# All you have to do is THINK!

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1 Mathematician s Notes BE A MATHEMATICIAN! All you have to do is THINK! Imagination is more important than knowledge. Albert Einstein We only think when confronted with a problem. John Dewey It s not that I m so smart it s just that I stay with problems longer. Albert Einstein Let s talk about math Math is about thinking! It is what mathematicians do first when presented with a problem to solve. Almost all of the problems that mathematicians solve begin with a real world problem to solve. You might think of them as word problems. Mathematicians think of word problems as world problems. The best tool you have for solving math problems is your brain! You can use your brain to think flexibly about math problems and whether your solutions make sense. There are many ways to solve problems. You need to be able to choose a way that works for you and you need to be able to explain and justify your method to others. Understanding how others solve math problems gives you more tools to solve them. As you become more skilled at solving problems, you will begin to choose the most efficient way to solve problems. Many people have been solving problems for a long time and so algorithms have developed for solving math problems. Algorithms are stepbystep procedures for solving problems. Once you understand why an algorithm works, you may want choose it as the most efficient way to solve the problems, but you may also have an efficient way that works for you better, depending upon the problem situation. Being able to choose different methods makes you a skilled mathematician, which prepares you to function in, and contribute to, the society in which you live. Math is everywhere in your life. It is beautiful, puzzling, and logical all at the same time. You can be a successful mathematician, even if you struggle with it. We learn and grow by engaging in and overcoming difficulties! You can be Mathematician! All you have to do is THINK! Math Notes Page 1 of 1

2 Mathematician s Notes It s all in the numbers: The Real Numbers! Pictures from Mathis fun.com. Number: a count or measurement They are really an idea in our minds. We write or talk about numbers using numerals such as "5" or "five". We could also hold up 5 fingers, or tap the table 5 times. These are all different ways of referring to the same number. There are also different types of numbers, such as whole numbers (1,2,3) decimals (1.48, 50.5), fractions (1/2, 3/8), and more. Numeral: A symbol or name that stands for a number. Examples 3, 49, and twelve are all numerals. Digit: A symbol used to make numerals. 0,1,2,3,4,5,6,7,8, and 9 are the ten digits used to make numerals. Types of Numbers: The Counting Numbers: the counting numbers have been around for thousands of years. We can use numbers to count: { } Zero: the idea of zero was not natural to early humans, but it represents a quantity, so it needed to be added to our number system. Not only does it represent a quantity, but it is a placeholder. 5 2 means 502 (5 hundreds, no tens, and 2 units). If you didn t have the zero to hold the place of the tens, then this number might look like 52. Whole numbers: include the Counting Numbers and Zero: Whole numbers: { } Negative Numbers: Around the 16 th century, mathematicians decided they needed negative numbers in order to make their algebraic solutions work. So, counting backwards from zero gives us negative numbers. A number less than zero is a negative number. Sometimes it s hard to understand how numbers can be negative. A simple example is temperature. We define zero degrees Celsius (0 C) to be when water freezes... but if we get colder we need negative temperatures. So 20 C is 20 below Zero So, negative numbers along with whole numbers become called Integers. Math Notes Page 2 of 2

3 Mathematician s Notes Vertical Number Line Integers Integers are the whole numbers and their opposites. The Integers include zero, the counting numbers, and the negative of the counting numbers, to make a list of numbers that stretch in either direction indefinitely. Zero is neither positive, nor negative and Numbers are opposites if they are the same distance from zero on either side of a number line. and When you add a number to its opposite, the sum is zero. A number line is helpful when operating with integers. Sometimes a horizontal numberline makes sense, but sometimes a vertical numberline makes more sense to use. Choose what is best for the problem you are trying to solve. Horizontal Number Line Rational Numbers: Numbers that can be written as a fraction. They include all the integers and all fractions. another category for numbers. If you have one orange and want to share it with someone, you need to cut it in half. You have just invented a new type of number! You took a number and divided it by another number to come up with. So now we need * *Remember the fraction bar means division. You cannot divide a number by zero. Since we now write division problems as: b cannot be zero. Irrational numbers: Irrational numbers are numbers that are not rational what? If the definitions above don t fit the number, then it is irrational. Irrational numbers: Numbers that are decimal fractions (decimals) that don t repeat in a pattern, or don t ever end. Nonperfect square roots: i.e. the π Is the symbol for Pi, it never ends or repeats We ll learn more about Pi in middle school, but we generally think of Real Numbers: All of these types of numbers make up the big category of real numbers. Imaginary Numbers: What?... Yes, there is a set of numbers called imaginary numbers, but we re going to wait until high school to learn about those Math Notes Page 3 of 3

5 Additive Inverse Property: A number plus its inverse, has a sum of zero. In addition a number s inverse is its opposite. The operation of Subtraction is the inverse of the operation of Addition. That means it undoes addition. This property allows us to subtract numbers. With Arithmetic: Vertical Number Line Or, you can offset the negative number by putting a parenthesis around it so it s easier to see. With Arithmetic: Opposites are numbers that are the same distance from zero on either side of a number line. Sometimes a horizontal number line makes sense to use, however sometimes a vertical number line makes more sense to use. You ve been moving back and forth on a number line since Kindergarten, it s the same here. Choose what is best for your situation. Horizontal Number Line The opposite of and the opposite of The Additive Inverse Property allows us to subtract REAL MATHEMATICIANS DON T SUBTRACT, THEY ADD THE OPPOSITE. The additive inverse property shows that subtraction gets the same result as adding the opposite. Remember: and Example: 4 positives take away 3 positives equals 1 positive And Example: 4 positives plus 3 negatives equals 1 positive 0001=1 Cancel We DON T CANCEL We MAKE ZEROES Math Notes Page 5 of 5

6 Multiplicative Inverse Property: Any number multiplied by its reciprocal equals one. This property allows us to divide numbers. 1 (any number divided by itself equals 1) ab ab With Arithmetic: 1 1 = (any number divided by itself equals 1) Reciprocal: A number multiplied by its reciprocal equals one. The reciprocal is the multiplicative inverse of the number. Examples of Reciprocals: and, because and 1 1 = *(any number divided by itself equals 1) *Some people call this canceling a number But it s not cancelling! It s making a one! And, because 4 groups of is 1. 1 Or, 1 1 = (any number divided by itself equals 1) We DON T CANCEL We MAKE ONES Cancel The multiplicative Inverse Property allows us to divide REAL MATHEMATICIANS DON T DIVIDE, THEY MULTIPLY BY THE RECIPROCAL. (In Middle School, this is how we write division) Cancel is a bad word to say in math class! You say that you are making a one or making a zero, but don t say that bad word! Cancel Math Notes Page 6 of 6

7 Zero Product Property: The product of any number and zero is zero With Arithmetic: And, this might seem like duh, but you are going to want to know about this for future algebra classes: If a times b is zero, then either a equals zero, or b equals zero So, if ab=0, then either a=0 or b=0, or both =0 So, if then because Can we talk about dividing by zero? It is impossible to divide a number by zero. If you try to use a calculator to divide by zero, it will get all upset and give you an E for error. Think! About it! Let say the problem is:, as we say in middle school. This means you have 8 pencils and you are going to divide them (pass them out) between 4 friends how many does each friend get? Well, duh, 2., as we say in middle school. This means you have 0 pencils and you are going to divide them (pass them out) between 4 friends how many does each friend get? Well, duh, 0. But., as we say in middle school. This means you have 4 pencils and you are going to divide them (pass them out) between 0 friends how many does each friend get? Well, uhhhhh..? You can pass out 4 pencils to zero people. Got it? Good! Math Notes Page 7 of 7

11 Absolute Value: The absolute value of a number is the measurement of its distance from zero. It tells how far a particular number is from zero. For Example: the number is 6 away from zero, but is also 6 away from zero, just on the other side. So the absolute value of and is 6. Since you don t measure distance in negative amounts, you just say that it is 6 away from zero, not away from zero. These are absolute value symbols. They are two vertical bars on either side of a number. so, and So, if you are using absolute value to add and subtract integers, you find the difference between the absolute values (ignore the signs) and then give the answer the sign of the number with greatest absolute value. Example: Change subtraction to addition of the opposite. There are more negatives than positives, the answer will be negative. Find the absolute Value: and The difference between 6 and 5 is 1 because There are more negatives than positive numbers, so the answer is negative. and The difference between 6 and 5 is 1 because So, because there are more negatives, the answer will is So, So, because there are more negatives, the answer will is So, With Tile Spacers: 5 positives combined with 6 negatives Equals You can also pair numerals to make zeroes 5 5 = 0 so, 5 5 With Tiles spacers would be really timeconsuming. So Try something like this: Same Result Take 16 negatives from the 36 negatives and pair them with the 16 positives add to the 20 remaining negatives to make a zero. Then or plus BAD WORD ALERT! Some people say cancel here, but really we are MAKING ZEROES Math Notes Page 11 of 11

12 Multiplying and Dividing Integers: Remember 3 rd grade? That s when you learned how to multiply. You learned that 4 groups of 3 pencils gave you a total of 12 pencils. Or:. And, if you had 12 pencils and divided them between you and 3 friends you would each get 3 pencils. Or:. You may remember that multiplication and division are inverse operations, they undo each other. Well, multiplication and division of Integers follows the same structure as multiplying and dividing whole numbers. Multiplication by a positive integer can be represented by combining groups of the same number: In both examples, the 4 indicates the number of groups of 3 (first example) and 3 (second example) to combine. and Division is the Inverse of Multiplication, so we can follow the patterns and divide. See if you notice any patterns when using numerals instead of tile spacers. If: Then: Multiplication by a negative integer can be represented by removing groups of the same number: Modeling with tile spacers: If you begin with an equal number of positive and negatives, that gives you a neutral field. So let s say you have 12 zeroes. If: Then: means remove four groups of 3. If you removed four groups of 3 positives, you would have 12 negatives left. So, If: Then: means remove four groups of 3. If you removed four groups of 3 negatives, you would have 12 positives left. So, If: Then: In all cases, if there are an even number of negative factors to be multiplied, the product is positive; if there are an odd number of negative factors to be multiplied, the product is negative. This pattern also applies when there are more than two factors. Multiply the first pair of factors, then multiply that result by the next factor, and so on,until all factors have been multiplied. =12 Math Notes Page 12 of 12

13 Making Ones, Giant Ones! Making ones is a huge part of being flexible with numbers and being successful in higher level Mathematics. The Identity Property of one and the equality properties help you make ones! Some people say the word cancel when they really mean that they are Making Ones. The Many Uses of the Giant One: Cancel 1) Renaming Fractions for Simplifying when Adding: (Equivalent fractions and Simplified Fractions) You can multiply a fraction, or any number, by one and the result is that same fraction, or number a) Example of the giant one when renaming fifths to something in tenths: Divide both denominator and numerator by the same divisor b) Example of the giant one when simplifying fractions: Multiply both denominator and numerator by the same factor c) Example of the giant one when changing mixed numbers to fractions: = d) Example of the giant one when changing fractions to mixed numbers: Math Notes Page 13 of 13

14 e) Example of the giant one when adding fractions: (This is for subtracting fractions too, but that is really adding the opposite of a number so, it s still adding.) Add: When adding fractions with different denominators, you can rename fractions with the same denominator. The Least Common Multiple (LCM) between 4 and 3 is 12, so you need to rename the fractions into twelfths by making equivalent fractions. Renaming fractions into the same units, makes it easier to talk about the total fraction... is easier to talk about than to say something like... h d 2) Example of the giant one when dividing exponents: some people call this canceling (which you already know is a bad word), but really we are making ones. Simplify: Cancel 3) Example of the giant one when solving algebraic equations: Divide both sides by 2 Multiply both sides by the reciprocal ( ) Simplify ( ) ( ) Simplify Simplify Simplify 9 Math Notes Page 14 of 14

15 Three Uses and Meanings of the Minus Sign: Many students get confused with problems which involve a minus sign. The first step is to figure out how the minus sign is being used and what it means in an expression or an equation. 1. Subtraction Operation: The problem is asking you to find the find the difference between two numbers. In other words, the problem is asking you to subtract one number from another. You can still think of subtraction as take away. (It s what you learned in first Grade) 6 take away 1 is 5 2. Negative number: Shows the location of a negative number on the number line. (It s what you learned in fifth Grade) Smaller numbers on the left Larger numbers on the right The dot identifies 3. The Opposite of.: This means the opposite of a number or quantity. It is sometimes the most difficult to figure out. Opposite numbers are the same distance from zero on a number line. With Words: o (You are going to learn more about this in middle school) The opposite of two is negative two With Symbols: o The opposite of ten is negative ten or o The opposite of is negative or Math Problems With the Minus Sign You Will See in Middle School: Problem With symbols Problem With Words Use /Meaning of the minus sign Or Five negatives take away one negative is four negatives. The negative number can have parenthesis, or not, these equations are equal. Three positives plus two negatives is one positive The opposite of.. negative two is two Subtraction Operation Identify a negative number First Minus sign: Opposite of Second Minus Sign: Identify a negative number Math Notes Page 15 of 15

16 Three Uses for a Variable that you ll see in Middle school: Variables are not just letters. They are symbols, or letters, that are used to represent quantities, either unknown or varying (changing). They have many different meanings depending on how they are used and what is the purpose. A variable can be a specific number, it can be used to explain (generalize) a pattern (a relationship between varying quantities), or a variable can be used as a placeholder for a formula. Here s some vocabulary you will need to know to understand how variables work: Variable: A quantity that can change, or that may take on different values. A letter or symbol representing such a quantity Quantity: Something that is measured. It can be: An exact number A number that varies (changes) Two quantities that vary (or change) together simplify: combine like terms as much as you can (adding, multiplying etc.) evaluate: substitute a number in for a variable and simplify the equations Equation: a mathematical sentence using an equal sign to separate two expressions You solve equations solve: find the value of the variable Expression: Common Variable Uses in Middle School: Unknown: Solving for one specific quantity a mathematical calculation using Generalize: To explain a pattern that works for numbers and/or variables, or other any stage in the pattern, and can mathematical operations predict future stages. don t have equal signs or an inequality Placeholder: Variables used in proven symbol mathematical formulas. You simplify or evaluate expressions. How are variables used? 1. As an unknown quantity: A variable is a letter, or other symbol, that is used to represent a specific number. An algebraic equation can be written and solved for one specific answer. Example: or This equation says that some quantity plus 2 is 10. The only answer in this case can be To generalize (explain) a pattern or situation: A variable can be used to show the relationships in a pattern that varies (changes). Example: Let s say that you are going to raise money for your soccer club by having a car wash. You are going to earn \$5 for each car that you wash. You are going to have to spend (a minus quantity) \$40 for supplies such as sponges, soap, towels, window cleaner, etc. Math Notes Page 16 of 16

17 You can write an expression to show how you might predict how much money you could earn, depending upon how many cars you wash. Quantities in the expression: Expression: This expression can be used to say \$5 times the number of cars washed, minus the \$40 for supplies will tell you the money earned for your soccer club. You can evaluate the expression for any number of cars you wash: If you wash 30 cars, you would substitute 30 for c in the expression and simplify the expression do the math Substitute Calculate multiplication You earn: Simplify If you wash 45 cars: You earn: Substitute Calculate multiplication Simplify You can use an equation to figure out how many cars you need to wash to earn a certain amount of money. You would use two variables. Equation: Quantities in the equation: Evaluate an equation: Let s say, you want to earn \$300 dollars, how many cars do you have to wash? You substitute 300 for the which represents the amount of money you want to earn, then solve the equation to find out how many cars to wash: THINK! What amount of money minus \$40 is \$300? So, we got that amount of money by washing cars at \$5 each how many cars did we wash? Or: You can add the \$40 to both sides of the equation because you are going to have to wash some cars to earn money to pay your coach back for the supplies. Next, divide the remaining amount by \$5 to get the number of cars you need to wash to earn \$300 for your club, and \$40 to pay for the supplies. Or: You can solve this problem using an Algorithm (stepbystep procedure) you can use. It is explained in more detail in the next pages. (see page 11) Math Notes Page 17 of 17

18 Solve the equation to find out how many cars you need to wash in order to earn \$300. Change a subtraction operation to the addition of the opposite. Make Zeroes: by adding 40 to each side. Calculate the addition. Simplify Make Ones: Divide each side by 5. Calculate the division. Simplify So, you would have to wash 68 cars in order to earn \$300 for your club. 3. As a Placeholder, as in a formula: Formulas use letters (variables) that stand given quantities in a formula. When you know a value for one of the variables, then you can substitute (plug in) that value in for the letter in the formula that is holding the place for that value. Example: The formula for finding the volume of a box is: V = bwh V stands for volume, b for base, w for width and h for height. When b=10, w=5, and h=4, then V = = 200 The letters a, b, and c in mathematical properties stand for any number, so don t let them scare you. Check out this one that you will learn in Algebra: The Quadratic Formula. is what you are trying to find out stand for values (numbers) that you will know, and can just plug (substitute) the values into the formula. Math Notes Page 18 of 18

21 Math Notes Order of Operations AF 1.3(6), AF 1.4(6), AF1.2(7) & AF 2.1(7) You may have learned to remember the steps for order of operations as PEMDAS, but now that you are in middle school it really should be GEMDAS. Grouping Symbols Parenthesis Square Root Symbol Division Bar And some other symbols that you will learn later G E M D A S Exponents Multiplication and Division in the order they occur From left to right Addition and Subtraction in the order they occur From left to right The order of operations is the conventional order in which to calculate number problems. It just gradually developed as the modern symbols for algebra and arithmetic developed. This order is determined according to the rules of math; i.e number properties and number relationships. One Way to remember the order of operations is to use an acronym: GEMDAS: Graciously Excuse My Dear Aunt Sally. Grouping Symbols; Exponents; Multiplication and Division (in the order they occur); Addition and Subtraction (in the order they occur) Representation 1: StepByStep (using an algorithm) Parenthesis (grouping) Exponents Multiplication & Division Addition & Subtraction Simplified Representation 2: You can simplify the operations between the plus and minus signs first 59 Math Notes Page 21 of 21

22 Math Notes Patterns and Properties for Exponents AF2.1 (7) A number in exponential form is written with a base and an exponent. When the exponential form is simplified, the result is a power of the base. Vocabulary: Base: is the number to be used as a factor Exponent: is the number of times to use the base as a factor Exponential form: 4 is the 3 is the Expanded form: Power: The process of using exponents is called "raising to a power", where the exponent is the "power". Exponential form Patterns for exponents: Expand it out Power What Patterns can we see in the chart? Any number to the zero power = 1 because it follows the pattern The base is the same The exponents increase and decrease in sequential order Going up the pattern multiplies the previous power by 2 (doubles it) Going down the pattern divides the previous power by 2, (halves it) The base to the power of 1 is equal to the base The base to the power of 0 equals 1 Negative exponents can be found by continuing down the pattern. They make fractions, not negative numbers. Negative exponents are related to positive exponents: Multiplying and dividing powers are related to each other:, so.. =8 = Properties for exponents (that come from the patterns of exponents): Raised to the Power of One: AF 2.2(7) Any number raised to the power of one equals that number. If there is no exponent written, we know that it is a power of one. With Arithmetic: Raised to the Power of Zero: AF 2.2(7) Any number raised to the power of zero equals 1. With Arithmetic: Some mathematicians disagree, for complicated reasons, but it is accepted that Math Notes Page 22 of 22

23 Math Notes Multiplying Exponents with the Same Base: AF 2.2(7) & ns 2.3(7) When in doubt, expand it out! When multiplying exponents with the same base, you add the exponents. With Arithmetic: Expand it out Dividing Exponents with the Same Base: AF 2.2(7) & ns 2.3(7) When in doubt, expand it out! When dividing exponents with the same base you can subtract the exponents. Or, you can make giant ones. Expand it out or With Arithmetic: Make ones: Finding a Power of a Power: AF 2.2(7) & ns 2.3(7) When in doubt, expand it out! With Arithmetic: Expand it out check Math Notes Page 23 of 23

24 Math Notes RESOURCES: Mathwords A to Z: o Conceptually accurate definitions Cool Math: o Good explanations about concepts and procedures in language students easily understand o Challenging Math Games Math is Fun: o This site has a great dictionary, with pictures and animations. o Fun games and puzzles Purple Math: o Great lessons for How do you really do this stuff? Good for higher level concepts. College Preparatory Math: o This is a great site for conceptual explanations of the mathematics. o The Parent guides are very helpful for parents, and students. Math Notes Page 24 of 24

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