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Math SATs- Algebra II

Welcome to a Math SAT Preparation Lesson. In this follow-up lesson to Basic Algebra we will go over some extra algebraic topics.

An example of an important numerical concept is dealing with averages. 1. Mode- choose the value that appears most often 2. Median- choose the medium answer from the range of values 3. Mean- add everything up and divide by the number of values

There are 3 types of averages:

For example, if my set of data is: 5,6,4,6,7, the mode is 6, my median is also 6, and my mean is 5.6.

It’s important to think of another concept when dealing with averages- rounding off.

The process of rounding off is very useful, since it saves you a lot of writing (even though the SATs are multiple choice). Always remember that if your number ends with a digit less than 5, you can erase that digit- for example, ... approximately. On the other hand, if your last digit is more than 5, you have to bump up the digit ahead of it by 1. For example, ..., approximately. If your number ends with a 5, you’re in trouble (unless there are other numbers after that 5, in which case you can still bump the previous digit up). There are different conventions used, but the one I like best is this: if your digit before the 5 is even, leave it as it is. If the digit before the 5 is odd, bump it up to an even digit. For example, , but . Don’t forget to use approximations correctly, answer questions to the correct amount of decimal places, and watch for those nasty 5’s.

Don’t forget those absolute values

Absolute value are a rare form of operation you’ll see at times. All it means is the positive value of whatever you’re looking at. So , and . When working with absolute values you have to remember that the inside can be either positive or negative, so if , .

And other insane operations! Can they do that?

Yeah, they can, and they do. You’ll get at least one question with a crazy symbol,say /\, defining a new operation. For example, a/\b=(a/b)+3. Don’t freak out,just translate whatever information you’re given into equations and operations you’re familiar with and you won’t have any problem dealing with those questions.

Another concept involves consecutive integers.

You may ask yourself, for example, if there are any 3 consecutive integers out there whose sum is 18. Well, let’s call the first integer , then the next ones are clearly and . Their sum is , and setting this equal to 18 gives . Thus, our digits are 5, 6 and 7. This method is useful for dealing with sequences of consecutive integers.

Now, let’s look at rates.

A rate is a measure of change over time. For example, speed is a measure of distance over time, so you should know that (for v-speed/velocity, d-distance and t-time). So, a simple example shows that if I’m traveling 10 miles in 2 hours, I’m going at a speed of 5 mph. Rates can also be added. For example, if I write a book in 2 days, my friend Mac writes it in 3 days and my other friend Bob writes a book in 6 days, if we work together we can write a book faster. But how much faster? Well, my rate is , Mac goes and Bob is , so together we have a rate of 1 book a day, thus together writing a book will take us one day.

When looking at such rates, it is important to remember conversions between such things as minutes and seconds, miles, feet and inches, etc. It is also important to remember directions- for example, if 2 objects are moving in perpendicular directions, the distance between them at a certain point of time can be calculated using the regular distance formula together with the Pythagorean Theorem for a right-angled triangle, the square of the hypotenuse equals the sum of the squares of the perpendicular sides.

Finally, rationality of numbers is an additional important concept.

While it may seem as if algebra is composed of mostly nice numbers, this is an illusion. Numbers that can be represented using fractions and decimals are called rational numbers, and are quite nice and easy to work with. However, some numbers, such as the square root of 2 and the well-known Pi cannot be represented with any sequential decimal order or a fraction. Such numbers are called irrational, and they have various different uses.

Note that these topics are more challenging than basic algebra ones and require slightly more thinking. So, if you think you’ve mastered your algebra, prove it by taking on the Algebra II test.

For a lot more algebra, check out the Welcome to Algebra Series in the Algebra community:

Part I

Part II

Part III

Part IV