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Absolute Value ABSOLUTE VALUE-- A number’s distance from zero. The absolute value of any number is always positive. 1. Absolute value is represented like this: | -4| = 4 and is read “the absolute value of negative four is four” meaning that negative four is four spaces away from zero. 2. For example: a. | -27| = 27 b. | -50| = 50 c. | 10| = 10 3. So…if you put the absolute value of these numbers is order from least to greatest you would get: 10, 27, 50 BUT if you put the numbers themselves in order it would be quite different: -50, -27, 10 Integers and Intervals o o o o o INTEGER – positive and negative whole numbers and zero. In other words: the counting numbers. This does not include decimals or fractions! (ie: is 14.5 an integer?) INTERVAL – a “piece” of the set of integers (show how to graph and write integers, what numbers the interval includes) My interval – timeline example (ie: [6 years old, 12 years old] I played softball) Notecard 1 Multiplication and Division of Integers Math notes: quick review of rules for multiplication and division of integers. Go over in detail adding and subtracting integers. -MULTIPLICATION AND DIVISION – for these two operations the rules are the same!!!! -Positive x/ positive = positive -Negative x/ negative = positive -Negative x/ positive = negative -ADDITION AND SUBTRACTION -The rules for these are more complicated…but lets begin by always reading these equations like a sentence (left to right) -First remember: a positive/plus sign indicates moving to the right on the number line-a negative/minus sign indicates moving to the left on the number line Notecard 2 on the Multiplication and Division of Integers HOWEVER--- when you are reading your sentence and you see a negative number to the right of your operation sign, it is like a “u turn” sign telling you to move the opposite way on the number line. 5+3=8 But---5 + -3 = 2 TRICK: with subtraction you can change the subtraction sign to an addition sign and change the number to the right of the operation sign to its opposite. Ie: 5 + -3 = 2 is the same as 5 – 3 = 2 Subtraction of POSITIVE AND NEGATIVE INTEGERS!!!!!!!! **Subtraction of positive and negative integers no longer exists!!!! 1. 2. change the subtraction sign into addition change the number to the right of the operation sign (the subtraction sign you turned into an addition sign) into its’ opposite. Ex: 5 – 4 = 1 is the same as 5 + (-4) = 1 Addition of Positive and Negative Integers If the numbers are the same sign, keep that sign for your answer and add the numbers. Ex: -6 + (-3) = - 9 or 8+3 = 11 1. 2. If the signs are different keep the sign of the number with the larger absolute value for your answer and subtract. Ex: -6 + 12 = 6 or -20 + 10 = -10 **A little song to help you remember!! Sing this to the tune of “Row Row Row Your Boat”: “Same sign add and keep, different signs subtract. Take the sign of the higher number then it will be exact!” Absolute Value ABSOLUTE VALUE-- A number’s distance from zero. The absolute value of any number is always positive. -Absolute value is represented like this: | -4| = 4 and is read “the absolute value of negative four is four” meaning that negative four is four spaces away from zero. For example: b. | -27| = 27 c. | -50| = 50 d. | 10| = 10 So…if you put the absolute value of these numbers is order from least to greatest you would get: 10, 27, 50 BUT if you put the numbers themselves in order it would be quite different: -50, -27, 10 PEMDAS This is an acronym for the order of operations. Order of operations is the order in which you complete problems with more than one operation. We will go deeper into this next week but for now know what each letter stands for: o P – parenthesis o E –exponents o M – multiplication o D – division o A – addition o S – subtraction Exponents Exponents are used to represent REPEATED MULTIPLICATION Here is how to write in exponential form: 3^5 3 is the base 5 is the exponent/ power Expanded form—writing it out 4^3 = 4 x4 x 4 = 64 Special case!!! **anything raised to the zero power is one Word Problem Strategies 1. 2. 3. 4. 5. 6. Read through at least once to get the main idea. What is the problem about? What is it asking? Read it a second time, underline important numbers and units. Identify what you are looking for Write out the key information you need to use in order to solve Determine what operation to use. How do we determine which operation to use? Set up an equation/expression to solve the problem Patterns and Sequences Example 1: Find the pattern: 1, 3, 6, 10, 15…. What is the rule? (add 2, add 3, add 4, add 5…etc.) What comes next? 21, 28… Example 2: Find the pattern: -20, -15, -10, -5, 0… What is the rule? (Add 5) What comes next? 5, 10… A sequence is an ordered set of numbers. When these numbers change by the SAME AMOUNT EACH TIME it is called an arithmetic sequence. Example 1 is NOT an arithmetic sequence while Example 2 is. Each number in a sequence is called a term. Variables and Algebraic Expressions Variable -- a letter or symbol that represents a quantity. Ex: x, y, z, #, b, etc. Algebraic Expression – and expression that contains one or more variables Ex: a+b 3d + 4x Expression vs. Equation An expression does not have an equal sign. An equation has an equal sign. Translating Words into Math Addition (combine) o Add, Plus, Sum, Total, Increased by, More than Subtraction (less) o Minus, Difference, Subtract, Less than, Decreased by, Take away Multiplication (put together groups of equal parts) o Product, Times, Multiply Division (separate into equal groups) o Quotient, Divide Mean, Mode, Median, Range Mean -- the average of a set of numbers 1. add al l the numbers in the set 2. divide the total by the amount of numbers in the set Mode – the number in the set that appears most often Range – the difference between the highest number and the lowest number in the set Median -- the number in the middle of the set 1. Order the numbers in the set from least to greatest 2. Take the number in the middle as your median 3. When the amount of numbers in the set is odd, this is easy. Ex. 4, 5, 7, 8, 19 the median of this set is 7 However, when the amount of numbers is even you must follow these rules: a. Order the numbers in the set from least to greatest b. Take the two numbers in the middle and find their average. The average of these two numbers in the median. Ex. 4, 5, 7, 9, 10, 20 7+9=16 16/2=8 the median of this set is 8 Integers: positive and negative whole numbers and zero Rational numbers: any number that can be expressed as a ratio of two integers. Ratio: comparing two quantities (we haven’t talked too much about this so don’t worry) So your thinking…what does that definition of rational numbers mean?! Basically---we’re talking about fractions and decimals people! 2 ,4 1 3 ,3 1 2, 3 4 5 . 6 7 8 9 Rounding Review When rounding we look at the number to the RIGHT of the number we are rounding to. If it is greater than or equal to 5, we round up. If if is less than 5, round down! Ex. 145.1 round to the nearest ten…… 150.0 Adding and Subtracting Decimals ALWAYS REMEMBER: When adding and subttacting decimals always line up the decimal points!! If there is a blank spot, insert a zero as a place holder. Money Baby! *Dealing with money is dealing with decimals! $money decimal 5 cents……………………………….0.05 20 dollars……………………………20.00 50 cents……………………………….0.50 2 dollars and 50 cents………………2.50 Multiplying Decimals Multiply the same as when you are using whole numbers! You do not have to line up the decimal points To place the decimal point add the number of decimal places in each number you are multiplying and move the decimal point that many places to the left in your answer. o Ex: 0.17- 2 decimal places X 3 0 decimal places 0.51 2 decimal places Dividing Decimals Dividend: the number to be DIVIDED Divisor: the number you are dividing by (the evil villain that chops numbers into smaller pieces) Step 1: write out the problem in long division form Step2: move the decimal point of the divisor right until you have a whole number Step3: move the decimal point of the dividend right the same number of positions! Step4: perform long division as usual, bringing the decimal straight up! Scientific Notation Scientific notation: a method for writing very LARGE or very SMALL numbers. **Scientific notation is used by both mathematicians AND scientists! (think of how big numbers get when talking about astronomy or cell biology for example) Scientific notation uses exponents, namely: powers of ten. What are powers of 10? 101 = 10 102 =100 103 = 1,000 104 = 10,000 Multiplying and Dividing by powers of 10 When multiplying by a power of ten, move the decimal point to the RIGHT the same number of places as there are zeros in the power of ten. When dividing by a power of ten, move the decimal point to the LEFT the same number of places as there are zeros in the power of ten. Synthesizing info for Scientific Notation Exponential Form 102 10-2 Expanded Form 10X10 1 = 1 102 100 Solution 100 0.01 Positive exponents make the number BIGGER. The decimal point moves to the RIGHT. Negative exponents make the number SMALLER. The decimal point moves to the LEFT. Let’s put what we know together so solve some problems in Scientific Notation: 6.0 x 103 = 6.0 x 1000 = 6,000.0 Positive exponents. The number being multiplied by the power of ten gets LARGER. The decimal point moves to the RIGHT. The exponent dictates how many zeros will be in the solution to the exponential form. The exponent also dictates how many places to the RIGHT the decimal point will move. 6.0 x 10-3 =6.0 x 0.001 = 0.006 Note the negative exponent. The number being multiplied by the power of ten gets SMALLER. The decimal point moves to the LEFT. The exponent dictates how many spaces behind the decimal there will be. The exponent also dictates how many places to the LEFT the decimal point will move (multiplying with negative exponents is like dividing with powers of ten!) Divisibility A number is divisible by another number if their quotient is a whole number with no remainder. o Ex. 46 is divisibly by 2 and 7 because: 42 ÷ 2 = 21 (whole number, no remainder) 46 ÷ 7 = 6 (whole number, no remainder) Any number greater than one is divisible by at least two numbers: one and itself. Numbers that are divisible by more than two numbers are called composite numbers. A prime number is divisible by only one and itself. Divisibility Rules A number is divisible by: 2 if the last digit is even 3 if the sum of the digits is divisibly by 3 4 if the last two digits form a number that is divisibly by 4 5 if the last digit is 0 or 5 6 if the number is divisible by both 2 and 3 9 if the sum of the digits is divisibly by 9 10 if the last digit is 0 Factors Whole numbers that are multiplied to find a product are called FACTORS of that product. (all the numbers that can be multiplied together to get a certain number) o Ex. 2 x 3 = 6 o 2 and 3 are factors of the number 6 o All the factors of 6 are: 1,2,3,6 A number is DIVISIBLE by its factors. o 6÷3=2 o 6÷2=3 o o o o PRIME FACTORIZATION – a number written as a product of its prime factors. You can use factors to write a number in different ways. The factorization of 12 = o 12 x 1 o 2x6 o 3x2x2 o 3x4 Note that 3 x 2 x 2 is the PRIME FACTORIZATION of 12 because it uses all prime factors to represent 12 1. 2. 3. 4. FACTOR TREES! Choose any two factors of the number you are factoring. Keep finding factors until each branch ends in a prime number The prime numbers you are left with are called the roots of the number you were factoring. Using exponents, write the number you were factoring as a product of its prime factors or roots. Fractions and Decimals fractions and decimals are known as RATIONAL NUMBERS. Decimals and fractions are often used to represent the same number. Ex. ½ = 0.5 Ex. 5/100 = 0.05 Decimals can be written as fractions OR as MIXED NUMBERS. MIXED NUMBER -- a number that contains and whole number and a fraction. Ex: 3 ½ Writing Decimals as Fractions Note Card 1 Identify the place value of the digit farthest to the right. Use that place value to determine the denominator. Decimal Fraction 0.23…………………..23/100 (twenty-three hundredths) 1.7…………………..1 7/10 (one and seven tenths) 0.004…………………..4/1000 (four thousandths) 1. 2. Writing Fractions as Decimals 1. It’s simply a division problem!! Ex. 1/5 = one divided by five = 1 ÷ 5 The numerator is the dividend The denominator is the divisor Writing Decimals as Fractions Note Card 2 Numerator Denominator Dividend Divisor Fraction Decimal ¾………..3 ÷ 4 = 0.75 5 2/3……2 ÷ 3 = 5.66 (with mixed numbers keep the whole number the same) TERMINATING DECIMAL – has a terminating (ending) number of decimal places. REPEATING DECIMAL – has a block of one or more digits that repeat continuously. Steps for REDUCING fractions: If the numerator and the denominator are both divisible by a common number, the fraction can be reduced. Find a common factor and divide both numerator and denominator by that factor Continue this process until the numerator and denominator have no common factors or until you get a prime number for the numerator or denominator Improper Fractions Improper Fraction-- a fraction where the numerator is bigger then the denominator. It is often best to convert improper fractions into mixed numbers! Step 1: Divide! Divide the numerator by the denominator. Use the answer for the whole number in the mixed number Step 2: Use the remainder as the numerator in the fraction of your mixed number. Step 3: Use the same denominator as the improper fraction Ex: Convert the improper fraction 16/5 into a mixed number= 3 1/5 Sometimes you’ll have to convert mixed numbers back into improper fractions as well!!! Ex: 5 1/3 = 16/3 LCM = Lowest Common Multiple The multiples of a number are what you get when you multiple it by other numbers. Just like the multiplication table. Multiples of 4: 4, 8,12, 20, 24 Multiples of 5: 5, 10, 15, 20, 25 The LCM of 4 and 5 would be 20, the smallest common multiple. The LCM becomes your common denominator when doing operations with fractions. Once you have a common denominator you can combine fractions! Yes! 2/3 + ¼ = 3/12 Converting Fractions To convert the fractions you must find what number you multiply the denominator by to get the denominator and multiply the numerator by that same number, giving you your new numerator. Step 3: Add or subtract the numerators. Step4: Reduce! Ordering Fractions and Percents Ordering Fractions: Strategies Draw a picture, picture it in your mind…if this works for you: GREAT! Find a common denominator Convert the fractions into decimals Convert the fractions into percents Percents Percent means “per one hundred” Ex. 8% means “8 per 100” or “8 out of 100” Percents *Because percent means “out of 100” 100% is 100 out of 100. That’s why 100% means “all” or “the whole thing.” If 24% means “24 out of 100”… As a fraction it would be: 24/100 As a decimal it would be: 0.24 Writing Decimals as Percents Method 1: Write the decimal as a fraction Convert the fraction to have 100 as the denominator Write the numerator with a percent sign Ex. 0.3= 30/100 30% Method 2: Multiply by 100 Ex. 0.3 x 100= 30 30% Writing Fractions as Percents Method 1 Convert the fraction to have 100 as the denominator Write the numerator with a percent sign Ex. 4/5 = 80/100 80% Method 2 Convert the fraction into a decimal Multiply by 100 Ex. 4/5 = 0.8; 0.8 x 100= 80 80% Ratios Intruments in the Baltimore Symphony Orchestra Violins 29 Violas 12 Cellos 10 Flutes 5 Total number of instruments: 56 You can copare the different groups using RATIOS. RATIO: a comparison of two quantities* using division. Think: ratios are like “fancy fractions” *quantity: amount EQUIVALENT RATIOS: Ratios that name the same comparison. You can find equivalent ratios by multiplying OR dividing both terms of a ratio by the SAME number (think: EQUIVALENT FRACTIONS) Proportions and Rates What is the scientific formula for water? H2O -This equation means there are two atoms of Hydrogen and one atom of Oxygen…when chemically bonded they make one molecule of water. What is the RATIO of hydrogen to oxygen? 2:1, 2 to 1, 2/1, two hydrogen atoms to one oxygen atom *No mater how many water molecules of water----hydrogen and oxygen will always be in the same RATIO---- 2:1 Using your knowledge of ratios and equivalent fractions, fill out the following chart: #of Water 1 2 3 4 Molecules Ratio: Hydrogen 2 Oxygen 1 *notice that all four of these fractions or RATIOS are equivalent! Proportion: an equation that shows two equivalent ratios. **Just like EQUIVALENT FRACTIONS! Using Percents! Using percents is a snap—PERCENTS ARE PROPORTIONS! % = IS 100 OF Ex: What is 12% of 50? What is 12% is 30? 12% of what is 10? 12% of 16 is what? Multiplying and Dividing Fractions!! The formula for multiplying fractions is easy: Multiply the numerators Multiply the denominators Remember to reduce! EX: 1/3 × 3/5 Multiply the numerators: 1 × 3 = 3 (3 is your numerator) Multiply the denominators: 3 × 5 = 15 (15 is your denominator) So… 3/15 is your answer…reduced, that’s 1/5 *if you are multiplying with mixed numbers…. you must convert them to improper fractions before multiplying!!!!!!!!!!! Dividing Fractions Note Card 1 With dividing fractions we have to know what a reciprocal is: Two numbers are reciprocals if their product is 1. To find the reciprocal of any fraction, switch the numerator and the denominator…FLIP IT! EX: the reciprocal of ¾ is 4/3 and ¾ × 4/3 = 1 Just like with multiplying fractions, you must first convert any mixed numbers into improper fractions!!!! Change the problem into a multiplication problem Find the reciprocal of the second number in the equation and multiply by the reciprocal Dividing Fractions- Note Card 2 Ex: ¾ ÷ ½ First rewrite the problem as a multiplication problem: ¾ × ½ Now find the reciprocal of the second number: the reciprocal of ½ is 2/1 Use the reciprocal to multiply: ¾ × 2/1 = Multiply the numerators: 3 × 2 = 6 (6 is the numerator) Multiply the denominators: 4 × 1 = 4 (4 is the denominator) The answer is 6/4, reduced is 1 ½