Utah State Standards for Mathematics: Grade 12

UT.1. Algebra 1: Students will expand number sense to understand, perform operations, and solve problems with real numbers.

1.1. Represent real numbers as points on the number line and distinguish rational numbers from irrational numbers.

1.1.a. Define a rational number as a point on the number line that can be expressed as the ratio of two integers, and points that cannot be so expressed as irrational.

1.1.b. Classify numbers as rational or irrational, knowing that rational numbers can be expressed as terminating or repeating decimals and irrational numbers can be expressed as non-terminating, non-repeating decimals.

1.1.c. Classify pi and square roots of non-perfect square numbers as irrational.

1.1.d. Place rational and irrational numbers on a number line between two integers.

1.2. Compute fluently and make reasonable estimates with rational and irrational numbers.

1.2.a. Simplify, add, subtract, multiply, and divide expressions with square roots.

1.2.b. Evaluate and simplify numerical expressions containing rational numbers and square roots using the order of operations.

1.2.c. Compute solutions to problems, represent answers in exact form, and determine the reasonableness of answers.

1.2.d. Calculate the measures of the sides of a right triangle using the Pythagorean Theorem.

UT.2. Algebra 1: Students will extend concepts of proportion to represent and analyze linear relations.

2.1. Represent and analyze the slope of a line.

2.1.a. Identify the slope of a line when given points, a graph, or an equation.

2.1.b. Identify horizontal and vertical lines given the equations or slopes.

2.1.c. Determine the effect of changes in slope or y-intercept in y = mx + b.

2.1.d. Determine and explain the meaning of slopes and intercepts using real-world examples.

2.2. Model and interpret problems having a constant rate of change using linear functions.

2.2.a. Write algebraic expressions or equations to generalize visual patterns, numerical patterns, relations, data sets, or scatter plots.

2.2.b. Represent linear equations in slope-intercept form, y = mx +b, and standard for, Ax + By = C.

2.2.c. Distinguish between linear and non-linear functions by examining a table, equation, or graph.

2.2.d. Interpret the slope of a linear function as a rate of change in real-world situations.

2.3. Represent and analyze linear relationships using algebraic equations, expressions, and graphs.

2.3.a. Write the equation of a line when given two points or the slope and a point on the line.

2.3.b. Approximate the equation of a line given the graph of a line.

2.3.c. Identify the x- and y-intercepts from an equation or graph of a line or a table of values.

2.3.d. Graph linear relations and inequalities by plotting points, by finding x- and y-intercepts, or by using the slope and any point on the line.

UT.3. Algebra 1: Students will develop fluency with the language and operations of algebra to analyze and represent relationships.

3.1. Simplify polynomials and the quotient of monomials.

3.1.a. Simplify and evaluate monomial expressions and formulas.

3.1.b. Add and subtract polynomials.

3.1.c. Multiply monomials by a polynomial.

3.1.d. Multiply binomials.

3.1.e. Simplify the quotient of monomials using positive exponents.

3.2. Solve and interpret linear equations and inequalities in various situations including real-world problems.

3.2.a. Solve single-variable linear equations and inequalities algebraically and graphically.

3.2.b. Solve real-world problems involving constant rates of change.

3.2.c. Solve equations for a specified variable.

3.2.d. Solve proportions that include algebraic first-degree expressions.

3.3. Solve and interpret pairs of linear equations and inequalities.

3.3.a. Solve systems of two linear equations graphically and algebraically with and without technology.

3.3.b. Determine the number of possible solutions for a system of two linear equations.

3.3.c. Graph a system of linear inequalities and identify the solution.

3.4. Factor polynomials with common monomial factors and factor simple quadratic expressions.

3.4.a. Find the greatest common monomial factor of a polynomial.

3.4.b. Factor trinomials with integer coefficients of the form x^2 + bx + c.

3.4.c. Factor the difference of two squares and perfect square trinomials.

3.5. Solve quadratic equations using factoring or by taking square roots.

3.5.a. Solve quadratic equations that can be simplified to the form x^2 = a where a is greater than or equal to 0 by taking square roots.

3.5.b. Solve quadratic equations using factoring.

3.5.c. Write a quadratic equation when given the solutions.

UT.4. Algebra 1: Students will understand concepts from statistics and apply statistical methods to solve problems.

4.1. Summarize, display, and analyze bivariate data.

4.1.a. Collect, record, organize, and display a set of data with at least two variables.

4.1.b. Determine whether the relationship between two variables is approximately linear or non-linear by examination of a scatter plot.

4.1.c. Characterize the relationship between two linear related variables as having positive, negative, or approximately zero correlation.

4.2. Estimate, interpret, and use lines fit to bivariate data.

4.2.a. Estimate the equation of a line of best fit to make and test conjectures.

4.2.b. Interpret the slope and y-intercept of a line through data.

4.2.c. Predict y-values for given x-values when appropriate using a line fitted to bivariate numerical data.

1.2.e. Prove lines parallel or perpendicular using slope or angle relationships.

1.3. Analyze characteristics and properties of triangles.

1.3.a. Prove congruency and similarity of triangles using postulates and theorems.

1.3.b. Prove the Pythagorean Theorem in multiple ways, find missing sides of right triangles using the Pythagorean Theorem, and determine whether a triangle is a right triangle using the converse of the Pythagorean Theorem.

1.3.c. Prove and apply theorems involving isosceles triangles.

1.3.d. Apply triangle inequality theorems.

1.3.e. Identify medians, altitudes, and angle bisectors of a triangle, and the perpendicular bisectors of the sides of a triangle, and justify the concurrency theorems.

1.4. Analyze characteristics and properties of polygons and circles.

1.4.a. Use examples and counterexamples to classify subsets of quadrilaterals.

1.4.b. Prove properties of quadrilaterals using triangle congruence relationships, postulates, and theorems.

1.4.c. Derive, justify, and use formulas for the number of diagonals, lines of symmetry, angle measures, perimeter, and area of regular polygons.

1.4.d. Define radius, diameter, chord, secant, arc, sector, central angle, inscribed angle, and tangent of a circle, and solve problems using their properties.

1.4.e. Show the relationship between intercepted arcs and inscribed or central angles, and find their measures.

1.5. Perform basic geometric constructions, describing and justifying the procedures used.

1.5.a. Investigate geometric relationships using constructions.

1.5.b. Copy and bisect angles and segments.

1.5.c. Construct perpendicular and parallel lines.

1.5.d. Justify procedures used to construct geometric figures.

1.5.e. Discover and investigate conjectures about geometric properties using constructions.

1.6. Analyze characteristics and properties of three-dimensional figures.

1.6.a. Identify and classify prisms, pyramids, cylinders and cones based on the shape of their base(s).

1.6.b. Identify three-dimensional objects from different perspectives using nets, cross sections, and two-dimensional views.

1.6.c. Describe the symmetries of three-dimensional figures.

1.6.d. Describe relationships between the faces, edges, and vertices of polyhedra.

4.1.d. Calculate or estimate the area of an irregular region.

4.1.e. Find the length of an arc and the area of a sector when given the angle measure and radius.

4.2.d. Solve problems involving geometric probability.

1.1.e. Simplify algebraic expressions involving negative and rational exponents.

2.2.e. Identify the domain and range of a function resulting from the combination or composition of functions.

2.4. Define and graph logarithmic functions and use them to solve problems in mathematics and real-world contexts.

2.4.a. Relate logarithmic and exponential functions.

2.4.b. Simplify logarithmic expressions.

2.4.c. Convert logarithms between bases.

2.4.d. Solve exponential and logarithmic equations.

2.4.e. Graph logarithmic functions.

2.4.f. Solve problems involving growth and decay.

3.3.d. Find the area of a sector in a circle using radian measure.

2.1.e. Write an equation with given solutions.

2.3.e. Relate the graphical representation of discontinuities and end behavior to the concept of limit.

3.2.e. Define a curve parametrically and draw parametric graphs.

4.1.f. Calculate probabilities in real problems using sampling distributions.