Algebra..... - Steps of Algebra... Expressions.. Conversion Benchmarks. - Customary & Metric.. - Fractions/ Decimals/ Percents Gallon Guy Diagram. Volume Coordinate Plane.. 0 Data Analysis & Statistics.. 0 Data Displays Decimals - Addition & Subtraction. Division.. Multiplication. Place Value Rounding. Divisibility Rules Exponents. Cubes & Roots.., Squares & Roots..., Factors & Multiples. - Greatest Common Factor (GCF). Least Common Multiple (LCM) Fractions - Add/Subtract Like Denominators. -8 Add/Subtract Unlike Denominators.. 8- Common Denominators -8 Division... Equivalent Fractions..... 8 Fraction Bars..... Improper Fractions... Jingle... Mixed Numbers.. Multiplication.. Simplifying Fractions. Geometry... - Angles... - Area & Perimeter... Circles.. Formula Equations Lines. Polygons.. Quadrilaterals. Transformations. Transversal. Triangles.. Integers -0 Math Properties.. Mathematical Practices (MP).. Multiplication Chart.. 8 Order of Operations.. Prime & Composite... - Composite Numbers.. Prime Numbers.. - Prime Factorization Probability.. Organized List/ Tree Diagram.. Ratios, Rates & Proportions. -0 Surface Area & Nets Three Dimensional Solids.. - Surface Area... Volume..... - Transformations. Whole Numbers Division.... Fact Families... Multiplication Place Value. Standard & Expanded Forms..
Whole Number Multiplication Single Digit Multiplication Multiplication, x 8 Double Digit Multiplication Triple Digit Multiplication Whole Number Division =,, x 0 + 0 0 0, 0 8 x 8 8 0 + 0 0, 8 Multiplication can be represented in the following ways: x * ()() Division Without Remainders Set up the problem: Divide: 0-8 - - 0 = Check your work by multiplying by : X Place Value Whole Numbers What is the name of the place with the digit of? - Thousands. What is the value of the? - Four thousand. Place Value: Decimals The digits to the right of the decimal point (.) are decimals. They all end with ths. What is the name of the place with the digit of 8? Hundredths. What is the value of the 8? Eight hundredths. Rounding Whole Numbers or Decimals (Right Round) Look to the right of the number to be rounded. - If that number is or greater, round up - If that number is or less, round down - Drop all digits to the right of the rounded number and fill in with zeros Ex:, - The digit to the right of the underlined number is, so round up. - Drop the remaining digits and fill with zeros, =,00 Ex:. - The digit to the right of the underlined number is, so round down. - Drop the remaining digits and fill with zeros
. =.0 or Ex: 0.8 - The digit to the right of the underlined number is, so round up. - Drop the remaining digits. 0.8 = 0. Standard/Expanded Notation Standard Form:,,0 Expanded Form:,000 + 00 + 0 + 0,000 +,000 + 00 + Fact Families Fact families are the groups of addition/subtraction and multiplication/division facts that form a family : Add/Subtract:,, + = + = = = Multiply/Divide:,, 0 = 0 = 0 0 = 0 = Exponents (Powers) An exponent tells the number of times a base is multiplied by itself. Anything raised to the zero power equals. º = Anything raised to the first power equals itself. Identity Property 8¹ = 8 Anything raised to the nd power is called squared. = base, = exponent = = Anything raised to the rd power is called cubed. = base, = exponent = =, Base The base is and is the answer. Figure out, to what power the base should be raised. x x x = So, Base = Roots Square Root: One of the two equal factors of a number. = Because = = Cubed Root: One of the three equal factors of a number. = Because = ³ = Order of Operations Left PEMDAS Right st : Parenthesis nd : Exponents rd : Multiplication OR Division th : Addition OR Subtraction PEMDAS. Do all computations within brackets, then parenthesis, if any.. Compute all the exponents.. Multiply or divide, in order that they are given, from left to right.. Add or subtract, in order that they are given, from left to right. Please Excuse My Dear Aunt Sally P + (0 ) + E + + M OR D + + A OR S + + + 0 8 Algebra Variable: A symbol used to measure a quantity that can change. Expression: A mathematical phrase that
contains operations, numbers and/or variables. Equation: A mathematical sentence that shows that two expressions are equivalent. Contains an equal (=) sign. Inverse Operation: The operation that reverses the effect of another operation (UNDO) Steps of Algebra. Write down the problem. Isolate the variable by doing the inverse operation on both sides. Cross out- on variable side. Draw Line. Drop down variable. Solve. Check Expressions Addition Words: x + Added to Plus Sum More than Subtraction Words: x - Subtracted Minus Difference Less than Take away Multiplication Words: x Times Multiplied By Product Groups of Division Words: x Divided by Quotient Into Decimals Adding/Subtracting Decimals Line up the decimal points. Annex (add) zeros if necessary Add or subtract as you would with whole numbers Remember to bring down the decimal point in the exact same spot into the answer. Ex:. + 8. =.0 (annexed zero) + 8.. Ex:. -. =. -.0 (annexed zero). Multiplying Decimals Multiply as you would with whole numbers. Ignore the decimals. DO NOT LINE UP THE DECIMALS. Count the total number of digits behind the decimals in the problem. That is how many places will be after the decimal in the product (answer). Ex:.. X + 0 0. Dividing Decimals by Whole Numbers:. 0.. - - 0-0 0 Bring up the decimal point into the quotient. Divide as you would with whole numbers. Annex zeros if necessary. Dividing Decimals by Decimals. 8. Change the divisor to a whole number by moving the decimal point to the right. Move the decimal point in the dividend the same number of spaces to the right. Annex zeros if necessary.. 8. Divide as you would with whole numbers. Remember to bring up the decimal point into the quotient.. 8 8. - - 0
Division with Decimal Remainders You won t know if there is a remainder until you do the problem. Set it up just like any other division problem. 0 0-8 - - 0 Since you won t leave the remainder as r, you need to annex up to zeros 0. 0. 0 0 0-8 - - 0 0 - =. 0-0 - This problem has a repeating decimal. Instead of writing., use the repeat bar over the digits that repeat, the in this case.. Terminating, Repeating, & Continuing Decimals Terminating: A decimal the ends on its own. Ex: 0. Repeating: A decimal in which one or more digits repeat infinitely. Ex: 0. = 0. Continuing: A decimal that neither terminates, nor repeats. Ex: 0.8 Factors To find all the factors of a number, make a T-Chart and write the number pairs on it, starting with. 8 8 8 Factors of 8:,,,,, 8,,,, 8 Greatest Common Factor (GCF) To find the greatest common factor, first list all the factors of each number: :,,, 8, :,,,,, 8,, The common factors of and are:,,, and 8. The greatest (largest) common factor of and is 8. Finding the GCF using the LADDER: Divide by prime numbers until no number can be divided evenly into both & 8 Multiply the numbers on the side: x x = 8 The GCF is 8. x 8 = and x 8 = Write in the distributive property form: o 8 ( + ) Prime & Composite Prime number: Has only two factors, and itself: ( = ) Composite number: A number that has three or more factors: 0 (,,, 0)
First Prime Numbers 8 0 8 0 8 0 8 0 8 0 8 0 8 0 8 80 8 8 8 8 8 8 8 88 8 0 8 00 Prime Factorization Factor Tree 8 8 - Write the number you are factoring at the top of the tree - Choose any pair of factors as branches. If either of these factors are not prime, you need to factor again - Continue until all the branches end in a prime number - Write only the prime factors in prime factorization form. x - You can only divide by prime numbers. Start with,,,, -Write the number you are factoring at the top of the ladder - See if you can divide it by. If so, write a two on the outside of the ladder, then write what the number divided by equals (8 =). - Continue this until the number cannot be divided by again. Go to three, and divide by three as many times as you can. - Continue with,, etc., if necessary, until you end up with a prime number. - Write the prime factors (excluding ) in prime factorization form. Least Common Multiple (LCM) Multiples of a number are the products of the number and other factors: = ; = 0; = Multiples of :, 0,, 0, To find the least common multiple of two numbers, first list out their multiples: :,,,,, 8,, :, 8,,, 0,, 8. First two common multiples:, Least (or lowest) Common Multiple: Converting Fractions to Decimals To change a fraction to a decimal, divide the numerator by the denominator. Annex zeros if needed. Write the fraction in lowest terms. 0.. 0 0-8 0-0 0 Converting Decimals to Fractions Decimals are fractions with a special set of denominators (tenths, hundredths, thousandths, etc.) and a special written form. To write a decimal as a fraction, say it aloud. You ll notice it sounds like a fraction:
Decimal: 0. Word name: nine tenths Fraction: Decimal: 0. Word name: forty-seven hundredths Fraction: Decimal:. Word name: three and two tenths Fraction: 0 00 0 Decimal:. Word name: five and twenty five hundredths Fraction: = 00 Converting Decimals & Percent using 00 MR. DL Decimal to percent Move the decimal two places to the right because you are multiplying by 00. Add the percent symbol. 0. = % 0. 0 = 0.% Percent to Decimal Move the decimal two places to the left because you are dividing by 00. If one is not present, add it to the end, then move it. Drop the percent symbol. % =. = 0..8% =. 8 = 0.8 Simplifying Fractions Whenever the numerator and denominator of a fraction can be divided by the same nonzero whole number (GCF) it can be reduced or written in lower terms. When the numerator and denominator can no longer be divided by the same nonzero whole number, it is in lowest terms or simplest form. EX: 0 Both can be divided evenly by which is the GCF of 0 &. 0 = = There is not a whole number that can be evenly divided into and, so is in lowest terms. Improper Fractions & Mixed Numbers Improper Fraction: a fraction where the numerator is larger than the denominator Mixed Number: a fraction with a whole number. Convert Improper Fraction to Mixed Number Divide the numerator by the denominator. If it divides evenly, then the answer is a whole number. If it does not divide evenly, keep the whole number and then the remainder becomes the new numerator and the denominator stays the same. = = = remainder Converting a Mixed Number into an Improper Fraction Using the order of operations, multiply the denominator by the whole number, then add the numerator. This becomes the new numerator. The denominator stays the same. = + = = Adding/Subtracting Fractions with LIKE Denominators Add or subtract the numerators. Write the new numerator, the denominator stays the same. Simplify when necessary. + = - = Common Denominators Common denominators may be found by different methods:
Multiply each fraction by denominator of the opposite fraction: EX: ; = = Find the LCM (least common denominator) of the denominators: :,, :,, Then, multiply the numerator by the number that you would need to multiply both the numerator and the denominator by to get the LCM as the new denominator. = = Equivalent Fractions Multiply the numerator and denominator by the same number to find equivalent fractions: = = 8 = = = = 8 Compare Fractions with LIKE Denominators If fractions have the same denominator, compare the numerators. greater because is than. because is less than. Ordering Fractions with UNLIKE Denominators Either find common denominators or convert to decimals, then put in order as requested. Comparing Fractions with UNLIKE Denominators One method is to get a common denominator by multiplying each fraction by the denominator of the opposite fraction, and then comparing numerators: 8 8 and = = <, so < Another method is to convert each fraction to a decimal, by dividing the numerator by the denominator, and then compare the decimals: and = 0. < = 0. < Adding or Subtracting Fractions with UNLIKE Denominators Since you need to have the same denominator to add or subtract fractions, multiply each fraction by denominator of the opposite fraction to find a common denominator. + = + = + = Adding or Subtracting Mixed Numbers Adding: Add the whole numbers, then add the fractions. If you do not have common denominators, you need to get common denominators before you may add. Write answer in lowest terms (reduce if necessary). 8 Subtraction: Do subtraction problems the same way. If you do not have common denominators, you need to get common + = = 8 8 8
denominators before you may subtract. + = + = + + = + = Multiplying Fractions Multiply the numerators. Multiply the denominators. Write the product in lowest terms (reduce) if necessary. = = If multiplying a whole number by a fraction, make the whole number a fraction by placing it over. = = = Dividing Fractions Invert (flip) the numerator and denominator of the second fraction (the right reciprocal). Change the operation to multiplication Multiply the numerators Multiply the denominators Write the product in lowest terms (reduce) if necessary. = = = If dividing a whole number by a fraction, or a fraction by a whole number, make the whole number a fraction by placing it over. Then follow the above steps. Fraction Jingle (By: Mrs. Mackey) I don t know what you ve been told, fractions are the way to go. Fractions, fractions, don t you know? Each operation has a different flow. (CHORUS) Sound off, /8 Knock it on down, / All the way down, /8, /, /8, /! (clap, clap) Adding and subtracting are so cool. It s quite easy, here s the rule. Change the denominators so they match. Then add the numerators, that s the catch. (CHORUS) Multiplication rules, we can name. The denominators aren t the same. Multiply the tops and then the bottoms. Simplify and then you got em! (CHORUS) Dividing fractions, that s the test. It s more confusing than the rest. We don t divide we multiply. By the right reciprocal, and that s no lie! Sound off, /8 Knock it on down, / All the way down, /8, /, /8, ooh rah! Rates, Ratios & Proportions Ratio: is a comparison of two quantities using division. Reduce when possible. Ratios can be compared as part to part or part to whole. Part to Part : to Part to Whole : to Rate: Compares quantities that have different units of measure. (Must be labeled) -liter bottle of soda costs $.8. rate = Price # of Liters = $.8 liters Unit Rate: The comparison to one unit. Unit rates make it easier to compare quantities. Divide both the numerator & denominator by the bottom number. $.8 liters = = $0. liter
Proportion: An equation that shows two equivalent ratios. 8 =? Integers 8 = Whole number counting numbers {,,,...}, zero {0}, and the negative of the whole number counting numbers {-, -, -,...} Cannot be fractions or decimals. Absolute Value: The distance of a number from zero on the number line. Numbers that are the same distance from 0 have the same absolute value. = - = - - 0 Distance can NEVER be negative. Opposites: Two numbers whose sum is 0. - is opposite to Same distance from zero on number line, but on opposite sides of zero. Zero (0) is its own opposite. Coordinate Plane Formed by a horizontal number line (x-axis) and a vertical number line (y-axis). The x-axis & y-axis intersect at (0, 0) called the origin. The coordinate plane is divided into quadrants - written in Roman numerals. Ordered Pair: An x and y coordinate that represents an exact point on the coordinate plane. (,-) represents an ordered pair. is the x coordinate & - is the y coordinate. Statistics & Data Analysis: Mean: (Average) Make everything into equal parts or fair share. Add the numbers and divide by the total number in the set:,,,,, = + + + + + = =.8 Mean =.8 - Round to the hundredths place if necessary. Mean Absolute Deviation (MAD): The average distance of all of the data values in a data set from the mean of the distribution. Median: (Middle Number) Place the numbers in ORDER. Find the middle number. Whack, Whack Median =,,,,,, =,,,,,, - If there isn t one middle number, find the median by adding the two middle numbers ( & ) and divide by.,,,,, =,,,,, + = =. Median =. Mode: (Most Frequent) Find the number that occurs most frequently. There can be more than one mode if two or more numbers occur most often,,,,,, = Mode:,,,,, = Modes: &,,,,, = No Mode Range: The difference between the greatest number and the least number:,,,,,, = = Range = Outlier: The number(s) that do not fit with the rest of the data. 0
Data Displays Box-and-Whisker Plot: A display that shows the distribution of values (median) in a data set separated into equal size groups. Number Summary - Minimum Value or Lower Extreme - Lower Quartile - Median - Upper Quartile - Maximum Value or Upper Extreme Interquartile Range (IQR): The difference between the values of the upper quartile (Q) and the lower quartile (Q). Histogram: A display that shows the distribution of numeric data. The range of data values, divided into intervals, is displayed on the horizontal axis. The vertical axis shows frequency in numbers or in percentages. The height of the bar over each interval indicates the count or percent of data values in that interval. The bars are fat & touching & are usually different colors. No key is needed. Line Plot/ Dot Plot: A way to organize data along a number line where the x s (or dots) above a number represent how often each value is mentioned. Probability The measure of how likely an event is to occur: - Impossible: 0% - Unlikely: % - As likely as not/ equally likely: 0% - Likely: % - Certain: 00% - Experimental Probability-the ratio of the number of times the event occurs to the total number of times the experiment is performed. P number of times the event occurs total number of trial Experiment- an activity involving chance that can have different results. Outcomes-the different results that can occur. Sample Space-the set of all possible outcomes Theoretical Probability- the ratio of the number of equally likely outcomes in an event to the total number of possible outcomes. P number of ways the event can occur total number possible outcomes Fair- An experiment with equally likely outcomes. Organized List/ Tree Diagram How many different sandwich combinations are possible with type of bread, meats and cheeses? H H B M M M H H H H Count the leaves (H s)= combinations Geometry Point: An exact location Line: A straight path that extends without end in opposite directions Ray: Has one endpoint and extends in only one direction Line Segment: Made of endpoints and all the points in-between Plane: A flat surface that extends without end in all directions Angles Obtuse Right Acute - 0 0-8 Straight Angle 80 Angle Relationships: Vertical Angles: Formed opposite each other when lines intersect (kissing angles). Opposite angles are equal.
Adjacent Angles: Side by side and have a common vertex and ray. They do not have to be congruent. Complimentary Supplementary angles whose angles whose measures have a measures have a sum of 0 sum of 80 Transversal 8 Corresponding Angles: = = Alternate Interior Angles: = = Alternate Exterior Angles: = = 8 Adjacent Angles: Supplementary Angles= Sum of 80 & & Classifying Lines: Parallel Lines: Lines in the same plane that NEVER intersect Perpendicular Lines: Intersect to form 0 angles or right angles Skew Lines: Lines that line in different planes. Neither parallel nor perpendicular Polygons: Triangle: sides Quadrilateral: sides Pentagon: sides Hexagon: sides Heptagon: sides Octagon: 8 sides Nonagon: sides Decagon: 0 sides Triangles Sum of angles are 80 Equilateral: All sides are equal Isosceles: sides are equal Scalene: No sides are equal Acute: All angles measure less than 0 Obtuse: angle measures greater than 0 Right: angle measures exactly 0 Quadrilaterals Parallelogram: Opposite sides are parallel & congruent. Opposite angles are congruent Rectangle: Parallelogram with right angles Rhombus: Parallelogram with congruent sides Square: Parallelogram with congruent sides and right angles Trapezoid: Quadrilateral with exactly parallel sides; may have right angles Transformations Moves a figure without changing its size or shape, so that the original figure and the transformed figure are always congruent Translation: A movement of a figure along a straight line (slide) Rotation: The movement of a figure around a point. Every quarter turn is 0 Reflection: When a figure flips over a line, creating a mirror image Line of Reflection: The line the figure is flipped over. Line Symmetry: A figure can be folded or reflected so that the parts of the figure match or are congruent. Tessellation: A repeating arrangement of one or more shapes that completely covers a plane with no gaps and no overlaps.
Perimeter Perimeter is the distance (outside area) around a figure: Add up all lengths as you go around perimeter is what you ve found! cm + cm + cm = 8 cm The perimeter of this figure is 8 centimeters Area The area of a figure is the number of square units inside the figure. Area of Squares & Rectangles & Parallelograms A = bh A = lw h b The area is square units, or u. If you were measuring in inches, it would be in. Night or day, day or night area equals base times height Area of Triangles: A= ½ bh or A= bh Area of triangles are easy to do base times height and divide by h h ½ ( ) = m Circles Center: The middle of the circle Radius: A line segment with one endpoint at the center of the circle and the other endpoint on the circle Chord: A line segment with both endpoints on a circledoes not have to go through the center of the circle, but can. Diameter: A chord that passes through the center of the circle. The length of the diameter is twice the length of the radius. Divides the circle into equal parts. Pi: () or. is the ratio of a circle's circumference to its diameter. A= Πr² Area of a Circle: A= Π r² A=. x ² A=. x A= 8. in² in Area & Perimeter Formulas Rectangle: o A=bh o A= lw o P= l + w Square: o A= s o A= bh o A= lw o P= s Parallelogram: o A= bh o P= l + w Rhombus: o A= bh o P= s Triangle: o A= bh o A= ½ bh o P= s + s + s Trapezoid: o A= ½ h (b + b ) o P= s + s + s + s Circle: o A= Πr² o C= Πd o C= Πr KEY: A = Area P = Perimeter b = base h = height l = length w = width s = sides Π = pi =. or r = radius d = diameter C = Circumference B= Area of base Ex: base = m height = m ( ) = m OR C= Πd OR C=. x 0 C=. in C= Πr C= x. x C= 0 x. C=. in
Three-Dimensional Solids Polyhedron: Threedimensional object, or solid figure, with flat surfaces. Face: Flat surfaces that make up polyhedrons. Edge: The side that faces share. Vertex: A point at which or more edges meet. (Corner) Vertex Prism: A polyhedron with congruent, parallel bases and other faces that are all parallelograms. Named for the shape of its bases. Cube Face Base Rectangular Prism Hexagonal Prism Triangular Prism Edge Cylinder: Has congruent, parallel bases, but bases of a cylinder are circular. Not a polyhedron because not every surface is a polygon. Bases: Prisms & Cylinders have congruent & parallel ends. Pyramid: Has polygon shaped base, and the other faces are triangles that come to a point. Named for the shape of its base. Square Pyramid Triangular Pyramid Cone: A circular base and a curved surface that comes to a point. Not a polyhedron because not every surface is a polygon. Volume Formulas Prisms: V=Bh Volume = Area of the Base x height cm V = Bh V = Rectangle Base (l w) V = ( x ) x V = x V = cm Cylinder V = r h Radius squared x height x. V = r h V = x x. V= x x. V = 0. m Square or Rectangular Pyramid V = Bh B= Find the area of the base x height cm cm cm V = Bh V = B (area of the base= l x w) B = x = 0 V = 0 x = 0 V = 0 = 0 ft Triangular Pyramid cm cm
V = Bh V = B (area of the base= ½ bh) B = ½ ( x ) = V = x = V = = cm Cone V = r h (area of the base- r ) Radius squared x. x height V = r h V = = x. x =.8 V =.8 = V =. yd Sphere V = r V = r V = radius cubed x. x V = = x. x = 08.08 =.0 in Nets - Dimensional pattern that can be folded into a - dimensional figure. Surface Area The area required to cover a - dimensional shape. Like wrapping a present. Fraction, Decimal & Percent Benchmarks Percent Fraction Decimal 0%.% % % 0% % % 0% %.% 0% 0% 0%.%.% % 80% 8% 8.% 00% 0 8 0 8 8 8.0 or 0.....0 or 0....0 or 0....0 or 0..0 or 0..0 or 0... or...80 or 0.8.8.8.0 or To convert to an equivalent decimal or a percent, multiply the decimal. x the numerator and round if needed. To make a percent, multiply the decimal x 00. EX: = (. x ) =. or. rounded. x 00 = % Squares & Square Roots ² ² ² ² ² ² ² 8² 8 ² 8 8 0² 00 00 0 ² ² Cubes & Cubed Roots ³ ³ 8 8 ³ ³ ³ 0³ 000 000 0
Math Properties Commutative Property of Addition- Add numbers in any order. + = + Commutative Property of Multiplication- Multiply numbers in any order. x = x Associative Property of Addition- When adding, group numbers together with parentheses. (+) + = + ( + ) Associative Property of Multiplication- When multiplying, group numbers together with parentheses. ( x ) x = x ( x ) Identity Property of Zero- The sum of any number and zero is equal to the number. + 0 = Identity Property of One- The product of any number and one is equal to the number. x = Property Customary of Zero- Volume The product of any cup number and 8 zero fl. oz is zero. gal x 0 = 0 qts gal 8 pts Distributive gal Property- cups Find the sum first qt and then multiply, cups or multiply each number in the pt cups sum and then add. x ( + ) = x + x = + 0 = OR x ( + ) = + 0 = ( + ) = + 0 is the GCF of & 0 Metric Prefixes Kilo = 000 Deci = 0. Hecto = 00 Centi = 0.0 Deka = 0 Milli = 0.00 Metric Weight/ Mass (GRAMS) kg 000 g kg,000,000 mg g 000 mg Customary/ Metric Conversions ( LENGTH) in. cm cm 0. in ft 0.8 cm m.8 ft mi. km km 0. mi m. in in 0.0 m m.0 yd Customary Time min 0 sec hr 0 min day hr wk days yr mo leap year days yr days yr wks Customary Length ft in yd in yd ft mi 80 ft mi 0 yd Customary Weight lb oz ton 000 lbs Customary/ Metric Conversions (CAPACITY) pt 0. L L. pt qt 0. L L.0 qt gal. L L 0. gal Customary/ Metric Conversions (MASS) oz 8. g g 0.00 oz lb 0. kg kg. lb
Customary Volume cup 8 fl. oz gal qts gal 8 pts gal cups qt cups pt cups
MUTIPLICATION TABLE x 0 8 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 8 0 0 8 0 8 0 0 8 0 0 8 0 8 0 8 0 0 0 0 0 0 0 0 8 0 8 0 0 8 0 8 8 0 8 0 8 80 88 0 8 8 0 08 0 0 0 0 0 0 0 0 0 80 0 00 0 0 0 88 0 0 8 0 8 08 0 0 8 0 0 0 8 0 8 8 0 8 0 0 0 0 0 0 0 80 8
0