 Home Mathematics Modeling mathematical ideas: developing strategic competence in elementary and middle school

# Developing Fraction Sense

Text Box 7.1 A Math Happening 7a: Unusual Baker

Miss Baker’s Sweetshop likes to cut the cakes differently each day of the week. On the order board, Miss Baker lists the fraction of the piece, and next to that, she has the cost of each piece. This week she is selling whole cakes for \$1.0 each. Determine the fractions of each piece of cake and how much each costs if the whole cake is \$10. ## LESSON STUDY VIGNETTE: THE UNUSUAL BAKER

Our fourth-grade state standard states, “Given a model, the student will write the decimal and fraction equivalents.”

Our Math Agenda: The students will

Understand fractions as parts of a whole.

• • Understand the conceptual meaning of the numerator and denominator.
• • Understand that the whole must be divided into equal-sized portions or fair shares.
• • Use fraction language: whole, halves, and fourths.
• •Set models show how subsets of the whole make fractional parts.
• • Recognize unit fractions.
• • Calculate the price for each piece of the whole.

This lesson was adapted from Teaching Children Mathematics, December 2011 and requires students to think about fractional parts of a whole and determine the value of various fractional parts when told that the value of the whole is \$10.00. In order to work through the examples, students have options of selecting manipulatives to model the problem, to draw pictures to demonstrate their thinking, or to work abstractly.

Math educators support the notion that “mathematical ideas can be represented in three ways: enactively (concrete representation), iconically (pictorial representation), and symbolically (written symbols)” and that “this idea of multiple representations suggests that children’s conceptual development evolves from concrete experiences to more abstract ones” (Cramer & Wyberg, 2009, p. 228). Offering students’ choices about how to display their thinking and making connections between various models is an important component of any math lesson that aims to deepen students’ mathematical knowledge. As Marshall, Superfine, and Canty (2010) state:

Translating and moving flexibly between representations is a key aspect of students’ mathematical understanding. Presenting students with opportunities to make connections between multiple representations makes math meaningful and can help students see the subject as a web of connected ideas as opposed to a collection of arbitrary, disconnected rules and procedures. Creating a learning environment in which students are encouraged to make connections among different representations has been a central feature of mathematics reform efforts for at least the past decade. (p. 39)

Students who have the opportunity to both create and make sense of multiple problem-solving strategies and solutions develop important foundational understanding. Too often making connection across representations is overlooked, but research has shown it to be a powerful part of the instructional process.

This lesson provides a relatable, real-world context for dividing a whole into fractional parts. Students are given the scenario in which a baker sells whole cakes for \$10 and are shown 6 cakes divided in different ways. They must then determine the cost for each of the pieces of cake that are depicted, with each of the 6 cakes being divided into increasingly complex ways. Sharp and Adams (2002) suggest that “teachers who nurture students’ construction of fractional knowledge by recognizing and utilizing personal knowledge (such as measuring cups or cake cutting) provide an appropriate foundation for fraction learning” (p. 334).

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