Needing some math help…

December 17, 2012 at 9:01 pm 5 comments

It has been 7 months since I last wrote because I have been super busy with big life changes. First of all, I got married. (I couldn’t possibly do justice to that here, but I had to mention it.) Then I moved from fifth grade to eighth grade. I have really loved the change both in spite of and because of the challenges that come with teaching a new curriculum. And perhaps most importantly, I’ve been busy with some pretty great math that has nothing to do with school:

Some good math

But I’m back because I couldn’t think of a better place to ask for math help. And I really need it.

Let me digress – I never really appreciated how important it is to deeply understand the math you’re teaching until this year. Every day something happens that makes me feel like I don’t know everything I should, and I think, “Gosh, pedagogy and teacher education are frickin’ important.” I have always been really good at math but that doesn’t mean that I didn’t get by on a little bit of rote memorization here and there. That is really hurting me now. Kids can’t just memorize rules in math because there would be too many of them. It would literally be impossible. Understanding concepts and chunking them together lightens the cognitive load. But how can a teacher do this for her kids when she can’t do it for herself? Which brings me to my question…

I’m struggling to explain why we simplify expressions differently depending on whether we are dealing with factors or addends. I can’t seem to put my finger on the underlying math concept that distinguishes the two. For example:

equation 1

The same concept pertains to simplifying radical expressions:

equation 2

And it pertains to scientific notation:

equation 3

I know that the ease of multiplication and division has to do with the commutative property. But how do you explain why it doesn’t work for addition and subtraction? I have used examples kids can solve with order of operations like (3×4)(2×5) and (3×4)+(2×5). But the main idea the students seem to come away with is “Something is wierd with addition and subtraction,” which is not really helpful.

I am writing because I am hoping that teachers who have taught this before or people who just simply enjoy thinking about math might be able to provide insight. I am especially interested in being able to explain the relationship between the examples I mentioned. I’d love an analogy or model that would allow me to really show the kids the basic concept that underlies it all. Frankly, I need an ah-ha moment.

I don’t know if this post will end up making me seem like an idiot or a smarty pants. But if nothing else, it highlights the depth to which teachers must understand their content and how difficult it is to get there on your own. So, come on, math geeks! I know there’s an explanation out there that my kids would love!

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Entry filed under: Math Content, Number Sense and Operations. Tags: .

Our Imaginary Field Trip I will break standards into objectives.

5 Comments Add your own

  • 1. Nellie  |  December 18, 2012 at 3:12 am

    So the question really is, why are fractions equivalent? What makes them represent the same quantity. a/b = na/nb because you are multiplying a/b by 1, namely, n/n. likewise for simplifying only one you are dividing by one, namely, n/n which may look like (1+x)/(1+x) or 5/5. So one is always looking for a common factor to the numerator and denominator to “create”, if you give me the liberty, a one–a n/n. 3/5 while we can write it as (2+1)/(4+1) doesn’t have a one to factor out but 3/6 does have a one…3(1)/3(2) = (3/3)(1/2)= 1(1/2)=1/2. Likewise,( x-2xy)/(3x+xy) does “have a one” namely, x/x. So (x-2xy)/(3x+xy)= (x)(1-2y)/(x)(3+y)= (x/x)[(1-2y)/(1+3y)]= (1-2y)/(1+3y). Hope this helped.

    Reply
  • 2. Nellie  |  December 18, 2012 at 11:43 am

    Upon waking I have more clarity. While this issue does deal with what equivalence means in terms of fractions, it may rest deeper at the identity elements for multiplication/division, namely, 1 or n/n, vs the identity element for addition/subtraction which is zero. If one adds or subtracts zero to/from a fraction, the resulting fraction is obviously equivalent to the original rational number. However if one adds or subtracts 1, namely, n/n, to/from a fraction the resulting rational number is NOT equivalent to the original. Similarly, multiplying/dividing by 1 or n/n results in an equivalent (identical) fraction but you wouldn’t mult/divide that fraction by zero because zero isn’t the identity element over mult/division and your result would either be zero or undefined.

    Reply
  • 3. Nellie  |  December 19, 2012 at 1:51 am

    Oh, Lisa about radical expressions and scientific notation…the reason is different. These are multiplicative processes. They both deal with exponents and the laws thereof which go back to basic multiplication. But you still can’t add or subtract unlike things. So you can add the sq rt of 2 plus 5 sq rt of 2 and get 6 of those suckers (sq rts of 2), but you can’t add a sq rt of 2 with a sq rt of 3 because they don’t have a common name. You can obviously approximate their sum but one is 2 to the 1/2 power and the other is 3 to the 1/2 power…it’s like trying to add dogs and apples.

    Reply
  • 4. Noya  |  January 2, 2013 at 10:50 pm

    I have been reading your blog in preparation for teaching my son math at home next semester. I love your dedication to conceptual teaching. I wish you were MY math teacher. I love how you rephrased the math standards as , “I will” statements and posted all of them on the wall. I would like to do something similar. Where do you suggest I begin? We are aligned with the common core. Did you just rephrase each standard listed there?

    Thanks

    Reply

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