Building Fraction Sense, by Michael Naylor

Manipulatives and visuals take center stage in the third installment of our series about fractions

During the school year, you'll want to give your students many opportunities to work with and discuss fractions. Be sure to provide manipulatives at every grade level. Manipulatives are not a "crutch" for students who have trouble with symbols, rather manipulatives are powerful tools for building flexible thinking. This month we have some tried-and-true activities for building fraction sense with drawings, models and manipulatives.

Potato Fractions (Grades K-2)
What kind of holiday feast would be complete without hot, buttery baked potatoes? Give each of your students three "potatoes" cut from brown construction paper. Ask your students to predict how many halves they can make by cutting the potatoes into two equal parts. Ask them to cut their potatoes, and then use the pieces to count: one half, two halves, three halves, etc. Have them reassemble their six halves into three potatoes, and ask how many halves they can make from four whole potatoes. Give them an additional potato and repeat the cutting and counting. Your students will be making sense of the numeric relationships between halves and wholes.

To complete the activity, ask students to select their own number of potato halves and glue them to a piece of paper. Help them to label their creations with a sentence like "Five whole potatoes = 10 half potatoes." Repeat the activity with other food items and create a fraction feast! Older children can cut the items into thirds or fourths.

Four snow-people demonstrate fractions. Two of the snow-people wear hats. Encourage the children to write statements like: "2 + 2 = 4, therefore two is one-half of four. One-half of the snow people are wearing hats."

Snow Person Fractions (Grades K-2)
Have your students draw four snow-people on a piece of paper. Ask them to draw hats on half of the snow-people, and then ask them to explain how they know how many is "half." Next have them draw faces on half of the snow-people, and continue with other decorations. You can challenge the students by telling them to add arms to one-fourth of the snow-people; be sure to have them explain how they know their fraction is correct.

This activity is easy to vary with different set sizes and different fractions, depending on the age and sophistication of your students.

Peppermint Sticks (Grades 3-5)
Give each of your students three strips of paper to represent peppermint sticks. Ask them to cut the sticks, dividing them equally among four friends. Students can glue the cut sticks to the top half of a piece of paper and write an explanation below of how they shared the sticks. Discuss the outcomes as a class; students will be surprised by all the different ways their classmates solved the problem.

You can vary this activity with different numbers of sticks and different numbers of friends to share with. How would your students share two pizzas with five friends?

Counting with Fractions (Grades 3-5)
Give your students manipulative pieces and have them count along with you: one third, two thirds, three thirds, four thirds, etc. Stop at different points in the count and ask the students to compare their fractional parts to whole numbers. Is five thirds more or less than one whole? More or less than two wholes? Ask how much more than a whole their pieces are, and how many additional pieces it would take to make wholes.

When they understand the concept, you can begin counting: One fourth, two fourths, three fourths, four fourths, whole. Five fourths, six fourths, seven fourths, eight fourths, whole.

Use several different fraction sizes and compare them. Which is greater: five fourths or five sixths? Why?

Fraction Remix (Grades 4-8)
Give your students a mixed fraction, such as two and two thirds, and ask them to create the fraction using just one kind of manipulative piece. Ask students to name the fraction as a single fraction (i.e. two and two thirds is the same as eight thirds). They should draw a picture to record their fraction (in the same manner as the peppermint sticks and snow-people) and write the fraction as both a mixed and a proper fraction. Repeat the activity with other mixed fractions.

Next, give your students proper fractions in which the numerator is greater than the denominator, such as five halves, seven fourths and eleven thirds. Have students model the fractions and rewrite them as mixed fractions.

The goal of this exercise is not to teach your students an algorithm for converting mixed fractions to proper fractions, but to reinforce concepts about fractional parts. Students will easily be able to create their own rules by thinking about how the fractions combine to make wholes – and by doing so, they'll make sense of mixed fractions.

Michael Naylor is a professor of math education at Western Washington University, Bellingham, WA.