Values of expressions and equivalent expressions — Mathematics, 11–13
Replace the letter by a number to find the value, and see that two different-looking expressions can tell the same story.
Swap the letter for a number
To find the value of an expression, replace the letter by a number and calculate. In 3n + 2 with n = 4: first 3 × 4 = 12, then 12 + 2 = 14. Careful: 3n means 3 × n, so with n = 4 it is 12, not “34”. Two expressions are equivalent when they give the same value for any number you put in place of the letter: they are two ways of looking at the same situation.
Two ways of counting the same thing
Picture n boxes with 3 pens each: there are 3n pens. Someone else sees each box as 2 pens in a bag plus 1 loose pen, and writes 2n + n. The two people wrote different things but are counting the same pens, so their expressions are equivalent. Each expression shows how its author thought. Seeing that helps you pick the easier one for a problem, and explain in your own words why two expressions always agree.
Testing 3n and 2n + n
Are 3n and 2n + n equivalent (n boxes, 3 pens each: 2 in a bag and 1 loose)? With n = 4: 3 × 4 = 12 and 2 × 4 + 4 = 8 + 4 = 12. With n = 7: 3 × 7 = 21 and 2 × 7 + 7 = 14 + 7 = 21. With n = 15: 3 × 15 = 45 and 2 × 15 + 15 = 30 + 15 = 45. Always equal, and the picture of the boxes says why: each box holds 2 + 1 = 3 pens. So 15 boxes have 45 pens, whichever way you count.
One matching value proves nothing
Trying a single value can fool you. Look at 2n + 3 and 5n: with n = 1 both give 5. But with n = 2 they give 7 and 10, so they are not equivalent. One different value is enough to show that two expressions are not equivalent; matching values only make you more confident, they do not prove it. Test several values, and always ask whether the meaning in the situation also explains the equality.
Spreadsheets and rules that grow
In a spreadsheet you write an expression once and drag it: you get its value for many numbers at a time, and you can compare two expressions column by column. It is handy for problems with a growing rule: someone who has already run 40 m and runs 4 m every second is 40 + 4n metres along after n seconds; with n = 10 that is 40 + 40 = 80 m. Seeing the values for several n lets you compare and decide.
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