GED LESSON Statistics: Measures of Central Tendency, Mean, Median and Mode

FULL PURPLE MATH ARTICLE HERE

Mean, median, and mode are three kinds of "averages". There are many "averages" in statistics, but these are, I think, the three most common, and are certainly the three you are most likely to encounter in your pre-statistics courses, if the topic comes up at all.

The "mean" is the "average" you're used to, where you add up all the numbers and then divide by the number of numbers. 

The "median" is the "middle" value in the list of numbers. To find the median, your numbers have to be listed in numerical order, so you may have to rewrite your list first. 

The "mode" is the value that occurs most often. If no number is repeated, then there is no mode for the list.

Find the mean, median, mode, and range for the following list of values: 13, 18, 13, 14, 13, 16, 14, 21, 13 The mean is the usual average, so: (13 + 18 + 13 + 14 + 13 + 16 + 14 + 21 + 13) ÷ 9 = 15 Note that the mean isn't a value from the original list. This is a common result. You should not assume that your mean will be one of your original numbers. The median is the middle value, so I'll have to rewrite the list in order: 13, 13, 13, 13, 14, 14, 16, 18, 21 There are nine numbers in the list, so the middle one will be the (9 + 1) ÷ 2 = 10 ÷ 2 = 5th number: 13, 13, 13, 13, 14, 14, 16, 18, 21 So the median is 14.   Copyright © Elizabeth Stapel 2004-2011 All Rights Reserved The mode is the number that is repeated more often than any other, so 13 is the mode. The largest value in the list is 21, and the smallest is 13, so the range is 21 – 13 = 8. mean: 15 median: 14 mode: 13 range: 8

Tables, Pictographs, Bar Graphs, Line Graphs and Circle Graphs on the GED Lesson

In statistics, pictograms are charts in which icons represent numbers to make it more interesting and easier to understand. A key is often included to indicate what each icon represents. All icons must be of the same size, but a fraction of an icon can be used to show the respective fraction of that amount.^[6]

For example, the following table:

Day	Letters sent
Monday	10
Tuesday	17
Wednesday	29
Thursday	41
Friday	18

can be graphed as follows:

Day	Letters sent
Monday
Tuesday
Wednesday
Thursday
Friday

Key:  = 10 letters

As the values are rounded to the nearest 5 letters, the second icon on Tuesday is the left half of the original.

GED Simple Interest (Introduction) Lesson

From Khan Academy

From FULL Wikipedia Article

Interest is money paid by a borrower to a lender for a credit or a similar liability. Important examples are bond yields, interest paid for bank loans, and returns on savings. Interest differs from profit in that it is paid to a lender, whereas profit is paid to an owner. In economics, the various forms of credit are also referred to as loanable funds.

When money is borrowed, interest is typically calculated as a percentage of the principal, the amount owed to the lender. The percentage of the principal that is paid over a certain period of time (typically a year) is called the interest rate. Interest rates are market prices which are determined by supply and demand. They are generally positive because loanable funds are scarce.

Interest is often compounded, which means that interest is earned on prior interest in addition to the principal. The total amount of debt grows exponentially, and its mathematical study led to the discovery of the number e.^[1] In practice, interest is most often calculated on a daily, monthly, or yearly basis, and its impact is influenced greatly by its compounding rate.

From Khan Academy

Five percent problems

GED LESSON: Unit Conversion Explanation and Practice

Convert Pounds to Ounces Video PRESS HERE

There are 16 ounces in 1 pound. To convert pounds into ounces MULTIPLY the number of pounds by 16.

Example:
How many ounces are there in 10 pounds?
Answer: 10 pounds X 16 (ounces per pound) = 160 ounces

Convert Gallons to Quarts, Pints, and Cups Video PRESS HERE

1. Convert Gallons to Quarts 
There are 4 quarts per 1 gallon. To convert gallons into quarts MULTIPLY  the number of gallons by 4.

Example
How many quarts are there in 10 gallons?
Answer: 10 gallons X 4 (quarts per gallon) = 40 quarts

2. Convert Gallons to pints
There are 8 pints in 1 gallon. To convert gallons into pints MULTIPLY the number of gallons by 8.

Example:
How many pints are there in 10 gallons?
Answer: 10 gallons X 8 (pints per gallon) = 10

3. Convert Gallons to cups
There are 16 cups in each gallon. To convert gallons to cups MULTIPLY the number of gallons by 16.

Example:
How many cups in 10 gallons?
Answer: 10 gallons X 16 (cups per gallon) = 160 cups

Converting Yards to Inches Video PRESS HERE

There are 36 inches in 1 yard. To convert yards to inches, MULTIPLY the number of yards by 36.

Example:

How many inches are there in 10 yards?
Answer: 10 yards X 36 (inches per yard) = 36 inches.

GED Lesson: Ratio and Proportions (Cross Multiplying)

Click here for practice problems

Cross-multiplication

From Wikipedia, the free encyclopedia

  (Redirected from Cross multiplying)

In mathematics, specifically in elementary arithmetic and elementary algebra, given an equation between two fractions or rational expressions, one can cross-multiply to simplify the equation or determine the value of a variable.

Given an equation like:

(where b and d are not zero), one can cross-multiply to get:

In Euclidean geometry the same calculation can be achieved by considering the ratios as those of similar triangles.

Contents

  [hide] 

• 1Procedure
• 2Use
• 3Rule of Three
• 4References
• 5Further reading

Procedure[edit]

In practice, the method of cross-multiplying means that we multiply the numerator of each (or one) side by the denominator of the other side, effectively crossing the terms over.

The mathematical justification for the method is from the following longer mathematical procedure. If we start with the basic equation:

we can multiply the terms on each side by the same number and the terms will remain equal. Therefore, if we multiply the fraction on each side by the product of the denominators of both sides—bd—we get:

We can reduce the fractions to lowest terms by noting that the two occurrences of  on the left-hand side cancel, as do the two occurrences of d on the right-hand side, leaving:

and we can divide both sides of the equation by any of the elements—in this case we will use d—getting:

Another justification of cross-multiplication is as follows. Starting with the given equation:

multiply by d/d = 1 on the left and by b/b = 1 on the right, getting:

and so:

Cancel the common denominator bd = db, leaving:

Each step in these procedures is based on a single, fundamental property of equations. Cross-multiplication is a shortcut, an easily understandable procedure that can be taught to students.

Use[edit]

This is a common procedure in mathematics, used to reduce fractions or calculate a value for a given variable in a fraction. If we have an equation like this, where x is a variable we are interested in solving for:

we can use cross multiplication to determine that:

SAMPLE PROBLEM

For example, let's say that we want to know how far a car will get in 7 hours, if we happen to know that its speed is constant and that it already travelled 90 miles in the last 3 hours. Converting the word problem into ratios we get

Cross-multiplying yields:

and so:

Note that even simple equations like this:

are solved using cross multiplication, since the missing b term is implicitly equal to 1:

Any equation containing fractions or rational expressions can be simplified by multiplying both sides by the least common denominator. This step is called clearing fractions.

Rule of Three[edit]

The Rule of Three^[1] is a shorthand version for a particular form of cross-multiplication, that may be taught to students by rote. It figures in the French national curriculum for secondary education.^[2]

For an equation of the form:

where the variable to be evaluated is in the right-hand denominator, the Rule of Three states that:

and then using cross-multiplication to calculate x:

Click here for practice problems

GED LESSON: Mindset and Math

FULL EDWEEK ARTICLE

"Having a positive mindset in math may do more than just help students feel more confident about their skills and more willing to keep trying when they fail; it may prime their brains to think better.

In an ongoing series of experiments at Stanford University, neuroscientists have found more efficient brain activity during math thinking in students with a positive mindset about math.

It's part of a growing effort to map the biological underpinnings of what educators call a positive or growth mindset, in which a student believes intelligence or other skills can be improved with training and practice, rather than being fixed and inherent traits.

"Our findings provide strong evidence that a positive mindset contributes to children's math competence," said Lang Chen, a Stanford University postdoctoral fellow in cognitive psychology and neuroscience. "Beyond the emotional or even motivational story of 'positive mindset,' there may be cognitive functions supporting the story."

GED LESSON: Introduction to Ratios

KHAN ACADEMY INTRODUCTION TO RATIOS VIDEO 

FULL WIKIPEDIA ARTICLE

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second.^[1] For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Thus, a ratio can be a fraction as opposed to a whole number. Also, in this example the ratio of lemons to oranges is 6:8 (or 3:4), and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers compared in a ratio can be any quantities of a comparable kind, such as objects, persons, lengths, or spoonfuls. A ratio is written "a to b" or a:b, or sometimes expressed arithmetically as a quotient of the two.^[2]When the two quantities have the same units, as is often the case, their ratio is a dimensionless number. A rate is a quotient of variables having different units. But in many applications, the word ratio is often used instead for this more general notion as well.^[3]

The ratio of numbers A and B can be expressed as:^[4]

• the ratio of A to B
• A is to B (followed by "as C is to D")
• A:B
• A fraction that is the quotient: A divided by B: , which can be expressed as either a simple or a decimal fraction.^[5]

The numbers A and B are sometimes called terms with A being the antecedent and B being the consequent.^[6]

The proportion expressing the equality of the ratios A:B and C:D is written A:B = C:D or A:B::C:D. This latter form, when spoken or written in the English language, is often expressed as

A is to B as C is to D.

A, B, C and D are called the terms of the proportion. A and D are called the extremes, and B and C are called the means. The equality of three or more proportions is called a continued proportion.^[7]

Ratios are sometimes used with three or more terms. The ratio of the dimensions of a "two by four" that is ten inches long is 2:4:10. A good concrete mix is sometimes quoted as 1:2:4 for the ratio of cement to sand to gravel.^[8]

For a mixture of 4/1 cement to water, it could be said that the ratio of cement to water is 4:1, that there is 4 times as much cement as water, or that there is a quarter (1/4) as much water as cement.