AHSGE NOTES Standard 1 – Objectives 1-4 The following slides have teacher’s notes and examples for understanding Standard 1, Objectives 1,2,3 and 4.

FOUNDATION INFORMATION To solve any problems, you must first have some background or foundation information to help you solve problems. Without this understanding, solving any problems will be difficult.

Standard 1-1 Order of Operations Foundation Information Absolute value of a number is the distance from zero – the distance is always positive. So the absolute value of 6 or -6 will be 6.

Absolute Value continued When simplifying problems with an absolute value sign – you must always take the absolute value first before solving the problem. However, if there is a mathematical operation inside the absolute value, you must first complete the operations(s), THEN take the absolute value of the answer. Remember the answer will be positive due to the absolute value sign.

Absolute Value Examples |-9| = 9 |8| = 8 | 2+ 3 | = |5| = 5 |7 – 50| = | -43| = | 7 – 12| = 2 + |-5| = = 7

Adding and Subtracting Integers Allow it is tempting to add and subtract numbers in your head, don’t do it! Use the calculator given for you to use on the grad exam. Too many minor mistakes can be made when doing math in your head and those minor mistakes becomes MAJOR deduction in points and may cause you to fail the test. So USE THE CALCULATOR!!!!!

Putting numbers into the Calculator When you put a negative number into the calculator, you first must input the number then make it negative. If you put the negative into the calculator first then the number, the negative sign will disappear and you will miss the problem on the test. Sometimes the authors of test will put a negative number in parentheses to draw your attention to the negative number so you will input it into the calculator correctly.

Examples (-2) + (-3) = -5 Calculator key strokes: 2 negative + 3 negative = 2 – ( -2) = 4 Calculator key strokes: 2 – 2 negative =

Multiplying and Dividing Integers Rules to remember: WHEN THE SIGNS ARE THE SAME, THE ANSWER IS POSITIVE. Positive X Positive = Positive Negative X Negative = Positive WHEN THE SIGNS ARE DIFFERENT, THE ANSWER IS NEGATIVE. Positive X Negative = Negative Negative X Positive = Negative

Examples 6 x -8 = x -5 = ▪ ∕ ▪ 4 = ▪ ∕ ▪ -4 = ▪ ∕ ▪ 4 = -3

Exponents A common mistakes students make is to multiply the base and the exponent to find the answer. 2⁴ 2 is the base and 4 is the exponent. Write the base then number of time represented by the exponent. 2 x 2 x 2 x 2 2⁴ = 16 not 8 why? 2⁴ = 2 x 2 x 2 x2 = 4 x 2 x2 = 8 x 2 = 16

Exponents continued (-3)⁴ = -3 x -3 x -3 x -3 = 81 Unlike a scientific calculator, you must put each number in, so be careful and work the problem, one step at a time.

Square Roots √ 49 = 7 or ±7 depending on the problem. If you are asked to find the √ 49 that is a single answer of 7. If you are asked to find the √ 49 in an equation, that is a double answer of both 7 and -7. Calculator steps: 49 √ You do not need to push =

A trick to solving complex square roots If you push the √ key and you see a decimal instead of a whole number that means the answer is not a simple number and the directions do not state to round, what do you do???? Look at the answers for help!Work smart not hard on this part of the test and let the answers work for you!!!!!

Examples of Square Roots √99 A. 9 √11B. 3 √33 C. 3 √11 Take the number outside the radical sign and square it, then multiply that answer by the number under the radical sign. A. 9 x 9 x 11 = 891 not 99 B. 3 x 3 x 33 = 297 not 99 C. 3 x 3 x 11 = 99 that is the answer!

Order of Operations To solve problems correctly, use this phrase to help you--- Please Excuse My Dear Aunt Sally Work all problems left to right. When there is both multiplication and division, work the operation that appears first working left to right. The same is for addition and subtraction.

Examples of Proper Order of Operation 4³ + 3 (4 + 6) – 3 ▪ 2 = ??????? 4³ - 3 (4 + 6) + 3 ▪ 2 4³ - 3 (10) + 3 ▪ (10) + 3 ▪ ▪

What if you have a fraction? Don’t let fractions scare you. You can work a fraction problem by doing ONE thing at a time. First solve the numerator – that is what on top of the fraction. Second solve the denominator – that is what on the bottom of the fraction. Last reduce, if possible the fraction. See the next slide for an example.

Fractional Example of O of O ² - 3 2² + 2(5-3) Solve the numerator (top) : ² - 3 = = 38 Solve the denominator (bottom): 2² + 2(5-3) = (2)= = 8 Simply 38/8 by reducing each by 2 Answer is 19/4

I hope this has helped you. This is the end of Standard 1 – 1 Order of Operations. Go back to the AHSGE page on my wikispace to see Standard 1 -2 Add and Subtract Polynomials.