Math Fundamentals 101: Master Math Basics from Scratch

Explore the 20 essential phrases for addition and subtraction, from add, plus, and sum to minus, difference, take away, and deduct, with examples.

Explore the five properties of addition and subtraction, including the commutative, associative, and identity properties for addition, and the identity property and the property of equality for subtraction.

Apply definitions of even and odd numbers to see how sums behave: even+even=even, odd+odd=even, and even+odd=odd. Explore doubling numbers and test examples from the ruler to reinforce these rules.

Learn to construct a number line with equal intervals extending left and right from zero, showing positive and negative values, and using opposites to explain comparisons.

Practice adding and subtracting positive and negative numbers using a number line. Learn turning minus signs into plus signs and two negatives into a plus, with a big practice problem.

Learn the traditional addition method by stacking numbers and aligning to the right. Carry when sums exceed nine and practice with 283 plus 52 to reinforce ones, tens, and hundreds.

Learn to subtract the traditional way by stacking numbers, aligning to the right, and borrowing from the left, illustrated with 357-182 and 205-57.

Learn a grouping method to add big numbers by breaking them into tens and ones, then sum the parts, as shown with 27 plus 13 equals 40.

Master big-number subtraction by adding to reach friendly end-in-zero checkpoints, using a number line to break digits into steps like 37 to 40 and 90 to 94.

Apply a fast zero-borrowing trick: subtract by reducing both numbers by one to avoid borrowing, then subtract from nine as shown in examples like 1000–582 and 80,000–5327.

Master six ways to write multiply, using x, dot, asterisks, and parentheses. Learn verbal forms like three by five, multiplied by, product, doubled, and percentages.

Explore the five properties of multiplication—commutative, associative, distributive, identity, and zero—through clear examples and visual Legos, showing consistent results across orders and groupings.

Discover how multiplication is simply repeated addition, illustrated with examples like five times two and six times seven using block models and rectangular layouts to reveal the total.

Use a two-step method to multiply integers with signs: ignore signs and multiply as positives, then adjust the result based on the even or odd number of negative signs.

Master absolute value as the nonnegative result of any number, illustrated by a washing machine. Solve inside the bars first, then apply the absolute value, even with negative signs outside.

Learn a quick pro tip for multiplying numbers that end in zeros by multiplying the front digits and appending zeros. Use this method for estimation and budgeting with large figures.

Learn the traditional multiplication method, aligning numbers to the right, carrying when needed, and solving three examples: three-by-one, three-by-two, and three-by-three digit problems.

Discover the box method for multiplying numbers by breaking them into place values, arranging a box, filling boxes with products, and summing to the final result.

Learn the line method for multiplication by drawing lines for each digit, counting line intersections inside bubbles, and carrying over when needed to find products like 13 × 21.

Understand division as splitting into equal parts, as shown by dividing six pencils into three cups; when numbers do not split evenly, you get a remainder or fractions and decimals.

Explore the key division terms and notation, including dividend, divisor, and quotient, and learn different ways to write six divided by three as fraction, slash, colon, or ratio.

Discover how division arises from repeated subtraction, and how to handle remainders (R) when the result isn't exact, as shown by 16 divided by three.

Explore why you can't divide by zero; view division as repeated subtraction and see why ten divided by zero remains undefined, addressing common infinity and zero misconceptions.

Learn a two-step method to divide integers: first treat all numbers as positive, then count negative signs to determine the final sign. Apply left-to-right for multiple divisions.

Master long division by learning the dividend, divisor, and quotient. Follow the divide, multiply, subtract, check, and bring down steps, grounded in repeated subtraction, to perform accurate calculations.

Master long division with remainders using the divide, multiply, subtract, check, and bring down steps, shown by 73 divided by 4 giving 18 with remainder 1.

Learn long division with two-digit divisors by applying the divide, multiply, subtract, check, and bring down steps, including strategies for combining digits and extending the dividend.

Learn the partial quotients method to replace long division with flexible multiples, solving 173 divided by 13 using 13 times 10, 13 times 2, and a final remainder.

Explore factors, multiples, and prime numbers through real-life examples, learn to factor numbers like 20 and 30, and understand that every whole number has at least two factors.

Master the process of testing for divisibility by identifying factors through step-by-step division, using 18 as an example to reveal factors like 1, 2, 3, 6, 9, and 18.

Learn patterns to determine divisibility of a number by 2 through 10. Use simple tests—last digit parity, digit sums, last digits, and the seven trick—with 1512 as a running example.

Explore prime numbers, natural numbers with exactly two factors, such as 3 and 17, and how they underpin finding the greatest common factor, the least common multiple, and cybersecurity encryption.

Explore how encryption uses prime numbers and factoring to protect bank accounts and online shopping, generating a secret large number from two primes and a hard-to-reverse password.

Play a number game to classify numbers as prime, composite, or neither, with primes having two factors, composites three or more, and one and zero as neither.

Master prime factorization by building factor trees to break numbers into prime factors, list bottom primes in ascending order, and reconstruct the original number by multiplying those primes.

Discover how to find the greatest common factor (GCF) by factoring, identifying common factors, and selecting the greatest, with examples such as 25 and 10 and 12 and 6.

Discover how to pick tile sizes by finding common factors of wall height and width, revealing the greatest common factor for a 16 by 16 inch tile with no gaps.

Identify how multiples arise by multiplying numbers by whole numbers, distinguish them from factors, and use six, eight, twelve, 24, and 120 as examples of infinite multiples versus finite factors.

Discover the least common multiple, or LCM, by listing multiples for numbers like four and six and identifying the smallest shared multiple, such as twelve.

Learn to find the greatest common factor and least common multiple using prime factorization and factor trees, identifying shared and unique primes across numbers.

Learn how fractions break a whole into equal parts, with the numerator on top and the denominator on bottom, using a pizza analogy and the one over eight example.

Explore fractions by identifying whether shaded regions form equivalent parts across shapes, distinguishing equal-area pieces from unequal ones, with practical examples of numerator and denominator.

Explore the five types of fractions—zero, proper, improper, whole numbers that can be written as fractions, and mixed numbers—and learn when to use each form in calculation and everyday contexts.

Master three common ways to write negative fractions, such as negative two over three, with the negative sign in front, on the numerator, or next to the denominator. Count the negative signs: even yields positive, odd yields negative, using a table analogy to visualize.

Explore equivalent fractions by comparing portions of a chocolate bar and cake plates. Learn how multiplying or dividing the numerator and denominator preserves value, with examples like 1/2 to 2/4.

Simplify fractions by finding the greatest common factor and dividing numerator and denominator to get the smallest equivalent fraction, with examples like 5/10 to 1/2 and 28/77 to 4/11.

Use a four-check mental checklist to simplify large fractions quickly, testing for zeros, fives, evenness, and digit sums divisible by three to find common factors.

Learn to add and subtract fractions with the same denominators using three steps: copy the denominator, add or subtract numerators, and simplify; illustrated with a cup of coffee and milk.

Learn to add and subtract fractions with different denominators by making denominators the same. Use the first method, the least common multiple, or multiplying denominators, with practical examples.

Learn the butterfly method for adding and subtracting fractions using bubble diagrams to multiply across, yielding quick results with large equivalents and subsequent simplification.

Master four fraction multiplication scenarios, simplify top and bottom numbers before solving, and cancel common factors to speed up calculation while counting negative signs to determine the final result.

Use circles to show six divided by two as three groups. Six divided by one half equals twelve groups, and fractions yield larger results.

Master dividing fractions by turning division into multiplication with the keep flip change method, flipping the second fraction, and performing pre-multiplication simplification.

Solve a battery problem using visual reasoning and fraction division: 3/4 full battery divided by 1/8 per day equals 6 days, illustrating drawing a battery and chunking into eighths.

Master a two-step technique to compare fractions quickly by multiplying across the two numbers, allowing you to decide which fraction is larger in seconds.

Learn to add and subtract three fractions by converting to a common denominator using the least common multiple, forming equivalent fractions, and simplifying the final result.

Multiply and divide three fractions using the KFC method—keep, flip, change—by converting divisions to multiplication, simplifying common factors, and tracking negative signs for a correct result.

Learn the difference between mixed numbers and improper fractions, when to use each, and how to convert between them for easy everyday calculations in cooking, travel, and more.

Convert improper fractions to mixed numbers and back, keeping the denominator constant, to simplify calculations and improve mental math.

Learn to add mixed numbers by breaking them into whole parts and fractions, then sum wholes and fractions or convert to improper fractions, using equivalent fractions for different denominators.

Master subtracting mixed numbers by handling two subtraction types, distributing the minus sign to both whole and fractional parts, borrowing when needed, and simplifying to the final mixed number.

Convert mixed numbers to improper fractions, multiply numerators and denominators, simplify with common factors, then convert the product back to a mixed number.

Convert mixed numbers to improper fractions, apply keep-flip-change to transform division into multiplication, simplify by common factors, multiply, then convert the result back to a mixed number.

Explore decimals by tracing their ten-based roots, learn the decimal point and whole versus fractional parts, and convert fractions like 1/10, 3/10, and 2/9 to decimals, including repeating decimals.

Explore decimal place values from ones to thousands and beyond, learning how each position powers ten and relates to tenths, hundredths, and thousandths, with decimals like 0.2, 0.08, and 0.007.

Master rounding decimals by applying place-value rules to hundredths and ones, using the digit behind to decide up or down, with examples like 6.495 to 6.50 and 9.99 to 5.00.

Learn what repeating decimals are, how to identify them, and how to write them with an overline, then convert these decimals to fractions with examples.

Master adding and subtracting decimals by lining up decimal points and stacking numbers. Use examples like 4.25 and 0.92 to reinforce alignment and place-value skills.

learn to multiply decimals by stacking numbers with the right alignment, multiplying as whole numbers, and reintroducing the decimal by counting decimal places for the final answer.

Use long division to convert remainders into decimals, illustrated by 19 divided by 8. Extend with a decimal point and zeros to reach 2.375, ending with zero when complete.

Learn to divide a decimal by a decimal using long division, clearing decimals with powers of ten, converting to whole numbers, and solving examples like 1.632 ÷ 5.1 to 0.32.

Convert fractions to decimals by checking if the denominator fits into ten, hundred, or thousand, using equivalent fractions for a quick result. Use long division for others, including repeating decimals.

Convert decimals to fractions in two steps: write the decimal part over its place value, then simplify to the lowest terms, using examples like 0.8, 0.35, and 11.09.

Understand how percentages are fractions over 100, how they relate to decimals, and why percent and percentage differ, with examples like 28% and 80%.

Convert decimals to percentages and back by moving the decimal point and multiplying or dividing by 100, with examples like 0.28 to 28% and 29.3% to 0.293.

Convert fractions to percentages by forming an equivalent fraction with a 100 denominator or using long division, illustrated with 2/5 = 40%, 15/200 = 7.5%, and 7/8 = 87.5%.

Master converting percentages to fractions by dividing by 100 and simplifying, with extra steps for decimals, then express results as fractions, improper fractions, or mixed numbers.

Learn how to find a percentage of a number by multiplying, using either fractions or decimals. Explore practical examples like calculating tips and converting results to mixed numbers or decimals.

Master mental math with the chopping board method, breaking percentages into 10%, 5%, and 1% to quickly compute tips and other percent problems.

Translate percentage word problems into math equations using the what toolbox. Solve for the unknown with fractions or long division, using examples like 25% of 80 and 24% of 50.

Learn to compute sale price from original price and discount percent, convert percentages to fractions, and determine the discount percent from original and sale prices.

Learn how simple interest affects loan costs for cars, mortgages, and student loans by applying the formula I = P R T to principal, rate, and time.

Learn to compute percent change from the original value to a new value using the change over original formula, or equivalently (new minus original) over original, for increases and decreases.

Master the reverse-percent trick using the commutative property of multiplication, converting percentages to fractions, with examples like 18% of 50 and 75% of 4.

Learn how to compute a percent of a percent by converting percentages to fractions or decimals, multiplying, and simplifying to 28% as in 35% of 80%, with practical examples.

Master exponent arithmetic by learning that exponents are repeated multiplication, visualize growth with paper folding, and apply exponential growth models to real-world phenomena like virus spread.

Master exponents by understanding the base, exponent, and power; learn how to write exponents with symbols, parentheses, and common phrases like squared and cubed.

Explore how any number to the zeroth power equals one and any number to the first power equals the base, with guidance on the zero-multiplication mistake.

Learn how to raise fractions to a power by multiplying the fraction or distributing the exponent across the numerator and denominator, using examples like (2/3)^3 and (1/2)^4.

Explore decimals with exponents by expanding the power, multiplying decimals without a calculator, and counting decimal places to place the decimal correctly; also convert decimals to fractions when helpful.

Learn how negative exponents become positive via reciprocals, turning 3^-2 and 2^-4 into fractions like 1/9 and 1/16. Explore decimal and mixed-number conversions.

Learn how to add and subtract powers by solving each power separately, then adding results; when bases and exponents match, express the sum as a multiple of the common power.

Master the rules for multiplying powers: add exponents with the same base, multiply bases with the same exponent, and handle cases where neither applies, including negative exponents.

Learn to multiply powers with different bases and exponents by expanding into prime factors and factoring common bases, yielding a single base with a combined exponent.

Divide powers with the same base by subtracting exponents, and when exponents match, divide the bases. Review also covers multiplying powers, cancellation, and checking results using a fraction bar.

Explains how exponents within exponents simplify using the power rule and the commutative property, with examples like two to the power three to the power five and negative exponents.

Explore compound interest and how it grows investments over time, contrasted with simple interest used for loans. Learn the formula and key terms like principal, rate, and compounding.