Integrated Math I (Honors)

Students broaden their understanding of equations and inequalities by solving one-variable equations with unknown coefficients, two-variable linear equations and inequalities, systems of equations, and nonlinear expressions and equations. They will expand their understanding of units to incorporate algebraic methods.

This course begins students' exploration of the world of functions. They will explore key concepts such as domain, range, roots, extrema, and end behavior. Furthermore, they will apply this newfound knowledge to study specific function types, namely exponential, absolute value, and piecewise functions, creating a solid foundation for analyzing more advanced functions in future courses.

Students will encounter polynomials for the first time. They will become proficient in terminology associated with one-variable polynomials and master fundamental polynomial operations such as addition, subtraction, multiplication, and factoring.

Building upon their prior understanding of sequences, students will learn to work with formal sequence notation, delve into recursive sequences, and master arithmetic sequences.

This course introduces various critical geometric concepts. Students will become adept at performing multi-step geometric transformations in the coordinate plane and will be able to describe their functional representations. They will also study essential geometric principles, such as congruence, the Pythagorean theorem, special right triangles, circles, and the distance formula.

For the first time, the concept of a rigorous mathematical proof is introduced. Students begin by exploring proof fundamentals, such as postulates, definitions, theorems, and the properties of congruence. Once mastered, they will use these ideas to prove important geometric theorems. Opportunities to demonstrate and develop their proof-writing skills are ongoing throughout this course.

Students will learn the fundamentals of linear correlation, emphasizing the critical distinction between correlation and causation.

By the end of this comprehensive course, students will have significantly enhanced their mathematical skills and knowledge, equipping them for more advanced studies in mathematics.

Equations & Inequalities

• Solving linear equations containing unknown coefficients.
• Using linear equations to model and solve real-world multi-step problems involving speed, distance, time, work, and mixtures.
• Solving compound inequalities, systems of inequalities, and graphing their solutions,
• Using inequalities to model situations given in context.
• Interval notation, including unbounded intervals and computing unions and intersections of intervals.
• Solving polynomial equations using the nth root method.

Two-Variable Equations & Inequalities

• Constructing equations of lines in point-slope and standard form and determining properties of lines given in these forms.
• Modeling with two-variable linear equations and analyzing and interpreting their graphs,
• Modeling physical phenomena using speed-time and distance-time graphs and using these models to deduce properties such as speed, acceleration, and distance traveled.
• Solving two-variable systems of linear equations using substitution and elimination, including cases with fractional and decimal coefficients.
• Determining when systems of equations have no solutions or infinitely many solutions and correctly applying consistency and dependency terminology.
• Relating the consistency and dependency of a linear system to the intersection of two lines.
• Modeling and solving contextual problems using systems of equations.
• Approximating the solution to a nonlinear system of equations using absolute errors and upper and lower bounds.
• Solving systems of two-variable linear inequalities.
• Using systems of inequalities to model situations in context and interpreting solutions in terms of feasible regions.

Units

• Performing unit conversions using algebraic techniques, including two-step conversions and converting between area units.
• Determining units associated with variables in formulas, including rates of change, and working with standard and non-standard units.
• Understanding and applying the concepts of direct and inverse variation, including modeling and solving real-world and mathematical problems.

Radicals & Rationals

• Manipulating and simplifying radical expressions using the rules of radicals.
• Simplifying simple rational expressions by factoring.
• Solving simple rational equations using various techniques.

Functions

• Interpreting and evaluating functions represented using algebraic notation, mapping diagrams, tables, and graphs.
• Determining whether a relation is a function by applying the vertical line test.
• Understanding the concepts of domain, range, roots, global extrema, and end behavior of functions and describing these properties given some simple functions.
• Identifying the intervals where a function is increasing or decreasing, including the distinction between strict and non-strict increase or decrease.

Absolute Value

• Applying the rules of absolute value.
• Solving simple absolute value equations and inequalities.

Exponents

• Simplifying algebraic expressions using the rules of exponents.
• Solving exponential equations with integer and fractional solutions.
• Constructing expressions and functions that model exponential growth and decay situations given in context.
• Comparing exponential and linear growth and decay.
• Graphing elementary growth and decay functions.

Polynomials

• Defining polynomials, the degree of a polynomial, monomials, binomials, and trinomials.
• Simplifying, adding, subtracting, and multiplying polynomials.
• Calculating a binomial expansion using Pascal's triangle.
• Factoring quadratic polynomials using greatest common factors, perfect squares, difference of squares, and grouping.

Sequences

• Calculate a sequence's terms explicitly or recursively and work with Fibonacci sequences.
• Solving problems with arithmetic sequences and using them to model situations in context.

Geometry

• Proving the vertical angles theorem.
• Proving the alternate and consecutive interior and exterior angle theorems and their converses.
• Proving the linear pair perpendicular theorem, the perpendicular bisector theorem, and their converses.
• Applying translations, reflections, dilations, stretches, and composite transformations to two-dimensional figures.
• Using functions to represent transformations in the coordinate plane and using them to find the image of an object.
• Explaining how rigid motions are used to demonstrate that two figures are congruent.
• Understanding and applying the ASA, AAS, SAS, SSS, and HL congruence criteria.
• Understanding and applying the reflexive, symmetric, and transitive properties of congruence.
• Proving that two triangles are congruent and applying CPCTC.
• Understanding and applying the Pythagorean Theorem and its converse.
• The 45-45-90 and 30-60-90 special right triangles.
• Calculating areas of 45-45-90, 30-60-90, and equilateral triangles.
• Understanding medians and centroids of triangles, applying the centroid ratio theorem, and finding the coordinates of a triangle's centroid.
• Applying the exterior angle theorem.
• Circle fundamentals: Radii, diameters, chords, interiors and exteriors, circumference, arcs, sectors, and segments.
• Calculating circumferences and areas of circles and circular sectors and solving related problems.
• Working with parallel and perpendicular lines in the coordinate plane.
• The distance formula, its derivation, and using it to find the shortest distance between a point and a line.
• Partitioning a directed line segment, solving related problems, and applying the midpoint formula.
• Calculating perimeters and areas of polygons in the coordinate plane.

Statistics

• Scatter plots, the linear correlation between two data sets, making predictions using trend lines, and distinguishing between correlation and causation.