Benchmark
7.1.1. Students represent rational numbers in equivalent forms and graph real numbers on a number line.
Indicators
1. Compare and order fractions, decimals, and percents, including use of scientific notation using hands-on materials, diagrams, and symbols. (7.1, 8.1)
2. Determine equivalent relationships between fractions, decimals, and percents. (7.1, PB 7.1)
3. Order and graph rational and irrational numbers on a number line. (PB 7.2)
4. 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)
Benchmark
7.1.2. Students simplify and evaluate numerical and variable expressions involving rational numbers, scientific notation,
and exponents.
Indicators
1. Perform operations with rational numbers. (7.2, 8.1, PB 7.4)
2. Formulate rules for basic operations with integers. (7.5)
3. Solve practical problems involving basic operations with integers. (7.5, PB 7.6)
4. Simplify numerical expressions using order of operations, mental math, and appropriate tools. (7.2, 8.1, PB 7.3)
5. Convert numbers written in scientific notation into standard notation. (7.1)
- Convert numbers written in standard notation into scientific notation. (7.1)
- Solve problems involving scientific notation. (7.1)
- Simplify expressions with positive and negative exponents.
(7.2, 8.1)
- Solve problems involving factors and multiples.
- Distinguish between simplifying numerical expressions and evaluating variable expressions.
11. Evaluate variable expressions for given replacement values of the variables. (8.4)
Benchmark
7.1.3. Students explain and apply properties of operations with real numbers and classify numbers into subsets of the real number system.
Indicators
1. Identify and explain, orally and in writing, the properties of operations with real numbers. (7.3)
· Associative properties for addition and multiplication
· Commutative properties for addition and multiplication
· Distributive property
· Identity elements for addition and multiplication
· Inverse elements for addition and multiplication
· Closure properties for addition and multiplication
· Multiplicative properties of zero
2. Apply properties of operations to justify the process of simplifying numerical variable expressions. (7.3)
3. Describe orally and in writing the relationship between the following subsets of the real number system, including the use of diagrams: natural numbers, whole numbers, integers, rational numbers, irrational numbers, and real numbers.
Benchmark
7.1.4. Students solve proportional reasoning problems including percent applications and scale drawings.
Indicators
1. Distinguish between proportional and non-proportional relationships.
2. Solve proportional reasoning problems that include rational numbers (whole numbers, fractions, and decimals) and percents. (7.6, 8.3)
3. Solve consumer application problems involving sales tax, discount, tip, and simple interest. (7.4, 8.3, PB 7.5)
4. Solve problems involving compound interest.
5. Solve problems involving percent of increase/decrease.
6. Solve practical problems that require using proportions, including scale drawings. (7.6, PB 7.7)
7. Solve problems that apply the slope of a line.
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Benchmark
7.2.1. Students solve problems involving one-, two-, and three-dimensional measurements.
Indicators
1. Verify the Pythagorean Theorem by 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)
3. Define key words related to measurement, including perimeter, area, volume, and surface area.
4. Estimate and find the area of polygons by subdividing them into rectangles and right triangles. (7.7)
5. Solve problems involving the perimeter and area of two-dimensional figures. (PB 7.8, PB 7.9)
6. Apply perimeter and area formulas in practical situations. (7.7)
7. Solve problems involving the surface area and volume of a prism, cylinder, cone, or pyramid. (8.7, PB 7.10, PB 7.11)
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Benchmark
7.3.1. Students solve problems involving geometric transformations.
Indicators
1. Represent transformations (reflections, translations, and rotations) by graphing the coordinates of the vertices of the transformed polygon. (7.13, 8.8, PB 7.15)
2. Dilate a geometric figure by a positive scale factor. (8.8)
3. Use key terms related to transformations, including image, preimage, rotation, reflection, translation, and dilation.
4. Identify real-world applications of transformations, such as tiling, fabric design, art, and scaling. (8.8)
Benchmark
7.3.2. Students solve problems involving angle relationships.
Indicators
1. Recognize and describe the relationships among pairs of angles, including vertical, complementary, supplementary, alternate interior, alternate exterior, and corresponding angles. (8.6)
2. Explain angle relationships when a pair of parallel lines is cut by a transversal.
3. Solve problems involving parallel lines cut by a transversal.
4. Solve problems involving vertical, complementary, and supplementary angles.
5. Develop a procedure for determining the sum of the measures of the interior angles of any polygon.
6. Develop a procedure to determine the measure of each interior angle of a regular polygon.
7. Develop a procedure to determine the number of diagonals of any polygon.
Benchmark
7.3.3. Students solve problems involving congruence and similarity.
Indicators
1. Recognize and describe attributes of congruent and similar figures.
2. Determine if geometric figures are similar. (7.11)
3. Write proportions to express the relationships between corresponding parts of similar figures. (7.11, PB 7.13)
4. Find the measures of sides of similar figures by using proportions.
Benchmark
7.3.4. Students classify and illustrate two- and three-dimensional figures.
Indicators
1. Illustrate triangles and describe their attributes orally and in writing, including scalene, isosceles, equilateral, acute, right, and obtuse.
2. Illustrate quadrilaterals and describe their attributes orally and in writing, including squares, rhombi, parallelograms, rectangles, and trapezoids.
3. Compare and contrast quadrilaterals. (7.9, PB 7.12)
4. Classify quadrilaterals using deductive reasoning and inference. (7.9)
5. Identify and draw polygons and describe their attributes orally and in writing, including triangles, quadrilaterals, pentagons, hexagons, heptagons, octagons, nonagons, and decagons. (7.10)
6. Draw nets for prisms, cylinders, cones, and pyramids.
7. Construct three-dimensional models given the top, side, and front views. (8.9)
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Benchmark
7.4.1. Students collect, organize, and display data.
Indicators
1. Choose methods for collecting data including sampling techniques and technologies. (7.16)
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, PB 7.18, PB 7.19)
5. Present data using graphing technologies.
Benchmark
7.4.2. Students analyze data for patterns and trends.
Indicators
1. Determine values for the extremes, quartiles, and ranges of data sets.
2. Identify gaps, outliers, and clusters in data sets.
3. Create and solve problems involving the measures of central tendency (mean, median, mode) and the range of a set of data. (7.16, PB 7.17)
4. Choose the best measure of central tendency that represents a set
of data.
5. Recognize and identify patterns and trends in data sets.
Benchmark
7.4.3. Students interpret data to make inferences and predictions.
Indicators
1. Make inferences and conjectures based on data analysis.
(7.18, 8.12)
2. Make comparisons and predictions using information in statistical graphs. (7.18, 8.12)
3. Write an interpretation of the data including the significance of gaps, outliers, and clusters.
4. Make predictions based on the correlation of data on a scatterplot.
Benchmark
7.4.4. Students solve probability problems with simple and compound events.
Indicators
1. Represent a sample space using a list, chart, or tree diagram.
2. Distinguish between dependent and independent events.
3. Demonstrate ways to solve problems involving dependent or independent events.
4. Express probability as a ratio, decimal, or percent.
5. Investigate and describe the difference between experimental and theoretical probability. (7.14)
6. Make predictions based on experimental and theoretical probability.
Benchmark
7.4.5. Students solve problems using counting principles.
Indicators
1. Identify and describe the number of possible arrangements of several objects using a tree diagram or the Fundamental (Basic) Counting Principle. (7.15, PB 7.16)
2. Determine the number of combinations of a group of objects.
3. Determine the number of permutations of a group of objects.
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Benchmark
7.5.1. Students solve multistep equations in one variable and translate and solve related word problems.
Indicators
1. Represent and demonstrate the steps in solving multistep equations using hands-on materials, diagrams, and algebraic sentences.
(7.22, 8.15)
2. Check the solution for a multistep equation by using substitution and simplification.
3. Write verbal expressions and sentences as algebraic expressions and equations. (7.20)
4. Use algebraic terms related to solving equations, including
variable, constant, expression, coefficient, term, equation and inequality.
5. Solve practical problems requiring the solution of multistep linear equations. (7.22)
6. Solve literal equations for a given variable.
Benchmark
7.5.2. Students solve and graph multistep inequalities in one variable.
Indicators
1. Represent and demonstrate steps in solving inequalities using hands-on materials, diagrams, and algebraic sentences. (7.22, 8.15)
2. Check the solution set for an inequality by using substitution and simplification.
3. Graph the solution set for an inequality on a number line.
Benchmark
7.5.3. Students represent linear functions using tables, graphs, and equations.
Indicators
1. Represent, analyze, and generalize a variety of patterns, including arithmetic and geometric sequences. (7.19, PB 7.20)
2. Graph ordered pairs that represent a function on a coordinate plane. (7.12, 7.19, 8.14, 8.16, PB 7.14)
3. Represent functional relationships as tables, graphs, rules, and words. (7.19, 8.14)
4. Graph functions using graphing technologies.
5. Use key terms related to functions, including domain, range, dependent variable, and independent variable. (8.18)
6. Recognize the slope and intercepts of a function when represented as a graph.
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Benchmark
7.6.1. 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
7.7.1. 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
7.8.1. 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
7.9.1. 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.
Benchmark
7.10.1. 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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