AP Biology: Math for Dummies - PowerPoint Presentation

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AP Biology: Math for Dummies

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Science Practices- The student can:

-use representations & models to communicate scientific phenomenon & solve scientific problems.

-

use mathematics appropriately.

-engage in scientific questioning to extend thinking or to guide investigations within the context of the AP course.

-plan & implement data collection strategies appropriate

to a scientific question.

5. -perform data analysis & evaluation of evidence.

6. -work with scientific explanations & theories.

7. -is able to connect & relate knowledge across various scales, concepts & representations in and across domains.

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Big Idea 1

The Process of Evolution Drives the Diversity and Unity of Life

Change in the genetic makeup of a population over time is evolution.

Organisms are linked by lines of descent from common ancestry.

Life continues to evolve within a changing environment.

The origin of living systems is explained by natural processes.

Hardy-Weinberg Equations

Probabilities

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Organisms use feedback mechanisms to regulate growth & reproduction & to maintain dynamic homeostasis.

Growth & dynamic homeostasis f a biological system

are influenced by changes in the system’s environment..

Many biological processes involved in growth, reproduction & dynamic homeostasis include temporal regulation & coordination.

Big Idea 2

Biological Systems Utilize Free Energy and Molecular Building Blocks to Grow, Reproduce and Maintain Dynamic Homeostasis

Growth, reproduction & maintenance of the organization of living systems require free energy & matter.

Growth, reproduction & dynamic homeostasis require that cells create & maintain internal environments that are different from their external environments.

Water Potential

Gibb’s Free Energy

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Big Idea 3

Living Systems Store, Retrieve, Transmit and Respond to Information Essential to Life Processes

Heritable information provides for continuity of life.

Expression of genetic information involves cellular & molecular mechanisms.

The processing of genetic information is imperfect & is a source of genetic variation.

Cells communicate by generating, transmitting

& receiving chemical signals.

Transmission of information results in changes

within and between biological systems.Chi squareGene Linkage

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Big Idea 4

Biological Systems Interact and These Systems and Their Interactions Possess Complex Properties

Interactions within biological systems lead to complex properties.

Competition & cooperation are important aspects of biological systems.

Naturally occurring diversity among & between components within biological systems affect interactions within the systems.

Population GrowthEnergy TransferPrimary Productivity

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Phase

Number

Percent spent in each phase

Interphase

52

Prophase/prometaphase

12

Metaphase

2

Anaphase

5

Telophase

1

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To measure the population density of monarch butterflies occupying

a

particular park, 100 butterflies are captured, marked with a small dot on a wing and then released. The next day, another 100 butterflies

a

re captured, including the recapture of 20 marked butterflies. What would you estimate the population to be?(100 x 100)/ 20 = 500

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2 x 2 = 4

6 x 4 = 24

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1

st

Law of Thermodynamics- energy cannot be created or destroyed, but it can change form.

18,000 energy accumulated as biomass; 12,000 going to the tree layer; 4,400 going to the shrub layer; 1,600 left, to go to the grass layer. 1,600 is 9% of 18,000 (1,600/18,000 x 100)

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Atmospheric pressure is the combined partial pressures of all of the gases that make up the atmosphere. At the summit of a high mountain, the atmospheric pressure is 380mm/Hg. The partial pressure of oxygen is 69mm/Hg. What percentage

Of the atmosphere is made up of oxygen at this altitude?

18%

69/380 = .18 = 18%

(the average partial pressure of oxygen at sea level is 21%)

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Use the Station 1 data to calculate the Primary Productivity of a water sample. Report your answer in units of

mg Carbon fixed/Liter

Station 1

4.2

mg O

2/L  0.698 = 2.9 mL O2/L2.9 mL O2/L  0.536= 1.6 mg Carbon fixed/L

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What is the mean rate of growth per day between day 5 and day 25? Record your answer to the nearest hundredth of a cm.

If this same rate of growth continues, how tall will the plant be on day 50? Record your answer to the nearest hundredth

of a cm.

18-3 = 15

15/20 days = .75 cm

.75 x 25 = 18.7518.75 + 18 = 36.75

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What is the water potential of a cell with a solute potential of -0.67

MPa

and a pressure potential of 0.43

MPa

? -.24MPa

-0.67 + 0.43

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You measure the total water potential of a cell and find it to be -0.24

MPa

. If the pressure potential of the same cell is 0.46

MPa

, what is the solute potential of that cell?

Since water potential is equal to the solute potential + the pressure potential, -0.24 MPa = 0.46 MPa + X. Solve for x= -0.7

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Trial

No Treatment

1% salt

3% salt

5% salt

7% salt9% salt147412528

24

5

2

46

423223

21

6

3

34

32

28

21

18

3

4

57

44

24

25

17

2

5

41

39

27

25

21

4

The purpose of a particular investigation was to see the effects of varying salt concentrations of nutrient agar and its effect on colony formation. Below are the results Determine the mean for each treatment and graph the results.

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A wind blown pollen grain with a solute potential of -3.0

MPa

has dried out

somewhat after blowing around in the wind. This has caused its turgor pressure

to go to zero. It lands on a flower stigma whose cells have a solute potential of -3.0 Mpa and a pressure potential of +1 Mpa. Which way will water flow? From the pollen grain to the stigma or the stigma to the pollen grain? Show how you deduced your answer.

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A population of ground squirrels has an annual per capita birth rate

o

f 0.06 and an annual per capital death rate of 0.02. Estimate the

n

umber of individuals added to (or lost from) a population of1,000 individuals in one year.dN/dt = B-D Change in population size/time = Birth rate – Death rate0.06 – 0.02 = 0.04 x 1000 = 40 individuals added per year

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In 2005, the United States had a population of approximately

295,000,000 people. If the birth rate was 13 births for every

1,000 people, approximately how many births occurred in the

United States in 2005?

295,000,000/1,000 = 295,000295,000 x 13 = 3,835,000

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Geneticists working in an agriculture lab wanted to develop a crop that combines the disease resistance of rye grain with the high crop yielding capacity of wheat grain. Rye grain has a diploid chromosome number (2

n

) of 14 and wheat grain has a diploid chromosome number of 42. The resulting grain is called triticale and is an

alloploidy

plant. How many chromosomes are found in the pollen grain of triticale?

Alloploidy results when two different plant species combine their diploid genome to make new and unique species. That would mean that this particular species would have 56 chromosomes. The cells in a pollen grain of would be haploid so the resulting number is 28.

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Cytosine makes up 38% of the nucleotides in a sample of DNA from an organism. Approximately, what percentage of the nucleotides in this sample will be thymine?

Remember Chargaff’s Law? In a DNA molecule the amount of

Cytosine must equal the amount of Guanine and the amount of

Adenine must equal the amount of Thymine. If 38% of the molecule Is Cytosine then 38% must be Guanine. That leaves 24% of the molecule to be made up of Adenine and Thymine. 12% of each.

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How many unique gametes could be produced through independent assortment by an individual with the genotype

AaBbCCDdEE

?

Aa

and Bb and

Dd can each make 2. CC and EE can only make 1.2 x 2 x 1 x 2 x 1 = 8

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What is the population standard deviation for the numbers: 75, 83, 96, 100, 121 and 125

?

Raise your hand if you need help.

Find the mean

Find the difference between the mean and each number (some will be + and some will be –)Square each difference (now they’re all +)Add them together

Take the square rootBam! That’s it.

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I hate standard deviations.

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A study was conducted on the island of Daphne Major in the Galapagos Islands by Peter and Rosemary Grant. This study lasted over 20 year s. The study investigated how the type of seeds available to the finches impacted the depth of their beaks. In years when rain and water were plentiful, the available seeds were smaller and easy to crack. In years experiencing drought, fewer seeds were produced, and the finches had to eat the larger, leftover seeds produced from previous years. During years of drought, birds with a greater beak depth had a selective advantage

.

Use the data above to determine the increase in the mean of the depth of the beak between the wet and dry years. Give your answer to the nearest hundredth of a millimeter.

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Experimental Design

P

ose

scientific

questions

Refine scientific questionsEvaluate scientific questions

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A population of beetles,

Tenebrio

molitor

, has been bred for several generations in tubs of Cheerios. The Cheerios have been their only food source. Could the beetles survive in something other than Cheerios?

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J

ustify

the selection of the kind of data

needed to answer a particular scientific question.

Design a plan for collecting data to answer a particular scientific question.

Collect data to answer a particular scientific question.Evaluate sources of data to answer a particular scientific question.

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A

nalyze

data

to identify patterns or relationships.

Refine observations and measurements based on data analysis.

Evaluate the evidence provided by data sets in relation to a particular scientific question.

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Models & Representations

1. Create a model

In a population of grasshoppers living in a deciduous forest,

there is a great diversity of color, allowing the insects to

b

lend with the colors of the forest. Light color grasshoppersare less abundant, since they are spotted more easily by birds.A prolonged drought in the area causes many plants to wither and die. Create a graph illustrating the frequency of the beginning population and your prediction of the change in frequency in response to the drought.

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2. Describe what is taking place

A dialysis tubing bag is filled with a mixture of 3% starch and 3% glucose and placed in a beaker of distilled water. After 3 hours, glucose can be detected in the water outside the dialysis tubing bag, but starch cannot.

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3. Refine Model

In the investigation described, how could you modify the experiment to determine the p

ermeability of the membrane to water?

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4. Use and Apply

The purpose of a particular investigation was to see the effects of varying

salt concentrations

of nutrient agar and its effect on colony formation. Below are the results Determine the mean for each treatment and graph the results.

 

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5. Re-express

Signal transduction pathways are regulatory mechanisms in living things.

Identify a signaling molecule and the response brought about in

an animal

a plant

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Forty flies were put into a choice chamber with two chambers. In one chamber there was a cotton ball soak with vinegar. The other chamber had nothing. After 20 minutes the number of flies were counted in both chambers. This was repeated four more times. Perform a chi-square analysis to determine if the difference between in the number of flies found in the two chambers is significant.

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76 124

Water Vinegar

100 100

-24 24

576 576

5.76 5.7611.52

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The allele for the hair pattern called “widow’s peak” is dominant over the allele for no “widow’s peak”. In a population of 100 individuals, 64 show the dominant phenotype. What is the frequency of the recessive allele?

.6

64 show the dominant phenotype,

So 36 show the recessive phenotype.

Since this is a population of 100, 36% Show the recessive phenotype

.36 = q2.6 = q

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In a certain population of deer on Fire Island, NY, the allele for a black spot behind the eye is dominant to the allele for no spot. After the hunting season, the percent of deer with no black spot is 15% and the population is in Hardy-Weinberg Equilibrium. What is the frequency for the allele for having no black spot, to the hundredths?

15% = .15 = q

2

q = .39

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The ability to taste PTC is due to a single dominant allele

(A).

You sampled 215 individuals in biology and determined that 150 could taste PTC and 65 could not. How many individuals in this population show the following genotype?

AA,

Aa, aa65/216 = .3 = q2q = .55p = .45(.45)(.45) = .2 = 20%20% x 215 = 43 AA

2(.45)(.55) = .495 = 50%50% x 215 = 107 Aa

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In a

dihybrid

cross between two

heterozygotes

, if you have 200 offspring, how many should show both dominant phenotypes

? 112This is a 9:3:3:1 ratio, with 9/16 showing both dominant phenotypes. 9/16 = .56 = 56%56% of 200 is 112

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In this genetic cross,

Aa

x

aa, there are 348 offspring. How many individuals are expected to have the dominant phenotype?

174; this is a 1:1 ratio, so 50% are expected to Have the dominant phenotype

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In a typical

Mendelian

monohybrid cross, two heterozygotes

produce 400 offspring. How many individuals are expected to have the recessive phenotype?100; this is a 3:1 phenotypic ratio

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A Cellular Biologist wants to double check that statement that cells spend 90 percent of their time in

Interphase

as compared to the various stages of Mitosis. She grows some

Allium

in her laboratory. She then takes one of the plants, cuts off the root tips, stains the DNA in the cells so as to be able to see the stages of the cell cycle. Her hypothesis states

“If cells spend 90 percent of their time in Interphase, then she should be able to calculate the relative time existing between Interphase and Mitosis based upon the cells counted in her specimen.” She counted 1000 cells from her preserved specimen under the microscope. Her data are shown below. Calculate the X2 to the nearest hundredth.

Stage of the Cell Cycle

Number of Cells Observed

Number of Cells Expected

Interphase

872

900

Mitosis

128

100

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In geckos, spots are dominant to the solid color. If the frequency In a population of 700 geckos, what percentage of the geckos would have spots, if the frequency of the recessive allele

is 0.2, and the population is in Hardy-Weinberg equilibrium?

96%

q= .2

p= .8Homozygous dominant (p2) = .64 (64%)Heterozygotes (2pq) = .32 (32%)

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The formula is easy: it is the square root

of the

Variance.

So now you ask, "What is the

Variance?“The Variance is defined as: The average of the squared differences from the Mean.

To calculate the variance follow these steps:Work out the Mean (the simple average of the numbers) Then for each number: subtract the Mean and square the result (the squared difference). 3. Then work out the average of those squared differences.

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14,000 (a) 35 (c)

180 (b) 100 (d)

You are starting with 87,400 kJ and simply subtracting to get the answers.

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Consider a field plot containing 200 kg of plant material. Approximately how many kg of carnivore production can be supported?

a. 200

b. 100

c. 20

d. 2

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Given the parents

AABBCc

x

AabbCc

, assume simple dominance and independent assortment. What proportion of the progeny will be expected to phenotypically

resemble the first parent?a. 1/4 b. 1/8 c. 3/4 d. 3/8