Making Sense Out of Fraction Word Problems

James Olsen, Ph.D. Western Illinois University

JR-Olsen@wiu.eduhttp://faculty.wiu.edu/JR-Olsen/wiu/

Western Illinois UniversityMathematics Teachers’ Conference

March 23, 2012

Outline• Brief History – My Life with PDF• The promise of the CCSS• The meanings of the operations, which can be

extended to operations on fractions.• Models For Fractions.• Meanings of the operations with fraction

models.• Word problems.• Quick example with a tape (length model)

6 ÷ 4 …or… ¼ of 6 …or… 6

4

Brief History – My Life with PFD• Mastery Quizzes in teacher education classes (16 objectives – see handout).• PFD Boot Camp web site – started.• Project got bigger. Two students: Hayley Moore and

Joseph Illichman helped me.• We wrote 216 PFD word problems.• Hayley, Joe, and I have presented in Florida and ICTM –

Springfield (2011). See Navigating the Landscape of Fraction, Decimal, and Percent Problems

• http://faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html

PFD = Percents Fractions & Decimals

Positive Potential for the CCSS to be Successful for Improving Student

Achievement• We’ve had “standards” for about 20 years. • We have pretty good research on teaching and learning.• (We are finally getting it) “Evidence shows clearly what

most people know intuitively: teachers matter more to student learning than anything else inside a school.”

• Focus, coherence, and understanding.• We know that understanding fractions is HUGE !

CCSS = Common Core State Standards

One of the Biggest Keys to Make CCSS a Success

• Teacher knowledge and understanding of mathematics.

Meanings (“situations”) of the Operations + – × ÷

• See pages 88 and 89 of the Common Core Mathematic Standards (handout).

• These meanings/situations should be used in early elementary to develop the four operations.

• We will use the same meanings/situations with fractions, mixed numbers, and decimals – word problems.

Subtraction isn’t just Take-Away !

Models For Fractions

• See Fraction Model Applet at NCTM

– Length (“tape”) model.– Area or Region models (rectangles or pies)– Set model.

• Demo the applet (next slide).

From CCSS Glossary:Visual fraction model. A tape diagram, number line diagram, or area model.

Assumptions• We believe to understand something well it should

be connected in the brain to many other ideas.• Number sense, operation sense, and fraction sense

should be gained continually over time.• Physical models and pictures are useful, but we need

to be moving toward the symbolic/abstract.

When you read the newspaper, there’s no bag of manipulatives hanging there to help you understand the percents.

How can we show ?

How can we show ?

How can we show 2 ½ × 3 ½

with an area model?

How can we show ?

Word Problems(Objective XI.)

Multiply fractions (and mixed numbers).

(see handout)

Word Problems(Objective XIII.)

Divide fractions (and mixed numbers).

(see handout)

Thank You

• Websites

• faculty.wiu.edu/JR-Olsen/wiu/• faculty.wiu.edu/JR-Olsen/wiu/BootCamp/landscape/front.html