Chapter+One

**Sec 1.1**

 * Place Value
 * Place value means that each digit has a value, or a multiplier, depending on which position it holds in the number. So, for example, if you have the number 33, the 3 on the left actually has a value of 30 because it is in the “tens” place compared to the 3 on the right, which only has a value of 3.
 * Roman numerals are not place value. Binary code is.
 * When written, place value numbers are grouped by three digits, separated by commas, each group has its own name. So, in the example of 555,444,333 the fives are said to be in the “millions” group and the fours in the “thousands” group.
 * Within each group, there are ones, tens, and hundreds. So in the millions group you would have millions, ten millions, and hundred millions.
 * When said or written, we say each group separately with the group name at the end except we don’t say the ones group name. 555,444,333 would be five hundred fifty-five million, four hundred forty-four thousand, three hundred thirty-three.
 * Don’t include “and” when you say or write a place value number unless it has a decimal point. “And” means the decimal point only.
 * When you are writing out a place value number using words, like I did above, be sure to put the commas in between each group of three, just like you would if you were writing digits: 54,677
 * When you write out a number with words, put a hyphen between the words between 21 and 99. So, it looks like forty-six.
 * Whole Numbers
 * The set of whole numbers includes zero and all of the positive “whole” numbers going on forever, meaning it doesn’t include fractions or decimal numbers.
 * It is written like this: W = {0,1,2,3…} The dots, called ellipses, means that it goes on forever in that direction.

**Sec 1.2**
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 * Integers
 * Pretend like you have a numberline that has 0 in the middle and arrows and numbers going in both the positive and negative directions. If you folded the paper in half right at zero, you would find that the same digits line up with each other. So, 6 and -6 would be the same distance from zero, but in opposite directions. Numbers that are like this are called “opposites”. That means that -5 is the opposite of 5 and 7 is the opposite of -7.
 * The set of integers includes all of the whole numbers, plus their opposites.
 * The set of integers is written as I = { . . . -2, -2, 0, 1, 2 . . . } The ellipses means that the numbers go on forever in both positive and negative directions.
 * This book starts by using a raised red dash to indicate a negative number and a regular blue dash for a minus sign. This is done so that people don’t confuse the two, but in reality they are the same thing and after a couple of chapters they just quit using the raised negative sign.
 * If a number has no sign in front of it, it means that it is positive.
 * Comparing Integers
 * On a numberline, the number to the left always has a lesser value.
 * 6 < 9 -3 < 5 -7 < -4
 * Absolute Value
 * Absolute Value means how far a number is from zero. It doesn’t matter if the number is positive or negative. So, the absolute value of 6 is 6 and the absolute value of -6 is still 6. Both are six “spaces” from zero.
 * Absolute Value is written as two straight lines with a number, or something representing a number between them. |5| means the absolute value of five.
 * Absolute value only works for whatever is in between the absolute value sign. So, |-5| = 5 but -|-5| = -5 (the absolute value of -5 is 5, but the minus sign outside of the absolute value sign still applies.
 * Absolute value will always be positive, or zero.

Sec 1.3

 * Adding Integers
 * “Sum” means the answer to an addition problem.
 * It is usually easy to use a numberline to add and subtract if the numbers are small, or confusing. You move right on the numberline toward the positive end if you are adding, and left towards the negative end if you are subtracting.
 * If the numbers to be added have the same sign (both are positive or both are negative), add the absolute values of the numbers and then attach the sign that they had. So, -4 + -5 = -9
 * If the numbers to be added have unlike signs (one is positive, one is negative), subtract the absolute value of the smaller number from the absolute value of the larger one, but use the sign of the larger number. So, 6 + -4 is really 6 – 4, which is 2, and it is positive because the larger number (6) was positive.
 * If there are more than two numbers to be added, add them two at a time, left to right.
 * Commutative Property of Addition
 * It doesn’t matter which direction numbers are added, they will get the same sum.
 * A + B = B + A
 * Only works if all of the numbers are being added.
 * Associative Property of Addition
 * I t doesn’t matter how you group numbers to be added (which order you do that in), it will get the same sum.
 * (A + B) + C = A + (B + C)
 * Only works if all of the numbers are being added.

Sec 1.4

 * Opposites are the numbers that are opposite from each other if the numberline was folded in half at zero. So, -5 and 5 are opposites.
 * Subtraction, the way it is done in this textbook, becomes addition of an opposite.
 * Basically you would re-write the subtraction problem to be the addition of an opposite, then use the rules for addition from section 1.3
 * 6 – 3 becomes 6 + -3
 * The book and I suggest making a pencil stroke to change the subtraction sign into an addition sign, then making one more stroke to change the number into its opposite (6 would become -6 and -7 would become +7).
 * If the first number is negative, you could change the order and the rules would still work: -4 + 7 could be rewritten as 7 – 4 (commutative property of addition)
 * “Difference” is the term meaning the answer to a subtraction problem.

Sec 1.5

 * Rounding Numbers
 * These rules are at bottom of page 28:
 * Put a line under the place to be rounded to.
 * Look at the next digit to right of the underlined place. If it is less than 5, the underlined digit stays the same. If it is 5 or more, the underlined digit goes up one. If it was a 9 in the underlined place, it would become a 0 and you would have to carry the 1 over one more place to the left.
 * After leaving the underlined digit the same or increasing it by one, change all of the digits to the right to be zeroes.
 * Front-end rounding
 * This follows the same rules as rounding to a place value, but you always us the first (largest) place.
 * It works the same for positive and negative numbers. You just leave the sign there.
 * This is usually used for estimating answers, which is one way to check your work.

Sec 1.6

 * Multiplication
 * Different ways to show multiplication: X • 6(4) (3)(2) xy
 * “Product” means the answer to a multiplication problem.
 * “Factors” are numbers or terms being multiplied together to come up with a product. It might be described as a number or term that evenly “goes into” another number or term.
 * Multiplying Integers
 * You could use the “minus, minus, plus” triangle.
 * If the signs are the same, the answer will be positive.
 * If the signs are different, the answer will be negative.
 * Multiplication Rules
 * Anything times zero = 0
 * Anything times one is itself
 * Commutative Property of Multiplication says you can multiply in different directions. So AB = BA
 * Associative Property of Multiplication says you can multiple in groups in different orders. So: A(BC) = (AB)C
 * Distributive Property is really handy when you are working with variables (letters represent unknown values). A(B + C) = A(B) + A(C)
 * Estimating Products
 * Front-end round both numbers, then multiply.
 * Don’t multiply first and then round!

Sec 1.7

 * Signs for division: / ÷ The bar like in a fraction. The square root sign that you use to do long division.
 * “Quotient” is the answer to a division problem.
 * Dividing Integers
 * The “minus, minus, plus triangle” that worked for multiplication also works for division.
 * If you are dividing integers with the same sign, the answer will be positive. If they have different signs, the answer will be negative.
 * Anything divided by itself equals one: 3 divided by 3 = 1
 * Anything divided by one equals itself: 3 divided by 1 = 3
 * Anytime 0 is divided by anything, the answer is zero: 0 divided by 3 = 0
 * Anything divided by zero is called “undefined.” 3 divided by 0 = undefined
 * Dividing pizza among friends – If you have zero slices to divide evenly among three friends, each gets zero, so 0 divided by 3 = 0. If, however, you have three slices to divide evenly among your friends, but you have zero friends, you will never be able to get rid of the slices, so the answer to 3 divided by 0 is undefined.
 * Estimating quotients
 * Front-end round the numbers first, then divide. It will be easier to cross out an equal number of zeroes on each term before dividing.
 * Long division with remainders
 * If you have a remainder 6, for example, it doesn’t mean .6 or 6 tenths. It means 6 over 15 in this case. When we get to fractions or decimals, the remainder will then become a fraction or a decimal number instead of a remainder.
 * Round up with story problems
 * How many busses are needed to get all 631 band members to band camp if each bus holds 65 people in addition to the driver? In this case, you need to round your answer up in order to get all of the people on the bus, even if there was only a remainder of 1.

Sec 1.8

 * Exponents
 * Exponents are repeated factors, the same number or variable being multiplied again and again. 5 • 5 • 5
 * Bases are the number or variable directly in front of the exponent. It is what is being multiplied by itself.
 * Usually don’t write the first power, so 61 is just written as 6
 * (-3) squared vs. -3 squared The exponent only affect the number or variable directly in front of it unless they are in parentheses. (-3)2 means negative 3 times negative 3, which would result in positive 9 for an answer. However, -32 means 3 times 3, with a negative sign in front, so the answer is negative 9.
 * Order of operations – PEMDAS
 * Parentheses first, from inside out. Do the innermost parentheses first, then move outward.
 * Absolute value signs are also a form of parentheses.
 * Fraction bars are understood parentheses [[image:file:///C:%5CDOCUME%7E1%5Cstaff%5CLOCALS%7E1%5CTemp%5Cmsohtml1%5C01%5Cclip_image004.gif width="72" height="36"]] Do the whole top, and then the whole bottom, then divide the results at the end.
 * Exponents next, again from inside out.
 * Multiply and divide are next. These can be done in either order (multiply or divide), but must be done left to right!
 * Add and Subtract are last. These can also be done in either order, but must also be done left to right!