Benchmark
8.1.1. Students simplify and evaluate expressions and solve problems involving rational numbers and percent.
Indicators
1. Simplify numerical expressions involving positive exponents and rational numbers using order of operations. (7.2, 8.1, PB 8.1)
2. Evaluate algebraic expressions using given replacement values of the variables. (8.4)
3. Simplify and evaluate expressions by using the laws of exponents.
4. Compare, order, and graph rational and irrational numbers on a number line including numbers written in scientific notation.
(7.1, 8.1, PB 8.2)
5. Identify a whole number between 0 and 225 as a perfect square or find the two consecutive integers between which the number’s square root lies. (8.5)
6. Solve practical problems involving whole numbers, integers, rational numbers, and percents including sales tax, discount, tip, simple interest, compound interest, percent of increase/decrease, and balancing a checkbook. (7.4, 8.3, PB 8.3)
7. Solve practical problems that require using proportions including scale drawings. (7.6, PB 8.3)
Benchmark
8.1.2. Students explain and apply properties of operations with real numbers and classify numbers into subsets of the real number system.
Indicators
- Identify and explain orally and in writing the properties of operations with real numbers. (7.3)
● Associative properties for addition and multiplication
● Commutative properties of addition and multiplication
● Distributive property
● Identity elements for addition and multiplication
● Inverse elements for addition and multiplication
● Closure properties for addition and multiplication
● Multiplicative property of zero
2. Apply properties of operations to justify the process of simplifying numerical and variable expressions. (7.3, 8.1)
3. Describe orally and in writing, including the use of diagrams, the relationship between the following subsets of the real number system: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers. (8.2)
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Benchmark
8.2.1. Students classify, illustrate, and solve problems involving two- and three-dimensional measurements.
Indicators
1. Verify the Pythagorean Theorem in solving problems, using diagrams, concrete materials, and measurement. (8.10)
2. Apply the Pythagorean Theorem to find the missing length of a side of a right triangle when given the lengths of the other two sides. (8.10, PB 8.11)
3. Describe, classify, and construct plane and solid figures including prisms, pyramids, cylinders, and cones.
4. Devise valid procedures to determine the surface area and volume of prisms, cylinders, pyramids, and cones.
5. Investigate and solve practical problems involving the surface area and volume of rectangular prisms, cylinders, cones and pyramids. (7.8, 8.7, PB 8.7, PB 8.8)
6. Construct a three-dimensional model, given the top, side, and front views. (8.9, PB 8.10)
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Benchmark
8.3.1. Students solve problems involving geometric transformations.
Indicators
1. Apply transformations (rotate or turn, reflect or flip, translate or slide, and dilate or scale) to geometric figures represented on a coordinate plane. (7.13, 8.8, PB 8.9)
2. Use key terms related to transformations, including image, preimage, rotation, reflection, translation, and dilation.
3. Identify and illustrate applications of geometric transformations such as tiling, fabric design, art, and scaling. (8.8)
Benchmark
8.3.2. Students solve problems involving angle relationships.
Indicators
1. Recognize, verify by measuring, and describe relationships among pairs of angles, including vertical, complementary, supplementary, alternative interior, alternative exterior, and corresponding angles. (8.6)
2. Measure and draw angles that are less than 360 degrees. (8.6)
3. Explain angle relationships when a pair of parallel lines is cut by a transversal.
4. Solve problems involving parallel lines cut by a transversal.
5. Solve problems involving vertical, complementary, and supplementary angles. (PB 8.6)
6. Develop a procedure to determine the sum of the measures of the interior angles of any polygon.
7. Develop a procedure to determine the measure of each interior angle of a regular polygon.
8. Develop a procedure to determine the number of diagonals of any polygon.
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Benchmark
8.4.1. Students collect, organize, and display data.
Indicators
1. Choose methods for collecting data including sampling techniques and technologies.
2. Organize and describe data using matrices and charts. (8.13)
3. Determine the most appropriate type of graph to display data.
4. Display data using frequency distributions, histograms, line plots, stem-and-leaf plots, box-and-whisker plots, bar graphs, circle graphs, line graphs, and scatter plots. (7.17)
5. Present data using graphing technologies.
Benchmark
8.4.2. Students analyze and interpret data.
Indicators
1. Identify patterns and trends in data sets.
2. Write an interpretation of the data including measures of central tendency, extremes, quartiles, ranges, and correlation. (PB 8.13)
3. Describe the significance of gaps, outliers, and clusters of data sets.
4. Make comparisons, predictions, inferences, and conclusions from information in statistical graphs. (7.18, 8.12, PB 8.14)
Benchmark
8.4.3. Students solve probability and counting problems.
Indicators
1. Represent the number of possible arrangements of objects using the fundamental counting principle, combinations, and permutations.
2. Solve probability problems involving dependent and independent events. (PB 8.12)
3. Analyze problem situations such as games of chance, board games, or grading scales and make predictions using knowledge of probability. (8.11)
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Benchmark
8.5.1. Students solve multistep equations and inequalities in one variable and translate and solve related word problems.
Indicators
1. Solve multistep equations and inequalities using hands-on materials, diagrams, and algebraic sentences. (7.22, 8.15, PB 8.17, PB 8.18)
2. Identify the properties of equality and properties of operations used to solve equations and inequalities.
3. Check the solution for a multistep equation or inequality by using substitution and simplification. (PB 8.4, 8.5)
4. Graph the solution set for an equation or inequality on a number line.
5. Translate word problems into algebraic equations and inequalities and solve. (7.20)
6. Use algebraic terms related to solving equations, including variable, constant, expression, coefficient, term, equation, and inequality. (7.21)
7. Solve literal equations for a given variable.
Benchmark
8.5.2. Students describe and represent relations using tables, graphs, equations, and word statements.
Indicators
1. Graph linear equations in two variables in the coordinate plane by using a table of ordered pairs. (8.16, PB 8.19)
2. Describe and represent relations and functions using tables, graphs, algebraic sentences, word statements, and graphing technologies. (7.19, 8.14, PB 8.15, PB 8.16)
3. Relate and compare tables, graphs, and rules as different forms of representations for relationships. (8.14)
4. Create and solve problems using proportions, formulas, and functions. (8.17, PB 8.20)
5. Use algebraic terms related to functions, including domain, range, independent variable, and dependent variable. (8.18)
6. Recognize the slope and y-intercept of linear equations written in slope-intercept form.
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Benchmark
8.6.1. Students recognize and apply mathematical models to real-world situations.
Indicators
1. Cite examples of mathematics use in real-world settings.
2. Describe the relationship between a mathematical model and a real-world situation.
3. Provide multiple solutions to a real-world problem.
4. Apply one solution path to multiple problems.
5. Construct and use mathematical models to represent real-world situations.
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Benchmark
8.7.1. Students use appropriate strategies to solve problems both within and outside mathematics.
Indicators
1. Formulate problems from a variety of mathematical situations.
2. Formulate problems from a variety of situations in other disciplines.
3. Explain, illustrate, and evaluate various strategies for solving problems.
4. Apply a specific strategy to a variety of problems.
5. Choose strategies appropriate to the problems.
6. Evaluate the appropriateness of the strategies for the problems.
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Benchmark
8.8.1. Students read, understand, and communicate mathematical ideas both orally and in writing.
Indicators
1. Use the symbols and vocabulary of mathematics to represent and describe mathematical ideas, generalizations, and relationships.
2. Ask questions pertinent to concepts and procedures written in textbooks and other mathematics references.
3. Write directions to accomplish a mathematical task.
4. Restate and explain both orally and in writing mathematical concepts and procedures read in textbooks or other mathematics references.
5. Explain both orally and in writing the mathematical meaning of information read in textbooks and other mathematics references.
6. Write mathematical analyses of information from a variety of sources.
7. Create webs, charts, graphs, and diagrams to communicate mathematical ideas.
8. Use technologies to communicate mathematical understanding.
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Benchmark
8.9.1. Students reason numerically and spatially and justify reasoning in a variety of mathematical situations.
Indicators
1. Evaluate reasonableness of numeric information in data sets.
2. Evaluate reasonableness of numeric results from solving problems and make necessary corrections.
3. Represent numeric and spatial relationships using hands-on materials and diagrams.
4. Translate numeric understanding of concepts and procedures into algebraic representation.
5. Translate geometric reasoning into algebraic representation.
6. Construct and defend valid mathematical assertions.
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Benchmark
8.10.1. Students use a variety of representations to illustrate mathematical concepts, skills, and processes.
Indicators
1. Use algebraic, geometric, and physical models to represent numeric procedures.
2. Use algebraic models to represent geometric relationships.
3. Use geometric models to represent algebraic relationships.
4. Make connections and apply mathematical models and processes to situations in other disciplines.
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