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Rational Number Project

 Fraction Operations and Initial Decimal Ideas Lesson 1: Overview Materials • Fraction Circles for students and teacher • Student Pages A, B, C, D Students review how to model fractions with fraction circles by ordering unit fractions, using 1-half as a benchmark to order two fractions , and comparing fractions close to one.

Translations:
• Symbolic to Manipulative to Verbal;
Symbolic to Verbal

Lesson 1

The RNP level 1 lessons support students’ development of informal ordering strategies. Four
informal ordering strategies have been identified: same numerator, same denominator,
transitive, and residual. These strategies are not symbolic ones, but strategies based on
students’ mental representations for fractions. These mental representations are closely tied to
the fraction circle model.

Same denominator: When comparing and students can conclude that is larger
because when comparing parts of a whole that are the same size (in this case ) then 4 of those
parts are bigger than 3 of them.

Same numerator: When comparing and , students can conclude that  is bigger because
5ths are larger than 6ths and four of the larger pieces will be bigger than 4 of the smaller pieces.
Students initially come this understanding by comparing unit fractions.

Transitive: When students use benchmark of ½ and one they are using the transitive property.
When comparing and students can conclude that is larger because is a little less
than ½ and is a little more than ½.

Residual: When comparing fractions and students can decide on the relative size of each
fraction by reflecting on the amount away from the whole. In this example, students can
conclude that ¾ is larger because the amount away from a whole is less than the amount away
from the whole for . Notice that to use this strategy students rely on the same numerator
strategy; they compare and to determine which of the original fractions have the largest
amount away from one.

Students who do not have experiences with concrete models like fraction circles or students
who may not have sufficient experiences with models to develop needed mental
representations to judge the relative size of fractions using these informal strategies make
consistent errors. On the next page we share with you examples of students’ errors based on
written test given to students after their RNP level 1 review lessons. We also share examples of
correct thinking among this group of sixth graders. In all questions students were asked to
circle the larger of the two fractions.

Misunderstandings

Students often focus on the denominator only after internalizing the relationship between the
size of the denominator and the size of the fractional part. To understand what a fraction
means, students need to coordinate the numerator and denominator - an idea the following
students did not do.

An underlying assumption when ordering fractions is that the units for both fractions must be
the same. Not realizing the unit needs to be the same is a common error as shown in this
student’s picture.

Whole number thinking also dominates students thinking when they first start working with
fractions. This is shown in different ways . Without models students might say that
because 6>3 and 8>4. But even after students use concrete models, their whole number
thinking may still dominate. In the following examples, note that some students determine the
larger fraction by deciding which fraction had the larger number of pieces. In other cases,
students look at the difference between numerator and denominator to identify the larger
fraction. In both instances, students have yet to focus on the relative size the fractional part
being examined. These students need more time with concrete models to overcome their whole
number thinking.

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