Describing relations with letters — Mathematics, 11–13
A letter stands for a number, so one short expression can say what a whole set of numbers or a whole relation has in common.
One letter for many numbers
In an expression, a letter holds the place of a number: it can be any number from a set. “n + 5” says “a number and 5 more”; “2 × n” says “twice a number” and is written 2n (between a number and a letter the × sign is left out). The same relation can be told three ways: in words (“Rui has 4 more stickers than Ana”), in a drawing (two rows of stickers, Rui's with 4 extra) and with letters (Ana n, Rui n + 4).
Why letters help
Describing “the even numbers” in words takes a sentence, and listing them (0, 2, 4, 6, 8…) never ends. With a letter you say it all at once: the even numbers are the numbers 2 × n, with n = 0, 1, 2, 3… A letter also lets you explain your thinking to others in a few symbols, and understand theirs: if one classmate writes n + 4 and another 4 + n, they can discuss whether they mean the same thing.
From even numbers to letters
Let us describe with a letter the set of even numbers. Drawing: a row of dots that splits into two equal rows with none left over. Words: an even number is twice another number. Letters: 2 × n, or 2n. With n = 0, 1, 2, 3, 4 we get 0, 2, 4, 6, 8. Is 58 even? 58 ÷ 2 = 29 and 2 × 29 = 58: yes. And 75? 75 ÷ 2 = 37.5, which is not a whole number: no. The other way round: “one more than three times n” is written 3n + 1, and with n = 4 it gives 3 × 4 + 1 = 13.
“5 more” is not “5 times”
Going from words to letters trips people up in two places. First: “5 more than n” is n + 5, not 5n; with n = 3 the first gives 8 and the second gives 15. Second: “3 fewer than n” is n − 3, not 3 − n. If you write the numbers in the order you hear them, you swap the relation. The test is to try it: if Ana has n = 10 stickers and Rui has 3 fewer, 10 − 3 = 7 makes sense; 3 − 10 gives no whole number you know.
Recipes, spreadsheets and programs
A recipe that needs 2 eggs per person can be written 2 × n, with n people. In a spreadsheet, the formula =2*A1 is the same idea: A1 is the “letter”, and the formula gives the result for whatever number you type there. In programs, variables play the same role. Whenever a rule repeats for many values, a letter saves you from writing it a thousand times.
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