Mathematics1 course

Arithmetic

Whole numbers, fractions, decimals and percentages: the number sense every later course relies on.

About Arithmetic

Arithmetic is the toolkit every later course quietly depends on: whole numbers, fractions, decimals, percentages and the order in which operations are carried out. Most mistakes students make in algebra and calculus are really arithmetic slips, such as dropping a negative sign or adding fractions without a common denominator.

The ideas are few but they are used constantly. Place value explains why decimals line up the way they do, factors and multiples explain how fractions simplify, and percentages are fractions with a denominator of 100. Once those connections are clear, a long list of rules turns into a handful of ideas that support each other.

Arithmetic shows up every day outside school: splitting a bill, comparing unit prices, working out a tip or a sale price, converting a recipe, and checking that a pay slip adds up. Fluency here also makes standardized tests faster, because less time goes into routine calculation and more into reading the problem.

Arithmetic Foundations

Operations, order of operations, fractions, decimals, ratios and percentages with worked practice. (Middle school)

Order of Operations

An expression like $8 - 2 \times 3$ only has one correct value because everyone agrees on the order in which operations are carried out. Without that agreement, the same line of symbols could mean 18 or 2, and calculators, spreadsheets and textbooks would disagree.

Fractions

A fraction $\frac{a}{b}$ names $a$ equal parts of a whole split into $b$ parts. Two fractions are equal when they describe the same amount, which is why multiplying the top and bottom by the same nonzero number never changes a fraction's value. That one idea powers simplifying, finding common denominators and comparing sizes.

Percentages

Percent means per hundred, so $35\%$ is the fraction $\frac{35}{100}$ or the decimal $0.35$ . Almost every percentage question is one of three forms: find a percent of a number, find what percent one number is of another, or find the original number before a percentage change.

Full Arithmetic Foundations course outline

How to study Arithmetic

Estimate before you calculate. If $48 \times 21$ should be close to $50 \times 20 = 1000$ , an answer of 108 is obviously wrong.
Write every step for fractions: find the common denominator, rewrite both fractions, then add the numerators.
Convert between forms on purpose. $\frac{3}{8} = 0.375 = 37.5\%$ is one number written three ways.
Check percent problems by working backwards: if a price rose 20% to $60, then $60 / 1.2 = 50$ was the original.

Before you start: No prerequisites beyond counting and the times tables.
Where it leads: Pre-algebra, where the same operations are applied to integers, exponents and unknowns.

Key terms in Arithmetic

Place value: The worth of a digit set by its position: in 4,072 the 7 means seven tens, so moving a digit one place left multiplies its value by ten.
Order of operations: The agreed sequence for evaluating expressions: brackets, then powers, then multiplication and division left to right, then addition and subtraction left to right.
Factor: A whole number that divides another exactly. The factors of 12 are 1, 2, 3, 4, 6 and 12; listing them in pairs keeps you from missing one.
Common denominator: A shared bottom number that lets fractions be added or compared, such as rewriting $\tfrac13$ and $\tfrac14$ as $\tfrac4{12}$ and $\tfrac3{12}$ .
Percent: A ratio out of 100. Finding 15% of 80 means $0.15 \times 80 = 12$ ; percent change compares the difference with the original amount.
Estimation: Rounding numbers before calculating to predict the size of an answer, which catches slipped decimal points and misplaced digits quickly.