1. 1 The Science and Mathematics of Natural Disasters Density of Liquids Mathematics: Day Seven

2. 2 (SMP) 2 Institute Staff 2006 July 10 – July 14 July 24 – July 28 Dr. Sherry Herron, Dr. Shelia Brown, Dr. Sharon Walker, Dr. David Beckett Science Dr. Myron Henry *, Mrs. Lida McDowell *, Mrs. Mary Peters Mathematics * Responsible for Katrina Mathematics Materials

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4. 4 Day Seven Mathematics Topics density, mass/weight ratios, unit rates, proportions measurement conversion scatter plots, line graphs TI-84 graphing windows linear functions, slope parts per hundred (pph or percent), parts per XXX notation

5. 5 Day Seven Partial Objectives from the 2007 Mississippi Mathematics Objectives (Revised) 5 th grade 4b (Convert units within a given measurement system to include length, weight/mass, and volume) 6 th grade 2c (Formulate algebraic equations to reflect a given situation), 4a (Convert units within a given measurement system to solve problems), 4d (Use scale factors to solve ratio and proportion problems), 4e(Predict and calculate the volume of prisms) 7 th grade 2c (Formulate algebraic expressions and equations to reflect a given situation and vice versa.), 4a (Convert from one unit to another, perform basic operations and solve real-world problems using standard (English and metric) measurements), 4c (Develop and justify geometric formulas for volume of prisms),4d (Solve problems involving scale factors using ratios and proportions) Pre-Algebra 2e (Graph linear equations using multiple methods) 2f (Given a linear graph, identify its slope), 2g (Determine slope from a graph and/or equation in slope-intercept or standard form), 4b (Develop, analyze, and explain methods for solving problems involving proportions) Algebra I 2d (Explain and illustrate how a change in one variable may result in a change in another variable), 2e (Graph and analyze linear functions), 5a (Draw conclusions and make predictions from scatter plots)

6. 6 University of Colorado researchers chart Katrina's Growth in Gulf Of Mexico – This image shows Hurricane Katrina’s maximum sustained wind speed after it crossed Florida and rode the warm Loop Current and Eddy Vortex up towards Louisiana.

7. 7 In the previous slide, researchers used sea surface height as a proxy for water temperature, as warmer water is less dense and therefore has a higher surface. How does the DENSITY of water affect ocean currents?

8. 8 is a multiplicative comparison of two real numbers or measures or quantities. A ratio is written Density: a ratio giving mass per ONE unit volume.

9. 9 is a ratio including units of measures in the two values being compared. The measuring unit is different for each quantity. is a rate for which one of the two values compared is ONE unit of measure or understood to be ONE unit of measure.

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11. 11 In the US system, the factors relating capacity to volume are not “nice numbers”. However, in the metric system, one liter (capacity) has a volume of exactly one cubic decimeter. In other words, a cubic decimeter has a capacity of one liter. So,

12. 12 Suppose you have measured volume and mass three times for each of alcohol, water, and salt water. Calculate the density of the liquids. Recall that 1 ml of water has a volume of 1 cm 3. LiquidVolume ( ml.) Mass (g.)Density Alcohol 20 ml 50 ml 80 ml __17 g__ __43 g___ __70 g___ ________ Water 20 ml 50 ml 80 ml 20 g___ __49 g___ __81 g___ ________ Salt Water 20 ml 50 ml 80 ml __21.5 g_ __53 g___ __85 g___ ________

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16. 16 Density of Liquids

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18. 18 Window Dimension: 63 pixels vertically (down); 95 Pixels horizontally (across) Pick (y min) = (-5) and (y max) = 85.

19. 19 ~ = alcohol + = water  = salt water

20. 20 Our task: to find the line graph from the mass and volume data for alcohol from our chart. Then find the slope of one line for alcohol and compare the line graph we have constructed with the linear function we have constructed. We should include units of measure where appropriate. LiquidSlope Alcohol Compare the slope values to the density values of the liquids.

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22. 22 Find the line graph for each of water and salt using the mass and volume values from the chart. Then find the slope of one line for each of water and salt water. Compare the line graphs and linear functions you have constructed LiquidSlope Water Salt Water Compare the slope values to the density values of the liquids.

23. 23 Using data from rows 8 and 9 of our chart, we derive the results below.

24. 24 Density is affected by salinity. We have seen that water with dissolved salts (higher salinity) is denser than water without dissolved salts (low or no salinity). We have also seen that a less dense substance will always lie above a more dense substance. When you have water of two different densities meeting, the lower density (less dense) water will move on top of the higher density (more dense water. The different densities actually cause the water to move, forming a density current. Density is affected by temperature. You have probably noticed while standing neck-deep in gulf water that water is colder toward the bottom and warmer on the top. The denser cold water lies on the bottom of the gulf or ocean floor and the less dense warm water lies near the surface. The different densities again cause the water to move, forming a density current. How does the DENSITY of water affect ocean currents?

25. 25 Parts per hundred (denoted by % and very rarely “pph”) – denotes one particle of a given substance for every 99 particles of a base substance (usually water). Parts per thousand (denoted by 0 / 00, and “ppt”) – denotes one particle of a given substance for every 999 particles of a base substance. This is roughly equivalent to one drop of ink in a cup of water, or one second per 17 minutes. Parts per million (denoted by “ppm”) – denotes one particle of a given substance for every 999,999 particles of a base substance. This is roughly equivalent to one drop of ink in a 40-gallon drum of water, or one second per 280 hours. Salinity is usually measured in “parts per thousand” (ppt), whereas traditional percentage is actually “parts per hundred” (pph). Parts per thousand or even parts per million are notations used where very low levels of concentration are significant. “Parts per XXX” is also used to measure levels of pollutants in the environment, or trace elements in forensics or in the Earth’s crust. Types of “parts per XXX” notation

26. 26 Find “parts per XXX” notation using the following metric measurements. thirty milligrams in a gram (by mass) five milligrams in a kilogram ( by mass) fourteen milliliters (or cm 3 ) in a cubic meter seven deciliters in a kiloliter

27. 27 DensityWater Properties More dense (higher density) Less dense (lower density) temperaturesalinity Cold water Warm water Salt water Fresh water