If a fraction is a piece of pie, how can we make students understand multiplying two pieces of pie?

–   Hung-Hsi Wu, Department of Mathematics, University of California

The Progressions Documents for the Common Core Standards for Mathematics describe the vertical sequences that helps students build conceptual understanding of math topics. The fractions document describes a shift in instruction that focuses on fractions as numbers, a recognized gap that has often prevented students from understanding operations with rational numbers.

In Grades 1 and 2, students begin to use fraction language to describe the partitioning of shapes into equal shares. The idea of a fraction is developed more formally in Grade 3, where the whole being partitioned can be a shape, a line segment, or any entity that can be subdivided and measured. The Common Core standards hold off the consideration of a whole as a set or collection of objects until Grade 4.

Beginning with unit fractions (numerator = 1), we next discuss that we can put unit fractions together and count them. There is no need for discussion of “proper” or “improper” fractions at this point; 5/3 is simply the quantity you get by combining together 5 parts when the whole is divided into 3 equal parts.

Throughout instruction, two key aspects of fractions are emphasized:

1) Specifying the whole

2) Explaining what is meant by “equal parts”

Initially students can use their idea of congruence (same size and shape) to understand equal shares, but they later come to understand that equal shares means equal measure, even if the appearance of the shares does not fit with their idea of congruence.

(see the full document for illustrations)

From the progressions document:

“The goal is for students to see unit fractions as the basic building blocks of fractions, in the same sense that the number 1 is the basic building block of the whole numbers; just as every whole number is obtained by combining a sufficient number of 1’s, every fraction is obtained by combining a sufficient number of unit fractions.”

The document goes on to describe the partitioning of a unit interval on the number line into equal lengths, and how this model reinforces the analogy between fractions and whole numbers:

“Just as 5 is the point on the number line reached by marking off 5 times the length of the unit interval from 0, so 5/3 is the point obtained in the same way using a different interval as the basic unit of length, namely the interval from 0 to 1/3.”

Equivalent fractions can be seen by partitioning number lines and noticing that many fractions label the same point on the line. Both area models and number lines continue to be used to generate the understanding in Grade 4 that multiplying or dividing both numerator and denominator by the same number generates an equivalent fraction. And the focus on unit fractions allows students to see that the meanings of the operations are the same for whole numbers and fractions.

Food for thought. Think about your own instruction in fractions; does it provide a foundation for students to see fractions as numbers that can be composed and decomposed, just like we do with whole numbers when we add, subtract, multiply, or divide? Or has your fraction instruction been isolated and focused on slices of pies? How will you shift your instruction to provide a more solid foundation for your students?

Read the full 13-page document and let me know your thoughts on the Questions/Comments/Solutions link at the top of the page. My bet is that you’ll find yourself truly understanding operations with fractions, perhaps for the first time. Just think of how much more you can give to your students with your new understanding!