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1. Fraction Model:
Investigating rational numbers using a virtual manipulative
Look in the Classroom
Mr. Hernandez knows that his students struggle when asked to compare and order groups of numbers in mixed forms. Although some can accurately convert numbers from one form to another, he knows they have merely memorized algorithms and need to strengthen their conceptual understanding of rational numbers. Mr. Hernandez decides to have his students explore the connection between the three number forms using the Fraction Model.
His goal for the activity is for students to be able to order sets of five numbers in mixed forms using a variety of strategies to determine the correct order. Also, as students explain their strategies for ordering the sets of numbers, an understanding of the part-whole relationship should be reflected in their explanations.
Mr. Hernandez starts class with a brief demonstration of the Fraction Model virtual manipulative. During the review, he shows how to operate each slider and initiates a discussion about how moving the sliders affects the values. Pausing with the sliders on 3/8, he asks:
Mr. H: What do you think I should do to with the sliders to make a larger number?
S1: Slide the top one to the right. (Mr. H demonstrates.)
Mr. H: How does [S1] know the number is larger?
Mr. Hernandez demonstrates on the Fraction Model in response to the answers from several students in the class. He directs the students' attention to each of the different representations and asks students to explain how that representation shows that the number is getting larger when he moves the top slider to the right. He stops operating the sliders and asks a new question:
Mr. H: We know that moving the top slider to the right increases the value of the number. Are there other ways to increase the value?
Mr. Hernandez sends the students to their own computers to use the Fraction Model in an attempt to answer the question. To help in this investigation, they are given a list of ten numbers, in various forms, and asked to order them from smallest to largest with the aid of the virtual manipulative. As the students complete their work, Mr. Hernandez initiates a new discussion.
Mr. H: Let's start by answering a question about the bottom slider. How does the bottom slider control the value of the number?
S4: Well, it's the opposite of the top. When you move it to the right, the number gets smaller, and to the left, it gets larger.
Mr. H: It looks like the denominator is getting bigger when I slide the bottom slider to the right. Is [S4] correct? Can anyone explain [S4]'s answer?
S5: He's right. It's the opposite of the top. When you dragged the bottom slider to the right, he got more pieces in the pie, but the number of pieces shaded stays the same. It's the same number of pieces of a pie that has more cuts, so that's less pie!
S6: When I was going to compare 4/82 and 4%, I knew I would set the top slider to 4 for both numbers and then adjust the bottom slider to 82 or 100. Since 100 is larger that 82, I knew, without even doing it, that 4/82 was the bigger fraction.
Mr. H: Explain how you knew it was larger.
S6: Well-it's the pieces thing that [S5] was talking about. If you have a pie that's cut into 82 pieces, each slice will be bigger than if you cut the same pie into 100 pieces. So if one guy has 4 pieces out of 82 total, he has more pie than another guy who has 4 pieces out of 100. You can compare fractions with common numerators as easy as you can with common denominators.
S7: You just have to remember that the bigger the denominator, the smaller each slice is.
As students suggest additional flexible strategies to compare rational numbers, Mr. Hernandez realizes that they are starting to develop a conceptual understanding of the part-whole relationship. For a final check, he gives each student a list of five numbers in mixed forms and asks them to order them without the aid of the Fraction Model. When the students are finished, he asks for an explanation of their strategies for determining whether 8/73 is larger than 6%.
S8: You know that 6/100, six percent, would be smaller than 6/73, so you know 8/73 is bigger than 6/100.
S9: Why?
S6: It's like what I said before about the common numerators. 6/73 is bigger than 6/100 and 8/73 is bigger than 6/73 so 8/73 is bigger than 6/100.
As the students continue with these kinds of explanations, Mr. Hernandez is convinced that he has achieved his goals for the lesson.
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