GLE-Math912
COLORADO MODEL CONTENT
STANDARDS FOR MATHEMATICS
9^{th}-12^{th} 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 real numbers, absolute value, and scientific notation using physical materials and technology in problem-solving situations.
High school students will:
1.1.1 demonstrate the relationships among subsets of the real number system, including
counting, whole, integer, rational, and irrational numbers, to one another
1.1.2 compare and order sets of real numbers
1.1.3 demonstrate the meaning of absolute value as distance on the number line
1.1.4 convert repeating decimals to fractions, and vice versa
1.1.5 use very large and very small numbers in real-life situations to solve problems (for
example, understanding the size of the national debt)
1.1.6 express numbers in scientific notation, and vice versa
1.2 Developing, testing, and explaining conjectures about properties of number systems and sets of numbers.
High school students will:
1.2.1 demonstrate that the field properties, including closure, commutative, associative,
distributive, identity, and inverse properties, apply to the real number system
1.2.2 verify conjectures about number theory concepts applied to the real number system (for
example, the sum of two odd numbers is even)
1.2.3 verify and apply the laws of exponents
1.3 Using number sense to estimate and justify the reasonableness of solutions to problems involving real numbers.
High school students will:
1.3.1 estimate, using appropriate techniques including rounding, solutions to problems
involving real numbers
1.3.2 determine and justify the reasonableness of solutions obtained using both estimation
and exact computations
1.3.3 appropriately apply strategies of estimation and/or exact computation in problemsolving
situations
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 Modeling real-world phenomena (for example, distance-versus-time relationships,
compound interest, amortization tables, mortality rates) using functions, equations,
inequalities, and matrices.
High school students will:
2.1.1 use matrices to model real-world problems
2.1.2 use a variety of functions, linear and nonlinear, to represent real-world and
mathematical relationships
2.1.3 use a variety of equations and inequalities to represent real-world and mathematical
relationships
2.1.4 use sequences to represent real-world applications
2.2 Representing functional relationships using written explanations, tables, equations, and graphs, and describing the connections among these representations.
High school students will:
2.2.1 express relations in a variety of forms (for example, numerical, graphic, verbal and
symbolic
2.2.2 convert from one form to another
2.2.3 describe a real-world situation using expressions, equations, inequalities, or matrices
2.2.4 interpret a graphical representation of a real-world situation
2.3 Solving problems involving functional relationships using graphing calculators and/or computers as well as appropriate paper-and-pencil techniques.
High school students will:
2.3.1 solve problems involving functions and relations using calculators, graphs, tables, and
algebraic methods
2.3.2 solve simple systems of equations and inequalities using algebraic or graphical methods
2.3.3 solve literal equations (for example, solve for p in the equation I = prt)
2.4 Analyzing and explaining the behaviors, transformations, and general properties of types of equations and functions (for example, linear, quadratic, exponential).
High school students will:
2.4.1 identify and interpret x- and y-intercepts in the context of a problem
2.4.2 recognize when a relation is a function and determine its domain and range
2.4.3 demonstrate horizontal and vertical translations on graphs of functions and their
meanings in the context of a problem
2.5 Interpreting algebraic equations and inequalities geometrically and describing geometric relationships algebraically.
High school students will:
2.5.1 graph solutions to equalities or inequalities in one- and two-dimensions
2.5.2 use the Pythagorean Theorem
2.5.3 express perimeter, area, and volume relationships of geometric figures algebraically
2.5.4 use algebraic equations to describe properties of geometric figures such as square,
rhombus, triangle, and parallelogram
2.5.5 describe properties of lines and segments (e.g., slope, length, midpoint) algebraically
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 Designing and conducting a statistical experiment to study a problem, and interpreting and communicating the results using the appropriate technology (for example, graphing calculators, computer software)
High school students will:
3.1.1 determine the type of data (that is, categorical or numerical) to be collected in the
design of a statistical study
3.1.2 determine the factors which may affect the outcome of the survey (for example, biased
questions or collection methods)
3.1.3 draw conclusions about a large population based upon a properly chosen random
sample
3.1.4 select and use an appropriate display to represent and describe a set of data (for
example, scatter plot, line graph, histogram)
3.2 Analyzing statistical claims for erroneous conclusions or distortions.
High school students will:
3.2.1 check a graph, table, or summary for misleading characteristics
3.2.2 recognize the misuse of statistical data in written arguments
3.2.3 describe how data can be interpreted in more than one way or be used to support more
than one position in a debate
3.2.4 describe how the responses to a survey can be affected by the way the questions are
phrased and/or by the reader’s bias
3.3 Fitting curves to scatter plots using informal methods or appropriate technology to
determine the strength of the relationship between two data sets and to make predictions.
High school students will:
3.3.1 graph data sets, create a scatter plot, and identify the control (independent) variable and
the dependent variable
3.3.2 determine a line of best fit from a scatter plot using visual techniques
3.3.3 identify the relationship (correlation) between variables as to direction and strength of
the correlation
3.3.4 predict values using the line of best fit
3.3.5 show how extrapolation may lead to faulty conclusions
3.3.6 use appropriate technology (for example, graphing calculator) as it relates to scatter
plots, regression lines, and correlation
3.3.7 recognize which model, linear or nonlinear, fits the data most appropriately
3.4 Drawing conclusions about distributions of data based on analysis of statistical summaries (for example, the combination of mean and standard deviation, and differences between the mean and median).
High school students will:
3.4.1 differentiate between mean, median, and mode and demonstrate the appropriate use of
each
3.4.2 use technology to find the standard deviation
3.4.3 recognize and classify various types of distributions (for example, bimodal, skewed,
uniform, binomial, and normal)
3.4.4 demonstrate how the mean and standard deviation affect the location and shape of the
normal curve
3.4.5 demonstrate how outliners might affect various representations of data, measures of
central tendency, and standard deviation
3.5 Using experimental and theoretical probability to represent and solve problems involving uncertainty (for example, the chance of playing professional sports if a student is a successful high school athlete).
High school students will:
3.5.1 determine the probability of an identified event using the sample space
3.5.2 distinguish between experimental and theoretical probability and use each appropriately
3.5.3 differentiate between independent and dependent events to calculate the probability in
real-world situations
3.5.4 use a complementary event to solve a problem
3.5.5 apply the addition rule or multiplication rule appropriately in probability problemsolving
situations
3.5.6 use a geometric model to represent probabilities (for example, the probability of hitting
the bull’s eye region in a target)
3.6 Solving real-world problems with informal use of combinations and permutations (for example, determining the number of possible meals at a restaurant featuring a given number of side dishes).
High school students will:
3.6.1 differentiate between and calculate permutations and combinations
3.6.2 apply the fundamental counting rule, a permutation, or a combination appropriately
3.6.3 determine probabilities of real-world problems using appropriate counting techniques
3.6.4 use tree diagrams, lists, and/or other methods to show outcomes
Standard 4:
Students use geometric concepts, properties, and relationships in problemsolving situations and communicate the reasoning used in solving these problems.
4.1 Finding and analyzing relationships among geometric figures using transformations (for example, reflections, translations, rotations, dilations) in coordinate systems.
High school students will:
4.1.1 describe and apply the properties of similar and congruent figures
4.1.2 solve problems involving symmetry and transformations
4.1.3 use coordinate geometry and/or tessellations to solve problems using geometric
transformations
4.1.4 describe cylinders, cones, and spheres that result from the rotation of rectangles,
triangles, and semicircles about a line
4.2 Deriving and using methods to measure perimeter, area, and volume of regular and
irregular geometric figures.
High school students will:
4.2.1 use the Pythagorean Theorem and its converse to solve real-world problems
4.2.2 use known properties and formulas of polygons to find areas of regular and irregular
figures
4.2.3 use known properties and formulas of geometric solids to find volumes and surface
areas of regular and irregular geometric solids
4.2.4 use known properties of geometric figures in real-world applications
4.3 Making and testing conjectures about geometric shapes and their properties, incorporating technology where appropriate.
High school students will:
4.3.1 make conjectures for properties of geometric figures and uses inductive and/or
deductive reasoning to verify those conjectures
4.3.2 use a formal process to prove geometric concepts (for example, direct, indirect,
paragraph, or verbal proofs, flow charts, or constructions)
4.4 Using trigonometric ratios in problem-solving situations (for example, finding the height of a building from a given point, if the distance to the building and the angle of elevation are known).
High school students will:
4.4.1 use right triangle trigonometry to solve real-world problems
4.4.2 use properties of special right triangles to solve real-world problems
4.4.3 recognize the relationship between slope and the tangent ratio
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 Measuring quantities indirectly using techniques of algebra, geometry, or trigonometry.
High school students will:
5.1.1 use appropriate measurements to solve problems indirectly (for example, find the
height of a flag pole using similar triangles)
5.1.2 use measurements to solve real-world problems involving rate of change (for example,
distance traveled using rate and time)
5.1.3 given the rate of change, model real-world problems algebraically or graphically
5.1.4 describe how changing the measure of one attribute of a geometric figure affects the
other measurements
5.2 Selecting and using appropriate techniques and tools to measure quantities in order to achieve specified degrees of precision, accuracy, and error (or tolerance) of measurements.
High school students will:
5.2.1 solve real-world problems involving multiple dimensions and express them using
appropriate units of measurements
5.2.2 given commonly-used multi-dimensional figures, decide what units and measurements
need to be taken and what instruments are necessary to achieve a specified degree of
accuracy
5.2.3 given a commonly-used three-dimensional figure, select the appropriate units of
measurement to determine volume and surface area
5.2.4 find the distance between a pair of points in a coordinate plane using the distance
formula
5.2.5 compare and contrast the concepts of precision, accuracy, and error of measurement
5.2.6 apply precision, accuracy and error of measurement to solve real-world problems
5.3 Determining the degree of accuracy of a measurement (for example, by understanding and using significant digits).
High school students will:
5.3.1 use and understand significant digits
5.3.2 in the context of a given problem, determine the accuracy required in the measurements
to produce an answer with the appropriate number of significant digits
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 ratios, proportions, and percents in problem-solving situations.
High school students will:
6.1.1 convert from one set of units to another (for example, feet/minutes to miles/hour)
6.1.2 solve a direct variation problem with proportions
6.1.3 compute percent increases and decreases
6.2 Selecting and using appropriate methods for computing with real numbers in problem solving situations from among mental arithmetic, estimation, paper-and-pencil, calculator, and computer methods, and determining whether the results are reasonable.
High school students will:
6.2.1 apply appropriate computational methods to solve multi-step problems involving real
numbers
6.2.2 apply inverse operations of arithmetic and algebraic operations to solve problems
involving real numbers
6.2.3 determine the reasonableness of an answer
6.2.4 solve problems involving very large and very small numbers using scientific notation
6.2.5 apply numerical, graphical, or symbolic methods to solve problems involving real
numbers and communicate with appropriate mathematical language
6.3 Describing the limitations of estimation and assessing the amount of error resulting from estimation within acceptable tolerance limits.
High school students will:
6.3.1 determine when estimation is an appropriate method to solve a problem and describe
what error might result from this estimate
6.3.2 demonstrate an appropriate upper/lower limit on an estimate