Algebra - should I allow or discourage this problem solving method?

[songsparrow]

Dd is currently working through solving simultaneous equations (Chapter 7 of Jacobs' Elementary Algebra).  The chapter has taught four methods of solving such equations: Addition, Subtraction, Graphing and Substitution.

 

However, she often solves problems using other strategies or combinations of strategies.  

 

A concrete example:  Today's problem:  A farmer has hens and rabbits.  Between the two there are 30 heads and 86 feet.  How many of each animal does the farmer have?  

 

She turned that into the following simultaneous equations: (x= hens, y=rabbits)

x + y = 30 

2x + 4y = 86

 

To solve it, she the changed the second equation into the equivalent equation: x + 2y = 43

 

Then she said that since x + y = 30 and x + 2y = 43, then y = 43 - 30 = 13.

 

She is correct about her reasoning - she has recognized legitimate relationships between the variables that she used to solve the equations.  I am just not sure if I should allow this or if I should insist that she use one of the four standard methods.  

Thoughts?

 

 

 

 

 

 

[vonfirmath]

That sounds like Substitution to me where she's doing part of it in her head because she can just see it.  Maybe make sure she knows how to do it all the way so she can show her work when necessary and go on?

[EKS]

I don't see how what she did isn't one of the methods she was taught.

 

She multiplied the second equation by 1/2.  Then she subtracted the first from the second.

[Btervet]

She used the Subtraction method, just didn't write down all the parts. Would it help to o encourage her to say how she knew y=43-30? Having her write it out explicitly might make the method more clear.

 

It is really important to learn all the different methods though. It's fine if she favors a certain strategy, but it's critical in higher math to be comfortable with all the methods.

[8filltheheart]

She used the elimination method. She simply did part of the math in her head and didn't show her work in a clear format. I would discourage her from presenting it that way.

 

In case you can't see what she did:

 

x + 2y = 43

-x - y = -30

. . . y= 13

[mom2bee]

She's just not showing all of her work. What she did is valid. Have her write out all steps for you if you can't follow her reasoning--the teacher/grader should be able to discern what the student was thinking and why.

 

   h +   r = 30  -- multiply by -1   to get:  -h + -r  = -30  

 2h + 4r = 86  -- multiply by 1/2 to get:   h + 2r  = 43

                                                              __________

                                                              0 +   r   = 13

 

So now you can back-sub 13 for r in the original equation

 

  h + 13 = 30

      - 13   -13

  h        = 17

 

So he has 17 hens, and 13 rabbits.

2(17) + 4(13) = 86

 

It all checks out.

 

She should be able to write out and explain any missing steps upon request though.

[justasque]

Today's problem:  A farmer has hens and rabbits.  Between the two there are 30 heads and 86 feet.  How many of each animal does the farmer have?

I often explain to my math students that textbooks *purposely*, when teaching a new method, use problems that are easy to solve in a way you already know or even in your head, so that you are more likely to relate the new method to the old, to be able to "see" how it works, etc. However, later on, perhaps later in the chapter or perhaps in a year or two, you will be expected to use the new method on problems that you *can't* do in your head or using the old method. At that point, if you haven't gone through the process of learning how the new method works and doing it often enough that it's in your mathematical tool box, then you're stuck.

 

I will often make up a problem that looks horribly complicated, with lots of variables and strange symbols and such, write it on the board, and quickly solve it with whatever the new method is, using lots of canceling and arm waving and such. Then I will explain that they too will be able to solve such a problem a few years hence, if they learn the new method on the easy-peasy equations they've been given in the current work.

 

It usually is a very effective answer to "but *why* do we have to do it this way"? question. In addition, it is an example of why one has to show one's work - because by writing down the work, you are training your brain to approach problems in a certain, logical way, that becomes second-nature; this skill is critical when the problems become way too large and complicated to do in one's head or in a slapdash intuitive way.

 

To relate this to your specific example, I would make sure that your dd understands the formal way to write whatever it is she is doing. In your example, I would have her write it this way (with or without the check - the important part is that she set up the subtraction problem to get to y = 43-30).:

 

x = number of hens

y = number of rabbits

x + y = 30 

2x + 4y = 86

 

2x + 4y = 86

X + 2Y = 43

 

x + 2y = 43

x + y = 30

------------

0x + y = 43-30

 

y = 13

There are 13 rabbits.

x + y = 30

x + 13 = 30

x = 30-13

x = 17

There are 17 hens.

 

Check:

x + y = 30

13 + 17 = 30

30 = 30 check!

 

2x + 4y = 86

2(17) + 4 (13) = 86

34 + 52 = 86

86 = 86 check!

[songsparrow]

Thanks everyone!  I see now that she was using the subtraction method, but she didn't explain it to me that way and interestingly she was not seeing the connection between what she did and the subtraction method as it was taught to her.  I discussed it with her and I think she's made the connection now.  I am working with her on trying to get her to show all of her work, but it's a challenge when she doesn't even seem to realize that she's doing steps in her head.  

 

On a related note - she also likes to make equivalent equations and then use the substitution method with them.  It wasn't taught that way in the book - making equivalent equations was only used with the addition or subtraction methods and not with the substitution method.  I pointed out that changing the equation(s) into equivalent equation(s) is unnecessary with the substitution method, and that means she's adding an extra step where a mistake could occur.  But she still has a tendency to do it quite often.  Is there any reason to discourage the use of equivalent equations in the substitution method?

 

 

[kiana]

On a related note - she also likes to make equivalent equations and then use the substitution method with them.  It wasn't taught that way in the book - making equivalent equations was only used with the addition or subtraction methods and not with the substitution method.  I pointed out that changing the equation(s) into equivalent equation(s) is unnecessary with the substitution method, and that means she's adding an extra step where a mistake could occur.  But she still has a tendency to do it quite often.  Is there any reason to discourage the use of equivalent equations in the substitution method?

 

No. 

[justasque]

Is there any reason to discourage the use of equivalent equations in the substitution method?

 

Nope. It's quite normal to take a problem and kind of poke around a bit, before moving forward on a straight path to the solution. It's *good* to be able to do this. Later down the line, she may encounter problems where the solution isn't immediately apparent; in those cases feeling free to play a bit with what you've got until something clicks and you see a clear path to the solution is a very good thing. What you don't want is a kid who cannot begin to write anything until they have the whole solution path already worked out; they are missing key skills they will need later on.

[creekland]

I like the way your dd can think through her problems!  Do make sure she knows how to use substitution even if she favors equivalent equations/subtraction.  It's common for kids to favor one method they "see" better.

 

When/if she gets to Calc (no reason to think she wouldn't with her reasoning abilities), substitution can be important.