QR STEM ProjectDr. Robert Mayes

Science and Mathematics Teaching Center

QR STEM is a funded by a Department of Education Mathematics and Science Grant (Project ID: 100150T2BA0).

Three categories of Quantitative Reasoning Quantitative Literacy: use of number and

arithmetic to quantify a context with the goal of understanding a phenomena so one can make informed decisions.

Quantitative Interpretation: ability to interpret a model of a given phenomena with the goal of understanding and making informed decisions; algebraic, geometric, and statistical modes.

Quantitative Modeling: ability to create a model of a phenomena with the goal of making predictions or discovering trends.

Categories Quantitative Literacy

Quantitative Interpretation Quantitative Modeling

Components Numeracy• Number Sense• Small/large Numbers• Scientific NotationMeasurement • Accuracy-precision• Estimation• Dimensional Analysis• UnitsProportional Reasoning • Fraction• Ratio• Percents• Rates/Change• ProportionsBasic Prob/Stats• Empirical Prob.• Counting• Central Tendency• Variation

Interpreting• tables• graphs• equations• science models• statistical plotsLogarithmic ScalesStatistics • Normal Distribution• Correlation• Causality

LogicProblem SolvingModeling• linear• polynomial• power• exponential• conceptual models• table or graphStatistics• Least Squares Fit• Inference• Hypothesis testing

QL has four major components that underpin the sciences: Numeracy Measurement Proportional Reasoning Descriptive Statistics and Basic Probability

Numeracy: ability to reason with numbers. logic and problem solving aspect of QR on the arithmetic

level includes having number sense, mastery of arithmetic

processes (addition, subtraction, multiplication, division), logic and reasoning with numbers, orders of magnitude, weights and measures

Number sense: awareness and understanding about what numbers are, their relationships, their magnitude, the relative effect of operating on numbers, including the use of mental mathematics and estimation (Fennel & Landis, 1994) includes the concepts of magnitude, ranking, comparison,

measurement, rounding, degree of accuracy, percents, and estimation

Science Examples: diameter of a hydrogen nucleus is approximately 0.000000000000001 meter while the total energy consumption in the United States is 100,000,000,000,000,000,000 joules

http://learn.genetics.utah.edu/content/begin/cells/scale/ Scientific Notation: alternative representation

Hydrogen nucleus? U.S. Energy Consumption?

Order of magnitude: How many orders of magnitude larger is U.S. energy consumption then a hydrogen nucleus?

15101 −×20101×

35

Three techniques for bringing numbers into perspective are estimation, comparisons, and scaling

Exemplar 1 (Bennett & Briggs, 2008): Provide an order of magnitude estimate of how much water U.S. citizens drink in a year.

Solution: Estimate that an average person drinks 3 10oz glasses of water a day. There are 365 days in a year, and the U.S. population is on the order 300 million or 3 x 108. So an estimate of water drunk is:

oz/year. There is approximately 0.3381 oz in a milliliter so an average person drinks 9.716 x 1012 mL/year. The metric prefaces are another numeracy skill that students must master for science, for example there are 1000 mL in a liter so 9.716 x 109 L/year.

Exemplar 2 (Bennett & Briggs, 2008): Annual world energy consumption is 5 x 1020 joules. Many people are not familiar with energy units like joules, so make a comparison to something that is familiar to get a perspective on world energy consumption.

Solution: One food Calorie is equivalent to 4,184 joules. A typical American uses 2,500 Calories of energy a day or 1.046 x 107 joules. So annual world energy consumption is approximately the same as the daily energy use of 4.78 x 1013 people.

Exemplar 3 (Bennett & Briggs, 2008): An atom has a diameter of about 10-10 meter. Provide a scale that puts this into perspective.

Solution: If we multiply the diameter by 1010

then we get 1 meter. So there are 10 x 109 or 10 billion atoms in a line along a meter stick. A centimeter is 1/100 of a meter so there are 108 atoms along a centimeter line, that is 100 million atoms.

Science requires careful comprehensive measurement of quantities such as distance, area, volume, discharge (1 acre-foot per day), mass, density, force, pressure, work, moment, energy, power, and heat.

Measuring is done with a variety of tools such as rulers, scales, inclinometers, spectrometers, and fluorometers.

Measurement is sometimes direct and at other times is calculated from other measures.

Fundamental characteristics ofmeasure are accuracy (how closethe measurement is to the actualvalue), precision (how refined the measure is), and error.

Exemplar 4 (Langkamp and Hull, 2007): The groundwater beneath a gasoline station was contaminated with methyl tert-butyl ether (MTBE), a gasoline additive used to increase gas mileage by increasing combustion. MTBE is also a cancer-causing agent. A groundwater sample was analyzed and MTBE measured 455 parts of MTBE per billion parts of water. The threshold measurement below which MTBE cannot be detected is 1 part per billion. What can we say about the accuracy and precision of these measures?

Solution: Precision is high since measure can determine 1 part per billion. Accuracy is unknown since we do not know the actual amount of MTBE in the water. Better to have accuracy first, then precision.

Exemplar 5 (Langkamp and Hull, 2007): The General Sherman sequoia tree in Sequoia National Park has an actual height of 83.82 meters. Using an inclinometer and trigonometry a park ranger gets a measure of 84.71. What is the error in the measure?

Solution: The absolute error is 84.71 m – 83.82 m = 0.89 meters. The relative error is 0.89 m/83.82 m = 0.0106 or as a percentage 1.06% error. Note there is no unit attached to this ratio since meters cancel.

Exemplar 6 (Langkampand Hull, 2007): What is the volume of water an average household uses each year to wash dishes?

Solution: In this case an estimate is about the best for which we can hope. Assume the household does dishes once a day and uses 6 to 10 gallons of water per wash. Average these values to get 8 gallons of water per wash. There are 365 days in a year, to make the calculation easier round to 400 days. Then calculate an estimated amount by paying attention to the units: .

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Exemplar 7 (Langkamp and Hull, 2007): A small well in a rural village produces 3.5 gallons per minute. What is this discharge in m3/hour?

Solution: Dimensional Analysis is the tracking of units when performing calculations. Requires student to understand ratios. Student can track the calculations they need to perform by tracking the units. Students must also have knowledge of units in measurement systems and conversions between units.

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Proportional Reasoning is a “form of mathematical reasoning that involves a sense of co-variation and of multiple comparisons, and the ability to mentally store and process several pieces of information” (Lesh, Post, & Northern, 1988).

Pivotal position in science - most common form of structural similarity, a critical aspect of recognizing similar patterns in two different contexts.

Underpins many of the QL components, including measurement, numeracy, and dimensional analysis.

The essential characteristic of proportional reasoning is to involve reasoning about the holistic relationship between two rational expressions (fractions, quotients, rates, and ratios).

Proportional reasoning is not the ability to employ the cross multiplication algorithm

Early phases in proportional reasoning involve additive reasoning (A – B = C – D) and multiplicative reasoning (A x B = C X D).

Traditional proportional reasoning involves relationships of the type A/B = C/D, where one of the values is unknown.

Karplus et al. (1983) views proportional reasoning as a linear relationship between variables such as y = mx, where the y-intercept is 0.

Proportional reasoning requires students to first understand fraction a/b, which at the most basic level is interpreted by students as comparing the part (numerator a) to the whole (denominator b) for like quantities.

Exemplar 8 (Langkamp and Hull, 2007): Determine the percentage of change in bacteria in a lake with an initial concentration of 720 colonies/liter and a final concentration a week later of 1,260 colonies/liter.

Solution: Percentage difference is calculated as

so we have

Note that fractions do not have units, since the comparison is to the same object so the units cancel. A basic QL skill is the ability to add, subtract, multiple, and divide fractions, as well as represent fractions as decimals and percents.

Exemplar 9 (Langkamp and Hull, 2007): The salinity of seawater is typically expressed as number of grams or extra ions in 1 liter of water. Seawater has about 35 grams of extra ions per liter of water. What is the salinity of common seawater in parts per thousand?

Solution: Ratio represents a relationship between two different quantities, focusing on part-to-part comparisons. Students must attain the conception of ratio before being able to set up equivalent ratios, one of the fundamental conceptions of proportional reasoning. Examples of ratios include parts per thousand or million, conversion factorssuch as 1000 cm3/1 liter, and normalizing data.

Exemplar 10 (Langkampand Hull, 2007): The Little Snake prairie dog colony in Colorado has 36,875 prairie dogs on 31,624 hectares, while the Wolf Creek colony has 20,009 prairie dogs on 3,174 hectares. Which colony is more robust?

Solution: Normalize the data to find the number per hectare so you can make a comparison on a common scale. The density of prairie dogs per hectare in the Wolf Creek colony is

while the Little Snake colony has only 1.17 pd/ha.

Exemplar 11 (Langkampand Hull, 2007): The total area of tropical forest in Congo is 278,797 km2 and in Zaire it is 1,439,178 km2. The protected tropical forest in Congo is 12,935 and in Zaire 93,160. What is the percentage of protected forest in Congo? What is the percentage of total forest in Congo to total forest in Zaire?

Solution: The percentage of protected forest in Congo is an example of percentage as a fraction, since we are comparing part-to-whole for like quantities:

The percentage of total forest is a part-to-part comparison, so it is an example of percentage as a ratio. Note we are still comparing like quantities.

Exemplar 12 (Langkamp and Hull, 2007): Carbon is stored in various reservoirs on Earth. The amount of carbon in these reservoirs is measured in petagrams(Pg), where 1 petagram is 1015 grams. The amount of carbon stored in fossil fuels is 3,700 Pg while that stored in vegetation is 2,300 Pg. Using fossil fuels as a referent, what is the percentage difference between carbon stored in fossil fuels and carbon stored in vegetation?

Solution:

So the carbon stored in vegetation is 37.8% less than that stored in fossil fuels.

Exemplar 13 (Langkampand Hull, 2007): In 1990 the forests of the world covered 3,510 million hectares. By 1995 world forests had decreased to 3,454 million hectares. How much forest will be lost by the year 2010?

Solution: A pre-proportional reasonermay use additive reasoning, calculating change by taking the difference in forest area without accounting for the years over which it occurs: 3,454 - 3,510 = -56 million hectors. If they disregard the years and consider this a yearly change they would grossly overestimate the amount of change. Calculating the rate of change per year requires finding the slope:

So over 20 years from 1990 to 2010 there will be a loss of

A proportion is an equivalence between two ratios: a/b = c/d. Many students can manipulate proportions to find the missing value, as in 2/5 = x/10, however this may indicate only rote use of the cross multiplying algorithm.

True proportional reasoning requires a perception of structural similarity; a conception of n times as many. If a student reduces A/B = C/D to P = C/D when solving, then they are not using possible structural relationships but are solving using algebra without regard to structure.

Exemplar 14 (Langkampand Hull, 2007): A capture-recapture method is used to determine the size of the rat population on an island. A sample of 250 rats is captured and tagged. They are released and allowed to mix back into the population. Sometime later a random sample of 500 rats is taken, and 21 are tagged. Estimate the total rat population.

Solution: A student may simply set up a proportion of tagged to total in the sample and tagged to total in the population

which using the cross multiplication algorithm gives so the population is about 5,952 rats. However such rote use of the cross multiplication algorithm may not indicate they are using proportional reasoning. To understand why the capture-recapture method works requires the student to use proportional reasoning. They understand that the population is about 24 times as much as the tagged rats and that this holds no matter how many rats are tagged in the original capture.

Student moves from a conception of proportional reasoning as equivalent ratios to a conception of linear direct variation y=kx.

Science contexts for proportional reasoning are often in the linear direct variation form.

Students may extend this to indirect variation y=k/x.

Requires an understanding of the underlying algebraic concepts of equivalence, variable, and transformations (structural similarity and invariance).

Exemplar 15 (Rockswold, 2002): Ozone in the stratosphere is measured in Dobson units, where 300 Dobson units is a midrange value that corresponds to an ozone layer 3 millimeters thick. In 1991 the reported minimum in the Antarctic ozone hole was approximately 110 Dobson units. A 0 Dobson measure would correspond to 0 thickness of the ozone layer. What is the direct variation coefficient for a model of the relationship between Dobson units and ozone layer thickness?

Solution: To show the relationship between proportional reasoning and direct variation, consider solving the problem by setting up ratios of thickness to Dobson units: 3mm/300Du and y mm/110 Du. Now we have been told there is a direct linear variation between the models, so the ratio or slope is constant, so we can set these ratios equal and solve for y, so y = 1.1 mm when the Dobson unit is 110. Take any ratio of thickness to units, call it y/x. Since the variation is linear we know that this general ratio is equal to the constant ratio of 3mm/300Du. Setting them equal gives the proportion: y/x = 3/300 then solving for y gives the direct variation from of y = (1/100) x where 1/100 or 0.01 is the constant of variation.

Probability is the chance of occurrence of an event, with the theoretical probability defined as:

Earth systems cannot be manipulated like dice to determine a theoretical probability. Often scientists can only estimate the probability through observations of the system. Empirical (experimental) probability is determining a probability based on observations or experiments.

Exemplar 16 (Langkamp and Hull, 2007): From 1900 to 1998, there were 26 years in which a major flood occurred on the Mississippi River. What is the probability of a major flood on the Mississippi River in any given year?

Solution: The event is a major flood in a given year, which occurred 26 times. The total outcomes is the number of years in which a major flood could have occurred which is 99 (must count the year 1900). So the empirical probability of a flood is 26/99 or approximately 26%.

Odds for an event are the ratio between the event occurring and the event not occurring. For Exemplar 16 the odds of having a major flood in a given year are 0.26/(1-0.26) = 0.26/0.74 or about 1 in 3.

Descriptive statistics allow us to summarize and describe data. The fundamental descriptive statistics are measures of the center of a distribution and measures of the spread in a distribution.

Measures of central tendency include the mean, median, and mode.

Variation is a measure of how much the data are spread out. These include range, quartiles, 5 number summary, and standard deviation.

Year Made

CRT Maker Lead (mg/L)

Year Made

CRT Maker Lead (mg/L)

Year Made

CRT Maker

Lead (mg/L)

90 Clinton 1.0 93 Toshiba 3.2 94 Zenith 21.584 Matsushita 1.0 84 Matsushita 3.5 77 Zenith 21.9

85 Matsushita 1.0 84 Sharp 4.4 87 NEC 26.6

87 Matsushita 1.0 98 Samsung 6.1 96 Orion 33.1

89 Samsung 1.0 95 Samsung 6.9 85 Sharp 35.286 Phillips 1.0 98 Chunghwa 9.1 92 Phillips 41.584 Goldstar 1.5 89 Panasonic 9.4 84 Quasar 43.594 Sharp 1.5 97 Toshiba 10.6 92 Toshiba 54.194 Zenith 1.6 87 NEC 10.7 85 Toshiba 54.597 Toshiba 2.2 98 Samsung 15.4 93 Panasonic 57.2

97 KCH 2.3 92 Chunghwa 19.3 89 Samsung 60.891 Chunghwa 2.8 97 Chunghwa 21.3 89 Hitachi 85.6

Solution: The median is the middle number when the data are arranged in ascending order. If there are an even number of data values then the median is the average of the two middle numbers. There are 36 data values so we average of the 18th and 19th value:

The mean is the sum of all the values divided by the number of values:

The mode is the most frequently occurring value, which is 1.0. The mode is not often used in analyzing scientific data.

Notice that the mean and median values differ significantly, so it does make a difference what measure of central tendency is reported. The CRT makers may report the median to argue that the amount of lead is not as high, while an environmental group would report the mean.

The simplest measure of variation is the range which is the difference between the largest and smallest values in the data set. For the CRT problem in Exemplar 17 the range is 85.6 – 1.0 = 84.6.

What are concerns with using range as measure of variation? While easy to calculate the range can be misleading, since one

outlier can make it appear the data set is more spread then it is. To avoid this one can use quartiles (values that divide the data set into quarters) and the 5 number summary – lowest value, lower quartile, median, upper quartile, and highest value. For the CRT problem the 5-number summary is 1, 1.9, 9.25, 29.85, 85.6. This indicates that the lowest 25% only varies from 1 to 1.9 while the upper quarter varies much more, from 29.85 to 85.6.

Standard deviation measures the average distance of all data values from their mean. So we find the deviation which is the distance of a value from the mean Square the deviation so that positive and negative deviations don’t cancel out

when adding them which could conceal spread Take the average of all deviations by summing them and dividing by one less

than the total number of data values (this is an adjustment for working with a sample rather than a population). The result is called variance.

But variance is in squared units and our original data is not squared, so we square root the variance to get standard deviation, which is in the same units as the original data. So the standard deviation for Exemplar 17 is:

Chebychev’s Rule: for any set of data at least 75% of the data lie within 2 standard deviations of the mean and at least 89% of the data lie within 3 standard deviations of the mean.

Any data value that lies 3 or more standard deviations from the mean is called an outlier and it is common practice to discard them from the data set.

Hitachi (1989) model with lead at a level of 85.6 mg/L is an outlier since the mean is about 19 and 3 times the standard deviation is 66: 19+66 = 85 < 85.6

We have restricted QL to the realm of number and arithmetic, but variation requires algebraic operations of taking roots or powers. So variation is at best on the border of QL and QI.

We have discussed it in the QL section because it is so commonly used to describe data.

Other basic statistics that are used in science which are on this border between QL and QI are z-scores (number of standard deviations a data point lies above or below the mean) and confidence intervals.