Many students who have difficulty with algebra do not see variables as a simplification. They see variables as adding to their problems not solving them. Let us try to solve a word problem using only words, no symbols. This method will be open to more than one attack. The purpose is to show the students the advantages of symbols over words.
In 3 years Mike will be twice as old as he was 8 years ago. Now old is he now?
In 3 years Mike will be twice as old as he was 8 years ago. In 3 years he will be 11 years older than he was 8 years ago. That difference, 11, is his age eight years ago. So he is 19.
If Mike is 19 now, he will be 22 in 3 years. Eight years ago he was 11. 22 is twice 11, so we are done.
Each reader may have a different way of reaching the solution, including trial and error. We must pay very close attention to each step. Nothing is automatic. We must ask and answer the question, “Why is that true?” after each sentence. You may grant my solution is correct, but feel my argument is beside the point, coincidental, or even fallacious. It surely in a “word” problem) Now look at the problem using symbols.
Let a variable stand for what we are looking for. What are we looking for? Mike’s age now. So let ~ = Mike’s age now. What other ages does the problem mention? Mike’s age 3 years from now, and Mike’s age 8 years ago. What do we do to someone’s age now to get their age 3 years from now? What do we do to someone’s age now to get their age 8 years ago? Three years from now he will be older so add 3 to his present age. Eight years ago he was younger so subtract 8 from his present age. Let us show this in an organized fashion.
Let x = Mike’s age now.
x+3 = Mike s age in three years.
x8 = Mike’s age eight years ago.
In 3 years he will be twice as old as he was 8 Years ago.
x + 3 2(x 8)
x +3 = 2x 16
19 = x
Same answer as before so the check remains the same.
Notice that the wordy solution requires more insight, while the symbolic solution requires more preparation. To solve the problem we translate it into symbols which make an equation for us to solve. We solve it using the properties of numbers.
The rhetorical method treats each problem as a unique special, case. The symbolic method allows us to see generalizations. In this case, age problems, we need remember only a few common sense ideas. When we talk of age in the future we add the years to the age now. When we talk of age in the past we subtract the years from the age now. Lastly, the difference between peoples ages remains constant throughout their lives.