GLE-Math5
COLORADO MODEL CONTENT
STANDARDS FOR MATHEMATICS
5th 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.
Fifth grade students will:
1.1.1 locate commonly-used positive rational numbers, including fractions, mixed numbers, terminating decimals through thousandths, and percents, on the number line
1.1.2 using concrete materials, demonstrate the meaning of integers
1.1.3 using concrete materials, demonstrate the equivalence of commonly-used fractions,
terminating decimals, and percents (for example, 7/10 = 0.7 = 70%)
1.1.4 pictorially, demonstrate the meaning of square numbers
1.2 Reading, writing, and ordering integers, rational numbers, and common irrational
numbers such as •2 and •5 and •.
Fifth 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 commonly-used proper fractions and terminating decimals using the symbols =, •, <, >
1.3 Applying number theory concepts (for example, primes, factors, multiples) to represent numbers in various ways.
Fifth grade students will:
1.3.1 identify factors, multiples, and prime composite numbers
1.3.2 write the prime factorization of whole numbers up to 50 (for example, 36 = 2 • 2 • 3 • 3)
1.3.3 relate exponential notation to repeated multiplication (for example, 81 = 3 • 3 • 3 • 3 = 81)
1.3.4 write whole numbers in expanded form without powers of ten (for example, 579 = 500 + 70 + 9 = (5 x 100) + ( 7 x 10) + (9 x 1))
1.3.5 demonstrate the divisibility rules for 2, 5, and 10
1.3.6 demonstrate an = a • a • ... • a, where ‘a’ and ‘n’ are counting numbers
1.4 Using the relationships among fractions, decimals, and percents, including the concepts of ratio and proportion, in problem-solving situations.
Fifth grade students will:
1.4.1 demonstrate the meaning of ratio in different contexts
1.4.2 use appropriate notation to express ratios, including a/b, a to b, and a: b
1.5 Developing, testing, and explaining conjectures about properties of integers and rational numbers.
Fifth grade students will:
1.5.1 demonstrate the commutative, associative, and identity properties for addition and
multiplication, and the multiplication property of zero for fractions
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 •.
Fifth grade students will:
1.6.1 estimate sums and differences of fractions and decimals using benchmarks (for
example, 5/6 + 7/8 must be equal to an amount less than 2, since each fraction is less
than 1)
1.6.2 estimate, using appropriate techniques, determine, and, then, justify the reasonableness of solutions to problems involving whole numbers
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.
Fifth grade students will:
2.1.1 represent, describe, and analyze patterns for relationships involving whole numbers and common proper fractions
2.1.2 recognize that a variable is used to represent an unknown quantity
2.2 Describing patterns using variables, expressions, equations, and inequalities in problemsolving situations.
Fifth grade students will:
2.2.1 solve problems from patterns involving whole numbers and common proper fractions 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).
Fifth grade students will:
2.3.1 in any functional relationship involving whole numbers and common proper fractions, describe how a change in one quantity affects the other
2.4 Distinguishing between linear and nonlinear functions through informal investigations.
Fifth grade students will:
2.4.1 graph discrete linear and nonlinear functions
2.4.2 match a description of a situation with its continuous graph
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).
Fifth grade students will:
2.5.1 solve problems involving linear relationships in whole numbers
2.5.2 solve simple linear equations with coefficients of 1 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.
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.
Fifth grade students will:
3.1.1 differentiate between categorical and numerical data
3.1.2 organize and display data using appropriate graphs, such as line, bar, circle, dot plots, frequency tables, and stem-and-leaf
3.1.3 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.
Fifth grade students will:
3.2.1 determine the mean of a set of data using manipulatives
3.2.2 informally distinguish between mean, median, and mode
3.2.3 determine the range of a set of data
3.2.4 given various displays of the same set of data (line, bar, circle, and stem-and-leaf),
determine which measure of central tendency is most evident
3.3 Evaluating arguments that are based on statistical claims.
Fifth grade students will:
3.3.1 critically evaluate line graphs, bar graphs, pictographs, or dot plots which do not begin at zero
3.4 Formulating hypotheses, drawing conclusions, and making convincing arguments based on data analysis.
Fifth grade students will:
3.4.1 distinguish between a census and a survey
3.4.2 explain why there may be differences in the data of two or more samples
3.5 Determining probabilities through experiments or simulations.
Fifth grade students will:
3.5.1 apply probability terms such as event, outcome, trials, and sample space
3.5.2 assign a number between 0 and 1, inclusive, to the probability of an event
3.5.3 perform 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.
Fifth grade students will:
3.6.1 predict the probability of independent compound events, such as tossing two coins or determining the gender of two children in a family, and conduct an experiment or
simulation to determine the probability
3.6.2 demonstrates that the sum of the probabilities equals one (as applied to the sample
space)
3.6.3 using one chance device, such as a number cube or a spinner, 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).
3.7.1 determine the number of outcomes of independent compound events, such as tossing two coins or determining the gender of two children in a family 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.
Fifth grade students will:
4.1.1 using a straight edge and a compass, paper folding, or computer software application, demonstrate the geometric constructions of copying a segment and copying an angle
4.1.2 build models of rectangular prisms including their nets
4.1.3 given a three-dimensional model built with cubes, draw the two-dimensional
orthogonal drawings (that is, the front view, right side view, and top view) and,
conversely, given orthogonal drawings, 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.
Fifth grade students will:
4.3.1 know that the measurement of an acute angle is less than 90o, a right angle is 90o, and an obtuse angle is greater than 90o
4.3.2 uses correct geometric symbols for lines, segments, rays, and angles
4.3.3 reason informally about properties of parallel lines, perpendicular lines, intersecting
lines, line segments, and rays
4.3.4 reason informally about properties (including lines of symmetry) of rectangles, squares, triangles (named by both lengths of sides and angles), and rectangular prisms
4.3.5 reason informally about congruence involving rectangles, squares, triangles, and
rectangular prisms
4.4 Solving problems using coordinate geometry.
Fifth grade students will:
4.4.1 set up a coordinate graph (include axes, origin, and scale) and use it to mark and read coordinate pairs in the first quadrant
4.1.2 from a scenario, choose the correct graph from given possible graph representations
4.1.3 given a distance, find pairs of points on the coordinate plane in the first quadrant
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.
Fifth grade students will:
4.5.1 solve problems involving perimeter and area of rectangles, squares, and triangles
4.5.2 solve problems involving volume of rectangular prisms
4.6 Transforming geometric figures using reflections, translations, and rotations to explore congruence.
Fifth grade students will:
4.6.1 use pattern blocks to tile a plane
4.6.2 show lines of symmetry of geometric shapes
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.
Fifth grade students will:
5.1.1 estimate the length of the sides and height of rectangles, squares, triangles, and
rectangular prisms
5.1.2 estimate the perimeter and area of rectangles, squares, and triangles
5.1.3 estimate the volume of rectangular prisms
5.1.4 continue to estimate and use the capacity, weight, and mass measurements from
previous grades
5.1.5 estimate measures of angles (for example, 30o, 45o, 60o, 90o, 120o, 150o, 180o)
5.2 Estimating, making, and using direct and indirect measurements to describe and make comparisons.
Fifth grade students will:
5.2.1 compares 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.
Fifth grade students will:
5.3.1 read and interpret scales on number lines, graphs, and maps
5.3.2 select the appropriate scale for a given problem
5.4 Developing and using formulas and procedures to solve problems involving measurement.
Fifth grade students will:
5.4.1 develop and use formulas for perimeter and area of rectangles, squares, and triangles using appropriate units
5.4.2 develop and use the formula for volume of rectangular prisms using appropriate units
5.5 Describing how a change in an object’s linear dimensions affects its perimeter, area, and volume.
Fifth grade students will:
5.5.1 describe how changes in one of the dimensions of a rectangle affects its perimeter and
area
5.5.2 using graph paper, demonstrate the changes in area of a rectangle having a constant
perimeter and variable side lengths
5.6 Selecting and using appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation.
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 rectangles, squares, triangles, and
rectangular prisms to the nearest inch and nearest centimeter
5.6.3 measure and draw angles using a protractor (for example, 30o, 45o, 60o, 90o, 120o, 150o, 180o)
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.
Fifth grade students will:
6.1.1 use appropriate notations of ratio such as a/b, a to b, and a:b
6.1.2 using concrete materials, determine commonly-used percentages (e.g., 25% and 50%) in real-world problems
6.2 Constructing, using, and explaining procedures to compute and estimate with whole numbers, fractions, decimals, and integers.
Fifth grade students will:
6.2.1 demonstrate order of operations with whole numbers
6.2.2 demonstrate with proficiency multiplication of whole numbers of three digits by two digits and three digits by three digits
6.2.3 demonstrate with proficiency division of whole numbers with a two-digit divisor
6.2.4 demonstrate equivalencies and simplification of proper fractions
6.2.5 using paper-and-pencil, demonstrate with proficiency addition and subtraction of
proper fractions and mixed numerals with common denominators and without
regrouping
6.2.6 using concrete materials, demonstrate addition and subtraction of mixed numerals with common denominators with regrouping
6.2.7 using concrete materials, demonstrate addition and subtraction of proper fractions with unlike denominators
6.2.8 demonstrate the inverse relationship of addition and subtraction of proper fractions and mixed numerals with common denominators
6.2.9 demonstrate how the value of a fraction changes as the denominator increases
6.2.10 demonstrate with proficiency addition and subtraction of decimals
6.2.11 demonstrate the inverse relationship of addition and subtraction of decimals
6.2.12 make change from any dollar denomination
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.
Fifth 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.
Fifth 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 using the four operations with whole numbers, 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