Here is an example that was recently brought to my attention by another tutor. Her 6th grade student was given this question:

How do you go about teaching this young student to solve this problem without using a system of equations?

First, we have to figure out what the information is telling us. I like to use tables, it leads nicely into functions later on, and organizes all of the information so nicely!

I can see right away that I could use comparison bars to represent two different subtraction equations. I see the amount of apples I started with, minus the amount I sold, equals the amount I have now; or x-82=a And I see the amount of pears I started with, minus the amount I sold, equals the amount I have now; or x-34=p.

But these equations are missing too much information to solve. So I need to try to figure out some of the other missing details. I used "x" for the start of both equations because the story told me that there were the same number of apples and pears to begin with. I can use that to look at the comparison given. The story said that I now have 4 times as many pears as apples. That means, for every 1 apple, I have 4 pears.

If I were doing a legit algebra problem I could use that information and replace the "p" in my second equation with 4x. BUT, we are only 6th graders and don't understand how to use a system of equations. So we have to think like scientists for a minute.

There are 3 facts about this problem that will help us solve it.

- BOTH fruits started with an equal quantity.
- Different amounts have been sold.
- The ratio of the remaining fruits is 1:4

For my low students, I would suggest the guess-and-test method to solve this. I

*that term and if there is ANYway possible you can avoid using and burning it into your young students minds, I really think that would be great. But it is a method that is taught. I'll just simply ask my students, "what numbers do you think will make this work?" It's kind of like looking at a broken bicycle and asking a child what will make that bike go again? Most students will likely pull a random number from thin air, with no rhyme or reason. Take it and work with it. I find that if you focus in on how to hypothesize at this point, you lose the math point.*__despise__
To help my more advanced students get this process started, I might focus in on the ratio of the remaining parts, and the difference in the amounts sold. These numbers give clues so the "guess" part of the problem solving can have a little more rhythm. It also can save A LOT of time on tests! Looking at the number tiles below, you can see an overlap. The difference in the remaining amounts is 3 parts (not to be confused with 3 pieces of fruit!), The difference between the amounts sold is 82-34=48. If we divide those 48 pieces of fruit into 3 parts, each one will be 16 pieces of fruit. We can use that 16 to solve now, without using that table and 5 or six "guesses" that may or may not end up leading us to the right answer.

If this has got you all feeling, "why do we even have to teach this way?" Trust me, I understand! Here's what I have been telling myself:

- These kids still need practice with their basic facts, none have them memorized, and NONE want to memorize them because they don't see the need. This tricks them into using those skills for a purpose! They get real life (sort of) practice using the basic facts and rules of math to solve problems.
- They are also getting practice working with numbers in a complex situation they might never encounter naturally before Algebra begins for real. This
ease their fears and provide some background knowledge for those teachers to capitalize on later.__should__ - Finally, they are problem solving, no matter what method is used, or how complex the steps are to get to the answer. Math is about finding a solution to a problem, based on what can be observed and what can be determined. This is a skill that transfers to every area of life.

Until next time!

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