Table of Contents Objective 1: Review (Natural Numbers)... 3 Objective 2: Reading and Writing Natural Numbers... 5 Objective 3: Lines: Rays, and Line Segments... 6 Objective 4: Comparing Natural Numbers... 9 Objective 5: Rounding Natural Numbers... 10 Objective 6: Adding Natural Numbers... 13 Objective 7: The Properties of Addition, Convenient Calculation, and Polygons... 14 Objective 8: Column Addition... 16 Objective 9: Subtracting Natural Numbers... 16 Objective 10: Column Subtraction... 18 Objective 11: Numerical Expressions with Parentheses... 19 Objective 12: Multiplying Natural Numbers... 20 Objective 13: Properties of Multiplication... 21 Objective 14: Column Multiplication by a One-Digit Number... 23 Objective 15: Column Multiplication by a Two-Digit Number... 24 Objective 16: Formulas for the Areas of Squares and Rectangles... 24 Objective 18: Dividing Round Numbers and Finding How Many times More/Less... 27 Objective 19: Division with a Remainder... 28 Objective 20: Long Division... 28 Objective 21: Order of Operations... 29 Objective 22: Fractions and Percentages... 30 Objective 23: Fractions and Division, Improper Fractions, and Comparing Fractions... 32 Objective 24: Mixed Numbers and Improper Fractions... 33 Objective 25: Trapezoids and Parallelograms... 34 Objective 26: Comparing Fractions... 36 Objective 27: Adding and Subtracting Fractions with Like Denominators... 36 Objective 29: Subtracting Mixed Numbers with Like Denominators... 38 Objective 30: Review: Factors, Prime, and Composite Numbers, GCFs... 39 Reasoning Mind Last Updated: 01/01/2014 1
Objective 31: Review: Multiples and LCMs... 40 Objective 32: The Equivalency Property of a Common Fraction; Reducing Fractions... 41 Objective 33: Bringing Fractions to Common Denominator... 42 Objective 34: Comparing Fractions with Different Denominators... 44 Objective 35: Adding and Subtracting Fractions with Unlike Denominators... 44 Objective 36: Adding and Subtracting Mixed Numbers... 45 Objective 37: Multiplying Common Fractions... 46 Objective 38: Dividing Fractions and Mixed Numbers... 48 Objective 39: Review: Angles... 49 Objective 40: Types of Triangles... 52 Objective 41: Decimals and Percentages... 55 Objective 42: Comparing Decimals... 55 Objective 43: Rounding Decimals... 55 Objective 44: Adding Decimals... 55 Objective 45: Subtracting Decimals... 56 Objective 46: Multiplying a Decimal by a Whole Number... 56 Objective 47: Multiplying a Decimal by a Decimal... 56 Objective 48: Dividing a Decimal by a Whole Number... 57 Objective 49: Dividing a Decimal by a Decimal... 57 Additional Vocabulary: (Previously Coordinate Plane)... 58 Additional Vocabulary (Previously Review: Polygon Basics)... 60 Additional Vocabulary... 62 Reasoning Mind Last Updated: 01/01/2014 2
Objective 1: Review (Natural Numbers) Natural Number The numbers we use to count. Natural Numbers are also called whole numbers. The first natural number is 1, and every next natural number is just the previous natural number plus 1. For example: 1, 2, 3, and so on. One's Place The digit in the one's place represents ones in a number. For example, in the number 21,845, the digit in the one's place is 5. Places In the decimal system, the value of a digit depends on its place. In a whole number, the rightmost digit is in the one's place. The next digit to the left is in the ten's place. Then, there is the hundred's place, thousand's place, ten-thousands' place, and so on. When we write decimals, the first place to the right of the decimal point is the tenth's place. Then comes the hundredths place, and so on. Sum The sum is the result of addition. The numbers we are adding are called summands. For example, in 56 + 30 = 86, 86 is the sum. Summand When we are doing addition, the numbers we add are called the summands. For example, in 56 + 30 = 86, 56 and 30 are the summands. Reasoning Mind Last Updated: 01/01/2014 3
Ten A ten is equal to 10 ones. It is written as "10." Thousand Thousands' Place A thousand is equal to 10 hundreds or 100 tens. It is written as "1,000". The digit in the thousands place represents thousands in a number. For example, in the number 21,845, the digit in the thousands' place is 1. Whole Number Whole number is another name for natural numbers. For example: 1, 2, 3, 4, 5, and so on. Reasoning Mind Last Updated: 01/01/2014 4
Objective 2: Reading and Writing Natural Numbers Hundreds' Place The digit in the hundreds' place represents the hundreds number. For example, in the number 21,845, the digit in the hundreds' place is 8. Period The digits of whole numbers are divided into groups, starting from the right, and putting a comma after every three digits. These groups are called periods. For example, the number 1,234,567,890 has four periods. Tens' Place The digit in the tens' place represents tens in a number. For example, in the number 21,845, the digit in the tens' place is 4. Tenths' Place In a decimal, the digit in the tenths' place represents tenths in a number. For example, in the decimal 6.351, the digit in the tenths' place is 3. Reasoning Mind Last Updated: 01/01/2014 5
Objective 3: Lines: Rays, and Line Segments Number Ray A ray with its endpoint labeled zero and called the origin, and with a given unit segment is called a number ray. Ray A straight curve that has exactly one endpoint and extends infinitely in one direction. Putting a point on a line divides it into two rays. Ruler A ruler is a tool we use to draw line segments and measure their lengths. Segment The part of a line between two points, called the endpoints of the segment; a segment includes both endpoints. Also called a line segment. Adjacent Segments Adjacent segments are segments that share and endpoint and have no other endpoint. Reasoning Mind Last Updated: 01/01/2014 6
Coordinates Coordinates are numbers used for recording the position of a point. We need one coordinate to locate points on a line, two coordinates to locate points on a plane, and three coordinates to locate points in a space. Curve A shape that has a length, but no width - all curves are infinitely thin. Lines, rays, line segments, and circles are all examples of curves. Reasoning Mind Last Updated: 01/01/2014 7
Endpoints A point at the end of a line segment or a ray is called an endpoint. Line segments have two endpoints and rays only have one. In the ray below, point M is the endpoint. Supplementary Rays Supplementary Rays are two rays that form a line. Reasoning Mind Last Updated: 01/01/2014 8
Objective 4: Comparing Natural Numbers Inequality An inequality is a mathematical statement that one equality is greater than or less than another. For example, a < b is an inequality stating that a is less than b. Number Line A number line is a line with a chosen origin, unit segment, and positive ray. Unknown An unknown is a quantity whose value isn't known. Usually, a letter is used to represent an unknown. For example, in x + 3 = 5, x is the unknown. Reasoning Mind Last Updated: 01/01/2014 9
Objective 5: Rounding Natural Numbers Absolute Value The distance from the origin on a number line to the point with coordinate x is called the absolute value of the number x. Approximate Value Approximate Value is the value that is close to, but not exact to, the real value. Often it is a number that is easier to work with than a real value. For example, 750,000 is an approximate value for the exact value of 748,362. Approximately Equal When one number is used as an approximate value for another value, these numbers are approximately equal. Diameter The diameter is the chord that goes through the center of a circle. Reasoning Mind Last Updated: 01/01/2014 10
Perimeter The length of the boundary of a shape in the plane. The perimeter of a polygon is the sum of the lengths of the polygon's sides. Perimeter = 55 Round Numbers A round number is a natural number ending with one or more zeros. Examples: 50, 1,400, 100 Rounding Rounding a natural number means replacing the number with a round number close to it. Decimals can also be rounded; when we round a decimal, we replace it with another decimal that is close to it and has fewer non-zero places after the decimal point. The number 642 can be rounded to 640. Reasoning Mind Last Updated: 01/01/2014 11
Rounding Down Rounding a natural number down means replacing the number with a round number close to it, but less than it. Decimals can also be rounded down; when we round a decimal down, we replace it with another decimal that is close to it, but less than it, and has fewer non-zero places after the decimal point. The number 764 can be rounded down to 760. Rounding Up Rounding a natural number up means replacing the number with a round number close to it, but greater than it. Decimals can also be rounded up; when we round a decimal up, we replace it with another decimal that is close to it, but greater than it, and has fewer non-zero places after the decimal point. The number 769 can be rounded up to 770. Square The word square has two meanings: 1) The number n 2 = n x n is called the square of n, n squared. 2) A rectangle whose sides all have equal length. Squares are often used as units of area. For example, a square meter is a square with sides having a length of 1. Reasoning Mind Last Updated: 01/01/2014 12
Objective 6: Adding Natural Numbers Addition When we put things together we can find the total with addition. We use the plus sign, +, for addition. For example, 4+2= 6. Equality A mathematical statement that two expressions are equal. All equalities can be recognized by the equality sign, =. For example, 9 + 4 = 13. Reasoning Mind Last Updated: 01/01/2014 13
Objective 7: The Properties of Addition, Convenient Calculation, and Polygons Algebraic Expressions An algebraic expression is a mathematically meaningful sequence of numbers, letters that stand for numbers, operation signs, and parentheses. Associative Property of Addition The associate property of addition is the property that changing the grouping of summands does not change the sum. In other words, for any three numbers, a, b, and c, (a + b) + c = a + (b + c) Commutative Property of Addition The commutative property of addition is the property that changing the order of the summands does not change the sum. In other words, for any two numbers a and b, a + b = b + a. For example, 3 + 5 = 5 + 3. Pentagon A pentagon is a polygon with five sides. Quadrilateral A quadrilateral is a polygon that has four sides. Squares, rectangles, and trapezoids are all quadrilaterals. Reasoning Mind Last Updated: 01/01/2014 14
Side of a Polygon A side of a polygon is one of the line segments that make up the boundary of the region. For example, line segment DE is one of the sides of this polygon. Zero Property of Addition In the zero property of addition, adding zero to a number does not change the number. In other words, for any number a, a + 0 = 0 + a = a. For example, 4 + 0 = 4. Reasoning Mind Last Updated: 01/01/2014 15
Objective 8: Column Addition Bar Graph A way to show data in picture form. In a bar graph, vertical or horizontal bars are used to show data. For example, if the data is about populations of different towns, the height of a bar above each town's name shows us how many people live in that town. Carry Over When we do column addition, we carry over when the sum of the digits in a column is a two-digit number. Column Addition Column addition is a method of addition in which you first line up the summands, and then add the digits in the ones' column, the tens' column, and so on. Objective 9: Subtracting Natural Numbers Minuend In subtraction, the number from which we are subtracting is Reasoning Mind Last Updated: 01/01/2014 16
the minuend. For example, in a - b = c, a is the minuend. Subtraction Subtraction is the operation of finding a summand from the sum and the other summand. Subtraction is the inverse operation to addition. For example, 20-12 = 8. Subtrahend In subtraction, the number that we subtract from the minuend. In a - b = c, b is the subtrahend. For example, in 78-14 = 64, 14 is the subtrahend. Reasoning Mind Last Updated: 01/01/2014 17
Objective 10: Column Subtraction Column Subtraction Column subtraction is a method of subtraction in which you line up the minuend and the subtrahend by places, like in column addition. Then you subtract ones from ones, tens from tens, and so on. Column subtraction for decimals is done similarly. Price The price is the amount of money one unit of a product is worth. For example, if one pencil costs 5 cents, then the price of pencils is 5 cents. Reasoning Mind Last Updated: 01/01/2014 18
Objective 11: Numerical Expressions with Parentheses Expression See numerical expression or algebraic expression. Example: 2 X 5 + 6³ Numerical Expression A numerical expression is a mathematically meaningful sequence of numbers, letters that stand for numbers, operation signs, and parentheses. For example, 2 X 5 + 6³. Parentheses Value of an Expression Parentheses are the two symbols ) and (. ( is called the opening parenthesis. ) is called the closing parenthesis. Parentheses group numbers. When evaluating an expression, we always do the operation inside the parentheses first. The result of evaluating an expression is called the value of the expression. For example, if we solve the expression (451 + 675) - 1,005, we will find that the value of the expression is 121. Reasoning Mind Last Updated: 01/01/2014 19
Objective 12: Multiplying Natural Numbers Multiplication Multiplication is one of the four basic arithmetic operations. Multiplication is the inverse operation to division. For example, 3 x 5. Product The product is the result of multiplication. The numbers we multiply are called factors. For example, in 2 x 6 = 12, 12 is the product. Reasoning Mind Last Updated: 01/01/2014 20
Objective 13: Properties of Multiplication Associative Property of Multiplication The associate property of multiplication is the property that states changing the grouping of factors does not change the product. In other words, for any three numbers, a, b, and c, (a x b) x c = a x (b x c) Commutative Property of Multiplication The commutative property of multiplication states that changing the order of the factor does not change the product. For example, mn = nm. Distributive Property of Multiplication Over Addition The distributive property of multiplication over addition states that to multiply a sum by a number, we can multiply each summand by the number and add the results. In other words, for any three numbers a, b, and c. Distributive Property of Multiplication Over Subtraction According to the distributive property of multiplication over subtraction, to multiply a difference by a number, we can multiply the minuend and subtrahend by the number and subtract the results. In other words, for any three numbers a, b, and c. One Property of Multiplication According to the one property of multiplication, the product of 1 and any number is that number. In other words, for any number a, a x 1-1 x a = a. For example, 7 x 1 = 1 x 7 = 7. Reasoning Mind Last Updated: 01/01/2014 21
Zero Property of Multiplication The zero property of multiplication states that the product of any number and zero is equal to zero. In other words, for any number a, a x 0= 0 x. For example, 8 x 0 = 0. Reasoning Mind Last Updated: 01/01/2014 22
Objective 14: Column Multiplication by a One-Digit Number Column Multiplication Column multiplication is a method of multiplication in which you line up the factors by places, like in column addition. Then you multiply the first number by the place value of each digit of the second number and add the results. Substitution Replacing letters in algebraic expression with numbers is called substitution. For example, when we replace p with 3 in (6 + p) + (15 + p), we say that we substitute 3 for p. Then, we get the numerical expression (6 + 3) + (15 + 3). Reasoning Mind Last Updated: 01/01/2014 23
Objective 15: Column Multiplication by a Two-Digit Number Times (more than) Times is another way to refer to the process of multiplication. Multiplication is one of the four basic arithmetic operations. Multiplication is the inverse operation to division. For example, 3 x 5. Objective 16: Formulas for the Areas of Squares and Rectangles Area Area is the measure of the total amount of surface on a plane that a shape takes up. To measure area, we use units such as square inches and square meters. Formula A formula is a mathematical statement describing a relationship between quantities expressed as numbers or letters. For example, the formula d = 2r describes the relationships between the diameter and the radius of a circle. Rectangle A rectangle is a quadrilateral whose angles are all right angles. Reasoning Mind Last Updated: 01/01/2014 24
Unit A unit is a standard used for measurement. Objective 17: Dividing Natural Numbers For example, inches and miles are units of length, and seconds and minutes are units of time. Division Division is the operation of finding a factor from a product and another factor. For example, 9 3 = 3. Division Property of One According to the division property of one, dividing a number by 1 does not change it. In other words, for any number a, a 1 = a. For example, 10 1 = 10. Divisor In division, the divisor is the number that we are dividing by. If b a = c, then a is the divisor. We also say that a is a divisor of b if a is a factor of b. For example, in 10 5 = 2, then 5 is the divisor. Factor The word factor has three meanings: 1) In multiplication, the numbers we are multiplying are called factors. 2) The number b is a factor of the number a if a is divisible by b. 3. To factor a natural number means to express it as a product of two or more natural numbers. For example, in 36 4 = 9. Four is a factor of 36. Quotient The quotient is the result of division. The number you are dividing is the dividend and the number you are dividing it by is the divisor. For example, in 18 6 = 3, 3 is the quotient. Reasoning Mind Last Updated: 01/01/2014 25
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Objective 18: Dividing Round Numbers and Finding How Many times More/Less Average Finding the average of a group of numbers means finding a number that is somehow "in the middle" of these numbers. There are several different averages. For example, the mean, median, and mode are all types of average. Mean The mean is the most common type of average. The mean of a group of numbers is their sum divided by the size of the group. For example, the mean of the numbers 3, 4, and 8 is (3 + 4 + 8) 3 = 15 3 = 5. The mean is also called the arithmetic mean. Reasoning Mind Last Updated: 01/01/2014 27
Objective 19: Division with a Remainder Remainder In division, the remainder is the number left over after the divisor has gone into the dividend as many times as it can. For example, 3 goes in 7 two times and no more: 7 = 2 3 + 1. So, 1 is the remainder. We write 7 3 = 2 R1. Objective 20: Long Division Long Division Long division is a method of dividing one number by another. Reasoning Mind Last Updated: 01/01/2014 28
Objective 21: Order of Operations Evaluating The process of finding the value of an expression. To evaluate a numerical expression, we just perform the operations in the expression. To evaluate an algebraic expression, we need to know what number each letter stands for, substitute these numbers for the letters, and then perform the operations. To evaluate 6 - b when b = 4 means to replace b with 4 and evaluate the numerical expression: 6 - b = 6-4 = 2. We substituted 4 for b in the expression. Order of Operations Rule The Order of Operations Rule states that in expressions with parentheses: do the operations in parentheses first, the do x and from left to right, then do + and - from left to right. Reasoning Mind Last Updated: 01/01/2014 29
Objective 22: Fractions and Percentages Center of a Circle Every point on a circle is the exact same distance from one particular point. This point is called the center of the circle. Chord A chord is line segment that connects two points of a circle. Circle A circle is a curve whose points are the same distance from one point, called the center of the circle. Common Fraction A common fraction is a fraction, written with a wholenumber numerator and denominator separated by a fraction bar. Reasoning Mind Last Updated: 01/01/2014 30
For example, 3/8 is a fraction. Compass The compass is the tool we use to draw circles. Even Number An even number is a natural number that can be divided by 2 with no remainder. 0, 2, 4, 6, 8, 10, 12, 14 and so on. Example: 8 2 = 4 There is no remainder when 8 is divided by 2, so 8 is even. Numerator In a common fraction, the number written above the fraction bar is the numerator. For example, in the fraction 3/8, 3 is the numerator. Quarter Quarter is the name of the fraction ¼. Percent One hundredth is an example of a percent. A percent is written with a % sign and is equal to a value out of 100. For example, 5% is equal to 5/100 and to the decimal.=0.05. 100% is equal fo 100/100 = 1. Reasoning Mind Last Updated: 01/01/2014 31
Objective 23: Fractions and Division, Improper Fractions, and Comparing Fractions Dividend In division, the number that is being divided is called the dividend. For example, if a b = c, a is the dividend. Improper Fraction A common fraction with a numerator greater than or equal to the denominator. For this reason, improper fractions are always greater than or equal to 1. For example, 25/2 is an improper fraction. Proper Fraction A proper fraction is a common fraction with a numerator less than the denominator. For this reason, proper fractions are always less than 1. For example, ¾ is a proper fraction. Ratio A ratio is another name for the quotient of two numbers. We normally say ratio when we are using the quotient to compare two numbers by division. For example, the ratio of 24 to 8 is 24 8 = 3. So, 24 is 3 times bigger than 8. Reasoning Mind Last Updated: 01/01/2014 32
Objective 24: Mixed Numbers and Improper Fractions Mixed Numbers Writing a mixed number is a way of writing numbers that are a sum of a natural number and a proper fraction. In a mixed number, the whole part is written first, followed by the fractional part. For example, the mixed number 1 2/3 equals 1 + 2/3. Whole Part Every fraction can be written as the sum of a whole number and a fraction less than 1; that whole number is called the whole part of a number. In other words, the whole part of a number is the difference between the number and its fractional part. For example, the decimal 35.92 can be written as 35 + 0.92. So, the whole part of the decimal is 35. In the mixed number 2 5/7, the whole part is 2. Reasoning Mind Last Updated: 01/01/2014 33
Objective 25: Trapezoids and Parallelograms Base of a Trapezoid The two parallel sides of a trapezoid are its bases. Degree The degree is the most common unit we use to measure angles. For example, if a straight angle is divided into 180 equal angles, we say that each of these angles ia a one degree angle, or it has 1 degree = 1, or it measures 1. Isosceles Trapezoid An isosceles trapezoid is a trapezoid where the two opposite sides that are not parallel have the same length. Parallel Lines On a plane, two lines that never cross are considered parallel lines. The lines below are parallel. Proportion A proportion is an equality of two ratios. We say it like this: "The ratio of 6 to 3 equals the ratio of 10 to 5." Or you can say, "6 is to 3 as 10 is to 5." On paper we write: 6/3 = 10/5. Protractor A protractor is the tool we use to measure angles. Reasoning Mind Last Updated: 01/01/2014 34
Trapezoid A trapezoid is a quadrilateral that has exactly two parallel sides. These two parallel sides are called the bases of the trapezoid. Reasoning Mind Last Updated: 01/01/2014 35
Objective 26: Comparing Fractions Like Denominators We say that two common fractions have like denominators if their denominators are equal. For example, 4/9 and 7/9 have like denominators. Objective 27: Adding and Subtracting Fractions with Like Denominators Cross Terms In a proportion, cross terms are the terms that lie diagonally across from each other. Fractional Expression A fractional expression is a fraction in which the numerator and/or denominator is not a whole number, but an expression. What is written above the bar in a fractional expression is called the numerator of the fractional expression. What is written below the bar in a fractional expression is called the denominator of the fractional expression. Objective 28: Adding Mixed Numbers with Like Denominators Equation An equation is an equality with one or more letters standing for unknown numbers. For example, 2 + x = 10 is an equation. Reasoning Mind Last Updated: 01/01/2014 36
Fractional Part Every fraction can be written as the sum of a whole number and a fraction less than 1, which is called the fractional part of the fraction. In other words, the fractional part of a number is the difference between the number and its whole part. For example, the decimal 35.92 can be written as 35 + 0.92. So, the fractional part of this decimal is 0.92. In the mixed number 2 5/7, the fractional part is 5/7. Reasoning Mind Last Updated: 01/01/2014 37
Objective 29: Subtracting Mixed Numbers with Like Denominators Borrowing When we do column subtraction, we sometimes have to subtract a larger digit from a smaller one. Then, we borrow from the next place. Difference The difference is the result of subtraction. For example, in 9-3 = 6, the difference between 9 and 3 is 6. Equation An equation is an equality with one or more letters standing for unknown numbers. For example, 2 + x = 10 is an equation. Fractional Part Every fraction can be written as the sum of a whole number and a fraction less than 1, which is called the fractional part of the fraction. In other words, the fractional part of a number is the difference between the number and its whole part. For example, the decimal 35.92 can be written as 35 + 0.92. So, the fractional part of this decimal is 0.92. In the mixed number 2 5/7, the fractional part is 5/7. Reasoning Mind Last Updated: 01/01/2014 38
Objective 30: Review: Factors, Prime, and Composite Numbers, GCFs Composite Number A composite number is a natural number that has factors other than 1 and itself. Note: 1 is neither a prime number nor a composite number. All other natural numbers are either prime or composite. The number 10 is a composite number. Divides We say that a divides b if b is divisible by a (that is, if a is a factor of b.) We can divide 18 by 3 with no remainder. 18 3 = 6. So we say 18 is divisible by 3, or 3 divides 16. We cannot divide 18 by 4 with no remainder: 18 4 = 4 R2. So we say 18 is not divisible by 4, 4 does not divide 18. Divisible We say that a is divisible by b if b is a factor of a. We can divide 18 by 3 with no remainder. 18 3 = 6. So we say 18 is divisible by 3, or 3 divides 16. We cannot divide 18 by 4 with no remainder: 18 4 = 4 R2. So we say 18 is not divisible by 4, 4 does not divide 18. Prime Number A prime number is a natural number that has exactly two divisors: 1 and itself. 1 is neither a prime number nor a composite number. All other natural numbers are either prime or composite. The first several prime numbers are 2, 3, 5, 7, 11, 13, 17, and 19, but there are also many more. Relatively Prime Numbers Two numbers are relatively prime if their GCF is 1. In other words, two numbers are relatively prime if they have no common factors, except for 1. Example: 25 and 26 share no common factors other than one. Therefore, they are relatively prime. Reasoning Mind Last Updated: 01/01/2014 39
Objective 31: Review: Multiples and LCMs Multiple A number is a multiple of a if it can be divided by a with no remainder. In other words, any number divisible by a is a multiple of a. For example, 5 is a multiple of 15 since 15 can be divided by 5 with no remainder. 15 3 = 5. Reasoning Mind Last Updated: 01/01/2014 40
Objective 32: The Equivalency Property of a Common Fraction; Reducing Fractions Cross Product Every proportion has two cross products - the cross product of the outer terms, and the cross product of the inner terms. GCD GCD is another name for Greatest Common Factor. GCF The GCF of a group of whole numbers is the largest natural number that divides each of the numbers in the group. The Equivalency Property of a Common Fraction If we multiply the numerator and the denominator of a given fraction by the same non-zero number, we get a fraction equal to the given fraction. The same applies if we divide the numerator and the denominator of a given factor by the same non-zero number. For example, 2/8 and 1/4 are equivalent, since they are the same number when reduced. Reasoning Mind Last Updated: 01/01/2014 41
Objective 33: Bringing Fractions to Common Denominator Complementary Factor A complimentary factor is a factor that is used to get a new denominator from the old one when converting fractions. The complementary factor can be found by dividing the new denominator by the old one. Irreducible Fraction If a fraction's numerator and denominator have no common factors other than 1. (in other words, if they are relatively prime numbers). We say that an irreducible fraction is in its lowest terms. Example: 2/3, 14/15, and 99/100 are some examples are irreducible fractions. LCM The LCM of a group of natural numbers is the smallest natural number that is divisible by each of the numbers in the group. For example, the least common denominator for fractions with denominators of 2 and 3 is 6. Least Common Denominator The least common denominator of a group of fractions is the least common multiple (LCM) of the fraction's denominators. For example, the least common denominator for fractions with denominators of 2 and 3 is 6. Odd Number An odd number is a natural number that cannot be divided by 2 with no remainder. When dividing any odd number by 2, the remainder is 1. Example: 9 2 = 4 R1. There is a remainder of 1 when 9 is Reasoning Mind Last Updated: 01/01/2014 42
divided by 2, so 9 is odd. Prime Factorization Prime factorization is writing a composite number as a product of only prime numbers. Any composite number has only one prime factorization: the factors can be written in a different order, but they are always the same. Example: The prime factorization of 24: 24 = 3 x 2 x 2 x 2. Reasoning Mind Last Updated: 01/01/2014 43
Objective 34: Comparing Fractions with Different Denominators Like Denominators We say that two common fractions have like denominators if their denominators are equal. For example, 4/9 and 7/9 have like denominators. Objective 35: Adding and Subtracting Fractions with Unlike Denominators Denominator In a common fraction, the number written below the fraction bar is the denominator. For example, in the fraction 3/8, 8 is the denominator. Fraction A fraction is a number that is possible to write as a/b, where a and b are whole numbers. Fractions represent parts of whole numbers. For example, if we divide a pizza into 8 equal parts and then eat the pizza, we will have eaten 3/8 of the pizza. You can also think of 3/8 as the quotient 3 8 (3 divided by 8). Reasoning Mind Last Updated: 01/01/2014 44
Objective 36: Adding and Subtracting Mixed Numbers Borrowing When we do column subtraction, we sometimes have to subtract a larger digit from a smaller one. Then, we borrow from the next place. Difference The difference is the result of subtraction. For example, in 9-3 = 6, the difference between 9 and 3 is 6. Equation An equation is an equality with one or more letters standing for unknown numbers. For example, 2 + x = 10 is an equation. Reasoning Mind Last Updated: 01/01/2014 45
Objective 37: Multiplying Common Fractions Base of a Prism In a prism, the two bases must be two opposite faces, which are polygons equal and parallel to each other. Cube The number n³ - n x n x n is called the cube of the number, or n cubed. The cube of n is just another name for the third power of n. For example, n³ = n x n x n. Prism A prism is a geometric solid that has a top base and a bottom base, where the two bases are opposite faces that are polygons equal and parallel to each other. All the faces of a prism are called sides. A prism is named after its bases. Reasoning Mind Last Updated: 01/01/2014 46
Rectangular Prism A rectangular prism is a prism whose bases are rectangles. Side Face of a Prism A side face of a prism is the face of a prism that is not a base of the prism. Volume The volume is the amount of space a solid takes up. It can be measured in units such as cubic centimeters and cubic inches. Reasoning Mind Last Updated: 01/01/2014 47
Objective 38: Dividing Fractions and Mixed Numbers Reciprocals Two numbers whose product is one are called reciprocals. Reasoning Mind Last Updated: 01/01/2014 48
Objective 39: Review: Angles Acute Angle An acute angle is smaller than a right angle. It measures less than 90 degrees. Angle An angle is the figure formed by two rays that have the same endpoint. Angle Bisector An angle bisector is a ray which has its endpoint at the angle's vertex and divides the angle into to equal parts. Drafting Triangle A drafting triangle is the tool that we can use to quickly draw right angles, or to determine if a given angle is acute, obtuse, or right. False A mathematical statement is false if it is not correct. Reasoning Mind Last Updated: 01/01/2014 49
For example, someone could write the equality 2 + 3 = 15. This equality is false because two plus three is five, not fifteen. If a mathematical statement is not false, then it is true. Obtuse Angle An obtuse angle is larger than a right angle but smaller than a straight angle. An obtuse angle measures more than 90 degrees, but less than 180 degrees. Right Angle One half of a straight angle is considered to be a right angle. Straight Angle A straight angle is an angle whose sides are supplementary rays. A straight angle measures 180 degrees. Supplementary Rays Supplementary Rays are two rays that form a line. True A mathematical statement is true if it is correct. Reasoning Mind Last Updated: 01/01/2014 50
For example, the equality 2 + 3 = 5 is true. If a statement is not true, it is false. Reasoning Mind Last Updated: 01/01/2014 51
Objective 40: Types of Triangles Acute Triangle An acute triangle is a triangle with three acute angles. Angle Bisector An angle bisector is a ray which has its endpoint at the angle's vertex and divides the angle into to equal parts. Angle of Rotational Symmetry For an object with rotational symmetry, this is the angle we have to rotate it by to get the same object back. Drafting Triangle A drafting triangle is the tool that we can use to quickly draw right angles, or to determine if a given angle is acute, obtuse, or right. Hypotenuse In a right triangle, the side opposite the right angle is the hypotenuse. In other words, it is the side that is not a leg. The hypotenuse is always the longest side in a right triangle. The unmarked side is the hypotenuse. Reasoning Mind Last Updated: 01/01/2014 52
Leg of a Right Triangle In a right triangle, one of the two sides that form the right angle is a leg of the right triangle. Sides a and b are both legs of the right triangle. Obtuse Triangle An obtuse triangle is a triangle with one obtuse angle. No triangle can have more than one obtuse angle. Perpendicular Perpendicular means to meet at a right angle. For example, two lines that meet at a right angle are called perpendicular lines. Right Triangle A right triangle is a triangle that has a right angle. Reasoning Mind Last Updated: 01/01/2014 53
Triangle A triangle is a polygon with three sides. Reasoning Mind Last Updated: 01/01/2014 54
Objective 41: Decimals and Percentages Hundred 100 ones, 10 ones. A hundred is written as 100. Decimal System The decimal system is the system that uses the digits 1, 2, 3, 4, 5, 6, 7, 8, 9, and 0 to write numbers. The decimal system is place-based, which means that the value of every digit in a number depends on its place in that number. For example,.7 = 7/10 because it is in the tenths place. Objective 42: Comparing Decimals Place Value The value of a digit depends on its place in a number. In 836,471, the digit 4 is in the hundred Objective 43: Rounding Decimals Rounding Decimals Decimals can be rounded. When we round a decimal, we replace it with another decimal that is close to it and has fewer non-zero places after the decimal point. Objective 44: Adding Decimals Hundredths' Place In a decimal, the digit in the hundredths' place represents hundredths in a number. For example, in the decimal 6.351, the digit 5 is in the hundredths' place. Reasoning Mind Last Updated: 01/01/2014 55
Objective 45: Subtracting Decimals Decimal Point The decimal point is the point separating the whole and fractional parts of a decimal. For example, in 3.5, the decimal point separates the whole number, 3, and the fractional part, 5. Pie Chart A pie chart is a way to show data. In a pie chart, a circle is divided into several parts, each showing a piece of data. Solution of an Equation For an equation with one unknown, a solution of an equation is a value for the unknown which turns the equation to a true equality. For example, if we solve the equation 2 + x = 10, then we will find that x = 8. Giving x the value of 8 turns the equation to a true equality, so x=8 is a solution of this equation. Objective 46: Multiplying a Decimal by a Whole Number Multiplying a decimal by a whole number Multiplying a decimal by a whole number means adding up summands, each of which is equal to that decimal. For example, If d is decimal and n is whole number, we write this as d n. Objective 47: Multiplying a Decimal by a Decimal Exponent An exponent is the power to which a number is being raised. Reasoning Mind Last Updated: 01/01/2014 56
For example, in the expression, 7³ (seven to the third power), 3 is the exponent. Objective 48: Dividing a Decimal by a Whole Number No new vocabulary Objective 49: Dividing a Decimal by a Decimal No new vocabulary Reasoning Mind Last Updated: 01/01/2014 57
Additional Vocabulary: (Previously Coordinate Plane) Axis of Symmetry The axis of symmetry is the line that splits an object with reflective symmetry into two symmetrical halves. Also called the line of symmetry. Intersecting Lines Intersecting lines are lines that cross. The point where they cross is called the point of intersection. Line A line is a type of curve. A line is straight, and it extends infinitely in both directions. Plane A plane is like the flat surface of a table, going on infinitely in all directions. We can draw lines, curves, and other curves and shapes on the plane. Reasoning Mind Last Updated: 01/01/2014 58
Quadrant The quadrant is the part of the plane between two perpendicular number rays with a common endpoint. The common endpoint. The common endpoint of the two perpendicular number rays is called the origin. Of the two perpendicular number rays, the horizontal one is the x-axis, and the vertical one is the y-axis. Reasoning Mind Last Updated: 01/01/2014 59
Additional Vocabulary (Previously Review: Polygon Basics) Acute Angle An angle smaller than a right angle. An acute angle measures smaller than 90. Closed Curve A closed curve is a curve that forms a loop with no beginning or end. Congruent Shapes Two shapes are congruent if one can be turned into the other by some combination of rotations, reflections, and translations. Diagonal A diagonal line is a line segment that connects two nonneighboring vertices of a polygon. (When the vertices are neighboring, the connecting them is a side of the polygon.) Reasoning Mind Last Updated: 01/01/2014 60
The line down the middle is the diagonal. Face One of the polygons that make up the surface of a polyhedron is called the face. For example, the faces of a rectangular prism are all rectangles. Open Curve An open curve is a curve that is not closed. Reasoning Mind Last Updated: 01/01/2014 61
Additional Vocabulary Apex of a Pyramid The apex of a pyramid is the point where all the sides of a pyramid meet. Base of a Pyramid All the faces of a pyramid except one are triangles that meet at one point. The other face is a polygon that is called the base of the pyramid. Center of Central Symmetry Center of Rotational Symmetry Central Symmetry Certain Event For an object with central symmetry, the center of rotational symmetry is also called the center of central symmetry. The point we have to rotate an object around to get the object back into its original position. To have a center of rotational symmetry, an object has to have rotational symmetry. An object has central symmetry if it can be rotated 180 to get the same object back. A certain event is an event that we know will happen for sure. For example, since we know that every math class eventually ends, the current math class ending is a certain event. Reasoning Mind Last Updated: 01/01/2014 62
Combinatorics Combinatorics is the branch of mathematics that involves counting how many ways there are to do something. For example, combinatorics could be used to answer the question, "How many different results can there be of rolling 3 dice?" Compatible Events Directly Proportional Quantities Two events that can happen at the same time are considered compatible events. Two quantities that are directly proportional if, when one increases a certain number of times, the other increase the same number of times are considered directly proportional quantities. For example: If we increase a length in feet a certain number of times, the length in inches increases a certain number of times. Therefore, the number of feet measured is directly proportional to the number of inches measured. We have direct proportionality. Disk A disk is a circle together with the surface inside it. Equally Likely Events Estimate Event Two or more events such that their chances of happening are the same are considered equally likely events. To estimate means to find a number that may not be the exact answer to a question, but is close enough. We call such a number an estimate of the exact answer. An event is something that happens, or may happen. An example of an event: a coin we throw lands heads up. Reasoning Mind Last Updated: 01/01/2014 63
Expanded Form A number written as the sum of the values of each of its places is written in the expanded form. For example, the expanded form of 764 is 700 + 60 + 4. Impossible Event Incompatible Events An impossible event is an event that can never happen. Two events that cannont happened at the same time are considered incompatible events. For example, getting all of the questions on a math test right and not passing the test are incompatible events. Integers Inverse Operations Positive integers, zero, and negative numbers are called integers. The number 0 is an integer, but it's neither positive nor negative. Inverse operations are operations that undo each other. For example, addition and subtraction are inverse operations: if we add 6 to a number, and then subtract 6 from the result, we will get the original number again. Multiplication and division are also inverse operations. Line Graph A line graph is a way to show data. A line graph has points that show pieces of data and line segments that connect the points. Reasoning Mind Last Updated: 01/01/2014 64
Line of Symmetry Median The line of symmetry is the line that splits an object with reflective symmetry into two symmetrical halves. The median is a common type of average. To find the median of a group of numbers, we line he numbers up in order from least to greatest and find the number in the middle. If, after lining the numbers up, there are two numbers in the middle, the median is the mean of the two middle numbers. Negative Integers Net of a Solid We call these numbers negative integers. We always write them with a minus sign. The set of all negative integers includes many more numbers than what is shown such as -6, -7, -8, and so on. It is a never ending list! Polyhedra can be unfolded along their edges into a flat surface made up of their faces. This flat surface is called the net of the solid. Octagon An octagon is a polygon with eight sides. Reasoning Mind Last Updated: 01/01/2014 65
Opposite Numbers Opposite numbers are considered to be two numbers that are written the same, except that one of them is negative (it has a minus sign) and the other is positive (it has no sign or a plus sign.) Positive Integers Another name for natural numbers is positive integers. Example: The set of all positive integers includes many more numbers than what is shown such as: 6, 7, 8 and so on. It is a never ending list! Property of Dividing a Number by Itself Proportion for Directly Proportional Quantities Pyramid The property of dividing a number by itself states that if we divide a non-zero number by itself, we get 1. In other words, for any non-zero number a, a a = 1. If two quantities a and b are directly proportional, and a increases by n times, then the ratio of a to (a n) is equal to the ratio of b to (b n). A pyramid is a geometric solid. All the faces of a pyramid except one must be triangles that meet at one point. The other face is a polygon and is called the base of the pyramid. A pyramid is named after its base. For example, if the base of the pyramid is a rectangle, the pyramid is called a rectangular pyramid. Reasoning Mind Last Updated: 01/01/2014 66
Radius The radius is the distance between the center of a circle and any point on the circle. A segment that connects the center with any point of the circle is also called the radius. Random Event Reflective Symmetry Reflex Angle Right Prism If, in identical circumstances, a repeated experiment could lead to an event either happening or not happening, then that event is called a random event. An object that can be split by a line so that one side is the mirror image of the other is said to have reflective symmetry. A reflex angle is an angle that is greater than a straight angle. A reflex angle measures more than 180, but less than 360. A right prism is a prism whose sides are all rectangles. In a right prism, the two bases are not only parallel but also aligned so that one is directly on top of the other; so, the sides are all perpendicular to the bases. Reasoning Mind Last Updated: 01/01/2014 67
Rotational Symmetry Side Face of a Pyramid If an object has rotational symmetry, then the object can be rotated by a certain angle around some point to get the same object back. That angle is called the angle of rotational symmetry, and the point is called the center of rotational symmetry. A side face of a pyramid is a face of a pyramid that is one of the triangles meeting at the apex. Sides of an Angle An angle is made up of two rays. Each one of these rays is called a side of the angle. Ray PS and Ray PT are the sides of this angle. Solid Symmetrical Tetrahedron A solid is a shape that takes up space and has volume. Cubes, balls, and rectangular prisms are examples of solids. An object is symmetrical if it can be rotated, reflected, and/or translated to get the same object back. A tetrahedron is another name for a triangular pyramid. Reasoning Mind Last Updated: 01/01/2014 68
Translation Unit Rate Moving an object without rotating it or changing its shape or size is considered translation. Translation is a type of transformation. The unit rate is a rate with a denominator of 1 unit. Examples: 25 miles per hour, and $6 per can are unit rates. Zero Property of Division The zero property of division states that zero divided by any number is zero. In other words, for any number a not equal to 0, 0 a = 0. For example, 0 4 = 0 Reasoning Mind Last Updated: 01/01/2014 69