Colorado Model Content Standards and Grade Level Expectations (Grade 6) 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. continue to locate commonly-used positive rational numbers, including fractions, mixed numbers, terminating decimals through thousandths, and percents, on the number line SE/TE: 1-1: Understanding Whole Numbers, 5-8; 1-2: Reading and Writing Decimals, 9-12; numbers, terminating decimals through thousandths, and percents, on the number line; 1-3: Comparing and Ordering Decimals, 13-16; 3-7: Comparing and Ordering Fractions, 148-151; 6-6: Percents, Fractions, and Decimals, 294-298 locate integers on the number line SE/TE: 10-1: Using a Number Line, 491-496 identify subsets of integers, including counting and whole numbers demonstrate the equivalence of commonlyused fractions, decimals, and percents pictorially, demonstrate the meaning of square roots of perfect square numbers through 100 SE/TE: 10-1: Using a Number Line, 491-496 SE/TE: 3-8: Fractions and Decimals, 153-156; 6-6: Percents, Fractions, and Decimals, 294-298 SE/TE: 2-8: Exponents, 99-104; 12-5: Exploring Square Roots and Rational Numbers, 616-619 1.2 Reading, writing, and ordering integers, rational numbers, and common irrational numbers such as 2 and 5 and π. read, write, and order positive rational numbers, including commonly-used fractions and terminating decimals through thousandths compare positive fractions and decimals using the symbols =,, <, > SE/TE: 1-1: Understanding Whole Numbers, 5-8; 1-2: Reading and Writing Decimals, 9-12; numbers, terminating decimals through thousandths, and percents, on the number line; 1-3: Comparing and Ordering Decimals, 13-16; 3-7: Comparing and Ordering Fractions, 148-151; 6-6: Percents, Fractions, and Decimals, 294-298 SE/TE: 1-3: Comparing and Ordering Decimals, 13-16; 3-7: Comparing and Ordering Fractions, 148-151; 3-8: Fractions and Decimals, 153-156 - 1 -
1.3 Applying number theory concepts (for example, primes, factors, multiples) to represent numbers in various ways. write the prime factorization of whole numbers in exponential form write whole numbers in expanded form with powers of ten write large whole numbers using scientific notation demonstrate the divisibility rules for 2, 3, 5, 6, 9, and 10 determine the greatest common factor and least common multiple of a pair of whole numbers SE/TE: 3-2: Prime Numbers and Prime Factorization, 123-126 SE/TE: 3-2: Prime Numbers and Prime Factorization, 123-126 SE/TE: Scientific Notation: 104 SE/TE: 3-1: Divisibility and Mental Math, 119-122 SE/TE: 3-3: Greatest Common Factor, 128-130; 3-6: Least Common Multiple, 143-147 1.4 Using the relationships among fractions, decimals, and percents, including the concepts of ratio and proportion, in problem-solving situations. represent fractions, decimals, and percents as ratios demonstrate the similarities and differences between ratios and fractions interpret and use ratios in different contexts to show the relative sizes of two quantities using appropriate notations, including a/b, a to b, a : b SE/TE: 6-1: Ratios, 269-272 SE/TE: 6-1: Ratios, 269-272 SE/TE: 6-1: Ratios, 269-272 1.5 Developing, testing, and explaining conjectures about properties of integers and rational numbers. demonstrate multiplication inverses of positive rational numbers SE/TE: 2-7: Solving Multiplication and Division Equations, 95-97 demonstrate that division by zero is undefined SE/TE: 2-7: Solving Multiplication and Division Equations, 95-97 - 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. estimate, using appropriate techniques, determine, and, then, justify the reasonableness of solutions to problems involving whole numbers and sums and differences of commonly-used fractions and decimals SE/TE: estimation: 17, 19-24, 26, 27, 28, 36, 42, 43, 62, 78, 84, 85, 86, 93, 105, 170, 171-174, 185, 186, 207, 226, 236, 239, 240, 248, 285, 292, 298, 303-305, 339, 380, 382, 434, 458, 550, 619 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 for relationships involving positive rational numbers use variables such as boxes, letters, or other symbols to describe a general rule and to solve problems SE/TE: 2-1: Describing a Pattern, 63-67; 2-4: Make a Table and Look for a Pattern, 80-82; patterns: 40, 42, 102, 103, 108, 122, 151, 198, 208, 229, 333, 399, 409, 451 SE/TE: 2-2: Variables and Expressions, 68-73; 10-8: Graphing Functions, 527-531 2.2 Describing patterns using variables, expressions, equations, and inequalities in problem solving situations. solve problems from patterns involving positive rational numbers using tables, graphs, and rules SE/TE: 2-1: Describing a Pattern, 63-67; 2-4: Make a Table and Look for a Pattern, 80-82; 10-8: Graphing Functions, 527-531; patterns: 40, 42, 102, 103, 108, 122, 151, 198, 208, 229, 333, 399, 409, 451 2.3 Analyzing functional relationships to explain how a change in one quantity results in a change in another. in any functional relationship involving positive rational numbers, describe how a change in one quantity affects the other SE/TE: 10-8: Graphing Functions, 527-531 - 3 -
2.4 Distinguishing between linear and nonlinear functions through informal investigations. graph discrete linear and nonlinear functions SE/TE: 10-8: Graphing Functions, 527-531 graph a continuous linear function for a given situation SE/TE: 10-8: Graphing Functions, 527-531 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). solve problems involving linear relationships in positive rational numbers solve simple linear equations with whole number coefficients by informal methods using manipulatives, tables, graphs, or technology SE/TE: 10-8: Graphing Functions, 527-531 SE/TE: Using Number Sense to Solve One-Step Equations, 84-89; 2-6: Solving Addition and Subtraction Equations, 90-94; 2-7: Solving Multiplication and Division Equations, 95-97; 4-6: Equations With Fractions, 196-199; 5-5: Solving Fraction Equations by Multiplying, 242-245; 12-1: Solving Two-Step Equations, 595-600 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, dot plots, frequency tables, stem-and-leaf, and histograms read, interpret, and draw conclusions from various displays of data SE/TE: Organizing and Displaying Data, 326-330; 7-4: Bar Graphs and Line Graphs, 335-339; 7-5: Circle Graphs, 341-345; 7-6: Using Spreadsheets to Organize Data, 347-350; 7-7: Stem-and-Leaf Plots, 352-355 SE/TE: 7-1: Mean, Median, and Mode, 322-325; 7-2: Organizing and Displaying Data, 326-330; 7-4: Bar Graphs and Line Graphs, 335-339; 7-5: Circle Graphs, 341-345; 7-6: Using Spreadsheets to Organize Data, 347-350; 7-7: Stem-and-Leaf Plots, 352-355; 7-8: Misleading Graphs and Statistics, 358-362 - 4 -
3.2 Displaying and using measures of central tendency, such as mean, median, and mode, and measures of variability, such as range and quartiles. determine the mean of a set of data by using an algorithm formally distinguish between mean, median, and mode given various displays of the same set of data (line, bar, circle, stem-and-leaf, and histograms), determine which measure of central tendency is most evident SE/TE: 7-1: Mean, Median, and Mode, 322-325 SE/TE: 7-1: Mean, Median, and Mode, 322-325 SE/TE: 7-1: Mean, Median, and Mode, 322-325 3.3 Evaluating arguments that are based on statistical claims. recognize a misleading display of data due to scaling SE/TE: 7-8: Misleading Graphs and Statistics, 358-362 critically evaluate biased sampling of a survey SE/TE: 11-3: Making Predictions from Data, 558-561 3.4 Formulating hypotheses, drawing conclusions, and making convincing arguments based on data analysis. demonstrate the meaning of random sampling and biased versus unbiased samples SE/TE: 11-3: Making Predictions from Data, 558-561 3.5 Determining probabilities through experiments or simulations. pictorially demonstrate the equivalence of probabilities as either a common fraction, decimal, or percent assigns 0% to an impossible event and 100% to a certain event performs experiments of independent compound events to estimate probability SE/TE: 11-1: Probability, 547-551; 11-2: Experimental Probability, 553-557 SE/TE: 11-1: Probability, 547-551 SE/TE: 11-1: Probability, 547-551; 11-2: Experimental Probability, 553-557; 11-7: Independent Events, 580-584 - 5 -
3.6 Making predictions and comparing results using both experimental and theoretical probability drawn from real-world problems. predict the probability of independent compound events, such as the sum of two number cubes, conduct an experiment or simulation to determine the probability, and assign the probability to all possible sums of two number cubes demonstrate that the sum of all probabilities of two number cubes equals one using two chance devices, such as two number cubes or two spinners, design a fair game, and an unfair game, and write the directions for each game SE/TE: 11-2: Experimental Probability, 553-557; 11-7: Independent Events, 580-584 SE/TE: 11-1: Probability, 547-551 SE/TE: 11-2: Experimental Probability, 554-555 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, such as the sum of tossing two number cubes by making a list or tree diagram SE/TE: 11-5: Tree Diagrams and the Counting Principle, 568-573 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 construction of an angle bisector build models of triangular prisms including their nets SE/TE: angle bisector; 385 SE/TE: 9-7: Three-Dimensional Figures and Spatial Reasoning, 462-466 - 6 -
given a three-dimensional model built with cubes, draw the orthogonal drawings (that is, the front view, right side view, and top view) and the foundation drawing (that is, the shape of the foundation, placement and the number of cubes that are built on this foundation) and, conversely, given the orthogonal and foundation drawing, build the model SE/TE: 9-7: Three-Dimensional Figures and Spatial Reasoning, 462-466 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. describe complementary and supplementary angles use correct geometric symbols for parallelism, perpendicularity, and triangles reason informally about the properties (including lines of symmetry) of parallelograms, rhombuses, and triangular prisms reason informally about congruence involving parallelograms, rhombuses, and triangular prisms SE/TE: 8-3: Special Pairs of Angles, 386-389 SE/TE: 8-1: Points, Lines, Segments, and Rays, 373-377; 8-4: Classifying Triangles, 392-396 SE/TE: 8-5: Exploring and Classifying Polygons, 397-400; 9-7: Three-Dimensional Figures and Spatial Reasoning, 462-466 SE/TE: 8-7: Congruent and Similar Figures, 405-409 4.4 Solving problems using coordinate geometry. identify the four quadrants of the coordinate plane set up a coordinate graph (include axes, origin, and scale) and use it to mark and read coordinate pairs in all four quadrants SE/TE: 10-6: Graphing in the Coordinate Plane, 518-521 SE/TE: 10-6: Graphing in the Coordinate Plane, 518-521 draw a graph from a given scenario SE/TE: 10-6: Graphing in the Coordinate Plane, 518-521; 10-8: Graphing Functions, 527-531 given a distance, find pairs of points on the coordinate plane separated by that horizontal or vertical distance SE/TE: 10-6: Graphing in the Coordinate Plane, 518-521 - 7 -
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 parallelograms and rhombuses solve problems involving volume of triangular prisms solve problems involving surface area of rectangular prisms SE/TE: 9-3: Perimeters and Areas of Rectangles, 440-445 SE/TE: 9-9: Volumes of Rectangular Prisms and Cylinders, 472-475 SE/TE: 9-8: Surface Areas of Prisms, and Cylinders, 467-471 4.6 Transforming geometric figures using reflections, translations, and rotations to explore congruence. tile a plane with polygons SE/TE: 8-9: Transformations, 415-419 demonstrate clockwise and counterclockwise rotation with 90, 180, and 270 turns using models, demonstrate the multiple transformations which occur to get from one congruent figure to the other, and give a written explanation of the transformations SE/TE: 8-9: Transformations, 415-419 SE/TE: 8-9: Transformations, 415-419 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 parallelograms and rhombuses estimate the perimeter and area of parallelograms and rhombuses SE/TE: 5-7: The Customary System, 250-252; 9-1: Metric Units of Length, Mass, and Capacity, 431-435 SE/TE: 9-3: Perimeters and Areas of Rectangles, 440, 443, 445; 9-4: Areas of Parallelograms and Triangles, 447, 458 estimate the volume of triangular prisms SE/TE: 9-9: Volumes of Rectangular Prisms and Cylinders, 472-475 estimate the surface area of rectangular prisms SE/TE: 9-8: Surface Areas of Prisms, and Cylinders, 467-471 - 8 -
continue to estimate and use the capacity, weight, and mass measurements from previous grades SE/TE: 5-7: The Customary System, 250-252; 9-1: Metric Units of Length, Mass, and Capacity, 431-435 estimate measures of angles SE/TE: 8-2: Angles, 379-383 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, and 5.6 SE/TE: 5-7: The Customary System, 250-252; 9-1: Metric Units of Length, Mass, and Capacity, 431-435 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 selects the appropriate scale for a given problem SE/TE: 7-2: Organizing and Displaying Data, 326-330; 7-4: Bar Graphs and Line Graphs, 335-339; 7-8: Misleading Graphs and Statistics, 358-362 SE/TE: 7-2: Organizing and Displaying Data, 326-330; 7-4: Bar Graphs and Line Graphs, 335-339 5.4 Developing and using formulas and procedures to solve problems involving measurement. develop and use formulas for perimeter and area of parallelograms and rhombuses using appropriate unit develop and use the formula for volume of triangular prisms using appropriate units SE/TE: 9-3: Perimeters and Areas of Rectangles, 440-445; 9-4: Areas of Parallelograms and Triangles, 446-459 SE/TE: 9-9: Volumes of Rectangular Prisms and Cylinders, 472-475 5.5 Describing how a change in an object s linear dimensions affects its perimeter, area, and volume. describe how changes in the base of a triangle affect its area when its height is constant describe how changes in one of the dimensions of a rectangular prism affects its volume SE/TE: 9-4: Areas of Parallelograms and Triangles, 446-459 SE/TE: 9-9: Volumes of Rectangular Prisms and Cylinders, 472-475 - 9 -
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 parallelograms and rhombuses to the nearest inch and nearest centimeter measure angles and draw complements and supplements, where possible, using a protractor SE/TE: 5-7: The Customary System, 250-252; 9-1: Metric Units of Length, Mass, and Capacity, 431-435 SE/TE: 5-7: The Customary System, 250-252; 9-1: Metric Units of Length, Mass, and Capacity, 431-435 SE/TE: 8-2: Angles, 379-383; Extension: 384-385; 8-3: Special Pairs of Angles, 386-389 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. demonstrate the equivalence of fractions, decimals, and percents using concrete materials, determine commonly-used percentages in real-world problems SE/TE: 3-8: Fractions and Decimals, 153-156; 6-6: Percents, Fractions, and Decimals, 294-298 SE/TE: 6-7: Finding a Percent of a Number, 299-302; 6-8: Estimating With Percents, 303-305 6.2 Constructing, using, and explaining procedures to compute and estimate with whole numbers, fractions, decimals, and integers. demonstrate order of operations including exponents with whole numbers choose the appropriate representation of the remainder in a division problem demonstrate equivalencies of mixed numerals and improper fractions SE/TE: 1-10: Order of Operations, 48-51 SE/TE: 660 SE/TE: 3-5: Mixed Numbers and Improper Fractions, 139-142 simplify fractions SE/TE: simplifying fractions: 175, 194, 218, 219, 268, 269, 294, 297, 553-10 -
using paper-and-pencil, demonstrate with proficiency addition and subtraction of fractions including mixed numerals using concrete materials, demonstrate multiplication and division of a common proper fraction and a whole number using concrete materials, demonstrate multiplication and division of proper fractions using concrete materials, demonstrate the meaning of multiplication and division of decimals by whole numbers SE/TE: 4-2: Fractions With Like Denominators, 175-179; 4-3: Fractions With Unlike Denominators, 180-183; 4-4: Adding Mixed Numbers, 185-189; 4-5: Subtracting Mixed Numbers, 190-194; 4-6: Equations With Fractions, 196-199; 4-7: Measuring Elapsed Time, 201-204 SE/TE: 5-1: Multiplying Fractions, 219-223; 5-3: Dividing Fractions, 230-234 SE/TE: 5-1: Multiplying Fractions, 219-223; 5-3: Dividing Fractions, 230-234; 5-5: Solving Fraction Equations by Multiplying, 242-245 SE/TE: 1-7: Multiplying Decimals, 35-38; 1-8: Multiplying and Dividing by 10, 100, and 1,000, 40-42; 1-9: Dividing Decimals, 43-47 demonstrate, by modeling, the inverse relationship of multiplication and division of common proper fractions SE/TE: 5-5: Solving Fraction Equations by Multiplying, 242-245 count change up to the amount given SE/TE: Test Prep: 29 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: 1-4: Estimating With Decimals, 19-24; 4-1: Estimating Sums and Differences, 171-174; 6-8: Estimating With Percents, 303-305 SE/TE: estimation: 17, 19-24, 26, 27, 28, 36, 42, 43, 62, 78, 84, 85, 86, 93, 105, 170, 171-174, 185, 186, 207, 226, 236, 239, 240, 248, 285, 292, 298, 303-305, 339, 380, 382, 434, 458, 550, 619 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 SE/TE: 1-6: Using a Problem-Solving Plan, 30-34 - 11 -
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 with sums and differences of common fractions and decimals, 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: 58-59, 114-115, 166-167, 214-215, 264-265, 316-317, 368-369, 426-427, 486-487, 542-543, 590-591, 634-635; Real-World Problem Solving: 10, 14, 22, 46, 66, 77, 97, 108, 126, 149, 151, 176, 183, 189, 207, 222, 224, 228, 252, 282, 295, 303, 327, 339, 347, 389, 394, 399, 400, 441, 447, 453, 480, 506, 508, 511, 514, 548, 572, 575, 584, 600, 611, 619 SE/TE: 1-5: Adding and Subtracting Decimals, 29; 4-2: Fractions With Like Denominators, 179; 4-3: Fractions With Unlike Denominators, 183 SE/TE: reasonableness of solutions: 25, 26, 27, 36, 43, 182, 186, 190, 191, 225, 237, 322, 457-12 -