Rational Number Project

Teaching Actions Warm Up Order these fractions from smallest to largest. Be prepared to explain your thinking.	Comments Your goal is to reinforce students’ informal order strategies and not to use common denominator rules . Here is an example: 1/10 is farthest from the whole. Five isn’t half of 12 yet. So it is under . Then 3/5. Then 3/4 and 14/15. Both are missing one piece but a fourth is a lot bigger than a fifteenth so it is missing a bigger piece.
Large Group Introduction 1. Start the lesson by asking student to share their responses to the warm up task. Record responses on the board and have students defend their answers. Encourage students to describe the mental images used to sort these fractions. 2. Questions to guide discussion: • Which is bigger 3/4 or 3/5? • Which fraction is just under 1/2? • Which fractions are just greater than 1/2? • Which fractions are close to one whole? • Why is 1/10 the smallest?
3. Spend some time ordering 3/4 and 14/15. • Which fraction is closer to 1? How do you know? • One student suggested the two fractions are equal because the both are one piece away from the whole. What do you think? • Verify your conclusion by modeling each fraction with the fraction circles.	Expect some students to struggle with 3/4 and 14/15. Misunderstandings to expect are: • Equal, as both are one away from the whole. • 3/4 is larger because the denominator is a lower number so it is going to have bigger pieces.
4. Test students understanding by asking them to order 4/5 and 8/9. Which is closer to a whole? Verify students’ responses by showing both fractions with fraction circles.	Behind this strategy is the ability to mentally represent both fractions using images of fraction circles. Here is an example of a 6th grader’s use of mental images to order 8/9 and 4/5. 8/9 is larger because if you get the fraction circles out, 1/5 is bigger than 1/9. So if you put 8/9 and 4/5, 8/9 would be bigger because it would have smaller pieces so there is going to be a small amount left and it’s going to be a bigger piece left for 4/5.
5. Summarize the main ideas from this lesson introduction: • You can judge the relative size of fractions by thinking about fraction circles. • Using 1/2 as a benchmark is helpful. • Thinking about how close a fraction is to one whole is also helpful when comparing fractions like 3/4 and 5/6 or 4/5 and 99/100. Small Group/Partner Work	Notice how this student uses his ability to order two unit fractions (1/5 and 1/9) to answer this.
6. Student pages A-D reinforce comparing fractions to 1/2, comparing fractions to unit fractions, and comparing fractions close to 1.
7. Present this problem: What do you think of this students’ reasoning? 6/8 is greater than 4/11 because with 6/8 you need 2 to get to the whole and with 4/11 you need 7.	This is a common error in student thinking. While the student gets a correct answer, her reasoning will not generalize to all fraction pairs. With fractions you have to consider the relative size of the amount away from one whole.
8. Does this student’s strategy work for and ?	Possible explanations are as follows: • Doubling the numerator gives you the size of the denominator if the fraction was equal to 1/2. • If the numerator is 7 then . Because 12ths are bigger than 14ths, . From this comparison .
9. To assess their understanding of using 1/2 as a benchmark ask this question: • Juanita said that because she knew that 6/12 equals 1/2 and 7/12 is more than 6/12. • Will said he knew because he doubled the numerator and saw that it was greater than the denominator. That made 7/12 bigger than 1/2. • I understand Juanita’s strategy but I don’t understand what Will meant. Can you help me? Can you use fraction circles to convince me his strategy is right?

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

Additional Notes to the Teacher

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.