Building Fraction Sense

The struggle here is real.

These two samples are from the same student. Makes me think about what we prioritize, and about who else is hiding. pic.twitter.com/ePCfiEVVWK

— Kent Haines (@KentHaines) September 13, 2016

A student has no idea where to place a fraction on a number line (because fractions aren't numbers, of course) but can convert to a mixed number like a champ.

My attempt to help out:




This applet gets at the heart of the things I've enjoyed working on lately.  The initial estimate offers very little help, but as the student progresses through, they have more references which allow the revisions to become more precise.  When my students worked with this applet, there were audible groans when I asked them to lower the lids on their computers as well as exclamations of "I got it!" when they moved closer to 0% error.

Here's a GeoGebra book that goes from estimating fraction to addition to multiplication.  I'm still working on division, but that should drop soon.

More Fraction Multiplication

My 7th grade students are in the middle of exploring fractions.  They are currently researching the question:

What are fractions and how can I add, subtract, multiply and divide them?

I'm not sure how much instruction I want to give here. Is this intuitive? What can I do to make this more usable?

 Applet with more instruction is here.

Update
John Golden:

@dcox21 @danielpearcy worth letting students adjust estimate afterfirst fraction is shown?
— John Golden (@mathhombre) November 8, 2012

Good idea.  Updated version.

The Comma

If [this], then [that].  We talk a lot about [this] and [that] in the math classroom. Teacher supplies [this], student responds with [that]. They even have names:  hypothesis and conclusion.  But, what about the comma?   All the power of the entire process is summed up with a tiny little "," that is all too often ignored.

No more.  It's time to give the comma a voice.

We were getting ready to add rational expressions and I wanted my students do have a workable rule for adding fractions with like and unlike denominators.  My goal was to develop the idea that when adding fractions with unlike denominators:

A lot of students don't see this very clearly.  They do what they do to jump from [this] to [that].  And most of the problems end up looking a lot like

with no real understanding of what's taken place between the hypothesis and conclusion.  And up to this point, no one has really cared because Johnny was able to find the correct answer on multiple choice scavenger hunts with a great deal of accuracy added fractions like a champ in 6th and 7th grade. However, when Johnny gets to algebra, and sees

for the first time, you'd think he's never worked with fractions before.


Time to talk about the comma.  It was actually a pretty simple adjustment to a simple question, but the conversations it generated made all the difference in the world.

This quickly became

See, the beauty here is that the process became the outcome.  The numbers become the variables and we get a good grip on how one-third plus two-sevenths becomes thirteen-twentyfirsts.  The abstract isn't so abstract and the easy part is swapping out the 1, 2, 3, and 7 for a, b, c, and d.

Mission accomplished.  Now, lets hope they remember it tomorrow.

To LCD or Not To LCD

Is finding the LCD necessary?  I know that it makes the numbers smaller when adding fractions, but does it help conceptually?

Would teaching fraction addition like this:  $\frac{a}{b}+ \frac{c}{d} = \frac{ad+bc}{bd}$

and then reducing later make adding polynomial fractions like  $\frac{x}{x+2}+ \frac{3x}{x-3}$ a bit more intuitive?

Does teaching LCD make an already difficult concept more difficult?  Or does it help?

Here are a couple of applets that illustrate the point.  I'm wondering which one would be more helpful for 5th - 7th grade kids.

With LCD and without LCD.

If you want the original GeoGebra files:  LCD, NO LCD

Fraction Multiplication

I've been spending some time lately thinking about how to help elementary school teachers teach math conceptually even if they don't consider themselves strong in math. I keep coming back to interactive applets (read: GeoGebra) because it's kind of a one-stop shop. Yesterday, I tried a few out with a 4th grade class and they responded very favorably. So, here are the latest two I've done on fraction multiplication.

Click on "File" and "Save" if you'd like a copy of either applet.



This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com





This is a Java Applet created using GeoGebra from www.geogebra.org - it looks like you don't have Java installed, please go to www.java.com


Please forgive the crowding, I had to minimize everything in order to fit them into the post.