K12_Math_Concepts_Structure.csv

concept,key_formulas,key_concepts,related_operations,related_topics,grade_level
Counting and Cardinality,,"Counting objects, comparing quantities",Counting,Number recognition,K
Number Recognition,,Identifying numerals 0–20,Counting objects,Number sense,K
Addition,+,Combining two groups to make a whole,Counting on,Subtraction,K
Subtraction,-,Taking away from a whole,Counting back,Addition,K
Basic Shapes,,Identifying circles, squares, triangles, rectangles,Counting sides,Geometry,K
Measurement Comparison,,Longer/shorter, heavier/lighter concepts,Comparing,Measurement,K
Patterns,,Recognizing and extending patterns,Sequencing,Number sense,K
Ordinal Numbers,,"First, second, third positions",Counting,Number recognition,K
Simple Data Collection,,Sorting and categorizing objects,Comparing,Data organization,K
Place Value,Decomposition by 10s,Understanding base-ten system,Addition,Subtraction,1
Two-Digit Addition,+,Adding numbers up to 100,Place value,Regrouping,1
Two-Digit Subtraction,-,Subtracting numbers up to 100,Place value,Regrouping,1
"Comparing Numbers",">, <, =","Greater than, less than, equal to",Place value,Number sense,1
Telling Time,,Reading clocks to hour and half-hour,Counting,Measurement,1
Measurement (Length),,Non-standard and standard units,Comparing,Measurement comparison,1
Basic Shapes and Attributes,,Defining and composing shapes,Counting sides,Basic shapes,1
Word Problems (Addition/Subtraction),,Real-world applications,Addition,Subtraction,1
Data Organization,,Simple graphs and charts,Counting,Statistics,1
Number Bonds,,Part-part-whole relationships for numbers to 10,Addition,Subtraction,1
Fact Families,,Relating addition and subtraction,Addition,Subtraction,1
Even and Odd Numbers,,Classifying numbers,Division by 2,Patterns,2
Skip Counting,,Counting by multiples,Addition,Multiplication,2
Three-Digit Numbers,,"Reading, writing, and comparing",Place value,Comparing numbers,2
Regrouping,,Carrying and borrowing,Addition,Subtraction,2
"Money","$, ¢","Coins, dollars, and counting money",Addition,Decimals,2
Time to 5 Minutes,,Reading analog and digital clocks,Skip counting,Telling time,2
Bar Graphs and Picture Graphs,,Creating and interpreting data,Counting,Data organization,2
Measurement Units,"in, ft, cm, m",Standard units of length,Measurement,Conversion,2
Arrays (Introduction),,Equal groups in rows and columns,Addition,Multiplication,2
Estimating Quantities,,Reasonable estimates for small numbers,Rounding,Number sense,2
Multiplication,×,Repeated addition,Arrays,Division,3
Division,÷,Repeated subtraction,Multiplication,Factors and multiples,3
Fractions,a/b,Part-to-whole relationships,Division,Decimals,3
Area and Perimeter,"P=2l+2w, A=lw",Measuring rectangles and squares,Multiplication,Measurement,3
Telling Time to the Minute,,Reading clocks and elapsed time,Addition,Time to 5 minutes,3
Measurement Conversions,,Converting within same system,Multiplication,Measurement units,3
Rounding,,To nearest 10 and 100,Place value,Estimation,3
Properties of Multiplication,"Commutative, Associative, Distributive",Understanding multiplication properties,Multiplication,Addition,3
Factors and Multiples (Basic),,Finding factors of numbers up to 100,Division,Multiplication,3
Data Interpretation,,Reading and creating scaled graphs,Multiplication,Bar graphs and picture graphs,3
Equivalent Fractions,a/b = c/d,Simplifying fractions,Multiplication,Ratio,4
Mixed Numbers,a b/c,Whole + fractional parts,Addition,Improper fractions,4
Decimals,"Place value tenths, hundredths",Relationship to fractions,Addition,Money,4
Rounding and Estimation,≈,Approximation to nearest place,Addition,Subtraction,4
Angle Measurement,°,Measuring angles with protractor,Addition,Geometry,4
Line Plots,,Creating and interpreting data,Fractions,Statistics,4
Factors and Divisibility,,Divisibility rules and factor pairs,Division,Prime numbers,4
Multiples and Least Common Multiple,LCM,Finding least common multiple,Multiplication,Fractions,4
Greatest Common Factor,GCF,Finding greatest common factor,Division,Factors and divisibility,4
Two-Dimensional Figures,,Properties of quadrilaterals and triangles,Classification,Geometry,4
Prime Numbers,,Numbers divisible only by 1 and itself,Division,Composite numbers,5
Order of Operations,PEMDAS,Sequence of operations,Multiplication,Parentheses,5
Fraction Addition and Subtraction,a/b ± c/d,Adding and subtracting unlike denominators,Multiplication,Equivalent fractions,5
Fraction Multiplication and Division,"(a/b) × (c/d), (a/b) ÷ (c/d)",Multiplying and dividing fractions,Multiplication,Fractions,5
Coordinate Graphing,"(x,y)",Plotting in first quadrant,Ordered pairs,Coordinate plane,5
Volume of Rectangular Prisms,V = l × w × h,Three-dimensional measurement,Multiplication,Volume,5
Measurement Conversion,,Converting between different units,Multiplication,Measurement conversions,5
"Decimal Operations","+, -, ×, ÷",Operations with decimals,Place value,Decimals,5
Exponents (Introduction),10ⁿ,Powers of 10 and basic exponent notation,Multiplication,Place value,5
Expressions,,Writing and evaluating numerical expressions,Order of operations,Variables,5
"Data and Graphs",,"Histograms, line graphs, stem-and-leaf plots",Data interpretation,Statistics,5
Integers,,Positive and negative numbers,Addition,Absolute value,6
Ratios and Proportions,a:b = c:d,Comparing quantities,Division,Percentages,6
Percentages,% = part/whole × 100,Per hundred relationships,Division,Decimals,6
Coordinate Plane,"(x,y)",Plotting points,Addition,Geometry,6
Absolute Value,|x|,Distance from zero,Integers,Number line,6
"Statistical Questions","mean, median, mode, range",Analyzing data distributions,Addition,Statistics,6
Rate and Unit Rate,rate = distance/time,"Speed, cost per unit, unit conversion",Division,Ratios and proportions,6
Expressions and Variables,,Writing and evaluating algebraic expressions,Substitution,Order of operations,6
One-Step Equations,"x + a = b, ax = b",Solving simple equations,Inverse operations,Variables,6
Area of Triangles and Polygons,A = ½bh,Area of triangles and composite shapes,Multiplication,Area and perimeter,6
Surface Area,SA = 2lw + 2lh + 2wh,Nets and surface area of prisms,Addition,Volume,6
Expressions and Equations,y = mx + b,Using variables,Addition,Algebraic manipulation,7
Inequalities,x > y,Comparing values,Subtraction,Number lines,7
Area,"A=lw, A=πr²",2D measurement,Multiplication,Perimeter,7
Angles and Triangles,Sum of angles = 180°,Triangle properties and angle relationships,Addition,Geometry,7
Circles,"C=2πr, A=πr²",Circumference and area,Multiplication,Pi,7
Scale Drawings,scale factor,Proportional reasoning,Multiplication,Ratios and proportions,7
Probability Basics,P(A) = favorable/total,Simple probability concepts,Division,Fractions,7
Proportional Relationships,y = kx,Direct proportion and constant of proportionality,Multiplication,Ratios and proportions,7
Percent Applications,,"Tax, tip, discount, markup",Multiplication,Percentages,7
Two-Step Equations,ax + b = c,Solving multi-step equations,Inverse operations,One-step equations,7
Volume,V=lwh,3D measurement,Multiplication,Surface area,8
Pythagorean Theorem,a²+b²=c²,Right triangle relationships,Square root,Distance formula,8
Slope,m=(y₂−y₁)/(x₂−x₁),Rate of change,Division,Linear equations,8
Exponents,aⁿ,Repeated multiplication,Multiplication,Powers,8
Systems of Equations,y = mx + b,Solving linear systems,Substitution,Equations,8
Scientific Notation,a × 10ⁿ,Working with very large and small numbers,Multiplication,Exponents,8
Transformations,,"Translations, rotations, reflections, dilations",Coordinate plane,Geometry,8
Functions (Introduction),y = f(x),Linear vs non-linear functions,Graphing,Slope,8
Scatter Plots,,Correlation and trend lines,Graphing,Statistics,8
Polynomials,ax²+bx+c,Combining like terms,Addition,Factoring,9
Quadratic Equations,ax²+bx+c=0,Parabolic relationships,Square roots,Vertex form,9
Functions,f(x),Input-output mapping,Substitution,Domain/range,9
Factoring,x²+bx+c=(x+m)(x+n),Breaking down expressions,Multiplication,Quadratics,9
Radical Expressions,"√x, ∛x",Simplifying and operations with radicals,Multiplication,Exponents,9
Rational Expressions,a/b,Simplifying rational expressions,Division,Fractions,9
Linear Systems (Methods),,Substitution, elimination, graphing methods,Systems of equations,Linear equations,9
Absolute Value Equations,|x| = a,Solving equations with absolute value,Absolute value,Equations,9
Square Root Functions,y = √x,Graphing and properties,Square roots,Functions,9
Exponential Growth and Decay,y = a(1±r)ᵗ,Real-world exponential problems,Exponents,Exponential functions,9
Trigonometry,"sinθ, cosθ, tanθ",Ratios of sides in right triangles,Division,Unit circle,10
Probability,P(A)=favorable/total,Chance of events,Multiplication,Statistics,10
Statistics,"mean, median, mode",Analyzing data sets,Addition,Probability,10
Circles (Advanced),"arc length, sector area","Arcs, sectors, circle theorems",Multiplication,Circles,10
Combinations and Permutations,"nCr, nPr",Counting principles,Multiplication,Probability,10
Systems of Inequalities,y > mx + b,Graphing and solving,Inequalities,Linear equations,10
Normal Distribution,z-score,Bell curves and standard scores,Statistics,Standard deviation,10
Conditional Probability,P(A|B) = P(A∩B)/P(B),Probability given conditions,Division,Probability,10
Expected Value,E(X) = Σ(x·P(x)),Probability and expected outcomes,Multiplication,Probability,10
Circle Geometry,,Inscribed angles and tangent lines,Angles,Circles (Advanced),10
Law of Sines,a/sinA=b/sinB,Triangle relationships,Division,Law of Cosines,11
Law of Cosines,c²=a²+b²−2abcosC,Non-right triangles,Multiplication,Trigonometry,11
Logarithms,logₐx=y ↔ aʸ=x,Inverse of exponents,Division,Exponential functions,11
Exponential Functions,y=a·bˣ,Growth and decay,Multiplication,Logarithms,11
Complex Numbers,a + bi,Operations with imaginary unit i,Addition,Quadratic equations,11
Conic Sections,"parabola, ellipse, hyperbola",Equations of conic sections,Quadratics,Circles,11
Sequences (Arithmetic/Geometric),"aₙ = a₁ + (n-1)d, aₙ = a₁rⁿ⁻¹",nth term and recursive formulas,Patterns,Sequences and series,11
"Series and Summation","Σ, Sₙ",Arithmetic and geometric series,Addition,Sequences,11
Inverse Functions,f⁻¹(x),Finding and graphing inverses,Functions,Composition,11
Rational Exponents,x^(m/n) = ⁿ√(xᵐ),Fractional exponents and radical form,Exponents,Radicals,11
Polynomial Division,,Long division and synthetic division,Division,Polynomials,11
Derivatives,dy/dx,Instantaneous rate of change,Subtraction,Calculus,12
Integrals,∫f(x)dx,Area under a curve,Addition,Antiderivatives,12
Sequences and Series,Σaₙ,Sum of patterns,Addition,Progressions,12
Matrices,[a b; c d],Array operations,Addition,Linear algebra,12
Standard Deviation,σ=√Σ(x−μ)²/n,Measure of spread,Subtraction,Variance,12
Limits,lim(x→a) f(x),Approaching values in calculus,Substitution,Derivatives,12
Vectors,⟨x,y,z⟩,Vector operations and applications,Addition,Linear algebra,12
Rational Functions,f(x) = p(x)/q(x),Asymptotes and graphing,Division,Rational expressions,12
Parametric Equations,"x = f(t), y = g(t)",Parametric curves and eliminating parameters,Substitution,Functions,12
"Polar Coordinates","r, θ",Converting between polar and rectangular,Trigonometry,Coordinate plane,12
L'Hôpital's Rule,lim f/g = lim f'/g',Evaluating limits of indeterminate forms,Limits,Derivatives,12
Number Writing (0-20),,Writing numerals correctly, number formation,Number recognition,Counting and cardinality,K
More/Less/Equal,,"Comparing quantities using more, less, equal",Comparing,Number sense,K
Sorting and Classifying,,Categorizing objects by attributes,Grouping,Data organization,K
Position Words,,"Above, below, beside, next to, in front, behind",Spatial reasoning,Geometry,K
Simple Addition Stories,+,Visual addition word problems within 5,Addition,Word problems,K
Calendar Skills,,"Days of week, months, seasons",Time,Patterns,K
Three-Addend Addition,+,Adding three numbers together,Addition,Mental math,1
Missing Addends,+,Finding unknown in equations (5 + ? = 8),Inverse operations,Fact families,1
Number Line Operations,"+, -",Using number lines for addition and subtraction,Visual math,"Addition, subtraction",1
Coin Recognition,,"Identifying penny, nickel, dime, quarter",Money,Counting,1
Half Hour Time,,Reading clocks at half-past,Time,Telling time,1
Equal Parts,,Introduction to halves and fourths,Fractions,Division,1
Tally Marks,,Counting and organizing with tallies,Counting by 5s,Data organization,1
Greater Than/Less Than Symbols,"<, >, =",Using comparison symbols correctly,Comparing numbers,Number sense,1
Place Value (Hundreds),,Understanding 3-digit place value deeply,Decomposing numbers,Three-digit numbers,2
Mental Math Strategies,"+, -","Adding 10, 100; subtracting 10 mentally",Place value,Number sense,2
Measurement (Inches/Centimeters),,Using rulers to measure length,Measurement,Units,2
Repeated Addition,+,Foundation for multiplication,Addition,Multiplication,2
Equal Groups,÷,Division concept introduction,Division,Multiplication,2
Shapes (2D Properties),,"Understanding sides, vertices, angles",Counting,Geometry,2
Line Plots (Simple),,Organizing measurement data on line plots,Data collection,Graphs,2
Counting Money,,Mixed coins (calculating totals),Addition,Money,2
Quarter Hours,,Time to 15-minute intervals,Time,Telling time,2
Number Patterns (100s chart),,Patterns in 100s chart,Skip counting,Patterns,2
Multiplication Tables (Focus),×,Fluency with 0-12 multiplication tables,Multiplication,Division,3
Division with Remainders,÷,Understanding and calculating remainders,Division,Multiplication,3
Fraction Comparison,"<, >, =",Comparing fractions to determine which is larger,Fractions,Equivalent fractions,3
Unit Fractions,1/n,"Understanding 1/2, 1/3, 1/4, etc.",Fractions,Equal parts,3
Liquid Measurement,,"Cups, pints, quarts, gallons",Measurement,Units,3
Weight Measurement,,"Ounces, pounds, grams, kilograms",Measurement,Units,3
Perimeter of Polygons,P = sum of all sides,Finding perimeter of various shapes,Addition,Area and perimeter,3
Scaled Picture Graphs,,Each symbol represents multiple units,Multiplication,Data interpretation,3
Scaled Bar Graphs,,Interpreting scales on bar graphs,Multiplication,Data interpretation,3
Frequency Tables,,Organizing categorical data,Counting,Data organization,3
Elapsed Time,,Time intervals and duration,Subtraction,Time,3
Temperature,,"Reading thermometers, Fahrenheit/Celsius",Measurement,Units,3
Multi-Digit Multiplication,×,2-digit × 2-digit multiplication,Multiplication,Place value,4
Long Division,÷,Multi-digit divisors,Division,Multiplication,4
Fraction Ordering,"<, >, =",Ordering multiple fractions,Comparison,Equivalent fractions,4
Fraction Models,,Visual representations (number lines, area),Fractions,Equivalent fractions,4
Decimal Place Value,,Understanding tenths and hundredths,Place value,Decimals,4
Decimal Comparison,"<, >, =",Comparing decimal values,Comparison,Place value,4
Protractor Use,,Measuring angles with protractor,Measurement,Angles,4
Symmetry,,Lines of symmetry in figures,Geometry,Shapes,4
Classifying Triangles,,"By angles (acute, right, obtuse) and sides (equilateral, isosceles, scalene)",Classification,Geometry,4
Coordinate Grids (First Quadrant),"(x, y)",Plotting points in first quadrant,Ordered pairs,Coordinate plane,4
Division of Decimals,÷,Dividing with decimal divisors,Division,Decimal operations,5
Powers of 10,10^n,Multiplying/dividing by powers of 10,Place value,Exponents,5
Fraction to Decimal Conversion,÷,Converting between fractions and decimals,Division,"Fractions, decimals",5
Percentage Basics,%,Understanding percent as per hundred,Fractions,Decimals,5
Three-Dimensional Figures,,"Identifying prisms, pyramids, cylinders, cones, spheres",Classification,Geometry,5
Cubic Units,units³,Understanding volume units,Volume,Measurement,5
Order of Operations (PEMDAS),PEMDAS,"Parentheses, exponents, multiplication/division, addition/subtraction",All operations,Expressions,5
Numerical Patterns,,Identifying rules and sequences,Patterns,Sequences,5
Mean (Average),sum ÷ count,Calculating and interpreting mean,Division,Statistics,5
Line Graphs,,Creating and interpreting trends over time,Data analysis,Graphs,5
Dividing Fractions,"÷, reciprocal",Using reciprocals to divide fractions,Division,Multiplication of fractions,6
Negative Numbers,,Number line with negative values,Integers,Absolute value,6
Operations with Integers,"+, -, ×, ÷",All four operations with positive and negative numbers,Integer rules,Number sense,6
GCF Applications,GCF,Using GCF to simplify fractions,Simplification,Greatest common factor,6
LCM Applications,LCM,Using LCM for adding fractions,Fraction addition,Least common multiple,6
Ratio Tables,a:b,Equivalent ratios in table format,Multiplication,Ratios and proportions,6
Double Number Lines,,Visual representation for ratios and proportions,Ratios,Proportional relationships,6
Percent of a Number,% × n,Finding percentages of quantities,Multiplication,Percentages,6
Box Plots,,Reading and creating box-and-whisker plots,Data analysis,Statistics,6
Histograms,,Frequency distributions with intervals,Data analysis,Bar graphs,6
Median and Mode,,Additional measures of center,Statistics,Mean,6
Interquartile Range (IQR),Q3 - Q1,Understanding spread in data,Subtraction,Box plots,6
Multi-Step Equations,,Solving complex algebraic equations with multiple operations,All operations,Two-step equations,7
Combining Like Terms,,Simplifying expressions by combining terms,"Addition, subtraction",Expressions,7
Distributive Property,a(b + c) = ab + ac,Expanding expressions using distribution,Multiplication,Properties of operations,7
Simple Interest,I = PRT,"Principal, rate, time calculations",Multiplication,Percentages,7
Percent Change,(new - old)/old × 100%,Percent increase and decrease,Percentages,Percent applications,7
Complementary and Supplementary Angles,"90°, 180°",Angle relationships and pairs,Subtraction,Angles,7
Volume of Cylinders,V = πr²h,Circular base volume calculations,Multiplication,Volume,7
Sample Space,,Listing all possible outcomes,Counting,Probability,7
Simultaneous Equations (Graphing),,Visual solution method for systems,Graphing,Systems of equations,8
Distance Formula,d = √[(x₂-x₁)² + (y₂-y₁)²],Finding distance between two points,Square roots,Pythagorean theorem,8
Volume of Cones and Spheres,"V = (1/3)πr²h, V = (4/3)πr³",Advanced 3D geometry volume,Multiplication,Volume,8
Bivariate Data,,Two-variable relationships and correlation,Data analysis,Scatter plots,8
Linear vs Non-Linear Functions,,Identifying function types from tables/graphs,Pattern recognition,Functions,8
Roots and Radicals,"³√, ⁴√",Cube roots and higher-order roots,Radicals,Exponents,8
Rational vs Irrational Numbers,,Number classification and properties,Number theory,Real numbers,8
Frequency Tables (Two-Way),,Categorical data with two variables,Data analysis,Statistics,8
Completing the Square,x² + bx + (b/2)²,Quadratic solving method,Algebra,Quadratic equations,9
Quadratic Formula,x = [-b ± √(b²-4ac)]/2a,Using formula to solve quadratics,Substitution,Quadratic equations,9
Parent Functions,,"Basic function families (linear, quadratic, etc.)",Function analysis,Functions,9
Function Transformations,"f(x) + k, f(x + h), af(x)","Shifts, stretches, reflections of functions",Graphing,Functions,9
Regression Lines,y = mx + b,Line of best fit for data,Statistics,Scatter plots,10
Standard Form to Slope-Intercept,Ax + By = C → y = mx + b,Converting between linear equation forms,Algebra,Linear equations,10
Unit Circle,x² + y² = 1,Trigonometry on the unit circle,Trig functions,Trigonometry,11
Trig Identities,sin²θ + cos²θ = 1,Pythagorean and reciprocal identities,Trigonometry,Trig functions,11
Arc Length and Sector Area,"s = rθ, A = (1/2)r²θ",Circle calculations with radians,Multiplication,Circles,11
Asymptotes,,"Vertical, horizontal, and oblique asymptotes",Limits,Rational functions,12
Continuity,,Function continuity concepts and definitions,Limits,Calculus,12
Chain Rule,dy/dx = dy/du × du/dx,Composite function derivatives,Differentiation,Derivatives,12