Number Sense

Recognize and write 0–100,000,000,000 as numerals and words; Roman numerals I–C; Place value: ten thousandths to hundred billions; comparing; expanded form; even/odd, positive/negative, prime/composite numbers; number line; expressions and equations; Part-whole relationships; inverse operations

Addition & Subtraction

Basic facts (fact families and other strategies); nine-digit column addition; Mental compensation; Properties: commutative, associative, identity; zero principle

Multiplication

Basic facts (fact families and other strategies); multiples; repeated addition; vertical form; one to three digits times two to four digits; renaming; factor trees; prime/composite numbers; GCF; LCM; Properties: commutative, associative, identity, zero, distributive

Division

Basic facts (fact families and other strategies); equal sets; measurement/partition; fraction form; equations; mental division by 10; two to four digits by one or two digits; remainders; mixed number and decimal quotients; decimal equivalent for a fraction

Equations

Inverse operations; compensation; simplify expressions; evaluate expressions; if-then statements; solve for an unknown

Pre-Algebra

Missing addend/subtrahend/minuend/factor; add, subtract, multiply, and divide integers; positive/negative numbers; Variables in expressions and equations; solve for a variable in an equation

Fractions

Part of a whole; part of a set; equivalent fractions; comparing; ordering; lowest terms; renaming improper fractions and mixed numbers; Add, subtract, multiply, and divide like and unlike fractions; cross-products; cancellation; pictorial representations or journal entries to express understanding of operations

Decimals

Read and write tenths, hundredths, thousandths; write as fractions and mixed numbers; comparing; ordering; renaming to thousandths; word forms; Add and subtract; Multiply a decimal by a whole number, by a decimal, and by a power of 10; Divide a decimal by a one-digit whole number; divide a whole number by a whole number with a decimal quotient; divide to rename a fraction as a decimal; divide a decimal by a power of 10

Geometry

Points; lines: parallel, intersecting, perpendicular; line segments; rays; symmetry; similar/congruent; constructions; angles: right, acute, obtuse; measure angles; sum of angles in a triangle = 180°; Plane figures: regular/irregular; polygons; triangles: scalene, isosceles, equilateral; quadrilaterals; parallelograms; transformations: translation, reflection, rotation; perimeter; area; Solid figures: sphere, cone, cylinder, polyhedron, cube, prism, pyramid; face, edge, vertex, curved surface; nets; surface area; volume; Circle: center point, radius, diameter, chord, central angles, circumference

Estimating

Round whole numbers and decimals to a given place; round fractions to the nearest half or whole; front-end estimation; Round to estimate a sum, difference, product, and quotient

Measurement

Length; capacity; weight; mass; Temperature: Fahrenheit; Celsius; Time: elapsed time; time zones; timeline; 24-hour clock; wages; Rename within the metric and customary systems to add, subtract, multiply, and divide

Problem Solving

Graphs; tables; charts; schedule; map skills; probability; money; Venn diagram; Using a problem-solving plan; multistep problems; problems with too little or too much information; group planning; Strategies: patterns; logic; guess and check; diagram/model; simpler problem; formulas; working backwards; writing an equation

Statistics & Graphs

Pictograph; bar/double bar graph; line/double line graph; histogram; circle graph; coordinate graph; Tables; charts; tallies; frequency table; line plot; stem-and-leaf plot; box-and-whisker plot; Mean, median, mode; range; frequency; scale; interval

Ratios, Proportions, Percents

Equivalent ratios; unit rate; scale drawings; map scale; Percents as fractions and decimals; writing fractions and decimals as percents; comparing percents to decimals and fractions; finding the percent of a number; determining the sales price given a discount as a percent; Proportionate geometric figures

Additional content

Introduction of speed, distance, and time; unit multipliers; comparing types of graphs, independent and dependent events; patterns; squares and square roots; Roman numerals through M (1,000); additional multistep problems

Whole numbers and decimals: ordering, rounding and comparing; operations and estimating; exponents; roots of perfect squares; approximating square roots; order of operations

Integers: numbers sets and subsets; opposite numbers and absolute value; ordering and rounding; operations; order of operations involving integers

Numerical and Algebraic Expressions: variables and expressions; properties of addition and multiplication; factoring common factors; combining like terms; adding linear expressions; properties of exponents; scientific notation

Fraction theory: divisibility; factors; prime and composite numbers; GCF and LCM; renaming fractions; rational numbers; comparing and ordering rational numbers

Fractions: operations; order of operations involving rational numbers; evaluating algebraic expressions involving rational numbers

Algebra: Writing equations; solving one- and two-step equations; solving equations with grouping symbols; solving one- and twostep inequalities

Ratios and proportion: ratios and rates; simplifying complex ratios; solving proportions; scale models; proportional relationships

Percents: proportions and percent; percent equations; enlargements and reductions; percent change; sales tax; discounts; sale price; simple interest

Measures: customary units of length, capacity, and weight; SI (metric) units of length, capacity, and mass; relating metric and customary units; converting rates

Introduction to Geometry: basic terms; measuring angles; pairs of angles; perpendicular and parallel lines; characteristics and area of triangles and quadrilaterals; exploring triangle theorems

Exploring area and volume: circles; areas of similar polygons; surface area of prisms, cylinders, and pyramids; volume of prisms and cylinders

Probability: simple events; theoretical and experimental probabilities; compound events: simulations

Statistics: populations and samples; descriptive measures (mean, median and mode); mean absolute deviation; comparing populations using line graphs, histograms, box-and-whisker plots, stem-and-leaf plot; visualizing data

Relations and functions: coordinate plane; functions and function rules; graphing linear functions; slope; sequences

Integers: absolute value; operations, exponents, roots, and their properties; order of operations

Expressions: evaluating and simplifying expressions; translating word phrases; scientific notation; estimating

Equations: solving one- and two-step equations; simplifying; solving linear inequalities; applying equations and inequalities

Rational expressions: prime factorization; GCF and LCM; rational numbers; decimal equivalents; converting repeating decimals to fractions; ratios and proportions; the real number system

Operations with rational numbers: evaluating and simplifying expressions; solving equations involving rational numbers; operations with scientific notation

Percents: solving percent equations; applying percents; discount, markup, tips, commission, simple and compound interest; percent change; scales

Applying equations: equations with variables on both sides; identities and contradictions; writing and solving equations and inequalities to solve problems; equations with powers, radical equations

Functions: illustrating relations; defining and graphing functions; slope; writing linear equations; proportional relationships; graphing linear inequalities

Systems of linear equations: solving linear systems by graphing, substitution, and elimination; special cases of linear systems

Geometry: angles; polygons; the Pythagorean Theorem; distance and midpoint formulas; congruence and similarity; translation, reflection, rotation, and dilation transformations

Perimeter, area, and volume: perimeter and circumference; areas of quadrilaterals, triangles, circles, and composite regions; ratios of lengths and areas of similar regions; surface areas and volumes of prisms, cylinders, pyramids, cones, spheres, and composite solids

Statistics and probability: measures of central tendency and variation; representing data; distribution tables; scatterplots and trend lines; two-way tables; probabilities of simple and compound events

Expressions: review of real numbers, number lines, absolute value, arithmetic operations of integers and rational numbers, exponents, using variables, algebraic expressions, and formulas, and order of operations

Solving equations: writing and solving linear equations, solving literal equations and proportions; real-world applications involving similar figures, percentages, money, motion, and mixtures

Solving inequalities: linear inequalities, including conjunctions and disjunctions; absolute-value equations and inequalities

Functions: representing relationships between data, using graphs, equations, and tables; direct and inverse variations; arithmetic sequences; graphing absolute value functions

Linear functions: graphs, slopes, and intercepts of linear equations; determining the equation of a line; parallel and perpendicular lines; correlation and lines of fit

Linear Systems: solving systems graphically, by substitution, and by elimination; applications of systems, graphing linear inequalities and systems of inequalities

Exponents: products, quotients, and powers of exponential expressions; scientific notation; graphing exponential functions; exponential growth and decay; geometric sequences

Polynomial Operations: classification, evaluation, operations, special patterns

Factoring Polynomials: common monomials, trinomials, special patterns

Radicals: simplification and operations with radicals; Pythagorean theorem, distance and midpoint formulas; solving radical equations and graphing radical functions; applications

Quadratic equations and functions: solving by factoring, taking roots, completing the square, and the quadratic formula; graphing parabolas and finding zeros; applications

Rational expressions and equations: simplification and operations with rational expressions; solving rational equations; applications (Each chapter includes a feature on sequences and graphing calculator tips.)

Biblical Worldview Shaping

• Design—Evaluating conclusions about design in the world based on geometric patterns
• Ethics—Applying geometric relationships to help others
• Foundations—Evaluating competing approaches to defining knowledge
• Modeling—Formulating a biblical view of geometrical modeling
• Reasoning—Applying reasoning to form true statements

Number and Quantity

• Vector and matrix quantities—Representing the magnitude and direction of vectors; Multiplying a vector by a matrix to produce another vector; Using matrices as transformations of the plane

Algebra

• Equations and inequalities—Rearranging formulas to emphasize different unknowns

Functions

• Linear, quadratic, and exponential models—Constructing linear and exponential functions
• Trigonometric functions—Relating radians and arcs on the unit circle; Relating trigonometric functions and the unit circle; Determining the values of sine, cosine, and tangent; Explaining the symmetry and periodicity of trigonometric functions by using the unit circle

Geometry

• Congruence—Defining foundational geometric terms; Representing and comparing transformations in the plane; Describing transformations that carry a polygon onto itself; Defining transformations; Graphing transformations; Determining and proving the congruence of figures; Relating triangle congruence and rigid motions; Proving theorems about lines and angles; Proving theorems about triangles; Proving theorems about parallelograms; Making geometric constructions
• Similarity, right triangles, and trigonometry—Verifying the properties of dilations; Determining the similarity of figures; Proving theorems involving similarity; Defining trigonometric ratios for acute angles; Solving right triangles by using the Pythagorean Theorem; Deriving the formula for the area of a triangle; Using the Law of Sines and Cosines
• Circles—Proving the similarity of all circles; Relating inscribed angles, radii, and chords; Constructing inscribed and circumscribed circles of a triangle; Proving properties of angles for a quadrilateral inscribed in a circle; Constructing tangent lines to the circle; Relating the length of an arc to the radius
• Expressing geometric properties with equations—Deriving the equation of a circle; Using coordinates to prove simple geometric theorems algebraically; Proving and using the slope criteria for parallel and perpendicular lines; Partitioning line segments; Using coordinates to compute perimeters and areas of polygons
• Using geometric measurement and dimension—Explaining circumference, area, and volume formulas; Using volume formulas to solve problems; Relating two-dimensional and three-dimensional objects
• Modeling with geometry—Using geometric shapes, their measures, and their properties to describe objects; Applying concepts of density in modeling situations; Applying geometric methods to solve design problems

Statistics and Probability

• Interpreting categorical and quantitative data—Using the mean and standard deviation to estimate population percentages

STEM

• Following the Engineering Design Process to solve a problem—Building and exploring Euclidean and non-Euclidean surfaces; Designing and building packaging to protect waffle cones in shipment

Biblical Worldview Shaping

• Foundations—Constructing a biblically faithful view of the discipline of mathematics
• Reasoning—Evaluating the role of reasoning in solving and verifying algebraic solutions
• Design—Describing with algebra God’s orderly design of the world
• Modeling—Analyzing the effectiveness and limitations of algebraic models
• Ethics—Using algebraic problemsolving to serve others

Number and Quantity

• The real number system
  • Explaining the meaning of rational exponents
  • Using the properties of exponents to rewrite expressions with radicals and rational exponents
  • Using the properties of rational and irrational numbers
• The complex number system
  • Finding and using conjugates of complex numbers
  • Solving quadratic equations with complex solutions
  • Extending polynomial identities to complex numbers
  • Proving the Fundamental Theorem of Algebra for quadratic polynomials
• Vector and matrix quantities
  • Finding the components of a vector
  • Solving problems involving quantities represented by vectors
  • Multiplying a vector by a scalar
  • Using matrices to represent and manipulate data
  • Multiplying matrices by scalars
  • Adding, subtracting, and multiplying matrices of appropriate dimensions
  • Explaining the operation properties used in adding and multiplying matrices
  • Multiplying a vector by a matrix to produce another vector
  • Using matrices as transformations of the plane

Algebra

• Expressions
  • Interpreting expressions that represent a quantity
  • Rewriting expressions on the basis of their structure
  • Factoring and solving quadratic expressions by factoring
  • Finding the value of quadratic expressions by completing the square
  • Simplifying and evaluating expressions with exponents
  • Finding the sum of a finite geometric series
  • Performing arithmetic operations on polynomials
  • Applying the Remainder Theorem to numbers and polynomials
  • Finding zeros of polynomials to use for graphing functions
  • Using polynomial identities to describe numerical relationships
  • Rewriting simple rational expressions in different forms
  • Performing operations on rational expressions
• Equations and inequalities
  • Creating and using equations and inequalities in one variable to solve real-world problems
  • Creating and graphing equations in two or more variables
  • Representing constraints by equations and inequalities or systems of equations and inequalities
  • Rearranging formulas to emphasize different unknowns
  • Explaining the progression of steps in solving a simple equation
  • Solving simple rational and radical equations in one variable
  • Solving linear equations and inequalities in one variable
  • Solving quadratic equations in one variable
  • Solving systems of linear equations
  • Representing a system of linear equations as a single matrix equation
  • Finding the inverse of a matrix
  • Representing and solving equations and inequalities graphically
• Linear equations
  • Solving equations and inequalities
  • Absolute value equations and inequalities
  • Distance on number lines
  • Word problems
  • Compound inequalities

Functions

• Interpreting functions
  • Relating domains and ranges
  • Using and interpreting function notation to evaluate functions
  • Relating sequences and functions
  • Graphing and interpreting graphs of functions
  • Calculating and interpreting a function’s average rate of change
  • Writing a function in equivalent forms
  • Comparing properties of functions expressed in different ways
• Building functions
  • Writing a function that describes a relationship between two quantities
  • Writing arithmetic and geometric sequences
  • Identifying the effect of transformations on graphs
  • Finding inverse functions
• Linear, quadratic, and exponential models
  • Modeling situations with linear functions and exponential functions
  • Constructing linear and exponential functions
  • Relating linear growth and exponential growth
  • Expressing and evaluating logarithms
  • Solving functions in real-world problems
• Trigonometric functions
  • Determining the values of sine, cosine, and tangent
  • Modeling periodic phenomena with trigonometric functions
  • Applying trigonometric identities to solve problems

Geometry

• Congruence
  • Graphing transformations
  • Relating triangle congruence and rigid motions
  • Proving theorems about triangles
  • Making geometric constructions
• Similarity, right triangles, and trigonometry
  • Determining the similarity of figures
  • Proving theorems involving similarity
  • Defining trigonometric ratios for acute angles
  • Relating the sine and cosine of complementary angles
  • Solving right triangles by using the Pythagorean Theorem
  • Deriving the formula for the area of a triangle
  • Proving and using the laws of sines and cosines
• Expressing geometric properties with equations
  • Deriving the equation of a parabola
  • Deriving the equations of ellipses and hyperbolas

Statistics and Probability

• Interpreting categorical and quantitative data
  • Representing data with plots on the real number line
  • Using appropriate statistics to compare center and spread
  • Interpreting differences in shape, center, and spread
  • Using the mean and standard deviation to estimate population percentages
  • Summarizing and interpreting data in two-way frequency tables
  • Representing and relating data on two variables on a scatterplot
  • Relating the slope and intercept to data
  • Computing and interpreting the correlation coefficient of a linear fit
  • Distinguishing between correlation and causation
• Making inferences and justifying conclusions
  • Evaluating random processes underlying statistical experiments
  • Relating randomization and the purposes of and differences among sample surveys, experiments, and observational studies
  • Estimating a population mean or proportion from a sample survey
  • Comparing two treatments using data from a randomized experiment
  • Evaluating reports based on data
• Conditional probability and the rules of probability
  • Distinguishing independence and conditional probability
  • Using independence and conditional probability to interpret data
  • Using the rules of probability to compute probabilities of compound events in a uniform probability model
• Using probability to make decisions
  • Calculating and using expected values
  • Using probability to evaluate the outcome of decisions

Technology

• Graphing calculator tips

STEM

• Following the Engineering Design Process to solve a problem
  • Designing a wind turbine
  • Creating a tornado simulation in a bottle

Analyzing functions

Relations; linear, quadratic, power, and piecewise functions; continuity, transformations, and operations of functions, parametric representations and modeling with functions

Radical, polynomial, and rational functions

Describing zeros, asymptotes, and end behavior of radical, polynomial, and rational functions and solving related equations and inequalities

Exponential and logarithmic functions

Graphing, applying properties, solving equations, and modeling

Trigonometric functions

Angle and arc measures; trigonometric functions in a right triangle, for other angles, and of real numbers; graphs of trig functions; inverse trig functions

Trigonometric identities and equations

Derive and verify identities; use identities to solve equations; derive and apply the law of sines and law of cosines

Vectors, polar graphs, and complex numbers

Describe and perform operations on 2-D and 3-D vectors; graph polar coordinates and equations; represent and perform operations on complex numbers in polar form

Systems and matrices

Using Gaussian elimination; determinants, and inverse matrices to solve systems of equations and inequalities; decomposing into partial fractions

Analytic geometry

Analyzing parabolas, ellipses, circles, and hyperbolas; rotated conics; parametric and polar representations

Sequences and series

Recursive and explicit formulas; arithmetic and geometric sequences; summations; the binomial theorem; mathematical induction

Descriptive statistics

Counting principles and basic probability; graphic representations; measures of central tendency and variability; normal distributions

Inferential statistics

Probability distributions; the central limit theorem; confidence intervals; hypothesis testing; research studies

Limits, derivatives, and integrals

Limit theorems; tangents to the curve; derivative theorems including product, quotient, and chain rules; area under a curve and integration; the fundamental theorem of calculus

Math skills

Fractions, decimals, integers; problem solving using proportions and percent; solving linear equations; negative exponents to prepare for finance formulas

Measurement

Customary and SI (metric) units; conversion within and between systems using dimensional analysis; perimeter, circumference, area, volume, and capacity

Income

Calculating hours worked from clock times, gross pay including overtime; payroll deductions, including FICA and federal withholding; buying and selling stocks and bonds, including brokerage fees

Budgeting

A standardized budget; an annual budget; reducing annual budget to monthly or weekly; emergency adjustments; revising the annual budget

Banking

Checking accounts and deposits; overdraft penalties and protection; service charges; reconciling a bank statement; simple interest; compounding interest; interest on savings using the minimum balance or daily interest methods; effective interest rate; savings programs with regular deposits

Borrowing

Simple interest loans; add-on loans and annual percentage rate; discount loans; amortized loans; how credit cards work; how interest is calculated and payments are applied to credit accounts

Transportation

Calculating the finance charge and monthly payment for a new car; depreciation; leasing costs including residual value, mileage penalty; car insurance; cost of gas, oil, and tires; scheduled maintenance and repair costs

Food

Store specials and coupons; unit prices; calculating freezer payback periods; consumer price index; calorie counting; finding the calories expended in activities

Clothing

Planning for seasonal buying; calculating discounts including multiple ones; online shopping; internet buying tips; savings from making clothing; retail marketing of clothing, returns, net profit and gross profit margin; markup rate and breakeven point for retailer

Housing

Renting; buying a house, including down payments, closing costs, points, and monthly payments; owner’s equity; building a house, including converting dimensions to scale for a set of plans; finding the area of rectangular lots in acres; calculating the amount of shingles needed from a house plan, allowing for pitched roofs

Maintaining a home

Real-estate tax based on millage rates; homeowner’s insurance; calculating utility charges for electricity, gas, water, and sewer; communications; calculating house repairs, maintenance, and home improvements

Life and health insurance

Mortality tables; term, whole-life, and universal life insurance; health insurance, including payout of benefits; Social-Security retirement benefits

Income taxes

General principles of calculating federal income taxes; 1040EZ, 1040A, and 1040, including extra schedules A and B and Form 2441 for childcare benefits

Vacations

Costs of food and lodging; cost of transportation; getting around at the site; travel times across time zones; economizing admissions