Scaling down the Solar system
The distance from the Sun to the Earth is about 150 million kilometers. The diameter of
the Sun is about 1.4 million kilometers and the diameter of the Earth is about 12,760
kilometers.
Put these numbers in perspective using a scale model of the solar system. Choose the
scale so that the objects and distances are of a reasonable size - your model should not
exceed the size of the hallway in Pearce.
•
•
•
What is the scale of your model? (use a ratio of two numbers)
The diameter of the Moon is 3400 km and its distance from the Earth is about
350,000 km. How large and how far from the Earth it would be in your model?
The distance from the Earth to the nearest stars besides the Sun is about 4.3
light-years. How far are these stars from the Earth in your model?
(1 light-year is the distance that light can travel in one year. The speed of light is
approximately 300,000 kilometers per second).
Timeline
The basic timeline of a 4.6 billion year old Earth:
3.8 billion years of simple cells (prokaryotes),
600 million years of simple animals,
65 million years since the non-avian dinosaurs died out,
200,000 years since humans started looking like they do today,
5000 years of human civilization (ancient Egypt).
If you wanted to create a timeline on which the distance from TODAY to the beginning of
human civilization was 1 mm, how far would the other events fall?
Two Views of a Tax Cut Case Study
These two graphs are different representations of the same tax cut – one called
Republican Math and one called Democratic Math.
Question 1. Describe what the data in each of these graphs represents. Can
both of these graphs be correct? Explain why or why not.
Question 2. Why is the percent decrease in taxes for families in the $10,000 to
$20,000 income bracket 3.4% while these families receive only 1.3% of the tax
cut? Give an example that shows with actual dollar amounts that this is possible.
Question 3. Can the amount of the tax cut be determined from the information
given in the two graphs? Why or why not?
Question 4. If one assumes the tax cut is $245 billion1, how much are taxes cut
(in dollars) for families in the $200,000 and over income bracket? How much
are taxes cut (in dollars) for families in the $20,000 to $30,000 income bracket?
Question 5. Assuming the tax cut of $245 billion and using your solutions to
question 4, what is the total amount of taxes paid by the families in the $200,000
and over income bracket? What is the total amount of taxes paid by families in
the $20,000 to $30,000 income bracket?
Question 6: A common argument that arises from issues related to tax cuts (as
indicated by the titles of the two graphs) proceeds as follows: Republicans allude
to the idea that taxes are cut uniformly for all income brackets, while Democrats
point to the fact that the bulk of the tax cuts go to the wealthy. How is this so?
Why did people in the $20,000-$30,000 bracket get a larger tax cut than the
$200,000 and over bracket (3.9% to 2.9%) yet the people in the $200,000 and
over bracket get a larger share of the tax cuts?
Adapted from National Numeracy Network
http://serc.carleton.edu/nnn/quantitative_writing/examples/24031.html
1
The agreement in Congress on the 1995 tax cut was $245 billion over seven years.
Odds with coins - exploring pattern
Write the sample space for flipping one coin: {
}
How many elements are in your sample space?
How many outcomes in your sample space satisfy the following events:
Getting 0 heads (0H):
Getting 1 head (1H):
Write the sample space for flipping two coins:
How many elements are in your sample space?
How many outcomes in your sample space satisfy the following events:
0H:
1H:
2H:
Write the sample space for flipping three coins:
How many elements are in your sample space?
How many outcomes in your sample space satisfy the following events:
0H:
1H:
2H:
3H:
Organize your answers in the following manner and observe the pattern. Try to
continue the pattern for 4, 5 and 6 coins without listing their sample space. (If
you don’t see the pattern, list them!)
1
1 coin:
1H
0H
2 coins:
0H
1H
2H
3 coins:
0H
1H
2H
3H
4 coins:
0H
1H
2H
3H
How many elements are there in the sample space for flipping 4 coins? 5 coins? n
coins?
What is the probability of getting two heads when flipping 6 coins?
4H
Find the following probabilities for these spinners
P(4) =
P(number greater than 5) =
1
P(9) =
2
3
P(number smaller that 10) =
P(5 OR 7) =
6
P(1 AND 2) =
4
P(even number) =
5
P(odd OR even number) =
The spinner A (left) is spun first, then spinner B (right).
P(double 2) =
2
1
P(2 followed by 4) =
2
P(odd number follow by even) =
1
3
P(exactly one even number) =
P( at least one 2) =
3
P(double 4) =
4
P(number less than 4 followed by a number less than 6) =
You pay $2 for spinning the spinner once and you can win up to $5 as shown in the
picture.
Would you play the game? Would you play it many times? Would you run it at a carnival
where you can expect plenty of people playing? Explain your reasoning.
2
0
4
5
0
2
Insurance company is calculating rates for the next year and determined that for every
10,000 policyholders:
•
•
•
•
•
•
1 is likely to have an accident for which the company will have to pay a
$200,000 claim
10 are likely to have an accident for which the company will have to pay a
$100,000 claim
20 are likely to have an accident for which the company will have to pay a
$50,000 claim
100 are likely to have an accident for which the company will have to pay a
$10,000 claim
1000 are likely to have an accident for which the company will have to pay a
$1,000 claim
The remainder of the policyholders will have no claims for the year.
1. Use these numbers to find out the minimal insurance premium each policyholder
must pay so that the company is able to cover all expected claims.
2. Discuss the similarities between running insurance business and running a spinner at
a carnival (Previous problem).
3. Explain why requiring every car owner to buy a car insurance makes sense from
mathematical point of view.