Eighth Grade

Number and Operation

Content Standard 1.0 The student will develop number and operation sense needed to represent numbers and number relationships verbally, symbolically, and graphically and to compute fluently and make reasonable estimates in problem solving.

Learning Expectations:

Accomplishments

8.1.1 Understand numbers, ways of representing numbers, relationships among numbers, and number systems.

1. recognize the place value of a given digit;
2. use exponents to express a monomial written in expanded form;
3. determine the square root of perfect squares;
4. use a variety of models to demonstrate the relationships within the real number system (e.g., Venn diagrams, webs);
5. work flexibly with fractions, decimals, and percents to solve problems;
6. compare and order fractions, decimals, and percents;
7. connect whole numbers, fractions, decimals, percents and integers to locations on the number line;
8. develop meaning for percents greater than 100 and less than one;
9. use appropriate mathematical language and symbols to express numerical relationships (e.g., <, >, =, ³ );
10. understand and use ratios and proportions to represent quantitative relationships;
11. recognize and use exponential, scientific, and calculator notation to represent large numbers in real-world situations;
12. demonstrate understanding of opposites, reciprocals, and integers;
13. identify the opposite of a rational number;
14. use concrete, pictorial, and symbolic representations of integers;
15. apply number theory concepts to solve problems (e.g., divisibility, factors, multiples, composite numbers, prime factorization, relatively prime).

8.1.2 Understand operations and how they relate to one another.

1. understand the meaning and effects of arithmetic operations with fractions, decimals, and integers;
2. apply the associative and commutative properties of addition and multiplication to simplify computations with integers, fractions, and decimals;
3. use the distributive property to simplify computations with integers, fractions, and decimals;
4. apply order of operations in computing with rational numbers;
5. understand and use the inverse relationships of addition and subtraction, multiplication and division, and squaring and finding square roots to simplify computations and solve problems.

8.1.3 Solve problems, compute fluently, and make reasonable estimates.

1. select and use appropriate methods and tools for computing with whole numbers, fractions, decimals, percents, and integers in problem-solving situations (e.g., mental computation, estimation, calculators, computers, paper and pencil);
2. develop and analyze procedures for computing with fractions, decimals, and integers;
3. develop and use strategies to estimate the results of rational number computations in real-world situations;
4. judge the reasonableness of the results of rational number estimates and computations;
5. solve one-step real-world problems involving whole numbers, fractions, decimals, and percents;
6. raise rational numbers to whole number powers;
7. solve multi-step real-world problems involving whole numbers, fractions, decimals, and percents;
8. develop, analyze, and explain methods for solving problems involving proportions (e.g., scaling, finding equivalent ratios);
9. calculate rates involving cost per unit to determine the best buy

Eighth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Algebra

Content Standard 2.0 The student will understand and generalize patterns as they represent and analyze quantitative relationships and change in a variety ofcontexts and problems using graphs, tables, and equations.

Learning Expectations:

Accomplishments

8.2.1 Understand patterns, relations, and functions.

1. represent, analyze, and generalize a variety of patterns with tables, graphs, words, and when possible symbolic rules;
2. develop understanding for arithmetic and geometric sequences;
3. relate and compare different forms of representation for a relationship;
4. identify functions as linear or nonlinear;
5. compare and contrast properties of functions from tables, graphs, or equations.

8.2.2 Represent and analyze mathematical situations and structures using algebraic symbols.

1. formulate multi-step equations that represent relationships and real-world situations;
2. develop meaning for intercept and slope;
3. use symbolic algebra to represent situations and solve problems;
4. use a variety of forms to represent linear relationships;
5. recognize and generate equivalent forms for simple algebraic expressions;
6. evaluate a first-degree algebraic expression given values for two or more variables;
7. solve one- and two-step linear equations involving integers;
8. use a variety of methods to solve real-world problems involving multi-step linear equations (e.g., manipulatives, technology, pencil and paper);
9. apply given formulas to solve real-world problems;
10. solve one-step linear inequalities;
11. identify the graph of a linear equation;
12. identify the graphical representation of the solution to a one-variable linear inequality;
13. develop understanding for particular values of patterns, relationships, and functions (e.g., x- and y- intercepts, slope, maximum and minimum values).

8.2.3 Use mathematical models to represent and understand quantitative relationships.

1. use a variety of representations to solve real-world problems (e.g., graphs, tables, equations).

8.2.4 Analyze change in various contexts.

1. compare linear relationships to non-linear relationships;
2. develop meaning for rate of change in real-world situations.

Eighth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Geometry

Content Standard 3.0 The student will develop an understanding of geometric concepts and relationships as the basis for geometric modeling and reasoning to solve problems involving one-, two-, and three-dimensional figures.

Learning Expectations:

Accomplishments

8.3.1 Analyze characteristics and properties of two- and three-dimensional geometric figures.

1. describe, classify, and understand relationships among types of two- and three-dimensional objects using their defining properties;
2. understand relationships among the angles (e.g., complementary, supplementary, interior, exterior, vertical, corresponding);
3. solve problems using angle relationships (e.g., complementary, supplementary, interior, exterior, vertical, corresponding);
4. determine the measure of an angle of a triangle given the measures of the other two angles;
5. understand relationships among the angles and side lengths of similar geometric figures;
6. determine congruence of line segments, angles, and polygons by direct comparison of given attributes;
7. develop understanding of the Pythagorean theorem.

8.3.2 Specify locations and describe spatial relationships using coordinate geometry and other representational systems.

1. graph points in the coordinate system.

8.3.3 Apply transformations and use symmetry to analyze mathematical situations.

1. describe sizes, positions, and orientations of shapes under transformations (e.g., rotations, translations, reflections, dilations);
2. relate symmetry and congruence to reflections about a line.

8.3.4 Use visualization, spatial reasoning, and geometric modeling to solve problems.

1. use appropriate tools and methods to draw geometric objects with specified properties, (e.g., side lengths, angle measure);
2. use two-dimensional representations of three-dimensional objects to visualize;
3. use visualization and spatial reasoning to solve real-world problems;
4. recognize and apply geometric ideas and relationships such as tessellations in areas outside the mathematics classroom (e.g., art, science, everyday life)

Eighth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Measurement

Content Standard 4.0 The student will become familiar with the units and processes of measurement in order to use a variety of tools, techniques, and formulas to determine and to estimate measurements in mathematical and real-world problems.

Learning Expectations:

Accomplishments

8.4.1 Understand measurable attributes of objects and the units, systems, and processes of measurement.

1. understand both metric and customary systems of measurement;
2. understand relationships among units and convert from one unit to another within the same system;
3. understand, select, and use units of appropriate size and type to measure angles, perimeter, areas, surface area, and volume.

8.4.2 Apply appropriate techniques, tools, and formulas to determine measurements.

1. use a variety of strategies to estimate length, perimeter, circumference, area, and volume;
2. select and apply techniques and tools to accurately measure length, perimeter, area, volume, and angles to appropriate levels of precision;
3. apply formulas and/or other strategies to solve problems involving perimeter, circumference of circles, and the area of triangles, parallelograms, trapezoids, and circles;
4. find area of complex and irregular shapes;
5. apply given formulas to find volume of selected prisms and cylinders;
6. compare and contrast the volumes of a variety of geometric solids;
7. solve problems involving rate/time/distance (i.e., d = rt);
8. solve problems involving scale factors using ratio and proportion;
9. solve real-world problems using the Pythagorean Theorem;
10. construct tables and graphs to represent rates of change;
11. find measures using proportional relationships and properties of similar figures;
12. determine the measure of angles by applying angle relationships (e.g., complementary, supplementary, interior, exterior, vertical, corresponding)

Eighth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Data Analysis & Probability

Content Standard 5.0 The student will understand and apply basic statistical and probability concepts in order to organize and analyze data and to make predictions and conjectures.

Learning Expectations:

Accomplishments

8.5.1 Formulate questions that can be addressed with data and collect, organize, and display relevant data to answer questions.

1. formulate questions, design studies, and collect real-world data for investigations using a variety of collection methods (e.g., random sampling, simulations);
2. select, create, and use appropriate graphical representations of real-world data (e.g., histograms, box plots, scatterplots).

8.5.2 Select and use appropriate statistical methods to analyze data.

1. find, use, and interpret measures of center and spread (e.g., mean, interquartile range);
2. develop meaning for frequency, distribution, and outliers;
3. discuss and understand the relationship between data sets and their graphical representations (e.g., bar graphs, line graphs, circle graphs, histograms, stem-and-leaf plots, box plots, scatterplots).

8.5.3 Develop and evaluate inferences and predictions that are based on data.

1. make conjectures and predictions based on data;
2. recognize misleading presentations of data;
3. develop meaning for lines of best fit;
4. determine an appropriate sample to test a hypothesis;
5. make conjectures to formulate new questions for future studies.

8.5.4 Understand and apply basic concepts of probability.

1. develop meaning of mutually exclusive events;
2. connect the symbolic representation of a probability to an experiment;
3. use a variety of methods to compute probabilities for compound events (e.g., multiplication, organized lists, tree diagrams, area models);
4. distinguish between theoretical and experimental probability;
5. find the probability of dependent and independent events

Eighth Grade Benchmarks

Performance Indicators State:

As documented through state assessment –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to

Performance Indicators Teacher:

As documented through teacher observation –

at Level 1, the student is able to

at Level 2, the student is able to

at Level 3, the student is able to