National Standards for Grade 12 - Mathematics

GRADE 12

1) Number sense

a) Write, rename, represent, or compare real numbers (e.g., pi , numerical relationships using number lines, models, or diagrams.

b) Represent very large or very small numbers using scientific notation in meaningful contexts.

c) Find or model absolute value or apply to problem situations.

d) Interpret calculator or computer displays of numbers given in scientific notation.

e) Order or compare real numbers, including very large or small real numbers.

2) Estimation

a) Establish or apply benchmarks for real numbers in contexts.

b) Make estimates of very large or very small numbers appropriate to a given situation by:
• identifying when estimation is appropriate or not,
• determining the level of accuracy needed,
• selecting the appropriate method of estimation, or
• analyzing the effect of an estimation method on the accuracy of results.

c) Verify solutions or determine the reasonableness of results in a variety of situations including scientific notation, calculator, and computer results.

d) Estimate square or cube roots of numbers less than 1,000 between two whole numbers.

3) Number operations

a) Perform computations with real numbers including common irrational numbers or the absolute value of numbers.

b) Describe the effect of multiplying and dividing by numbers including the effect of multiplying or dividing a real number by:
• zero, or
• a number less than zero, or
• a number between zero and one, or
• one, or
• a number greater than one.

c) Solve application problems involving numbers, including rational and common irrationals, using exact answers or estimates as appropriate.

4) Ratios and proportional reasoning

a) Use proportions to model problems.

b) Use proportional reasoning to solve problems (including rates).

c) Solve problems involving percentages (including percent increase and decrease, interest rates, tax, discount, tips, or part/whole relationships).

5) Properties of number and operations

a) Solve problems involving factors, multiples, or prime factorization.

b) Use prime or composite numbers to solve problems.

c) Use divisibility or remainders in problem settings.

d) Apply basic properties of operations.

e) Provide a mathematical argument about a numerical property or relationship.

1) Measuring physical attributes

a) Compare objects with respect to length, area, volume, angle measurement, weight, or mass.

b) Estimate the size of an object with respect to a given measurement attribute (e.g., area).

c) Select or use appropriate measurement instrument to determine or create a given length, area, volume, angle, weight, or mass.

d) Solve mathematical or real-world problems involving perimeter or area of plane figures such as or composite figures.

e) Solve problems involving volume or surface area of rectangular solids, cylinders, prisms, or composite shapes.

f) Solve problems involving indirect measurement such as finding the height of a building by comparing its shadow with the height and shadow of a known object.

g) Solve problems involving rates such as speed or population density.

2) System of measurement

a) Select or use appropriate type of unit for the attribute being measured such as length, area, angle, time, or volume.

b) Solve problems involving conversions within the same measurement system such as conversions involving square inches and square feet.

c) Estimate the measure of an object in one system given the measure of that object in another system and the approximate conversion factor. For example:
• Distance conversion: 1 kilometer is approximately e of a mile.
• Money conversion: U.S. dollar is approximately 1.5 Canadian dollars.
• Temperature conversion: Fahrenheit to Celsius

d) Determine appropriate size of unit of measurement in problem situation involving such attributes as length, area, or volume.

e) Determine appropriate accuracy of measurement in problem sit­uations (e.g., the accuracy of each of several lengths needed to obtain a specified accuracy of a total length) and find the measure to that degree of accuracy.

f) Construct or solve problems (e.g., floor area of a room) involving scale drawings.

GRADE 12

1) Dimension and shape

a) Use two-dimensional representations of three-dimensional objects to visualize and solve problems involving surface area and volume.

b) Give precise mathematical descriptions or definitions of geometric shapes in the plane and in three-dimensional space.

c) Draw or sketch from a written description plane figures (e.g., isosceles triangles, regular polygons, curved figures) and planar images of three-dimensional figures (e.g., polyhedra, spheres, and hemispheres).

d) Describe or analyze properties of spheres and hemispheres.

2) Transformation of shapes and preservation of properties

a) Recognize or identify types of symmetries (e.g., point, line, rotational, self-congruences) of two- and three-dimensional figures.

b) Give or recognize the precise mathematical relationship (e.g., congruence, similarity, orientation) between a figure and its image under a transformation.

c) Perform or describe the effect of a single transformation on two- and three-dimensional geometric shapes (reflections across lines of symmetry, rotations, translations, and dilations).

d) Describe the final outcome of successive transformations.

e) Justify relationships of congruence and similarity, and apply these relationships using scaling and proportional reasoning.

3) Relationships between geometric figures

a) Apply geometric properties and relationships in solving multi-step problems in two and three dimensions (including rigid and non-rigid figures).

b) Represent problem situations with geometric models to solve mathematical or real-world problems.

d) Use the Pythagorean theorem to solve problems in two- or three-dimensional situations.

c) Describe and analyze properties of circles (e.g., perpendicularity of tangent and radius, angle inscribed in a semicircle).

d) Analyze properties or relationships of triangles, quadrilaterals, and other polygonal plane figures.

e) Describe or analyze properties and relationships of parallel, perpendicular, or intersecting lines, including the angle relationships that arise in these cases.

4) Position and direction

a) Describe the intersection of two or more geometric figures in the plane (e.g., intersection of a circle and a line).

b) Visualize or describe the cross section of a solid.

c) Represent geometric figures using rectangular coordinates on a plane.

d) Use vectors to represent velocity and direction.

5) Mathematical reasoning

a) Make, test, and validate geometric conjectures using a variety of methods including deductive reasoning and counterexamples.

GRADE 12

1) Data representation

Histograms, line graphs, scatter plots, box plots, circle graphs, stem and leaf plots, frequency distributions, and tables.

a) Read or interpret data, including interpolating or extrapolating from data.

b) For a given set of data, complete a graph and then solve a problem using the data in the graph (histograms, scatter plots, line graphs).

c) Solve problems by estimating and computing with univariate or bivariate data (including scatter plots and two-way tables).

d) Given a graph or a set of data, determine whether information is represented effectively and appropriately (bar graphs, box plots, histograms, scatter plots, line graphs).

e) Compare and contrast the effectiveness of different representations of the same data.

2) Characteristics of data sets

a) Calculate, interpret, or use mean, median, mode, range, interquartile range, or standard deviation.

b) Recognize how linear transfor­mations of one-variable data affect mean, median, mode, and range (e.g., effect on the mean by adding a constant to each data point).

c) Determine the effect of outliers on mean, median, mode, range, interquartile range, or standard deviation.

d) Compare two or more data sets using mean, median, mode, range, interquartile range, or standard deviation describing the same characteristic for two different populations or subsets of the same population.

e) Given a set of data or a scatter plot, visually choose the line of best fit and explain the meaning of the line. Use the line to make predictions.

f) Use or interpret a normal distri­bution as a mathematical model appropriate for summarizing certain sets of data.

g) Given a scatter plot, make decisions or predictions involving a line or curve of best fit.

h) Given a scatter plot, estimate the correlation coefficient (e.g., Given a scatter plot, is the cor­relation closer to 0, .5, or 1.0? Is it a positive or negative correlation?).

3) Experiments and samples

a) Identify possible sources of bias in data collection methods and describe how such bias can be controlled and reduced.

b) Recognize and describe a method to select a simple random sample.

c) Make inferences from sample results.

d) Identify or evaluate the charac­teristics of a good survey or of a well-designed experiment.

4) Probability

a) Analyze a situation that involves probability of independent or dependent events.

b) Determine the theoretical probability of simple and compound events in familiar or unfamiliar contexts.

c) Given the results of an experiment or simulation, estimate the probability of simple or compound events in familiar or unfamiliar contexts.

d) Use theoretical probability to evaluate or predict experimental outcomes.

e) Determine the number of ways an event can occur using tree diagrams, formulas for combinations and permutations, or other counting techniques.

f) Determine the probability of the possible outcomes of an event.

g) Determine the probability of independent and dependent events.

h) Determine conditional probabil­ity using two-way tables.

i) Interpret probabilities within a given context.

GRADE 12

1) Patterns, relations, and functions

a) Recognize, describe, or extend arithmetic, geometric progressions, or patterns using words or symbols.

b) Express the function in general terms (either recursively or explicitly), given a table, verbal description, or some terms of a sequence.

c) Identify or analyze distinguishing properties of linear, quadratic, inverse (y = k/x) or exponential functions from tables, graphs, or equations."

d) Determine the domain and range of functions given various contexts.

e) Recognize and analyze the general forms of linear, quadratic, inverse, or exponential functions (e.g., in y = ax + b, recognize the roles of a and b).

f) Express linear and exponential functions in recursive and ex­plicit form given a table or verbal description.

2) Algebraic representations

a) Translate between different rep­resentations of algebraic expressions using symbols, graphs, tables, diagrams, or written ­descriptions.

b) Analyze or interpret relationships expressed in symbols, graphs, tables, diagrams, or written descriptions.

c) Graph or interpret points that are represented by one or more ordered pairs of numbers on a rectangular coordinate system.

d) Perform or interpret transformations on the graphs of linear and quadratic functions.

e) Use algebraic properties to develop a valid mathematical argument.

f) Use an algebraic model of a situation to make inferences or predictions.

g) Given a real-world situation, determine if a linear, quadratic, inverse, or exponential function fits the situation (e.g., half-life bacterial growth).

h) Solve problems involving exponential growth and decay.

3) Variables, expressions, and operations

a) Write algebraic expressions, equations, or inequalities to represent a situation.

b) Perform basic operations, using appropriate tools, on algebraic expressions (including grouping and order of multiple operations involving basic operations, ex­ponents, roots, simplifying, and ­expanding).

c) Write equivalent forms of algebraic expressions, equations, or inequalities to represent and explain mathematical relationships.

4) Equations and inequalities

a) Solve linear, rational, or quad­ratic equations or inequalities.

b) Analyze situations or solve problems using linear or quadratic equations or inequalities sym­bolically or graphically.

c) Recognize the relationship between the solution of a system of linear equations and its graph.

d) Solve problems involving more advanced formulas [e.g., the volumes and surface areas of three dimensional solids; or such formulas as: A = P(1 + r)t, A = Pert]."

e) Given a familiar formula, solve for one of the variables.

f) Solve or interpret systems of equations or inequalities.