Building Foundations for Mathematics Conceptual Understanding of Fractions

Building Conceptual Understanding “For students in the upper elementary grades and even middle school, fractions present a considerable challenge. The area of fractions is where students often give up trying and resort instead to rules. This lack of understanding is then translated into untold difficulties with fraction computation, decimal and percent concepts, the use of fractions in measurement and ratio and proportion concepts.” - Van de Walle and Lovin 2006

Understanding Fractions Outcomes: Understand important fraction concepts Explore the pitfalls of fractions Understand the progression of fractions Experience decomposition with fractions

Early Fraction Concepts Big Ideas Fractional parts are equal shares or equal-sized portions of a whole or unit (and object or a collection of things). More abstractly, the unit is counted as 1. Fractional parts have special names that tell how many of that size part are needed to make the whole. The more fractional parts used to make a whole, the smaller the parts. The denominator of a fraction indicates: what number the whole has been divided in is a divisor (so the numerator is a multiplier) Two equivalent fractions are two ways of describing the same amount by using different sized fractional parts.

Why do students have a difficult time making sense of fractions? As a table group brainstorm all of the misconceptions and common errors students have and make about fractions.

Learning Pitfalls of Fractions Thinking of a fraction as a pair of numbers rather than a representation of a specific amount. Applying whole-number knowledge inappropriately to fractions: 1/8 > 1/5 Inattention to the location of 1 on the number line. Counting the visible parts in one group to find the denominator, rather that counting the number of groups. Counting the parts shaded and not shaded and assign numerator and denominator to those quantities.

Flexibility with Fractions “Flexibility—with fractions as well as with whole numbers—gives students a strong advantage as they move into subsequent grades that build on fraction concepts. But more important, I believe that when students connect their flexibility with whole numbers to other concepts—such as decimals, fractions, and negative integers—the story of math begins to unravel, sparking meaning and, thus, confidence and excitement.” Matt Haber, founder of Problem Solved! Innovative Learning for the 21st Century NCTM August 17, 2015

What Should Students Know? Fractions aren’t just between zero and one; they live between all the numbers on the number line A fraction can have many different names. There are more strategies than just “finding a common denominator” for comparing and ordering fractions. Fractions can be ordered on a number line just like whole numbers. The thinking involved when placing fractions on a number line can be symbolized algebraically.

How is the quantity 2/10 read? Language of Fractions How is the quantity 2/10 read?

Language of Fractions 2/10 is: 2 parts of 10 parts 2 things compared to 10 things 2 things divided into 10 parts two tenths of one

Fractions A number in the form Numerator Denominator

The denominator can never be equal to 0. Fractions The denominator can never be equal to 0. 12 Does not exist! =

A fraction with a numerator of 0 equals 0. Fractions A fraction with a numerator of 0 equals 0. = = 4 156

Progression of Fractions https://gfletchy.com/progression-videos/

Fraction Sense “Fraction sense implies a deep and flexible understanding of fractions that is not dependent on any one context or type of problem. Fraction sense is tied to common sense: Students with fraction sense can reason about fractions and don’t apply rules and procedures blindly; nor do they give nonsensical answers to problems involving fractions.” Beyond Pizzas & Pies, by Julie McNamara and Meghan M. Shaughnessy.

Decomposition with fractions

Number Talk with fractions How can you decompose 6 8 ? Is the fraction closer to 0, 1 2 , or 1? 3 8 6 7 1 5 Can you see 1 2 ? Can you see 1 2 differently?

½ and some more…. Describe the following fractions using “1/2 and _____” 3/4 7/8 2/3 5/6 1 1/4

Table Talk What would your students understand about fractions if they did these tasks regularly? How can you build in regular fraction practice (outside of the fraction chapter) to develop conceptual understanding?

Understanding Fractions Outcomes: Understand important fraction concepts Explore the pitfalls of fractions Understand the progression of fractions Experience decomposition with fractions