UK MA 201 - Chapter 6 Notes for Instructors
Course Ma 201-
Pages 4

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Chapter 6Notes for InstructorsContentThis focus of this chapter is the rational number system. Although the chapter containsonly three sections, you should allot a full week for section 6.2. Students tend to have moredifficulty with fractions than they do with any other topic covered in this class.ManipulativesWe have several types of manipulatives that can be used to model fractions includingFrax PaxTMSquares, Cuisenairerrods, pattern blocks, and colored counters (the circularpieces that are yellow on one side and red on the other).• It is possible to use some of the Frax PaxTMto model multiplication with the areamodel. This model will be discussed momentarily. If you do this, you may need tospecify that your unit is not the large square. If you want to use the large square as theunit, you could combine several boxes of Frax PaxTMif one of your factors is greaterthan one.• The Cuisenairerrods can be used to do an activity similar to the Cooperative Inves-tigation shown on page 348 of the textbook. Questions 2 and 3 of this investigationwere both challenging and enlightening for the students.• The pattern blocks can be used to construct colored region models for fractions likethose shown on page 347 and in problems 8, 10, and 25 of Section 6.1. You couldalso write questions similar to those in the Cooperative Investigation on page 348 thatinvolve pattern blocks instead of fractions strips.• The colored counters can be used to construct set models of fractions.Notes and Suggestions:Notes on Section 6.1: The Basic Concepts of Fractions and Rational Numbers• It is imperative that students know the three things which must be well defined inorder to interpret the meaning of any fractionab. (See the bulleted items on page 346.)• Some of the students in this course have a lot of difficulty reading and understandingdefinitions. To encourage them to read definitions carefully, I like to ask, “Is76insimplest form?” Most of my students thought that it needed to be converted to amixed number, but this is not so. Because the GCD(7, 6) = 1 and 6 is positive, thefraction76is in simplest form. (See the definition on page 352.)• I think it is important to emphasize that you can find a common denominator withoutnecessarily finding a least common denominator for a set of fractions. Moreover, any1common denominator will often be sufficient for solving a certain problem. For exam-ple, if you need to add two fractions, it is sufficient to find any common denominatorfor the two fractions. Finding the least common denominator is, in many cases, moreof a stylistic issue than anything else.• I think the Order Relation on the Rational Numbers is best understood by finding acommon denominator for the pair of fractionsabandcd. I found that it was a worthwhileexercise to have students explain the Order Relation on the Rational Numbers by usingcommon denominators. Without this explanation, the inequality ad < bc has littlemeaning to students.Notes on Section 6.2: The Arithmetic of Rational Numbers• Students should be able to demonstrate whyab+cd6=a+cb+dfor given values of a, b, c,and d. For example, we can use the Frax Pax Squares to see that12+136=25since thearea covered by the12rectangle and the13rectangle together is much larger than thearea covered by two of the15rectangles.• Since common denominators are required for the addition of fractions and subtractionis defined according to the missing addend model, it follows that common denominatorsare required for the subtraction of fractions.• I really like the way that the area model can be used to multiply fractions. To under-stand this model, students need to understand that the unit is the 1×1 square becauseit has an area of 1 square unit. When you have established this, the other informationneeded to interpret the fractional answer (see the bulleted items on page 346) followseasily. Two examples are given below.– Example 1:12×23To compute12×23using the area model, you need a rectangle of dimensions12by23. To obtain such a rectangle, you must divide the unit square into six equalpieces as shown below. Two these small pieces compose the12by23rectangle.Hence the area of the12by23rectangle is26square units and12×23=26.– Example 2: 413× 135To compute 413× 135using the area model, you need a rectangle of dimensions 413by 135. To obtain such a rectangle, you must divide each unit square into fifteenequal pieces as shown below. Note that the diagram below contains ten units.2The 413by 135rectangle is composed of four unit squares and forty-four of thesmall rectangles. Since fifteen of the small rectangles can be used to construct aunit, we actually have six units and fourteen of the small rectangles. Thus thearea of the 413by 135is 61415square units, and 413× 135= 61415.Questions 11–13 in Section 6.2 test students’ understanding of the area model formultiplication.• Students should be able to explain the Invert and Multiply Algorithm for Division ofFractions by using the missing factor model for division.Notes on Section 6.3: The Rational Number System• Most of the properties in this section should flow easily from the corresponding prop-erties for integers. Of course, we now have multiplicative inverses for nonzero rationalnumbers and closure under division of nonzero rational numbers.• The only really new idea in this section is the Density Property of Rational Numbers,so I think this section can be covered quickly. Still I think it is important to havestudents identify the properties they use. Questions similar to Problems 9 and 11 inSection 6.3 test this ability.Worksheets3At this point in the course, I moved my class to a workshop oriented class. I ran theclass very much like I would have run a Math Excel workshop. I did this because I thinkthe students learn the material better when they actually solve the problems themselves. Idid not want my students to simply mimic me. I wanted them to understand the problemsand their solutions. Each student became an “expert” for a small subset of the problemson a worksheet. I partitioned the worksheet problems into small subsets of problems andrandomly assigned the problems to the students. (For example, the students might draw asubset from a hat.) Each student was required to write clear correct solutions for his or hersubset. These solutions were placed in a folder in the Math Library for all the students to see.Students were also required to do the other


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UK MA 201 - Chapter 6 Notes for Instructors

Course: Ma 201-
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