GLE-Math6
COLORADO MODEL CONTENT
STANDARDS FOR MATHEMATICS
6th Grade Level Expectations
Standard 1:
Students develop number sense and use numbers and number relationships in problem-solving 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.
Sixth grade students will:
1.1.1 continue to locate commonly-used positive rational numbers, including fractions, mixed
numbers, terminating decimals through thousandths, and percents, on the number line
1.1.2 locate integers on the number line
1.1.3 identify subsets of integers, including counting and whole numbers
1.1.4 demonstrate the equivalence of commonly-used fractions, decimals, and percents
1.1.5 pictorially, demonstrate the meaning of square roots of perfect square numbers through
1.2 Reading, writing, and ordering integers, rational numbers, and common irrational
numbers such as •2 and •5 and •.
Sixth grade students will:
1.2.1 read, write, and order positive rational numbers, including commonly-used fractions
and terminating decimals through thousandths
1.2.2 compare positive fractions and decimals using the symbols =, •, <, >
1.3 Applying number theory concepts (for example, primes, factors, multiples) to represent numbers in various ways.
Sixth grade students will:
1.3.1 write the prime factorization of whole numbers in exponential form (for example, 36 =
2 2 • 3 2)
1.3.2 write whole numbers in expanded form with powers of ten (for example, 579 = 500 +
70 + 9 = 5 x 100+ 7 x 10 + 9 x 1)
1.3.3 write large whole numbers using scientific notation (for example, 246,000,000 = 2.46 x
108; 2.46 x 108 = 246,000,000)
1.3.4 demonstrate the divisibility rules for 2, 3, 5, 6, 9, and 10
1.3.5 determine the greatest common factor and least common multiple of a pair of whole
numbers
1.4 Using the relationships among fractions, decimals, and percents, including the concepts of ratio and proportion, in problem-solving situations.
Sixth grade students will:
1.4.1 represent fractions, decimals, and percents as ratios
1.4.2 demonstrate the similarities and differences between ratios and fractions
1.4.3 interpret and use ratios in different contexts (e.g. batting averages, miles per hour) to
show the relative sizes of two quantities using appropriate notations, including a/b, a to
b, a : b
1.5 Developing, testing, and explaining conjectures about properties of integers and rational numbers.
Sixth grade students will:
1.5.1 demonstrate multiplication inverses of positive rational numbers (for example, 1/9 • 9 = 1)
1.5.2 demonstrate that division by zero is undefined
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 •.
Sixth grade students will:
1.6.1 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
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.
Sixth grade students will:
2.1.1 represent, describe, and analyze patterns for relationships involving positive rational
numbers
2.1.2 use variables such as boxes, letters, or other symbols to describe a general rule and to
solve problems
2.2 Describing patterns using variables, expressions, equations, and inequalities in problemsolving situations.
Sixth grade students will:
2.2.1 solve problems from patterns involving positive rational numbers using tables, graphs,
and rules
2.3 Analyzing functional relationships to explain how a change in one quantity results in a change in another (for example, how the area of a circle changes as the radius increases, or how a person’s height changes over time).
Sixth grade students will:
2.3.1 in any functional relationship involving positive rational numbers, describe how a
change in one quantity affects the other
2.4 Distinguishing between linear and nonlinear functions through informal investigations.
Sixth grade students will:
2.4.1 graph discrete linear and nonlinear functions
2.4.2 graph a continuous linear function for a given situation
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).
Sixth grade students will:
2.5.1 solve problems involving linear relationships in positive rational numbers
2.5.2 solve simple linear equations with whole number coefficients by informal methods
using manipulatives, tables, graphs, or technology
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.
Sixth grade students will:
2.3.1 organize and display data using appropriate graphs, such as line, bar, circle, dot plots,
frequency tables, stem-and-leaf, and histograms
2.3.2 read, interpret, and draw conclusions from various displays of data
3.2 Displaying and using measures of central tendency, such as mean, median, and mode, and measures of variability, such as range and quartiles.
Sixth grade students will:
3.2.1 determine the mean of a set of data by using an algorithm
3.2.2 formally distinguish between mean, median, and mode
3.2.3 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
3.3 Evaluating arguments that are based on statistical claims.
Sixth grade students will:
3.3.1 recognize a misleading display of data due to scaling
3.3.2 critically evaluate biased sampling of a survey
3.4 Formulating hypotheses, drawing conclusions, and making convincing arguments based on data analysis.
Sixth grade students will:
3.4.1 demonstrate the meaning of random sampling and biased versus unbiased samples
3.5 Determining probabilities through experiments or simulations.
Sixth grade students will:
3.5.1 pictorially demonstrate the equivalence of probabilities as either a common fraction,
decimal, or percent
3.5.2 assigns 0% to an impossible event and 100% to a certain event
3.5.3 performs experiments of independent compound events to estimate probability
3.6 Making predictions and comparing results using both experimental and theoretical
probability drawn from real-world problems.
Sixth grade students will:
3.6.1 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
3.6.2 demonstrate that the sum of all probabilities of two number cubes equals one
3.6.3using 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
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).
Sixth grade students will:
3.7.1 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
Standard 4:
Students use geometric concepts, properties, and relationships in problemsolving 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.
Sixth grade students will:
4.1.1 using a straight edge and a compass, paper folding, or computer software application,
demonstrate the geometric construction of an angle bisector
4.1.2 build models of triangular prisms including their nets
4.1.3 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
4.2 Describing, analyzing, and reasoning informally about the properties (for example,
parallelism, perpendicularity, congruence) of two- and three-dimensional figures; and
4.3 Applying the concepts of ratio, proportion, and similarity in problem-solving situations.
Sixth grade students will:
4.3.1 describe complementary and supplementary angles
4.3.2 use correct geometric symbols for parallelism, perpendicularity, and triangles
4.2.3 reason informally about the properties (including lines of symmetry) of parallelograms,
rhombuses, and triangular prisms
4.2.4 reason informally about congruence involving parallelograms, rhombuses, and
triangular prisms
4.4 Solving problems using coordinate geometry
Sixth grade students will:
4.4.1 identify the four quadrants of the coordinate plane
4.4.2 set up a coordinate graph (include axes, origin, and scale) and use it to mark and read
coordinate pairs in all four quadrants
4.4.3 draw a graph from a given scenario
4.4.4 given a distance, find pairs of points on the coordinate plane separated by that
horizontal or vertical distance
4.5 Solving problems involving perimeter and area in two dimensions, and involving surface area and volume in three dimensions.
Sixth grade students will:
4.5.1 solve problems involving perimeter and area of parallelograms and rhombuses
4.5.2 olve problems involving volume of triangular prisms
4.5.3 solve problems involving surface area of rectangular prisms
4.6 Transforming geometric figures using reflections, translations, and rotations to explore congruence.
Sixth grade students will:
4.6.1 tile a plane with polygons
4.6.2 demonstrate clockwise and counterclockwise rotation with 90o, 180o, and 270o turns
4.6.3 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
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.
Sixth grade students will:
5.1.1 estimate the length of the sides and height of parallelograms and rhombuses
5.1.2 estimate the perimeter and area of parallelograms and rhombuses
5.1.3 estimate the volume of triangular prisms
5.1.4 estimate the surface area of rectangular prisms
5.1.5 continue to estimate and use the capacity, weight, and mass measurements from
previous grades
5.1.6 estimate measures of angles
5.2 Estimating, making, and using direct and indirect measurements to describe and make comparisons.
Sixth grade students will:
5.2.1 compare the estimates and direct measurements obtained in benchmarks 5.1, 5.4, and
5.6
5.3 Reading and interpreting various scales including those based on number lines, graphs, and maps.
Sixth grade students will:
5.3.1 read and interpret scales on number lines, graphs, and maps
5.3.2 selects the appropriate scale for a given problem
5.4 Developing and using formulas and procedures to solve problems involving measurement.
Sixth grade students will:
5.4.1 develop and use formulas for perimeter and area of parallelograms and rhombuses
using appropriate unit
5.4.2 develop and use the formula for volume of triangular prisms using appropriate units
5.5 Describing how a change in an object’s linear dimensions affects its perimeter, area, and volume.
Sixth grade students will:
5.5.1 describe how changes in the base of a triangle affect its area when its height is constant
5.5.2 describe how changes in one of the dimensions of a rectangular prism affects its volume
5.6 Selecting and using appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation.
Sixth grade students will:
5.6.1 select and use the appropriate units and tools to measure to the degree of accuracy
required in a particular problem
5.6.2 measure the length of the sides and heights of parallelograms and rhombuses to the
nearest inch and nearest centimeter
5.6.3 measure angles and draw complements and supplements, where possible, using a
protractor
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 realworld problems.
Sixth grade students will:
6.1.1 demonstrate the equivalence of fractions, decimals, and percents
6.1.2 using concrete materials, determine commonly-used percentages in real-world
problems
6.2 Constructing, using, and explaining procedures to compute and estimate with whole numbers, fractions, decimals, and integers.
Sixth grade students will:
6.2.1 demonstrate order of operations including exponents with whole numbers
6.2.2 choose the appropriate representation of the remainder in a division problem
6.2.3 demonstrate equivalencies of mixed numerals and improper fractions
6.2.4 simplify fractions
6.2.5 using paper-and-pencil, demonstrate with proficiency addition and subtraction of
fractions including mixed numerals
6.2.6 using concrete materials, demonstrate multiplication and division of a common proper
fraction and a whole number
6.2.7 using concrete materials, demonstrate multiplication and division of proper fractions
6.2.8 using concrete materials, demonstrate the meaning of multiplication and division of
decimals by whole numbers
6.2.9 demonstrate, by modeling, the inverse relationship of multiplication and division of
common proper fractions
6.2.10 count change up to the amount given
6.3 Developing, applying and explaining a variety of different estimation strategies in problemsolving situations, and explaining why an estimate may be acceptable in place of an exact answer.
Sixth grade students will:
6.3.1 determine from real-world problems whether an estimated or exact answer is acceptable
6.3.2 use estimation techniques before performing operations
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, paper-and-pencil, calculator, and computer methods, and determining whether the results are reasonable.
Sixth grade students will:
6.4.1 determine whether information given in a problem-solving situation is sufficient,
insufficient, or extraneous
6.4.2 given a real-world problem-solving situation, use the correct operation and appropriate
method (mental arithmetic, estimation, paper-and-pencil, calculator, or computer) to
solve the problem
6.4.3 given a math sentence with sums and differences of common fractions and decimals,
create and illustrate a real-world problem
6.4.4 in a problem-solving situation, determine whether the results are reasonable and justify
those results with correct computations