Rational Number Project Home Page Heller, P., Post, T., Behr, M., & Lesh, R. (unpublished). The effect of two context variables on qualitative and numerical reasoning about rates.
The Effect of Two Context Variables on Qualitative and

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Tables and Figures

 Table 1 Table 2 Table 3 Table 4 Table 5

 Figure 1 Figure 2 Figure 3 Figure 4

 Table 1 Some Rate Types Found in Textbook Problems Definition Examples Distribution: divide something equally among people, groups, or objects. cookies per person acres of land per family electric charge per electron Packing: spread something evenly over a spatial dimension books per foot of shelf space mass of aluminum per cubic centimeter Package Size: count or measure the amount of something in a "package" (when things are not necessarily spread evenly over a spatial dimension). candies per box ounces per bottle electrons per neutral iron atom Exchange: exchange or trade one kind of thing for another kind of thing. (buying goods or services) money earned per week (salary) Mixture: mix two (or more) things together into some whole, or separate a whole to its constituent parts. lemonade mix per glass of water questions right per questions wrong on a test molar concentration of acids Speed: how fast or slow an object moves or an event takes place. gallons of water emptied from a tank per hour speed of light Consumption/Production: how efficiently something is consumed (used up) or produced (made). gallons of oil burned by a furnace per hour electrical energy used per hour (power) Scaling: enlarge or shrink something billions of dollars per mm on a graph of national debt inches per mile map scale Conversion: convert a quantity from one unit of measurement into a different unit of measurement. centimeters per inch square feet per acre pounds per dollar

This table appeared in Heller, P, Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989). Proportional Reasoning: The effect of two context variables, rate type and problem setting, Journal of Research in Science Teaching, 26(3): 205-220.
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 Table 2 Examples of Parallel Items on the Rational Number and Context Test Rational Number Test Context Testa Missing-Value Problems Place a number in the box that will make the two fractions equal. 4/20 = 12/[ ] Steve and Mark drew the same size maps of their classroom. Steve drew the 20 foot long windows as 4 inches long on his map. How far apart are two desks that Mark drew as 12 inches apart on his map? Numerical-Comparison Problems Circle the smaller fraction. If they are equal, circle both fractions.   6/24     2/6 Alice and Jenny hammered (equally spaced) nails into different boards. Alice hammered a line of 24 nails into a board 6 ft long. Jenny hammered a line of 6 nails into a board 2 ft long. On which board are the nails hammered closer together? a) Alice b) Jenny c) Their nails are spaced exactly the same. d) There is not enough information to tell. Fraction Test Context Testa Directional Questions What will happen to the fraction 7/8 if the top number gets smaller and the bottom number gets bigger? QRb: If Nick mixed less green tint with more white paint than he did yesterday, his green paint would be a) Fraction gets bigger a) a darker shade. b) Fraction gets smaller b) a lighter shade. c). Fraction stays the same c) exactly the same shade. d) There is not enough information to tell. d) There is not enough information to tell. What will happen to the fraction 3/4 if the top number gets bigger and the bottom number gets bigger? QC: Nancy drove more miles than Kathy. Nancy drove for more time than Kathy. Who was the faster driver? a) Fraction gets bigger a)Nancy b) Fraction gets smaller b)Kathy c) Fraction stays the same c)They drove at exactly the same speed. d) There is not enough information to tell. d) There is not enough information to tell. a Each version of the Context Test used only one rate type. For illustrative purposes, four different rate types have been shown on this table. b QR stands for qualitative-rate-change question and OC qualitative-comparison question.
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 Table 3 Rate Types and Problem Settings Used in This Study Problem Setting Rate Type More Familiar Less Familiar Speed running laps driving cars Variable 1 distance (laps) distance (miles) Variable 2 time (minutes) time (hours) Mixture mixing lemonade mixing paint Variable 1 concentrate (teaspoons) 1 tint (drops) Variable 2 water (ounces) white paint (ounces) Scaling making classroom map reading city map Variable 1 length (inches) length (inches) Variable 2 distance (feet) distance (miles) Density movie lines nails in board Variable 1 objects (people) objects (nails) Variable 2 line (yards) board (feet)
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 Table 5 Means and Standard Deviations for Rational Number Groups on Qualitative Reasoning and Numerical Reasoning Scales Score Rational Number Ability and Rate Type Groups Directional Scale (range 0 - 8) Numerical Scale (range 0 - 8) Low Ability Speed n = 67 4.30 (1.95)* 3.46 (1.97) Density n = 59 4.91 (1.76) 2.88 (1.87) Mix n = 61 4.51 (1.56) 2.33 (1.62) Scale n = 58 2.33 (1.51) 2.50 (1.84) Medium Ability Speed n = 80 5.49 (1.51) 5.23 (1.93) Density n = 85 5.72 (1.48) 4.83 (2.08) Mix n = 95 5.14 (1.40) 4.54 (2.04) Scale n = 75 2.54 (1.74) 3.73 (1.97) High Ability Scale n = 85 6.39 (1.25) 6.41 (1.60) Speed n = 84 6.67 (1.24) 6.89 (1.24) Density n = 72 6.08 (1.15) 6.65 (1.58) Mix n = 92 3.64 (1.98) 5.45 (1.86) * Standard deviations
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FIGURE 1

 Figure 1. Context group means for seventh and eighth grade students on the directional scale.
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FIGURE 2

 Figure 2. Context group means for seventh and eighth grade students on the numerical scale.
 (top) Figure 3 Figure 3. Rate-type group means on the numerical scale for three levels of rational number ability.
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FIGURE 4

 Figure 4. Graphs of the percent of students who solved correctly (a) missing-value problems, (b) numerical-comparison problems, and (c) directional questions on the context test versus the percent of students who solved the numerically and/or structurally equivalent problem correctly on the rational number test.
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