🔢 Arithmetic & Number Sense

111 topics

• Counting & Cardinality (Beginner) – Assigning one number word to each object to find how many there are.
• Place Value (Beginner) – The value of a digit depends on its position: ones, tens, hundreds and so on.
• Number Line (Beginner) – A line where numbers are placed in order at equal spacing to compare and add them.
• Comparing Numbers (Beginner) – Use <, > and = to say which number is smaller, larger or equal.
• Ordering Numbers (Beginner) – Arranging numbers from least to greatest or the reverse.
• Even and Odd Numbers (Beginner) – Even numbers divide by 2 with no remainder; odd numbers leave remainder 1.
• Skip Counting (Beginner) – Counting forward by a fixed step such as 2s, 5s or 10s; the root of multiplication.
• Addition (Beginner) – Combining two quantities into a total called the sum.
• Subtraction (Beginner) – Taking one quantity away from another to find the difference.
• Properties of Addition (Beginner) – Addition is commutative, associative and has identity 0.
• Multiplication as Repeated Addition (Beginner) – Multiplying a×b means adding a copies of b.
• Multiplication Tables (Beginner) – Memorised products from 1×1 to 12×12 give speed in all later arithmetic.
• Properties of Multiplication (Beginner) – Multiplication is commutative, associative, has identity 1 and zero property.
• Division (Beginner) – Splitting into equal groups or finding how many groups fit; inverse of multiplication.
• Division with Remainder (Beginner) – When b does not divide a exactly, a = bq + r with 0 ≤ r < b.
• Fact Families (Beginner) – Related addition/subtraction or multiplication/division facts using the same three numbers.
• Zero and Identity (Beginner) – Zero is the additive identity; one is the multiplicative identity; division by zero is undefined.
• Rounding (Beginner) – Replacing a number with a nearby simpler one to a chosen place value.
• Estimation (Beginner) – Finding an approximate answer quickly to check reasonableness.
• Mental Math Strategies (Beginner) – Use compensation, doubling, splitting and friendly numbers to calculate in your head.
• Order of Operations (Beginner) – Evaluate brackets, then exponents, then multiplication/division, then addition/subtraction (PEMDAS/BODMAS).
• Word Problems (Beginner) – Translate a story into an arithmetic expression and solve it step by step.
• Written Addition Algorithm (Beginner) – Add digits column by column carrying ten to the next place.
• Written Subtraction Algorithm (Beginner) – Subtract column by column borrowing from the next place when needed.
• Long Multiplication (Intermediate) – Multiply by each digit of the multiplier, shift by place value, then add partial products.
• Long Division (Intermediate) – Divide, multiply, subtract and bring down repeatedly to get the quotient.
• Factors and Multiples (Intermediate) – Factors divide a number exactly; multiples are its products with whole numbers.
• Divisibility Rules (Intermediate) – Quick tests: by 2 last digit even, by 3 digit-sum divisible by 3, by 5 ends 0/5, by 9 digit-sum divisible by 9.
• Prime Numbers (Intermediate) – A prime has exactly two positive divisors: 1 and itself.
• Composite Numbers (Intermediate) – Numbers greater than 1 with more than two factors.
• Prime Factorization (Intermediate) – Every integer above 1 is a unique product of primes.
• Greatest Common Divisor (Intermediate) – Largest number dividing two numbers exactly; found with prime factors or Euclid.
• Least Common Multiple (Intermediate) – Smallest positive number that is a multiple of both numbers.
• Euclidean Algorithm (Intermediate) – Repeatedly replace (a,b) by (b, a mod b) until the remainder is 0.
• Exponents (Powers) (Intermediate) – Repeated multiplication of the same base.
• Square Numbers and Roots (Intermediate) – A square is n×n; a square root undoes squaring.
• Cube Numbers and Roots (Intermediate) – A cube is n×n×n; the cube root undoes it.
• Negative Numbers (Intermediate) – Numbers below zero; they extend the number line to the left.
• Integers (Intermediate) – The set …, −2, −1, 0, 1, 2, … closed under addition, subtraction and multiplication.
• Adding Integers (Intermediate) – Same signs add and keep the sign; different signs subtract and take the sign of the larger magnitude.
• Multiplying Integers (Intermediate) – Same signs give positive; different signs give negative.
• Absolute Value (Intermediate) – The distance of a number from zero on the number line.
• Fractions Basics (Intermediate) – A fraction a/b names a parts out of b equal parts of a whole.
• Equivalent Fractions (Intermediate) – Multiply or divide numerator and denominator by the same non-zero number.
• Simplifying Fractions (Intermediate) – Divide numerator and denominator by their GCD.
• Comparing Fractions (Intermediate) – Use a common denominator or cross-multiplication to compare.
• Adding Fractions (Intermediate) – Convert to a common denominator then add numerators.
• Subtracting Fractions (Intermediate) – Common denominator, then subtract numerators.
• Multiplying Fractions (Intermediate) – Multiply numerators and denominators.
• Dividing Fractions (Intermediate) – Multiply by the reciprocal of the divisor.
• Mixed Numbers (Intermediate) – A whole number plus a proper fraction; convert to improper fractions to calculate.
• Reciprocals (Intermediate) – The reciprocal of a/b is b/a; their product is 1.
• Decimals (Intermediate) – Place-value numbers using tenths, hundredths, thousandths… after the point.
• Comparing and Ordering Decimals (Intermediate) – Compare digit by digit from the left starting at the largest place value.
• Adding and Subtracting Decimals (Intermediate) – Line up decimal points, then add or subtract as with whole numbers.
• Multiplying Decimals (Intermediate) – Multiply ignoring the points, then place the point using total decimal places.
• Dividing Decimals (Intermediate) – Shift the point in both numbers until the divisor is whole, then divide.
• Fractions to Decimals (Intermediate) – Divide numerator by denominator; the result terminates or repeats.
• Repeating Decimals (Advanced) – Decimals with a repeating block are rational and convertible to fractions.
• Percentages (Intermediate) – A percent is a fraction out of 100.
• Percentage of a Quantity (Intermediate) – Multiply the quantity by the percent as a decimal.
• Percentage Change (Intermediate) – Compare new and old values relative to the old value.
• Discounts and Sales Tax (Intermediate) – Apply percent decrease or increase to price.
• Simple Interest (Intermediate) – Interest earned on the principal only.
• Compound Interest (Advanced) – Interest earned on principal and past interest.
• Ratios (Intermediate) – A comparison of two quantities by division.
• Proportions (Intermediate) – Two equal ratios; solve using cross-multiplication.
• Rates and Unit Rates (Intermediate) – A ratio comparing different units, reduced to per-one.
• Direct Proportion (Intermediate) – When one quantity doubles, the other doubles too.
• Inverse Proportion (Advanced) – When one quantity doubles, the other halves.
• Scale Factor and Maps (Intermediate) – Ratio between drawing size and real size.
• Unit Conversion (Intermediate) – Multiply by conversion factors equal to 1 to change units.
• Length, Mass and Capacity (Beginner) – Metric and imperial units to measure size, weight and volume.
• Time and Clocks (Beginner) – Reading analog/digital time, elapsed time and 24-hour clocks.
• Money Arithmetic (Beginner) – Adding, subtracting and making change with currency.
• Averages: Mean (Intermediate) – Sum of values divided by their count.
• Averages: Median and Mode (Intermediate) – Median is the middle value; mode the most frequent.
• Powers of Ten (Intermediate) – Multiplying by 10ⁿ shifts the decimal point n places.
• Scientific Notation (Intermediate) – Write numbers as a×10ⁿ with 1 ≤ a < 10.
• Significant Figures (Intermediate) – Digits that carry meaning about the precision of a measurement.
• Rational Numbers (Advanced) – Numbers expressible as p/q with integers p, q, q≠0.
• Irrational Numbers (Advanced) – Real numbers that cannot be written as a fraction, such as √2 and π.
• Real Numbers (Advanced) – All rational and irrational numbers; the continuous number line.
• Square Roots of Non-squares (Advanced) – Approximate or simplify surds like √50 = 5√2.
• Rationalizing Denominators (Advanced) – Remove roots from denominators by multiplying by a conjugate.
• Laws of Exponents (Advanced) – Rules for multiplying, dividing and raising powers.
• Negative and Zero Exponents (Advanced) – a⁰ = 1 and a⁻ⁿ is the reciprocal of aⁿ.
• Fractional Exponents (Advanced) – Roots written as powers.
• Binary Numbers (Advanced) – Base-2 system using digits 0 and 1; the language of computers.
• Number Bases (Advanced) – Representing numbers with place values that are powers of a base b.
• Hexadecimal and Octal (Advanced) – Base-16 and base-8 representations used in computing.
• Roman Numerals (Beginner) – An additive-subtractive numeral system using I, V, X, L, C, D, M.
• Perfect Numbers (Advanced) – A number equal to the sum of its proper divisors.
• Triangular Numbers (Intermediate) – Numbers of dots forming triangles.
• Figurate Numbers (Advanced) – Square, pentagonal and other numbers arranged as shapes.
• Arithmetic Sequences (Intro) (Intermediate) – A list where each term adds a constant difference.
• Geometric Sequences (Intro) (Intermediate) – A list where each term multiplies by a constant ratio.
• Patterns and Rules (Beginner) – Spotting how a sequence grows to predict later terms.
• Fibonacci Numbers (Intermediate) – Each term is the sum of the two before it.
• Pascal’s Triangle Basics (Intermediate) – A triangle where each entry is the sum of the two above it.
• Casting Out Nines (Advanced) – Check arithmetic using digit sums modulo 9.
• Modular Clock Arithmetic (Advanced) – Numbers wrap around after reaching the modulus.
• Approximating Square Roots (Advanced) – Use the Babylonian method to refine guesses.
• Lattice and Peasant Multiplication (Advanced) – Alternative multiplication algorithms using grids or halving and doubling.
• Ancient Number Systems (Intermediate) – Egyptian, Babylonian and Mayan ways of writing numbers.
• Egyptian Fractions (Expert) – Writing a fraction as a sum of distinct unit fractions.
• Continued Fractions Intro (Expert) – Nested fractions that give best rational approximations of reals.
• Estimation with Benchmarks (Intermediate) – Compare to 0, 1/2 and 1 to judge fraction size quickly.
• Number Sense Puzzles (Intermediate) – Magic squares, cryptarithms and logic puzzles that train flexible thinking.
• Complex Word Problems (Advanced) – Multi-step problems that combine ratio, percent and rate.
• Financial Math Basics (Advanced) – Budgets, loans, depreciation and savings using percent and exponents.