Pathways
Explore connected math skills from kindergarten through precalculus. Each link opens a skill with its problem types and connections.
- Counting objects to 20
- Counting by ones and by tens
- Numerals through 20
- Comparing two groups by matching
- Ordering numbers through 10
- Adding within 10
- Subtracting within 10
- Number bonds to 10
- Making 10
- Addition and subtraction facts within 5
- Teen numbers as a ten and ones
- Comparing length, weight, and capacity
- Sorting and counting
- Naming shapes
- Composing shapes
- Building and drawing closed shapes
- Join and separate: pictures, equations
- Adding within 20
- Subtracting within 20
- Equations for addition and subtraction stories
- Adding three numbers
- The meaning of the equal sign
- Finding a missing number
- Reading and writing tens and ones
- Compare two-digit numbers
- Add two-digit numbers
- Ten more and ten less
- Measuring length with equal units
- Telling time to the half-hour
- Reading a categorical graph
- Halves and quarters of a whole
- Defining attributes of shapes
- Counting on and counting back
- Shapes made by putting shapes together
- Turnaround addition facts
- Hundreds, tens, and ones
- Skip-counting
- Comparing three-digit numbers
- Adding within 100
- Subtracting within 100
- Regrouping within 1,000
- Adding several two-digit numbers
- Two-step word problems
- Addition and subtraction facts
- Odd and even numbers
- Rectangular arrays
- Measuring length with a ruler
- Sums and differences on a number line
- Telling time to five minutes
- Coins and bills
- Line plots of lengths
- Picture graphs and bar graphs
- Halves, thirds, and quarters
- Sides, angles, and faces
- Adding or subtracting 10 or 100 mentally
- Explaining why a strategy works
- Multiplication as equal groups
- One-digit multiplication facts
- Division as sharing and grouping
- Quotients from related facts
- Missing factors and quotients
- Equal multiplication and division equations
- Properties of addition and multiplication
- Multiplication and division word problems
- Multi-operation word problems
- Patterns in the operations
- Rounding whole numbers
- Adding and subtracting within 1,000
- Multiplying by a multiple of 10
- Unit fractions of a whole
- Fractions shown by a model
- Fractions on a number line
- Recognizing equivalent fractions
- Comparing fractions
- Finding elapsed time
- Mass and liquid volume
- Reading scaled picture and bar graphs
- Line plots with fractional lengths
- Area of a rectangle
- Perimeter of a polygon
- Classifying quadrilaterals
- Fractions of an area
- Measuring area by counting squares
- Equal equations with the four operations
- Additive and multiplicative comparison
- Multi-step word problems
- Factors, multiples, primes, and composites
- Patterns that follow a rule
- Place value of large whole numbers
- Adding and subtracting whole numbers
- Multiplying with partial products
- Dividing by a one-digit number
- Generating equivalent fractions
- Comparing fractions with benchmarks
- Adding and subtracting like fractions
- Adding and subtracting mixed numbers
- A fraction times a whole number
- Tenths and hundredths as decimals
- Comparing decimals to hundredths
- Money with decimals
- Converting to a smaller unit
- Area and perimeter problems
- Measuring angles
- Parallel lines, perpendicular lines, and symmetry
- Adding and subtracting from a line plot
- Frequency tables and scaled bar graphs
- Powers of 10 and place value
- Decimal place value through thousandths
- The standard multiplication algorithm
- Dividing by a two-digit number
- The four operations with decimals
- Adding and subtracting unlike fractions
- A fraction as division
- Multiplying fractions
- Dividing with unit fractions
- Reading a numerical expression
- Graphing in the first quadrant
- Two patterns on one graph
- Volume in unit cubes
- Volume of a rectangular prism
- Categories and subcategories of shapes
- Profit, cost, and taxes
- Converting within one measurement system
- Fraction operations on a line plot
- Understanding ratios
- Finding a unit rate
- Equivalent ratios
- Percent of a number
- Dividing fractions
- Fluent whole-number division
- Fluent decimal arithmetic
- Greatest common factor and least common multiple
- Positive and negative numbers
- Ordering numbers and absolute value
- Coordinates in four quadrants
- Whole-number powers
- Expressions with variables
- Equivalent expressions
- Solving one-step equations
- Inequalities on a number line
- Dependent and independent variables
- Area of triangles and other polygons
- Volume with fractional edge lengths
- Side lengths from coordinates
- Surface area of prisms and pyramids
- Asking a statistical question
- Mean of a data set
- Mean and median
- Dot plots, histograms, and box plots
- Describing spread and outliers
- Center, spread, and shape of a distribution
- Unit rates with fractions
- Representing a proportional relationship
- Percent increase and decrease
- Income tax, budgets, and net worth
- Adding and subtracting signed numbers
- Multiplying and dividing signed numbers
- Word problems with signed numbers
- Simplifying complex fractions
- Expanding and factoring linear expressions
- Solving two-step equations
- Solving inequalities
- Scale drawings of lengths and areas
- When given measures make a triangle
- Cross-sections of solids
- Circumference and area of a circle
- Supplementary, vertical, and adjacent angles
- Area and volume of composite figures
- Median and spread of signed data
- Samples and the populations they represent
- Comparing two data distributions
- Probability of one event
- Probability of compound events
- Solving absolute-value equations
- Choosing units and a sensible precision
- Structure of an expression
- Rearranging a formula
- Linear equations from constraints
- Systems that model two constraints
- Domain and range
- Evaluating a function
- Piecewise and absolute-value functions
- Rules for an arithmetic sequence
- Rules for a geometric sequence
- Adding, subtracting, and multiplying polynomials
- Factoring quadratic expressions
- Solving quadratic equations
- Completing the square
- The quadratic formula
- Graphs of quadratic functions
- The minimum of an upward-opening quadratic
- Building an exponential model
- Residuals and correlation
- Values a rational expression excludes
- Naming the quantities in a model
- Why each equation step is valid
- Linear equations with letter coefficients
- Why elimination keeps the same solutions
- A graph as the solution set
- Regions defined by inequalities
- Average rate of change
- Comparing center, spread, and shape
- Association in a two-way table
- Reading slope and intercept in context
- Association is not causation
- Telling rationals from irrationals
- Approximating roots
- Laws of exponents
- Working in scientific notation
- Slope of a line
- Linear equations with variables on both sides
- Systems of linear equations
- What makes a relation a function
- Comparing functions
- Linear models of a situation
- Reading increase and decrease from a graph
- Translations, reflections, and rotations
- Dilations and similarity
- Angle sums and exterior angles
- The Pythagorean theorem
- Distance in the coordinate plane
- Volume of cylinders, cones, and spheres
- Scatter plots and trend lines
- Two-way tables
- What slides, flips, and turns keep the same
- Precise definitions in geometry
- Writing and critiquing a geometric proof
- Congruence criteria for triangles
- Proving triangles similar
- Constructions with compass and straightedge
- Sine, cosine, and tangent
- Side ratios of special right triangles
- Theorems about circles
- Arc length and sector area
- Equations of circles
- Proofs that use coordinates
- Modeling with geometry
- Finding a conditional probability
- Independence and mutual exclusion
- Why every circle is similar to every other
- The incircle and the circumcircle
- A tangent from a point outside a circle
- What a transformation keeps
- Symmetries of a polygon
- Constructing the image of a figure
- Proofs about lines and angles
- Proofs about triangles
- Proofs about parallelograms
- Regular polygons inscribed in a circle
- Why the circle and solid formulas hold
- Cavalieri's principle
- Volumes of cylinders, cones, pyramids, spheres
- Cross-sections and solids of revolution
- A parabola as points equally far from a focus and a line
- Slopes of parallel and perpendicular lines
- Dividing a segment in a given ratio
- Perimeter and area from coordinates
- Modeling density
- Designs that meet geometric constraints
- How a dilation changes a figure
- Theorems about similar triangles
- Sine and cosine of complementary angles
- Unions, intersections, and complements
- The addition rule for probability
- Rational exponents and radicals
- Equations with one square root
- Solving radical equations
- Calculating with complex numbers
- Dividing polynomials
- Zeros and factors of a polynomial
- Operations on rational expressions
- Rational equations with one denominator
- Solving rational equations
- Where a line meets a parabola
- Solving a linear and a nonlinear equation together
- How many times a line meets a curve
- Shifts, stretches, and reflections of graphs
- Composition of functions
- Finding an inverse function
- Logarithms as inverses of exponentials
- Properties of logarithms
- Solving exponential and logarithmic equations
- Sums of a finite geometric sequence
- Radians and the unit circle
- Period, amplitude, and midline
- The normal distribution
- Studies, experiments, and samples
- Margin of error and what a sample supports
- Using expected value to compare choices
- Which number systems stay closed
- Proving a polynomial identity
- Quadratic, rational, and exponential constraints
- Approximating a solution from a graph
- Comparing functions given in different forms
- Linear growth against exponential growth
- Choosing a periodic model
- Checking a chance model by simulation
- A margin of error from simulated samples
- Testing a difference by randomization
- Reading a statistical report critically
- End behavior of a polynomial
- Asymptotes, holes, and intercepts
- Inverse trigonometric functions
- Pythagorean and angle-sum identities
- Solving trigonometric equations
- The laws of sines and cosines
- Vectors and their components
- Operations on matrices
- The complex plane
- Conic sections and their equations
- Motion described by a parameter
- Polar and Cartesian coordinates
- Checking a model against evidence
- The conjugate of a complex number
- Distance and midpoints in the complex plane
- Factoring over the complex numbers
- Fundamental theorem of algebra
- Velocity and force as vectors
- Data arranged in a matrix
- Which matrix rules hold
- Inverse of a matrix
- Matrices as transformations of the plane
- The binomial theorem
- A system written as a matrix equation
- Exact values at special angles
- Odd, even, and periodic trig functions
- Area from two sides and the included angle
- Deriving the laws of sines and cosines
- The multiplication rule for probability
- Permutations and combinations
- Defining a random variable
- A distribution from a sample space
- A distribution built from frequencies
- A procedure that gives each outcome a fair chance
- Comparing uncertain strategies