## Wednesday, July 22, 2015

### Topic list for comprehensive final on 7/23

You get two 8.5" x 11" pages of notes, front and back, which means four pages total.

These are the possible topics for the 100 point test.
Logical operators, and ^, or v, not ~
Decimals and percents, changing fractions into repeating decimals and vice versa.
Scales other than percents
Scientific notation and rounding to significant digits.
Roman numerals
Fractions in lowest terms
Prime factorizations and all factors
Time: adding subtracting, converting
100 coin problems
simultaneous equations
bels/decibels and Richter scale
Classifications of triangles given angles
Classifications of triangles given sides
Heron’s formula
Area of a triangle with points (0, 0), (a, b) and (c, d)
Distance between points
Pythagorean Theorem
Sets, Venn diagrams, contingency tables and probabilities
Slope of a line and lines with undefined slope
Point-slope form, slope-intercept form
Sum and product problems
Stats: mean, median, mode, five number summary, IQR, outliers
standard deviation, z-scores, z-scores to proportions less than a z-score, reverse lookup table, proportions between two z-scores for a normally distributed set

If something says "formula" or "form", You can expect that formula to be on the test. For things that take methods, like changing from Richter to amount of energy or looking up z-scores, those will not be explained, so they should be in your notes.

## Tuesday, July 21, 2015

### Finding percentiles from the z-score list. Finding the proportion at an exact number on a list of a normally distributed set

Finding percentiles from the z-score list.
One of the things we can do with the z-score to proportion list is find a z-score that corresponds to the cut-off point for a particular percentile. Here are two examples at the 8th percentile and the 80th percentile.

The 8th percentile. because 8 < 50, we will look on the negative z-score side of the sheet. What we want to find is the two positions on the table where the values go from .08xx to .07xx. you will find then in the -1.4 row at the first two positions.

-1.40 -> 0.0808 (8 above 0.0800)
-1.41 -> 0.0793 (7 below 0.0800)

Because 7 < 8, -1.41 counts as the closest to the two, but we should check to see if the average -1.405 would be better. We look at far/close which is 8/7 ~= 1.142857... and if it's less than 3 (it is), we will use the average,so the z-score for the 8th percentile is -1.405.

The 80th percentile. because 80 > 50, we will look on the positive z-score side of the sheet. What we want to find is the two positions on the table where the values go from .79xx to .80xx. you will find then in the 0.8 row in the middle.

0.84 -> 0.7995 (5 below 0.8000)
0.85 -> 0.8023 (23 above 0.8000)

Because 5 < 23, 0.84 counts as the closest to the two, but we should check to see if the average 0.845 would be better. We look at far/close which is 23/5 = 4.6 and if it's more than 3 (it is), we will use the closest instead of the average,so the z-score for the 80th percentile is 0.84.

Finding the proportion at an exact number on a list of a normally distributed set. Consider the SAT values for average mu_x = 500 and sigma_x = 100. SAT scores are always round to the nearest 10, so if we want to find out what percentage of scores are at 600, we have to look at the proportion at 605 and 595.

Raw score 605 become the z-score (605-500)/100 = 105/100 = 1.05, which corresponds to 0.8531

Raw score 595 become the z-score (595-500)/100 = 95/100 = 0.95, which corresponds to 0.8289.

We subtract the small proportion from the big, .8531 - .8289 = .0242. This means about 2.42% of SAT takers will get exactly 600 on one section of the SAT.

## Thursday, July 16, 2015

### Finding x and y when we are given their sum and product

If we think about a rectangle, the sum of the two adjacent sides is half the perimeter and their product is the area. If we are given the sum and product, can we find the two side lengths? This is the original problem that was being considered when the quadratic formula was derived nearly 900 years ago, though many civilizations before this had something like the idea that is about to be presented.

x + y = sum
xy = product

Let's call half the sum a, which stands for average. Since we are given the sum, a is a known quantity as well. It might be that x = y = a, but generally what is true is we can find a number d, which stands for difference such that

x = a + d
y = a - d

This means the product is the difference of squares, (a + d)(a - d) = a² - d².

Using this, here is how we solve a given problem.

Problem: The sum is 14 and the product is 24.

Solution method: The average a = 7, so we need to solve for d.

(7 + d)(7 - d) = 7² - d² = 49 - = 24

Subtract 24 from each side to get

25 - = 0

Add d² to both sides to get

d² = 25.

Take the square root to get d = 5. We use the numbers 7 and 5 in the following way.

7 + 5 = 12
7 - 5 = 2

12 + 2= 14
12 x 2 = 24

With this problem, you could have used guessing and checking to try to find two numbers that added up to 14 (13 and 1, 12 and 2, 11 and 3, no wait, go back to 12 and 2) where the product was 24. let's do one where guessing and checking isn't an option.

Problem: The sum is 15 and the product is 45.

Solution: The average is 7.5 and a² = 56.25. So 56.25 - = 45.

Subtracting, we get 11.25 - = 0 or d² = 11.25 = 45/4.

Taking square roots, we get d = sqrt(45/4), which simplifies to 3sqrt(5)/2, which rounds to 3.354. Our two numbers are

7.5 + 2sqrt(5)/2 and 7.5 - 2sqrt(5)/2, which round to 10.854 and and 4.146.

More problems of this type.

Problem: The sum is 20 and the product is 40.

Problem: The sum is 40 and the product is 20.

Problem: The sum is 10 and the product is 40.

Problem: The sum is 40 and the product is 10.

## Wednesday, July 8, 2015

### Prep for midterm 2

Midterm 2 will be based on homeworks 4 through 7 and the biographies of the five British logicians, Boole, Babbage, De Morgan, Dodgson and Lovelace. The most recent posts on this blog from the June 28th post about Richter scale up through the logician biographies will be of the most use in studying.

## Thursday, July 2, 2015

### Links to practice problems for distance between two points and area of a triangle defined by three points, one of which is (0, 0)

Here is the link to the practice problems.

More practice
Here are four points. Distances should be given in simplified square root form and approximated to the nearest thousandth,

(0, 0), (6, 3), (10, 1), (-4, -5)

a) Distance from (0, 0) to (6, 3)
b) Distance from (0, 0) to (10, 1)
c) Distance from (0, 0) to (-4, 5)
d) Distance from (6, 3) to (10, 1)
e) Distance from (6, 3) to (-4, -5)
f) Distance from (10, 1) to (-4, -5)

g) the area of the triangle defined by the points (0, 0), (6, 3) and (10, 1)
h) the area of the triangle defined by the points (6, 3), (10, 1) and
(-4, -5)