by C. Elkins, OK Math and Reading Lady
The fractions focus today will be on some basic concepts that students should understand before they work to compare them, determine equivalent fractions, simplify them, use mixed fractions, or add / subtract them. I am including a FREE copy of my Fraction Basics reference guide (click here), along with a photo of an anchor chart I made for a fourth grade class.
I have been rereading a book I love about fractions called “Beyond Pizzas and Pies, 1st Edition.” It has great examples of children’s misconceptions about fractions and lessons on how to try to remediate them. A recurring theme in the book is that while kids can learn “tricks” to help them solve fraction problems, they often do little to help students conceptualize what fractions are. Here’s a link to Math Solutions regarding this book: Beyond Pizzas and Pies (2nd Edition) Following are five examples from the book that made an impact on me and my teaching (which I will go into more detail about on future posts).
- To compare fractions, students often forget that the size of the whole makes a difference. Consider this problem: “Jack ate 1/2 of a pizza from Pizza Hut. James ate 3/4 of a pizza from Mario’s. Who ate more pizza?” Most children would pick James because 3/4 is more than 1/2. BUT, the story did not mention the size of the pizzas. What if James was a personal pan size and Jack’s was a large? The objects or sets being compared must be the same size whole.
- When comparing these two fractions (5/6 and 3/4) students often quote “the larger the denominator, the smaller the fraction” and then say that “5/6 is smaller than 3/4 because it has a larger denominator.” BUT that quote is only part of the concept. The numerator must also be considered as well as the size of the whole. Think about it like this: 3/4 is 1/4 away from a whole. 5/6 is 1/6 away from a whole. Since 1/6 is a smaller part than 1/4, then 3/4 is the smaller fraction. Picture this using a bar model or number line (see the basic concepts chart).
- Students often get confused when we tell them to multiply (or divide) a fraction by a/a to get an equivalent fraction, because they have always been told that multiplication results in a larger amount and division results in a smaller amount. They can learn to follow the “rule” but until it is paired with a visual example (see my basics chart – page 1 at the bottom), they don’t really get that multiplying (or dividing) a fraction by a/a is multiplying/dividing it by 1 (which of course doesn’t change the amount, but just reduces or increases the size of the parts).
- Students often draw circles (hence the book name “Beyond Pizzas and Pies”) to show all of their fraction problems, when a bar, number line, or set might actually represent the problem better.
- To compare or estimate fractions, students often just need to consider whether the fraction is less than half or more than half, but they don’t often know how to determine the benchmark fraction 1/2 or its equivalence. Some often resort to the “butterfly” method or finding common denominators when it isn’t necessary.
See the beginning of the post for a FREE copy of the above 2-page Fraction guide. Stay tuned for Fractions: Part 4.