GLE-Math8
COLORADO MODEL CONTENT
STANDARDS FOR MATHEMATICS
8th 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.
Eighth grade students will:
1.1.1 locate rational numbers and commonly-used irrational numbers on the number line (for
example, -7/2, -2.48, 0, 15/16, •2, •)
1.1.2 demonstrate the equivalence of fractions, terminating decimals, and percents of positive
and negative rational numbers
1.1.3 distinguish between the sets of rational and irrational numbers
1.1.4 determine the two consecutive whole numbers between which the square root of a
whole number lies (for example, •72 lies between 8 and 9)
1.1.5 pictorially, demonstrate the meaning of commonly-used irrational numbers
1.2 Reading, writing, and ordering integers, rational numbers, and common irrational
numbers such as •2 and •5 and •.
Eighth grade students will:
1.2.1 read, write, and order rational numbers and commonly-used irrational numbers
1.2.2 compare rational numbers and commonly-used irrational numbers using the symbols =,
•, <, >
1.3 Applying number theory concepts (for example, primes, factors, multiples) to represent numbers in various ways.
Eighth grade students will:
1.3.1 write and use appropriately negative powers of ten (for example, 1/102 = 10-2)
1.3.2 write rational numbers in expanded form with negative powers of ten (for example,
579.42 = 5 x 100 + 7 x 10 + 9 x 1 + 4 x 10-1 + 2 x 10-2)
1.3.3 write very small rational numbers in scientific notation (for example, .00036 = 3.6 x
10-4)
1.3.4 demonstrate the meaning of an, where ‘a’ is any rational number and ‘n’ is a counting
number
1.4 Using the relationships among fractions, decimals, and percents, including the concepts of ratio and proportion, in problem-solving situations.
Eighth grade students will:
1.4.1 apply proportional reasoning to solve problems
As students in grades 5-8 extend their knowledge, what they know and are able to do includes
1.5 Developing, testing, and explaining conjectures about properties of integers and rational numbers.
Eighth grade students will:
1.5.1 demonstrate properties for rational numbers, including closure
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 •.
Eighth grade students will:
1.6.1 estimate, using appropriate techniques, determine, and, then, justify the reasonableness
of solutions to problems involving positive and negative rational 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.
Eighth grade students will:
2.1.1 represent, describe, and analyze patterns with rational numbers
2.2 Describing patterns using variables, expressions, equations, and inequalities in problemsolving situations.
Seventh grade students will:
Eighth grade students will:
2.2.1 solve problems from patterns involving 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).
Eighth grade students will:
2.3.1 in any functional relationship involving rational numbers, describe how a change in one
quantity affects the other
2.3.2 in a linear function, explain the meaning of slope as a rate of change
2.3.3 identify independent and dependent variables
2.4 Distinguishing between linear and nonlinear functions through informal investigations.
Eighth grade students will:
2.4.1 graph discrete linear and nonlinear functions
2.4.2 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
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).
Eighth grade students will:
2.5.1 translate written expressions or equations to algebraic expressions or equations, and
vice versa
2.5.2 using formal methods, solve one-step linear equations involving rational numbers
2.5.3 solve linear equations involving integers with variables and constants on both sides of
the equation
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.
Eighth grade students will:
3.1.1 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-andleaf,
histograms, scatter plots, box-and-whiskers
3.1.2 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.
Eighth grade students will:
3.2.1 state the purpose of using measures of central tendency and variability with data sets
3.2.2 create sets of data with the same mean and different ranges and compare the variability
3.2.3 in a problem-solving situation, select the most appropriate display and measure of
central tendency to solve the problem
3.3 Evaluating arguments that are based on statistical claims.
Eighth grade students will:
3.3.1 determine the improper computation of percent increase or decrease
3.3.2 recognize a misleading display of data which arises from area and volume models
3.4 Formulating hypotheses, drawing conclusions, and making convincing arguments based on data analysis.
Eighth grade students will:
3.4.1 display, analyze, and draw conclusions from a given set of data or student-generated set
of data
3.5 Determining probabilities through experiments or simulations.
Eighth grade students will:
3.5.1 perform experiments of simple independent and dependent events to estimate
probability
3.5.2 perform experiments to estimate the probability of complementary events
3.6 Making predictions and comparing results using both experimental and theoretical
probability drawn from real-world problems.
Eighth grade students will:
3.6.1 determine the probability of independent, dependent, and complementary events with
replacement and without replacement
3.6.2 analyze games of chance to determine whether they are fair or unfair; if unfair, rewrite
the rules of the game to make it fair
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).
Eighth grade students will:
3.7.1 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)
3.7.2 use Pascal’s triangle to determine how many and which outcomes occur for
independent compound events with exactly two outcomes
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.
Eighth grade students will:
4.1.1 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
4.1.2 build models of three-dimensional oblique solids
4.2.2 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
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.
Eighth grade students will:
4.3.1 identify and use correct notation for triangle congruence of Side-Side-Side, Side-Angle-
Side, and Angle-Side-Angle
4.3.2 reason informally about the relationships among angles formed by two lines cut by a
transversal and two parallel lines cut by a transversal
4.3.3 reason informally about the sum of the measures of the angles of a triangle equaling
180o
4.3.4 reason informally about the properties of the special right triangles, 30o-60o-90o and
45o-45o-90o
4.3.5 continue to reason informally about the sides and angles of congruent and similar
polygons
4.3.6 demonstrate proportional reasoning to indirectly determine lengths of segments of
similar polygons
4.4 Solving problems using coordinate geometry.
Eighth grade students will:
4.4.1 enlarge figures on a coordinate plane by rational scale factors
4.4.2 reduce figures on a coordinate plane by rational scale factors
4.4.3 determine the percent increase or decrease of perimeter and area of the enlargement or
reduction of squares, rectangles and triangles
4.4.4 describe the relationship of more than two points on the coordinate plane
4.4.5 given a distance, find pairs of points on the coordinate plane separated by that distance
4.4.6 determine the distance between a pair of points in the coordinate plane
4.5 Solving problems involving perimeter and area in two dimensions, and involving surface area and volume in three dimensions.
Eighth grade students will:
4.5.1 solve problems involving perimeter and area of trapezoids
4.5.2 solve problems involving volume of square pyramids and cones
4.5.3 solve problems involving surface area of cylinders
4.6 Transforming geometric figures using reflections, translations, and rotations to explore congruence.
Eighth grade students will:
4.6.1 determine the scale factor for dilations to illustrate similarity
4.6.2 create Escher-type tessellations to illustrate congruence
4.6.3 state the coordinates to describe the reflection of a figure across the x- and y-axes
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.
Eighth grade students will:
5.1.1 estimate the length of the sides and height of trapezoids
5.1.2 estimate the perimeter and area of trapezoids
5.1.3 continue to compare the perimeter and area of transformed geometric figures
5.1.4 estimate the volume of square pyramids and cones
5.1.5 estimate the surface area of cylinders
5.1.6 continue to estimate and use the capacity, weight, and mass measurements from
previous grades
5.1.7 estimate measures of angles
5.2 Estimating, making, and using direct and indirect measurements to describe and make comparisons.
Eighth grade students will:
5.2.1 compare the estimates and direct measurements obtained in benchmarks 5.1, 5.4, and
5.6
5.2.2 demonstrate proportional reasoning to indirectly determine lengths of segments of
similar polygons
5.3 Reading and interpreting various scales including those based on number lines, graphs, and maps.
Eighth 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
5.3.3 construct scale drawings
5.4 Developing and using formulas and procedures to solve problems involving measurement.
Eighth grade students will:
5.4.1 develop and use formulas for the perimeter and area of trapezoids using appropriate
units
5.4.2 develop and use the formula for volume of square pyramids and cones using
appropriate units
5.4.3 develop and use the Pythagorean Theorem
5.4.4 use the relationships in 30-60-90 and 45-45-90 triangles to solve problems
5.5 Describing how a change in an object’s linear dimensions affects its perimeter, area, and volume.
Eighth grade students will:
5.5.1 describe how changing the radius of a circle affects the circumference and area
5.5.2 describe how changing the height or radius of the base of a cylinder affects the volume
5.6 Selecting and using appropriate units and tools to measure to the degree of accuracy required in a particular problem-solving situation.
Eighth 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 trapezoids to the nearest sixteenth inch
and nearest millimeter
5.6.3 using a protractor, measure angles of two lines cut by a transversal and angles of two
parallel lines cut by a transversal
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.
Eighth grade students will:
6.1.1 compute percent of increase or decrease in real-world problems
6.1.2 apply proportional reasoning in problem-solving situations (for example, scale,
similarity, percentage, unit pricing, simple interest, and rate)
6.2 Constructing, using, and explaining procedures to compute and estimate with whole numbers, fractions, decimals, and integers.
Eighth grade students will:
6.2.1 demonstrate order of operations with rational numbers
6.2.2 demonstrate the meaning of the four basic operations of rational numbers
6.2.3 using paper-and-pencil, demonstrate with proficiency computation of rational numbers
6.2.4 demonstrate the inverse relationship of addition and subtraction of rational numbers
6.2.5 demonstrate the inverse relationship of multiplication and division of rational numbers
6.2.6 demonstrate multiplication of rational numbers as repeated addition
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.
Eighth 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.
Eighth 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 rational 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