Writing Algebraic Expressions

Most algebra questions start life as a sentence in English, so you need a dictionary. For adding, look for more than, sum, increased by, total and altogether: six more than n is n plus 6. For subtracting, look for less than, decreased by, take away, fewer and difference. And here is the trap that catches almost everybody. Six less than n is n minus 6, not 6 minus n. The words arrive in one order but the algebra comes out in the other, because you start at n and take six away from it. If you are ever unsure, test with a real number: at n equal to 10, six less than n should be 4, and n minus 6 gives exactly that.

Now the multiplying and dividing words. Double means two lots, so double n is 2n. Triple means three lots, giving 3n. Times, lots of, and product all mean multiply, so the product of 4 and n is 4n. Going the other way, half of n is n over 2, a third of n is n over 3, and shared equally between, split between and quotient all mean divide. Real situations use these constantly. If one pen costs 3 dollars, then k pens cost 3k dollars, because you pay 3 for each one. If m marbles are shared equally between 4 friends, each friend gets m over 4. Notice the unknown can be the price or the count.

Many expressions need two steps, and the order of those steps changes the answer. Take double n, then add 3. Step one gives 2n, step two adds 3, so the expression is 2n plus 3. Now flip it: add 3 to n, then double. Step one gives n plus 3, and step two doubles that whole thing, so you must wrap it in brackets and write 2 times bracket n plus 3. Brackets are how algebra says do this bit first. To see that they really are different, put n equal to 10 into each. Ten doubled is 20, plus 3 is 23. But ten plus 3 is 13, and doubled that is 26.

Word problems feel harder than they are, because the first move is always the same: decide what the letter means and write it down. Suppose Mia is m years old. That sentence has already given you the letter, so m is Mia's age in years. Her brother is 3 years older, which is 3 more than m, so his age is m plus 3. Her mother is twice Mia's age, and twice means two lots, so her age is 2m. Now test the whole thing with a real number. If Mia were 12, her brother would be 15 and her mother 24, and every expression agrees. Money stories work identically: Ana has d dollars and spends 12, leaving d minus 12.

Shapes are a great place to practise, because the picture tells you exactly what to add. Perimeter just means the distance all the way around, so walk around the edge and write down each side as you pass it. For a rectangle with sides x and 7, that walk gives x plus 7 plus x plus 7, which is two x sides and two sevens, so the perimeter is 2x plus 14. Check it at x equal to 5: the expression gives 24, and adding 5, 7, 5 and 7 also gives 24. A triangle with three equal sides of a has perimeter a plus a plus a, which is 3a. One short expression describes every triangle of that shape.