Writing Expressions from Patterns and Words

A variable is a letter that stands for a number that can change or that you don't know yet, like $h$ for hours or $n$ for a stage number. A term is a number, a variable, or numbers and variables multiplied together; terms are separated by $+$ and $-$ signs. The number multiplying a variable is its coefficient: in $4h$ the coefficient is $4$. A term with no variable, like $15$, is a constant, because its value never changes. An expression is a combination of terms, such as $15+4h$. It has no equals sign. Once you write $15+4h=35$, you have an equation: a statement that two things are equal, which you can solve (that is the job of a later lesson). Two hidden coefficients trip people up: $x$ means $1x$, so its coefficient is $1$, and $-x$ means $-1x$, so its coefficient is $-1$.

Look at the tile pattern below. Stage 1 uses 5 tiles, stage 2 uses 8 and stage 3 uses 11. Instead of drawing stage 50, look at how each stage is built. Every stage has 2 tiles that never change, plus 3 rows of tiles whose length matches the stage number. So stage $n$ has 3 rows of $n$ tiles, which is $3n$ tiles, plus the 2 fixed ones: $3n+2$ tiles in total. The coefficient $3$ is the number of tiles added each stage, and the constant $2$ is the part that stays the same.

When you can't see the structure in a picture, a table still works. Step 1: find how much the count changes from one stage to the next. If the change is the same every time, that change is the coefficient of $n$. Step 2: work backward one stage to find "stage 0": subtract the change from the stage 1 value. That number is the constant. For 5, 8, 11 the change is 3, and stage 0 would be $5-3=2$, giving $3n+2$. Step 3: test the expression on a stage you already know. This method works only when the change is constant (a linear pattern). The square numbers 1, 4, 9, 16 grow by 3, then 5, then 7, so no expression of the form "coefficient times $n$ plus constant" fits them; they need $n^2$.

People solved problems about unknown amounts long before anyone wrote $x$. Clay tablets from Babylon (in present-day Iraq), nearly 4000 years old, contain problems we would now solve with equations, written entirely in words. Around 820 CE (sources give dates between about 813 and 833), the Persian scholar Muhammad ibn Musa al-Khwarizmi, working at the House of Wisdom in Baghdad, wrote a book often called The Compendious Book on Calculation by Completion and Balancing. Its Arabic title contains the word al-jabr, "restoring" or "completion": moving a subtracted quantity to the other side of an equation so that it becomes added. It also contains al-muqabala, "balancing": cancelling equal amounts from both sides. When the book was translated into Latin, al-jabr became our word algebra, and his own name, written in Latin as Algoritmi, gave us the word algorithm. He still wrote everything in words. One of his best-known problems, paraphrased in modern English, says that a square together with ten of its roots (a "root" meant the unknown number itself) makes thirty-nine. Today we write it as $x^2+10x=39$.

• About 1800 BCE, Babylon: word problems about unknown quantities on clay tablets, solved with step-by-step recipes.
• Probably the 3rd century CE, Alexandria: Diophantus uses short abbreviations for an unknown in his work Arithmetica (his exact dates are uncertain).
• Around 820 CE, Baghdad: al-Khwarizmi describes completion and balancing as general methods, the start of algebra as its own subject.
• 1557, Wales and England: Robert Recorde introduces the equals sign, two parallel lines of the same length.
• Late 1500s to 1637, France: François Viète uses letters for both known and unknown quantities, and René Descartes popularizes $x$, $y$ and $z$ for unknowns.