Being able to solve proportions is important in seventh grade math. In a Common Core classroom, the emphasis is on making sense of the solution method, rather than on memorizing a given procedure.
One general rule for whether your child is making sense of his solution is this: Does he know what each number means along the way?
Consider this question: you ran 5 miles in 40 minutes yesterday. If you run at the same rate, do you have time for a 3‐mile run right now? In order to answer this question, a student could figure out the time it should take him to run 3 miles. He might write this proportion:
In the spirit of keeping track of the meaning of numbers as you work, here are two ways a typical seventh grader might solve this proportion. In each case, you look at what meanings the student might have for the numbers.
Unit rate:
This means that you run 1 mile every 8 minutes (on average) if you ran 5 miles in 40 minutes. One mile every 8 minutes is a unit rate. Using this unit rate, you must run 3 miles in 24 minutes.
In this solution, every number has a meaning. The fraction
is a rate that compares miles to minutes. You can think of this as "1 mile for every 8 minutes" or as "one‐eighth of a mile per minute." Either way, you can multiply 8 by 3 because running 3 miles takes 3 times as long as running 1 mile. Because
is 24, it should take 24 minutes to run three miles. This strategy builds on strategies for working with equivalent fractions and ratios, which are already familiar to students.
Cross‐multiply and divide: Multiplying 40 by 3 gives 120. (This is the cross‐multiplication part.) Dividing 120 by 5 gives 24, so you can run 3 miles in 24 minutes.
In this solution, it's not at all obvious what the 120 means in the context of running. Forty minutes times 3 miles equals 120‐minute miles? This doesn't make sense. And then why divide 120 by 5? Without a meaning for the 120, it's hard to say why you're dividing.
A typical seventh grader can work out the first solution, given time to think. This solution makes sense, and it's based on other things that he knows well. A seventh grader would be unlikely to come up with the second solution alone.
In a Common Core classroom, teachers encourage students to develop strategies, such as the first solution, rather than telling students solution methods that don't make sense to them.