two coordinates for two pairs of points, determine whether the line through the first pair of points intersects the line through the second pair.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 4 Benchmarks 8-10 Benchmarks D

▪ Use Algebraic Representations (organizer)

4. Extend the uses of variables to include covariants where y depends on x.

Performance Skills:

▪ Compare and contrast dependent/independent variables.

▪ Solve an equation for y in an equation.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 5 Benchmarks 8-10 Benchmarks D

▪ Use Algebraic Representations (organizer)

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

Add, subtract, and multiply polynomials.

Performance Skills:

▪ Use physical models to add and subtract monomials and polynomials.

▪ Use physical models to multiply a polynomial by a monomial.

- Algebra tiles.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 6 Benchmarks 8-10 Benchmarks E

▪ Use Algebraic Representations (organizer)

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.

Performance Skills:

▪ Graph an equation using slope-intercept form.

▪ Describe how slope affects the direction and steepness of a line.

Eg.

Positive slope

Negative slope

Zero slope

Undefined (no) slope ↨

▪ Interpret slope as a constant rate of change in real world situations.

▪ Interpret meaning of y-intercept of real-world situations.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 7 Benchmarks 8-10 Benchmark D, F

▪ Use Algebraic Representations (organizer)

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

8.EE5 Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

8.EE6 Use similar triangles to explain why the slope m is the same between any two distinct points on a non-vertical line in the coordinate plane;

Performance Skills:

▪ Develop an equation/inequality to represent and solve a real-world situation.

▪ Use graphs and tables to represent and solve real-world situations.

▪ Compare a distance-time graph to a distance-time equation to determine which of two moving objects has greater speed.

▪ Derive the equation y=mx for a line through the origin and equation y=mx + b for a line intercepting the vertical axis at b.

▪ Represent inequalities on a number line or a coordinate plane.

Example: x > 3

2 3 4 5

Example: x > 3

1 2 3

**For example: As a salesperson, you are paid $50 per week plus $3 per sale. This week you want your pay to be at least $100. Write an inequality for the number of sales you need to make, and describe the solutions.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 8 Benchmarks 8-10 Benchmark D

▪ Use Algebraic Representations (organizer)

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

Performance Skills:

▪ Recognize conventional ways to write algebraic expressions.

▪ Reduce algebraic expressions into simplest terms.

▪ Write, simplify, and evaluate an algebraic expression to generalize a situation.

▪ Use a given formula to solve problem/situations.

▪ Use formulas in problem-solving situations.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 9 Benchmarks 8-10 Benchmark F

▪ Use Algebraic Representations (organizer)

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

Performance Skills:

▪ Keep the same.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 10 Benchmarks 8-10 Benchmark H

▪ Use Algebraic Representations (organizer)

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

Performance Skills:

▪ Define 2 by 2 systems of linear equations.

▪ Solve 2 by 2 systems of linear equations graphically and by substitution.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 11 Benchmarks 8-10 Benchmark H

▪ Use Algebraic Representations (organizer)

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

Performance Skills:

Same as above

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 12 Benchmarks 8-10 Benchmark G

▪ Use Algebraic Representations (organizer)

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

Performance Skills:

▪ Define/identify quadratic eaquations.

▪ Solve simple quadratic equations graphically.

Eg.

Set up a table of values to find points.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 13 Benchmarks 8-10 Benchmark J

▪ Use Algebraic Representations (organizer)

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

Use slope-intercept form in graphing a linear equation.

Performance Skill:

▪ Define slope, midpoint, distance.

- Use formulas to find slope, midpoint, distance from given points.

▪ Recognize slope as the ratio that describes the tilt of a line.

▪ Calculate the slope of a line.

▪ Recognize the y-intercept of a line is the point at which the line crosses the y-axis

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 14 Benchmarks 8-10 Benchmark I

▪ 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.

Performance Skill:

▪ Differentiate and explain types of changes in mathematical relationships such as:

- Linear vs. non-linear.

- Continuous vs. non-continuous.

- Direct variation vs. inverse variation.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

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.

Performance Skill:

▪ Describe and compare how changes in an equation in slope intercept form affect the graph.

Eg.

Change in coefficient of x → slope

Change in constant → y - int

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Indicator 16 Benchmarks 8-10 Benchmark J

▪ Analyze Change (organizer)

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

Performance Skills:

▪ Same as above.

Pre-Algebra Level 8

2011-2012

Course of Study

Patterns, Functions, and Algebra (Continued)

Vocabulary

Simple Variable Expressions

Linear/Nonlinear Progressions

Points in the Coordinate Plane

Simplified

Inverse Operations

Distance-Time Relationships

Function

Term

Coefficient

Constant

Arithmetic Sequence/Progression

Geometric Sequence/Progression

Inequality

Like Terms

Variable

nth Term

Covariants

Monomials

Polynomials

Symbolic Algebra (Equations + Inequalities)

Slope

y-Intercept

Simple Quadratic Equations

Non-Continuous

Direct/Inverse Variation

Slope

Distance

Ordered Pairs

Coefficient

Interest Compounded

Nonlinear Growth Pattern

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

Pre-Algebra Level 8

2011-2012

Course of Study

Data 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.

Mathematics Benchmarks

By the end of the 8-10 program:

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.

No Grade Level Indicators for Benchmarks K

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 1 Benchmarks 8-10 Benchmark A

▪ Data Collection (organizer)

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

Performance Skill:

Use, create, and interpret scatterplots and graphs

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 2 Benchmarks 8-10 Benchmark B

▪ Data Collection (organizer)

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.

Performance Skill:

▪ Determine which graphical representation is the most appropriate for an identified purpose.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 3 Benchmarks 8-10 Benchmark B

▪ Data Collection (organizer)

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

Performance Skill:

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

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 4 Benchmarks 8-10 Benchmark D

▪ 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).

Performance Skills:

▪ Calculate and analyze the most appropriate measure(s) of central tendency (mean, median, mode) for a given data.

▪ Calculate and compare measures of spread (range, quartile, interquartile range).

▪ Evaluate the effect of an outlier on the above measures.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 5 Benchmarks 8-10 Benchmark C

▪ Statistical Methods (organizer)

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

Performance Skill:

▪ Compare the impact of extremes on mean, median, and mode.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 6 Benchmarks 8-10 Benchmark F

▪ Statistical Methods (organizer)

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

a. Construct and interpret scatter plots for bivariate measurement data to investigate patterns of association between two quantities.

b. Describe patterns such as clustering, outliers, positive or negative association, linear association, and nonlinear association.

c. Know that straight lines are widely used to model relationships between two quantitative variables.

Performance Skill:

▪ Discuss the relationship of points in a scatterplot and approximate line of best fit.

▪ Explain that for scatter plots that suggest a linear association, informally fit a straight line, and formally assess the model fit by judging the closeness of the data points to the line.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 7 Benchmarks 8-10 Benchmark G

▪ Statistical Methods (organizer)

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

Performance Skill:

▪ Identify different ways of selecting samples.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 8 Benchmarks 8-10 Benchmark E

▪ Statistical Methods (organizer)

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

Performance Skill:

▪ Compare data from two or more samples to determine how sample selection can influence results.

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

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Indicator 9 Benchmarks 8-10 Benchmark F

▪ Statistical Methods (organizer)

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

Performance Skill:

▪ Analyze data and substantiate interpretation of graphs.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

8.SP 3

▪ Statistics and Probability

Use the equation of a linear model to solve problems in the context of bivariate measurement data, interpreting the slope and intercept.

Performance Skills:

▪ In linear model for a biology experiment, interpret a slope of 1.5 cm/hr as meaning that an additional hour of sunlight each day is associated with an additioanl 1.5 cm in mature plant height.

Pre-Algebra Level 8

2011-2012

Course of Study

Data Analysis and Probability (Continued)

8.SP 4

▪ Statistics and Probability

Understand that patterns of association can also be seen in bivariate categorical data by displyain frequencies and relative frequencies in a two-way table.

Construct and interpret a two-way table summarizing data on two categorical variables collected from the same subjects.

Use relative frequencies calculated for rows or columns to describe possible association between the two variables.

Performance Skills:

▪ Collect data from students in your class on whether or not they have a curfew on schol nights and whether or not they have assigned chores at home. Is there evidence that those who have a curfew also tend to have chores?

Pre-Algebra Level 9

2011-2012

Course of Study

Data Analysis and Probability (Continued)

Vocabulary

Box-and-Whisker Plots

Stem-and-Leaf Plots

Measure of Spread-

Quartile

Interquartile

Outliers (Inclusion or Exclusion of)

Misuses of Statistical Data

Probabilities

Compound Events

Tree Diagrams

Area Models

Scale

Interval

Range

Central Tendency

Samples

Sample Selection

Sample Space

Theoretical Probability

Experimental Probability

Extremes

* MEPCV

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

Pre-Algebra Level 8

2011-2012

Course of Study

Math Processes

The Math Processes Benchmarks should be embedded in daily mathematics teaching. There are no specific grade level indicators for the Math Processes Benchmarks.

K-12 Mathematics Benchmarks

By the end of the 5-7 program:

A. Clarify problem-solving situation and identify potential solution processes; e.g., consider different strategies and approaches to a problem, restate problem from various perspectives.

B. Apply and adapt problem-solving strategies to solve a variety of problems, including unfamiliar and non-routine problem situations.

C. Use more than one strategy to solve a problem, and recognize there are advantages associated with various methods.

D. Recognize whether an estimate or an exact solution is appropriate for a given problem situation.

E. Use deductive thinking to construct informal arguments to support reasoning and to justify solutions to problems.

F. Use inductive thinking to generalize a pattern of observations for particular cases, make conjectures, and provide supporting arguments for conjectures

G. Relate mathematical ideas to one another and to other content areas; e.g., use area models for adding fractions, interpret graphs in reading, science and social studies.

H. Use representations to organize and communicate mathematical thinking and problem solutions.

I. Select, apply, and translate among mathematical representations to solve problems; e.g., representing a number as a fraction, decimal or percent as appropriate for a problem.

J. Communicate mathematical thinking to others and analyze the mathematical thinking and strategies of others.

K. Recognize and use mathematical language and symbols when reading, writing and conversing with others.

Pre-Algebra Level 8

2011-2012

Course of Study

Math Processes (Continued)

Grade Level Indicators

Specific grade-level indicators have not been included for the mathematical processes standard because content and processes should be interconnected at the indicator level. Therefore, mathematical processes have been embedded within the grade-level indicators for the five content standards.

Vocabulary/Strategies

Problem-Solving Strategies

▪ Choose a method of computation

▪ Draw a diagram

▪ Guess and check

▪ Look for a pattern

▪ Make a list

▪ Make a model

▪ Solve a simple problem

▪ Use logical reasoning

▪ Work backwards

▪ Eliminate possibilities

▪ Reasonable answers

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).

Pre-Algebra Level 8

2011-2012

Course of Study

Math Processes (Continued)

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

Other Stated Verbs in the Indicators:

Demonstrated Apply

Use Perform

Simplify Draw

Solve Generalize

Develop Create

Represent Justify

Select Construct

Convert Compute

Estimate

-----------------------

Real

Numbers

Irrational

Rational

N

W

Integers

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