Colorado Model Content Standards and Grade Level Expectations (Grade 8) Standard 1: Students develop number sense and use numbers and number relationships in problemsolving situations and communicate the reasoning used in solving these problems. 1.1 Demonstrating meanings for integers, rational numbers, percents, exponents, square roots, and pi ( ) using physical materials and technology in problem-solving situations. locate rational numbers and commonly-used irrational numbers on the number line demonstrate the equivalence of fractions, terminating decimals, and percents of positive and negative rational numbers distinguish between the sets of rational and irrational numbers determine the two consecutive whole numbers between which the square root of a whole number lies pictorially, demonstrate the meaning of commonly-used irrational numbers SE/TE: 1-3: Integers and Absolute Value, 16-21; 4-3: Comparing and Ordering Rational Numbers, 196-200 SE/TE: 4-2: Equivalent Forms of Rational Numbers, 190-195; 6-1: Fractions, Decimals, and Percents, 303-308 SE/TE: 4-2: Equivalent Forms of Rational Numbers, 190-195; 4-3: Comparing and Ordering Rational Numbers, 196-200; 4-8: Exploring Square Roots and Irrational Numbers, 221-225 SE/TE: 4-8: Exploring Square Roots and Irrational Numbers, 221-225 SE/TE: 4-8: Exploring Square Roots and Irrational Numbers, 221-225 1.2 Reading, writing, and ordering integers, rational numbers, and common irrational numbers such as 2 and 5 and π. read, write, and order rational numbers and commonly-used irrational numbers compare rational numbers and commonlyused irrational numbers using the symbols =,, <, > SE/TE: 1-3: Integers and Absolute Value, 16-21; 4-2: Equivalent Forms of Rational Numbers, 190-195; 4-3: Comparing and Ordering Rational Numbers, 196-200 SE/TE: 4-3: Comparing and Ordering Rational Numbers, 196-200 - 1 -
1.3 Applying number theory concepts (for example, primes, factors, multiples) to represent numbers in various ways. write and use appropriately negative powers of ten write rational numbers in expanded form with negative powers of ten write very small rational numbers in scientific notation demonstrate the meaning of an, where a is any rational number and n is a counting number SE/TE: 7-1: Scientific Notation, 365-369 SE/TE: 7-1: Scientific Notation, 365-369 SE/TE: 7-1: Scientific Notation, 365-369 SE/TE: 1-1: Algebraic Expressions and the Order of Operations, 5-10 1.4 Using the relationships among fractions, decimals, and percents, including the concepts of ratio and proportion, in problem-solving situations. apply proportional reasoning to solve problems SE/TE: 5-4: Solving Proportions, 258-263; 5-5: Similar Figures and Proportions, 265-269; 5-6: Similarity Transformations, 271-276; 5-7: Scale Models and Maps, 278-281; 5-8: Similarity and Indirect Measurement, 282-286; 5-9: The Sine and Cosine Ratios, 287-291 1.5 Developing, testing, and explaining conjectures about properties of integers and rational numbers. demonstrate properties for rational numbers, including closure SE/TE: 4-2: Equivalent Forms of Rational Numbers, 190-195; 4-3: Comparing and Ordering Rational Numbers, 196-200; 4-4: Adding and Subtracting Rational Numbers, 201-204; 4-5: Multiplying and Dividing Rational Numbers, 205-210 - 2 -
1.6 Using number sense to estimate and justify the reasonableness of solutions to problems involving integers, rational numbers, and common irrational numbers such as 2, 5, and π. estimate, using appropriate techniques, determine, and, then, justify the reasonableness of solutions to problems involving positive and negative rational numbers SE/TE: estimation: 10, 26, 31, 42, 71, 143, 250, 251, 260, 281, 314, 316, 327, 328, 351, 451, 475, 492, 498, 503, 507, 509; reasonableness of solutions: 13-14, 79, 98, 104, 123, 206, 250, 304, 316, 321, 327-328, 339 Standard 2: Students use algebraic methods to explore, model and describe patterns and functions involving numbers, shapes, data, and graphs in problem-solving situations and communicate the reasoning used in solving these problems. 2.1 Representing, describing, and analyzing patterns and relationships using tables graphs, verbal rules, and standard algebraic notation. represent, describe, and analyze patterns with rational numbers SE/TE: 8-4: Solve a Simpler Problem and Look for a Pattern, 426-428; 10-5: Making Predictions From Scatter Plots, 557-562; 12-1: Sequences, 637-641; 12-2: Functions, 643-647; 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656; 12-6: Quadratic and Other Nonlinear Functions, 663-667 2.2 Describing patterns using variables, expressions, equations, and inequalities in problem solving situations. solve problems from patterns involving rational numbers using tables, graphs, and rules SE/TE: 12-2: Functions, 643-647; 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656; 12-6: Quadratic and Other Nonlinear Functions, 663-667 2.3 Analyzing functional relationships to explain how a change in one quantity results in a change in another. in any functional relationship involving rational numbers, describe how a change in one quantity affects the other SE/TE: 12-2: Functions, 643-647; 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656; 12-6: Quadratic and Other Nonlinear Functions, 663-667 - 3 -
in a linear function, explain the meaning of slope as a rate of change SE/TE: 12-2: Functions, 643-647; 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656 identify independent and dependent variables N/A 2.4 Distinguishing between linear and nonlinear functions through informal investigations. graph discrete linear and nonlinear functions SE/TE: 12-2: Functions, 643-647; 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656; 12-6: Quadratic and Other Nonlinear Functions, 663-667 graph and distinguish between continuous linear and nonlinear functions, such as, y = 3x + 2, y = x2, and y = x3, either by creating a table or using technology SE/TE: 12-3: Graphing Linear Functions, 648-652; 12-4: Writing Rules for Linear Functions, 653-656; 12-6: Quadratic and Other Nonlinear Functions, 663-667 2.5 Solving simple linear equations in problem-solving situations using a variety of methods (informal, formal and graphical) and a variety of tools (physical materials, calculators and computers). translate written expressions or equations to algebraic expressions or equations, and vice versa using formal methods, solve one-step linear equations involving rational numbers solve linear equations involving integers with variables and constants on both sides of the equation SE/TE: 1-1: Algebraic Expressions and the Order of Operations, 5-10; 2-5: Draw a Diagram and Write an Equation, 85-88; 5-3: Write an Equation, 254-257; 6-7: Write an Equation, 338-340; 7-5: Write an Equation, 388-391; 12-7: Write an Equation, 668-671 SE/TE: 2-1: Solving One-Step Equations, 61-66; 2-2: Solving Two-Step Equations, 69-72; 3-6: Using Graphs of Equations, 147-150 SE/TE: 3-6: Using Graphs of Equations, 147-150 - 4 -
Standard 3: Students use data collection and analysis, statistics, and probability in problem-solving situations and communicate the reasoning and processes used in solving these problems. 3.1 Reading and constructing displays of data using appropriate techniques (for example, line graphs, circle graphs, scatter plots, box plots, stem-and-leaf plots) and appropriate technology. organize and display data using appropriate graphs, such as line, bar, circle (using ratios to determine degrees and draw with protractors), dot plots, frequency tables, stem-and-leaf, histograms, scatter plots, box-and-whiskers read, interpret, and draw conclusions from various displays of data SE/TE: 10-1: Displaying Frequency, 533-538; 10-3: Stem-and-Leaf Plots, 546-550; 10-4: Boxand-Whisker Plots, 552-566; 10-6: Circle Graphs, 563-567; 10-7: Choosing an Appropriate Graph, 569-574 SE/TE: 10-1: Displaying Frequency, 533-538; 10-2: Reading Graphs Critically, 540-544; 10-3: Stem-and-Leaf Plots, 546-550; 10-4: Boxand-Whisker Plots, 552-566; 10-5: Making Predictions From Scatter Plots, 557-562; 10-6: Circle Graphs, 563-567; 10-7: Choosing an Appropriate Graph, 569-574 3.2 Displaying and using measures of central tendency, such as mean, median, and mode, and measures of variability, such as range and quartiles. state the purpose of using measures of central tendency and variability with data sets create sets of data with the same mean and different ranges and compare the variability in a problem-solving situation, select the most appropriate display and measure of central tendency to solve the problem SE/TE: 1-6: Using Integers With Mean, Median, and Mode, 33-38 SE/TE: 1-6: Using Integers With Mean, Median, and Mode, 33-38 SE/TE: 1-6: Using Integers With Mean, Median, and Mode, 33-38 3.3 Evaluating arguments that are based on statistical claims. determine the improper computation of percent increase or decrease recognize a misleading display of data which arises from area and volume models SE/TE: related material: 10-2: Reading Graphs Critically, 540-544 SE/TE: related material: 10-2: Reading Graphs Critically, 540-544 - 5 -
3.4 Formulating hypotheses, drawing conclusions, and making convincing arguments based on data analysis. display, analyze, and draw conclusions from a given set of data or student-generated set of data SE/TE: 10-1: Displaying Frequency, 533-538; 10-2: Reading Graphs Critically, 540-544; 10-3: Stem-and-Leaf Plots, 546-550; 10-4: Boxand-Whisker Plots, 552-566; 10-5: Making Predictions From Scatter Plots, 557-562; 10-6: Circle Graphs, 563-567; 10-7: Choosing an Appropriate Graph, 569-574 3.5 Determining probabilities through experiments or simulations. perform experiments of simple independent and dependent events to estimate probability perform experiments to estimate the probability of complementary events SE/TE: 11-4: Theoretical and Experimental Probability, 606-610 SE/TE: 11-4: Theoretical and Experimental Probability, 606-610; 11-5: Independent and Dependent Events, 611-615 3.6 Making predictions and comparing results using both experimental and theoretical probability drawn from real-world problems. determine the probability of independent, dependent, and complementary events with replacement and without replacement analyze games of chance to determine whether they are fair or unfair; if unfair, rewrite the rules of the game to make it fair SE/TE: 11-4: Theoretical and Experimental Probability, 606-610; 11-5: Independent and Dependent Events, 611-615 SE/TE: 11-4: Theoretical and Experimental Probability, 610; 11-5: Independent and Dependent Events, 612 3.7 Using counting strategies to determine all the possible outcomes from an experiment (for example, the number of ways students can line up to have their picture taken). determine the number of outcomes of independent compound events by using the fundamental counting principle (for example, if one choice occurs in m ways and the second choice occurs in n ways, then the number of ways for them to occur together is m x n) SE/TE: 11-1: Counting Outcomes, 591-594 - 6 -
use Pascal s triangle to determine how many and which outcomes occur for independent compound events with exactly two outcomes SE/TE: 11-4: Theoretical and Experimental Probability, 606-610 Standard 4: Students use geometric concepts, properties, and relationships in problem-solving situations and communicate the reasoning used in solving these problems. 4.1 Constructing two- and three-dimensional models using a variety of materials and tools. using a straight edge and a compass, paper folding, or computer software application, demonstrate the geometric constructions of a perpendicular to a point on a line segment, a perpendicular to a line from a point not on the line segment, and triangle congruence of Side- Side-Side, Side-Angle-Side, and Angle-Side- Angle build models of three-dimensional oblique solids given a three-dimensional model built with cubes, use isometric paper to draw the isometric drawing (that is, a drawing that shows the corner view and the top or bottom view), the orthogonal drawings (that is, the front view, right side view, and top view) and the foundation view (that is, the shape of the foundation, placement and the number of cubes that are built on this foundation) and, conversely, given the drawings, build the models SE/TE: 8-9: Constructions, 454-458 SE/TE: 9-2: Drawing Views of Solids, 476-480 SE/TE: 9-2: Drawing Views of Solids, 476-480; 9-3: Nets and Solids, 481-485 4.2 Describing, analyzing, and reasoning informally about the properties of two- and three-dimensional figures; and 4.3 Applying the concepts of ratio, proportion, and similarity in problem-solving situations. identify and use correct notation for triangle congruence of Side-Side-Side, Side-Angle- Side, and Angle-Side-Angle reason informally about the relationships among angles formed by two lines cut by a transversal and two parallel lines cut by a transversal SE/TE: 8-3: Congruent Polygons, 420-425 SE/TE: 8-2: Angles and Parallel Lines, 413-418 - 7 -
reason informally about the sum of the measures of the angles of a triangle equaling 180o reason informally about the properties of the special right triangles, 30o-60o-90o and 45o- 45o-90o continue to reason informally about the sides and angles of congruent and similar polygons demonstrate proportional reasoning to indirectly determine lengths of segments of similar polygons SE/TE: 8-6: Angles and Polygons, 436-439 SE/TE: 8-6: Angles and Polygons, 436-439 SE/TE: 8-3: Congruent Polygons, 420-425 SE/TE: 5-5: Similar Figures and Proportions, 265-269; 5-7: Scale Models and Maps, 278-281 4.4 Solving problems using coordinate geometry. enlarge figures on a coordinate plane by rational scale factors reduce figures on a coordinate plane by rational scale factors determine the percent increase or decrease of perimeter and area of the enlargement or reduction of squares, rectangles and triangles describe the relationship of more than two points on the coordinate plane given a distance, find pairs of points on the coordinate plane separated by that distance determine the distance between a pair of points in the coordinate plane SE/TE: 5-6: Similarity Transformations, 271-276 SE/TE: 5-6: Similarity Transformations, 271-276 SE/TE: 5-6: Similarity Transformations, 271-276 SE/TE: 3-1: Graphing Points, 117-121 SE/TE: 3-1: Graphing Points, 117-121 SE/TE: 3-1: Graphing Points, 117-121 4.5 Solving problems involving perimeter and area in two dimensions, and involving surface area and volume in three dimensions. solve problems involving perimeter and area of trapezoids solve problems involving volume of square pyramids and cones SE/TE: 8-7: Areas of Polygons, 441-446 SE/TE: 9-7: Volumes of Pyramids and Cones, 506-510 - 8 -
solve problems involving surface area of cylinders SE/TE: 9-4: Surface Areas of Prisms and Cylinders, 488-493 4.6 Transforming geometric figures using reflections, translations, and rotations to explore congruence. determine the scale factor for dilations to illustrate similarity create Escher-type tessellations to illustrate congruence state the coordinates to describe the reflection of a figure across the x- and y-axes SE/TE: 5-6: Similarity Transformations, 271-276 SE/TE: Tessellations: 440 SE/TE: 3-9: Reflections and Symmetry, 163-168 Standard 5: Students use a variety of tools and techniques to measure, apply the results in problem-solving situations, and communicate the reasoning used in solving these problems. 5.1 Estimating, using, and describing measures of distance, perimeter, area, volume, capacity, weight, mass, and angle comparison. estimate the length of the sides and height of trapezoids SE/TE: 5-5: Similar Figures and Proportions, 265-269 estimate the perimeter and area of trapezoids SE/TE: 8-7: Areas of Polygons, 441-446 continue to compare the perimeter and area of transformed geometric figures estimate the volume of square pyramids and cones SE/TE: 5-6: Similarity Transformations, 271-276 SE/TE: 9-7: Volumes of Pyramids and Cones, 506-510 estimate the surface area of cylinders SE/TE: 9-4: Surface Areas of Prisms and Cylinders, 488-493 continue to estimate and use the capacity, weight, and mass measurements from previous grades SE/TE: 5-2: Choosing and Converting Units, 248-252 estimate measures of angles SE/TE: 8-2: Angles and Parallel Lines, 413-418 - 9 -
5.2 Estimating, making, and using direct and indirect measurements to describe and make comparisons. compare the estimates and direct measurements obtained in benchmarks 5.1, 5.4, and5.6 demonstrate proportional reasoning to indirectly determine lengths of segments of similar polygons SE/TE: 5-2: Choosing and Converting Units, 248-252 SE/TE: 5-5: Similar Figures and Proportions, 265-269; 5-8: Similarity and Indirect Measurement, 282-286 5.3 Reading and interpreting various scales including those based on number lines, graphs, and maps. read and interpret scales on number lines, graphs, and maps select the appropriate scale for a given problem SE/TE: 5-7: Scale Models and Maps, 278-281; 10-2: Reading Graphs Critically, 540-544; 10-7: Choosing an Appropriate Graph, 569-574 SE/TE: 10-7: Choosing an Appropriate Graph, 569-574 construct scale drawings SE/TE: 5-7: Scale Models and Maps, 278-281 5.4 Developing and using formulas and procedures to solve problems involving measurement. develop and use formulas for the perimeter and area of trapezoids using appropriate units develop and use the formula for volume of square pyramids and cones using appropriate units SE/TE: 8-7: Areas of Polygons, 441-446 SE/TE: 9-7: Volumes of Pyramids and Cones, 506-510 develop and use the Pythagorean Theorem SE/TE: 4-9: The Pythagorean Theorem, 226-231; 290, 296 use the relationships in 30-60-90 and 45-45-90 triangles to solve problems SE/TE: 8-6: Angles and Polygons, 436-439 5.5 Describing how a change in an object s linear dimensions affects its perimeter, area, and volume. describe how changing the radius of a circle affects the circumference and area SE/TE: 8-8: Circumferences and Areas of Circles, 447-452 - 10 -
describe how changing the height or radius of the base of a cylinder affects the volume SE/TE: 9-6: Volumes of Prisms and Cylinders, 500-505 5.6 Selecting and using appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation. select and use the appropriate units and tools to measure to the degree of accuracy required in a particular problem measure the length of the sides and heights of trapezoids to the nearest sixteenth inch and nearest millimeter using a protractor, measure angles of two lines cut by a transversal and angles of two parallel lines cut by a transversal SE/TE: 5-2: Choosing and Converting Units, 248-252 SE/TE: 5-5: Similar Figures and Proportions, 265-269 SE/TE: 8-2: Angles and Parallel Lines, 413-418 Standard 6: Students link concepts and procedures as they develop and use computational techniques, including estimation, mental arithmetic, paper-and-pencil, calculators, and computers, in problem-solving situations and communicate the reasoning used in solving these problems. 6.1 Using models to explain how ratios, proportions, and percents can be used to solve real world problems. compute percent of increase or decrease in real-world problems apply proportional reasoning in problemsolving situations (for example, scale, similarity, percentage, unit pricing, simple interest, and rate) SE/TE: 6-5: Percent of Change, 326-330; 6-6: Markup and Discount, 331-336 SE/TE: 5-4: Solving Proportions, 258-263; 5-5: Similar Figures and Proportions, 265-269; 5-6: Similarity Transformations, 271-276; 5-7: Scale Models and Maps, 278-281; 5-8: Similarity and Indirect Measurement, 282-286; 5-9: The Sine and Cosine Ratios, 287-291; 6-3: Percents and Proportions, 314-318; 6-4: Percents and Equations, 320-325; 6-5: Percent of Change, 326-330; 6-6: Markup and Discount, 331-336; 6-8: Simple and Compound Interest, 342-347 - 11 -
6.2 Constructing, using, and explaining procedures to compute and estimate with whole numbers, fractions, decimals, and integers. demonstrate order of operations with rational numbers demonstrate the meaning of the four basic operations of rational numbers using paper-and-pencil, demonstrate with proficiency computation of rational numbers demonstrate the inverse relationship of addition and subtraction of rational numbers demonstrate the inverse relationship of multiplication and division of rational numbers demonstrate multiplication of rational numbers as repeated addition SE/TE: 1-1: Algebraic Expressions and the Order of Operations, 5-10 SE/TE: 4-4: Adding and Subtracting Rational Numbers, 201-204; 4-5: Multiplying and Dividing Rational Numbers, 205-210 SE/TE: 4-4: Adding and Subtracting Rational Numbers, 201-204; 4-5: Multiplying and Dividing Rational Numbers, 205-210 SE/TE: 2-1: Solving One-Step Equations, 61-66; 2-2: Solving Two-Step Equations, 69-72; 2-6: Solving Inequalities by Adding or Subtracting, 90-95 SE/TE: 2-1: Solving One-Step Equations, 61-66; 2-2: Solving Two-Step Equations, 69-72; 2-7: Solving Inequalities by Multiplying or Dividing, 97-101 SE/TE: 4-5: Multiplying and Dividing Rational Numbers, 205-210 6.3 Developing, applying and explaining a variety of different estimation strategies in problem solving situations, and explaining why an estimate may be acceptable in place of an exact answer. determine from real-world problems whether an estimated or exact answer is acceptable use estimation techniques before performing operations SE/TE: Real-World Snapshots: 56-57, 112-113, 180-181, 238-239, 298-299, 360-361, 402-403, 466-467, 528-529, 586-587, 632-633, 694-695 SE/TE: estimation: 10, 26, 31, 42, 71, 143, 250, 251, 260, 281, 314, 316, 327, 328, 351, 451, 475, 492, 498, 503, 507, 509-12 -
6.4 Selecting and using appropriate methods for computing with commonly-used fractions and decimals, percents, and integers in problem-solving situations from among mental arithmetic, estimation, paperand-pencil, calculator, and computer methods, and determining whether the results are reasonable. determine whether information given in a problem-solving situation is sufficient, insufficient, or extraneous given a real-world problem-solving situation, use the correct operation and appropriate method (mental arithmetic, estimation, paperand-pencil, calculator, or computer) to solve the problem given a math sentence using the four operations with rational numbers, create and illustrate a real-world problem in a problem-solving situation, determine whether the results are reasonable and justify those results with correct computations SE/TE: Real-World Snapshots: 56-57, 112-113, 180-181, 238-239, 298-299, 360-361, 402-403, 466-467, 528-529, 586-587, 632-633, 694-695 SE/TE: Real-World Problem Solving: 12, 18, 21, 34, 66, 72, 69, 85, 120, 121, 137, 145, 189, 191, 220, 224, 275, 278, 286, 290, 316, 321, 325, 333, 366, 386, 390, 395, 408, 412, 426, 442, 474, 479, 493, 508, 543, 544, 561, 570, 597, 615, 618, 623, 645, 647, 655, 682 SE/TE: 4-4: Adding and Subtracting Rational Numbers, 201-204; 4-5: Multiplying and Dividing Rational Numbers, 205-210 SE/TE: reasonableness of solutions: 13-14, 79, 98, 104, 123, 206, 250, 304, 316, 321, 327-328, 339-13 -