Rational Number Project Home Page Behr, M., Harel, G., Post, T., & Lesh, R. (1992). Rational number, ratio and proportion. In D. Grouws (Ed.), Handbook of Research on Mathematics Teaching and Learning (pp. 296-333). NY: Macmillan Publishing.

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 FIGURE 14- 1 FIGURE 14- 2 FIGURE 14- 3 FIGURE 14- 4 FIGURE 14- 5 FIGURE 14- 6 FIGURE 14- 7 FIGURE 14- 8 FIGURE 14- 9 FIGURE 14- 10 FIGURE 14- 11 FIGURE 14- 12 FIGURE 14- 13 FIGURE 14- 14 FIGURE 14- 15 FIGURE 14- 16 FIGURE 14- 17 FIGURE 14- 18
TABLE 14-2

 FIGURE 14-1 The construct theory of rational numbers, overview of semantic analysis - interpretations of three-fourths.
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 1. (0 0 0 0 0 0 0 0) 1(8-unit). We could have started with any unit with 4n objects where n is any natural number. 2. (00 / 00 / 00 / 00) The 8-unit is partitioned into 4 parts. 3. ((00)(00)(00)(00)) Each part is unitize as 1(2-unit), and a unit-of-units --(1/4(4(2-unit)s-unit)) -- is formed. 4. (()()()()) Each 2-unit is (1/4(4(2-unit)s-unit)) and can be reunitized as a unit-of-units: (1/4(4(2-unit)s-unit)-unit). Note that each (2-unit) becomes a special type of a (1/4-unit). 5. (( ) ()) Three (1/4(4(2-unit)s-unit)-unit)s are shaded. FIGURE 14-2 Three-fourths as parts of a discrete unit-whole leads to the interpretation that three-fourths is 3(1/4-unit)s.
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 FIGURE 14-3 Three-fourths as parts of a continuous unit-whole leads to the interpretation that three-fourths is 3(1/4-unit)s.
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 1. (0 0 0 0 0 0 0 0) 1(8-unit). We could have started with any unit with 4n objects where n is any natural number. 2. (00 / 00 / 00 / 00) The 8-unit is partitioned into 4 parts. 3. ((00)(00)(00)(00)) Each part is unitize as 1(2-unit), and a unit-of-units --(1/4(4(2-unit)s-unit)) -- is formed. 4. (()()()()) Each 2-unit is (1/4(4(2-unit)s-unit)) and can be reunitized as a unit-of-units: (1/4(4(2-unit)s-unit)-unit). Note that each (2-unit) becomes a special type of a (1/4-unit). 5. ( ()) Three (1/4(4(2-unit)s-unit)-unit)s unitized to (3/4(4(2-unit)s-unit)-unit). FIGURE 14-4 Three-fourths as a composite part of a discrete unit-whole leads to the interpretation of three-fourths as (3/4(4(n-unit)s)-unit).
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 1. (0000) (0000) (0000) [*] [*] [*] [*] 3(4-unit)s 4[1-unit]s 2. (0/0/0/0) (0/0/0/0) (0/0/0/0) [*] [*] [*] [*] Each (4-unit) is partitioned into 4 parts. 3. ()(ØØØØ)() [*] [*] [*] [*] The 4 objects in each (4-unit) are each unitized as 1/4(4-unit) to give four 1/4(4-unit)s. 4. [*] [*] [*] [*] The first four 1/4(4-unit)s are distributed equally among the 4[1-unit]s; this gives another 1/494-unit)/[1-unit]. 5. Ø    Ø    Ø   Ø          [*] [*] [*] [*] The second four 1/4(4-unit)s are distributed equally among the 4[1-unit]s; this gives another 1/4(4-unit)/[1-unit]. 6. Ø    Ø    Ø   Ø          [*] [*] [*] [*] The third four 1/4(4-unit)s are distributed equally among the 4[1-unit]s; this gives another 1/4(4-unit)/[1-unit]. 7. The three(1/4(4-unit))s are accumulated to 3(1/4(4-unit)-unit)s/[1-unit]. FIGURE 14-5 The partitive division of 3 ÷ 4, based on discrete quantity, leads to the interpretation that three-fourths is 3(1/4-units) in one measure space per one unit of quantity from another measure space.
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 1. 3(1-unit)s 4[1-unit]s 2. Each (1-unit) is partitioned into 4 parts. 3. Each of the 4 parts in each (1-unit) is unitized as 1(1/4-unit) to give 4(1/4-unit)s. 4. The first 4(1/4-unit)s are distributed equally among the 4[1-unit]s. This gives 1(1/4-unit)/[1-unit]. 5. The second 4(1/4-unit)s are distributed equally among the 4[1-unit]s to give a second 1(1/4-unit)/[1-unit]. 6. The third 4(1/4-unit)s are distributed equally among the 4[1-unit]s. This gives a third 1(1/4-unit)/[1-unit]. 7. The three 1(1/4-unit)s/[1-unit] are accumulated to 3(1/4-unit)s/[1-unit]. FIGURE 14-6 The partitive division of 3 4, based on continuous quantity leads to the interpretation that three-fourths of 3(1/4-units) of quantity in one measure space per one unit of quantity in another measure space.
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 FIGURE 14-7 The partitive division of 3 ÷4 that leads to the interpretation that three-fourths is 3(1/4-units) in one measure space per one unit of quantity in another measure space.
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 FIGURE 14-8 The partitive division of 3 ÷ 4 based on discrete quantity with 3 interpreted as 1(3-unit) to give an interpretation of three-fourths as one-fourth of a (3-unit) (a unit-of-units) in one measure space per one unit of quantity in another measure space: 3/4 = 1/4(3(4-unit)s-unit)/[1-unit].
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 FIGURE 14-9 Partition of three candy bars as a (3-unit), which shows 3/4 as 1/4(3-unit).
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 FIGURE 14-10 A partition of three circles conceptualized as 1(3-unit) based on successive halving of units.
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 FIGURE 14-11 Quotitive division of 3(1-unit) by 4{1-units}/[1-unit], which leads to the interpretation that three-fourths is 1[3/4-unit].
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 FIGURE 14-12 The mathematics-of-quantitiy model of the quotitive division of 3 ÷ 4, which leads to the interpretation that three-fourths is on (3/4-unit).
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 FIGURE 14-13 Quotitive division of 3(1-unit) by 4{1-units}/[1-unit], which leads to the interpretation that 3/4 is 3/4[1-unit].
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 FIGURE 14-14 A hierarchical tree-structure for problem 1.
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 FIGURE 14-15 Matrix representation of Dienes' mathematical and variability principles applied to rational-number concepts.
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 FIGURE 14-16 Lesh's model for translations between modes of representation (adapted from Bruner).
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 FIGURE 14-17 Diagrams to represent two interpretations of the perceptual-variability principles in an instructional context.
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 FIGURE 14-18 A diagram of how Dienes' variability principles can be extended to an instructional setting for teacher education.
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TABLE 14-2. Design of Part-Whole Instructional Materials

Lesson
Embodiment
Activity

 1 Color-coded circular pieces Name pieces Compare sizes Observe that as size decrease, number to make whole increases Observe equivalence 2 Color-coded rectangular pieces Name pieces Compare sizes Observe that as size decrease, number to make whole increases Observe equivalence 3 Color-coded circular and rectangular pieces Observe similarities and differences between circular and rectangular pieces Translate between circular and rectangular pieces 4 Color-coded circular and rectangular pieces Attach unit fraction names (one-fourth, one-fifth) to parts of whole Work with equivalent sets of fractions 5 Paper folding with circles and rectangles Attach fraction names to shaded parts of folded regions Learn names for unit and proper fractions Associate names with color-coded parts and with shaded parts of folded regions Note similarities and differences 6 Cuisenaire rods Attach fraction names to display of rods Note fractions as sums of unit fractions Compare display with colored pieces and paper folding Investigate real-world problems 7 All materials from Lessons 1 to 6 Review, using all four embodiments Identify proper fractions orally, in written form, symbolically, and pictorially Translate from one mode to another 8 Chips Review division as partitioning (18 ÷3 = 6 represents 18 chips, 3 groups, 6 in each group) Represent fractions by covering equal-sized groups of chips Associate fractions with amount covered (3 of 4 groups covered is 3/4) 9 All materials from Lessons 1 to 7 Translate to any mode, using any embodiment, a fraction represented with physical objects, orally, or with a written symbol 10 Color-coded pieces Represent improper fractions using pieces Paper folding Chips Translate between improper and mixed number notation orally and in writing Translate representation to paper folding Translate representation to chips 11 Number line Associate whole number, fraction, and mixed numbers with points on the number line Convert improper fraction to whole or mixed numbers Determine equivalence Add fractions with same denominators 12 All materials from Lessons 1 to 11 Using pieces, chips, rods, or pictures of parts of a unit, construct the unit 13 Chips Paper folding Represent a model for multiplication of fractions Generalize to algorithm with product of numerators and product of denominators
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