Rational Number Project Home Page Heller, P., Ahlgren, A., Post, T., Behr, M., & Lesh, R. (1989, March). Proportional Reasoning: The Effect of Two Context Variables, Rate Type and Problem Setting. Journal for Research in Science Teaching, 26(1), 205-220.

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Tables and Figures

 Table I Table II Table III Table IV Table V Table VI Table VII

 Figure 1 Figure 2

 Table I 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.unit prices (buying goods or services) money earned per week (salary) Mixture: mix two (or more) things together into some whole, or separate a whole ito 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

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 Table II 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) Exchange buying gum buying records Variable 1 No. pieces gum No. records Variable 2 price (cents) price (dollars) Consumption gas mileage of truck oil burning furnace Variable 1 distance (miles) volume (gallons) Variable 2 volume (gallons) time (hours)

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 Table III Data Used in Numerical Proportional Reasoning Problems Person 1 Person 2 Problem Format V1 V2 V1 V2 Missing Value #1 4 20 12 ? #2 8 24 ? 6 #3 3 12 ? 36 Numerical Comparison #1 8 32 2 10 #2 3 15 9 45 #3 6 24 2 6 Note. V1 and V2 are the two variables in the problem setting (for example, number of pieces of gum and price in cents).

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 Table IV Direction of Changes in the Numerator and Denominator of the Rates in the Qualitative Questions Problem Format Numerator Denominator Qualitative Rate Change #1 less more #2 less less #3 same more #4 same less #5 more less Qualitative Comparisons #1 more more #2 same less #3 more more #4 more less #5 same more

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 Table V Means and Standard Deviations for Context Groups on Proportional and Directional Reasoning Scales Score Context Group (Ratio type, and Setting) Proportional Reasoning (range 0 - 6) Directional Reasoning (range 0 - 8) Exchange more familiar 4.23 (1.71) 6.85 (1.48) less familiar 4.38 (1.64) 6.53 (2.21) Speed more familiar 3.95 (1.89) 6.28 (1.75 less familiar 3.40 (1.81) 5.35 (2.26) Comsumption more familiar 3.54 (1.89) 6.00 (2.19) less familiar 2.97 (1.44) 4.00 (1.96)

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 Table VI Analysis of Variance for Directional Reasoning Score Source MS df F P Main effects 68.7 5 20.1 <.001 Rational no. level 74.9 2 22.0 <.001 Ratio type 54.6 2 16.0 <.001 Setting 84.3 1 24.7 <.001 2-way interactions 4.2 8 1.2 .29 Rational no. x Ratio type 2.2 4 0.6 .64 Rational no. x Setting 0.6 2 0.2 .84 Ratio type x Setting 13.4 2 3.9 .02 3-way interaction 2.4 4 0.6 .65 Error 3.4 219

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 Table VII Analysis of Variance for Proportional Reasoning Score Source MS df F P Main effects 41.5 5 17.9 <.001 Rational no. level 75.3 2 32.5 <.001 Ratio type 25.2 2 10.9 <.001 Setting 6.5 1 6.5 .10 2-way interactions 2.7 8 1.2 .32 Rational no. x Ratio type 1.4 4 0.6 .66 Rational no. x Setting 3.6 2 1.6 .21 Ratio type x Setting 4.3 2 1.8 .16 3-way interaction 0.9 4 0.4 .82 Error 2.3 219

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 Fig. 1 1a. X is similar to Y. 1b. X is necessary, but not sufficient, for Y. 1c. X is sufficient, but not necessary, for Y. 1d. X is similar to Y, but restricted variance gives appearance of 1c. Fig. 1. Some possible relationships of two scales.

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Fig. 2.

Fig. 2. Context group means for directional and proportional reasong tests.

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