Saturday, February 21, 2015

Fly on the Math Teacher's Wall: Squashing Fraction Misconceptions

The irony is not lost on me. Fractions, the math concept I most struggled with in elementary school, is now one of my favorites to teach. In this Blog Hop, my math blogging friends and I will be exploring fraction misconceptions. Here we go...

After years of operating with whole numbers, it's new territory to see fractions and understand the what numerator and denominator mean. What do each of those numbers really mean?

In this example, we'll look at an egg carton. First, we'll consider what the whole this case the whole is the entire egg carton.

Look at the following examples and ask yourself:

1. What does the string show?
2. What do the tile show?


The string shows how many parts our whole is divided into (our denominator) and the tile show how many of those parts have been filled (our numerator.)

A common misconception results when students look at the pieces in the model without taking the meaning of numerator/denominator into consideration. For example, in the first photo above, a student might say that they have 6 pieces, so it's 6/2. Most students, however, can readily tell you that the top example shows one-half, so a bit of probing (Where do you see the 1 in 1/2? Where do you see the 2 in 1/2?) helps to reestablish context.

In a similar example, I've heard students struggle with the question, "What fraction of a dollar is a nickel?"

Many students will answer "one-fifth" because they are thinking of 5 cents; if it has a 5 in it, it must be 1/5.

I like to pull out Money Value Pieces and again revisit the concept of numerator and denominator. We first talk about what our "whole" is: 100 cents.

I might ask, "What does 1/5 look like on our model?" Since we've explored numerator and denominator, they know that the whole would be broken into five portions:

It doesn't take long for someone to say, "One-fifth of a dollar is 20 cents!" (They can check this using the dime piece, a ten strip.) Then, using the nickel model, they explore how many pieces it would take to cover the dollar. "Twenty! So a nickel is 1/20 of a dollar!"

Students need many opportunities to explore the concepts of numerator and denominator using a variety of manipulatives and visual models. (More love2learn2day examples here.) I ask them to record their thinking in a variety of venues: math notebooks, class anchor charts, and video productions. In this ScreenChomp example, you'll hear a pair of students explain the meaning of numerator and denominator; notice that they use more than one visual to explain their thinking.

To continue on the Fraction Misconceptions Blog Hop, please visit my friend Jamie at Miss Math Dork!

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