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MATH TIPS

MATH TIPS. for PARENTS. NUMBER PROPERTIES THE OPERATION CALLED ADDITION. Associative Property of Addition:. Changing the grouping of the terms (addends) will not change the sum (answer in addition) . In Arithmetic: (5 + 3) + 2 = (3 + 2) In Algebra: (a + b) + c = a + (b + c).

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MATH TIPS

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  1. MATH TIPS for PARENTS

  2. NUMBER PROPERTIESTHE OPERATION CALLED ADDITION

  3. Associative Property of Addition: • Changing the grouping of the terms (addends) will not change the sum (answer in addition). In Arithmetic:(5 + 3) + 2 = (3 + 2)In Algebra:(a + b) + c = a + (b + c)

  4. Commutative Property of Addition • Changing the order of the numbers (addends) will not change the sum (answer in addition).In Arithmetic:8 + 4 = 4 + 8In Algebra:a + b = b + a

  5. Identity Property of Addition • Zero added to any given number (given addend), the sum will equal the given number (given addend). • In Arithmetic:6 + 0 = 6In Algebra:a + 0 = a

  6. Inverse Operation of Addition • Subtraction undoes the operation called addition.In Arithmetic:If 7 + 4 = 11, then11 - 7 = 4 and11 - 4 = 7In Algebra:a + b = c, thenc - a = b and c - b = a

  7. THE OPERATION CALLED SUBTRACTION

  8. Inverse Operation of Subtraction • Addition undoes the operation called subtraction.In Arithmetic:If16 - 9 = 7, then9 + 7 = 16 and 7 + 9 = 16In Algebra:c - b = a, thenb + a = cand a + b = c

  9. THE OPERATION CALLED DIVISION

  10. Inverse Operation of Division • Multiplication undoes the operation called division.In Arithmetic:If 48 / 8 = 6, then8 x 6 = 48and 6 x 8 = 48In Algebra:c / b = a, thenb x a = canda x b = c

  11. THE OPERATION CALLED MULTIPLICATION

  12. Associative Property of Multiplication • Changing the grouping of the factors will not change the product (answer in multiplication).In Arithmetic:(5 x 4) x 2 = 5 x (4 x 2) In Algebra:(a x b) x c = a x (b x c) or (ab) c = a (bc)

  13. Commutative Property of Multiplication • Changing the order of the factors (multiplicand and multiplier) will not change the product (answer in multiplication).In Arithmetic:6 x 9 = 9 x 6 In Algebra:a x b = b x a or ab = ba

  14. Identity Property of Multiplication • The product (answer in multiplication) and 1 is the original number.In Arithmetic:7 x 1 = 7In Algebra:a x 1 = a or a • 1 = a

  15. Multiplication Property of Zero • The product (answer in multiplication) of any number and zero is zero.In Arithmetic:9 x 0 = 0In Algebra:a x 0 = 0 or a • 0 = 0Multiplication is repeated addition.8 x 4 = 8 + 8 + 8 + 8

  16. Distributive Property of Multiplicationover Addition or Subtraction • Multiplication by the same factor may be distributed over two or more addends. This property allows you to multiply each term inside a set of parentheses by a term inside the parentheses. *In many cases this is an excellent vehicle for mental math.In Arithmetic:OVER ADDITION 5(90 + 10) = (5 x 90) + (5 x 10)OVER SUBTRACTION 5(90 - 10) = (5 x 90) - (5 x 10)In Algebra:OVER ADDITION a(b + c) = (a x b) + (a x c) ora(b + c) = ab + acOVER SUBTRACTION a(b - c) = (a x b) - (a x c)

  17. GLOSSARYofMATHEMATICAL TERMS

  18. Add/Addend/Addition/Array ADDTo put one thing, set or group with another thing, set or group. ADDENDNumbers to be added.Example: 12 + 23 = 25 a + b + c = abc ADDITIONThe operation of putting together two or more numbers, things, groups or sets.Example: 8 + 2 + 4 = 14 is an addition problem ARRAYAn orderly arrangement of persons or things, rows and columns. The number of elements in an array can be found by multiplying the number of rows by the number of columns.Example: * * * * * * * * * * * * * * * * * * 3 x 6 = 18

  19. Associative Property of Addition-Multiplication/Attribute ASSOCIATIVE PROPERTY OF ADDITIONThe way in which three numbers to be added are grouped two at a time does not affect the sum.Example: 3 + (5 + 6) = (3 + 5) + 6 3 + 11 = 8 + 6 14 = 14 ASSOCIATIVE PROPERTY OF MULTIPLICATIONThe way in which three numbers to be multiplied are grouped two at a time does not affect the product.Example: 3 x (2 x 6) = (3 x 2) x 6 3 x 12 = 6 x 6 36 = 36 ATTRIBUTEA quality that is thought of as belonging to a person of thing. Characteristics; such as, size, shape, color and/or thickness.

  20. Average/Axis AVERAGEA number found by dividing the sum (total) of two or the sum (total) of two or more quantities by the number of quantities. The average of 86, 54, 9 and 93 is 68.STEP 1 STEP 2 86 68is the average 54 How many addends? 4) 272 39 Quantity is 4 - 24 + 93 32272sum or total - 32 0 AXIS (axes)Horizontal and vertical number lines in a number plane.

  21. Bar Graph/Braces Colors the Class Likes BAR GRAPHA picture in which number informationis shown by means of bars of different lengths. BRACESBraces are symbols { }. They are used to list names of numbers (elements) of a set.Example: { Pauline, April, Joni, Jackie}is a set of secretaries. {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}is a set of the days of the week. {1, 2, 3, 4, 5, 6, 7, 8, 9}is a set of counting numbers from 1 to 9. 25 20 15 10 0

  22. Capacity/Cardinal Number/Centigrade/Cent/Centimeter CAPACITYThe amount that can be held in a space. CARDINAL NUMBERA number that tells how many there are.Example: There are five squares CENTIGRADEDivided into one hundred degrees (100%). On the centigrade temperature scale, freezing point is at zero degrees (0%). The boiling point water is at one hundred degrees (100º)* Celsius scale is the official name of the temperature CENTA coin of the United States and Canada. One hundred cents make a dollar. CENTIMETERA unit of length in the metric system. A centimeter is equal to one hundredths of a meter or .39 of an inch.

  23. Century/Closed Figure/Closure CENTURYA period of one hundred years. CLOSED FIGUREA geometric figure that entirely encloses part of the plane. CLOSUREA property of a set of numbers such that the operation with two or more numbers of that set results in a number of the set.Example: In addition and multiplication with counting numbers, the results is a counting numbers. 2 + 4 = 6; 2 x 4 = 8 Thus, the counting numbers are closed under these two operations. In subtraction, if 4 is subtracted from 2, the result (-2) is not a counting number. Also in dividing a 2 by 4, the results (1/2) is not a counting number. Thus, the counting numbers are not closed with respect to subtraction and division.

  24. Combine/Common/Common Factor/Common Multiple COMBINETo put (join) together. COMMONBelonging equally to all. COMMON FACTORA common factor of two or more numbers is a number which is a factor of each of the numbers.Example: 8 = {1, 2, 4, 8} 32 = {1, 2, 4, 8, 16, 32} 1, 2, 4 and 8 are the common factors of 8 and 32 COMMON MULTIPLEA common multiple of two or more numbers is a number which is a multiple of each of the numbers.Example: 12 = {12, 24, 36, 48, 72, 84, 96, 108, 120} 15 = {15, 30, 45, 60, 75, 90, 105, 120, 135, 150} 60 and 120 are the common multiples

  25. Commutative Property of (Addition)(Multiplication)/Compare/Composite Number COMMUTATIVE PROPERTY OF ADDITIONThe order of two numbers (addends) may be switched around and the answer (total, sum) is the same.Example: 7 + 4 = 11 and 4 + 7 = 11; therefore, 7 + 4 = 4 + 7 COMMUTATIVE PROPERTY OF MULTIPLICATIONThe order of two numbers (factors) may be switched around and the answer (total product) is the same.Example: 8 x 6 = 48 and 8 x 6 = 48; therefore, 8 x 6 = 6 x 8 COMPARETo study, discover and/or find out how persons or things are alike or different. COMPOSITE NUMBERA number which has factors other than itself and one.Since 16 = 1 x 16, 2 x 8 and 4 x 4, it is a composite number.

  26. Conditional Sentence/Congruent Figure/Conjecture/Conjunction CONDITIONAL SENTENCE (In logically thinking)A sentence of the form “if. . ., then. . .?Example: If 6 x 7 = 42 and 7 x 6 = 42, Then 42  6 = 7 and 42  6 = 7 CONGRUENT FIGUREGeometric shapes consisting of the same shape and size.Example: 8 x 6 = 48 and 8 x 6 = 48; therefore, 8 x 6 = 6 x 8 CONJECTUREA guess resulting from an experiment.Example: 2, 4, 6, 8, 10 are even numbers; therefore, even numbers must have 0, 2, 4, 5, or 8 in the ones’ place. CONJUNCTION (In logically thinking)A two-part sentence joined by “and” to form true parts.Example: 1/4 + 1/4 = 2/4 = 1/2

  27. Coordinates/Counting Number/Decade/Decimal COORDINATESTo numbers, an ordered pair, used to plot a point in a number plane. COUNTING NUMBER (Natural Numbers)To numbers, an ordered pair, used to plot a point in a number plane.Example: 1, 2, 3, 4, 5. . . *There is no longest number. Counting numbers are infinite. DECADEA period of ten years. DECIMALNames the same number as a fraction when the denominator is 10, 100, 1000. . . It is written with a decimal point.Example: .75

  28. Decimal System/Diagonal/Degree/Denominator DECIMAL SYSTEMA plan for naming numbers that is based on ten is called a decimal system of numeration. The Hindu-Arabic system is a decimal system. DIAGONALA straight line that connects the opposite corners of a rectangle.Example: DEGREEA unit of angle measurement. DENOMINATORIn 3/5 the denominator is 5. It tells the number of equal parts, groups or sets the whole was divided.

  29. Difference/Digit/Disjoint Sets DIFFERENCEThe number which results when one number is subtracted from another is called the difference. It is a missing addend in addition.Example: 7 - 4 = 3 the difference is 3 DIGITAny one of the basic numerals, 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, is a digit.The numeral 12 is a two-digit numeral and the numeral 354 is a three digit numeral. DISJOINT SETSSets that have no members in common are disjoint sets.Example: Set A = {a, b}, Set B {1, 2, 3}. Sets A and B are disjoint

  30. Distributive Property of Multiplication over Addition/Divide/Dividend DISTRIBUTIVE PROPERTY OFMULTIPLICATION OVER ADDITIONMultiplication by the same factor may be distributed over two or more addends.Example: 3 x (6 + 4) = (3 x 6) + (3 x 4) = 18 + 12 = 30 DIVIDETo separate into equal parts, pieces, groups or sets..Example: x x x x x x x x x x 10  2 = 5 DIVIDENDA number that shows the total amount to be separated into equal parts, groups of sets by another number.Example: 100  25 = 4, the dividend is 100

  31. Divisible/Divisor/Element/Element of a Set/Empty Set DIVISIBLECapability of being separated equally without a remainder.Example: 18 is divisible by 1, 2, 3, 6, 9 and 18 DIVISORA number that tells what kind of equal parts, groups or sets the dividend is to be separated. ELEMENTA member of a set. ELEMENT OF A SETA member of a set. EMPTY SETThe set which has no members. The number of the empty set is zero. A symbol for the empty set is { }.

  32. Equal/Endpoint/Equal Sets/Equal Sign EQUALA relationship between two expressions denoting exactly the same or equivalent quantities.Example: The two expressions 2 + 6 and 3 + 5 are said to be equal because they both denote exactly the same quantity. ENDPOINTA point at the end of a line segment or ray. EQUAL SETSTwo sets with exactly the same things, elements or members.Example: A = {1, 2, 3} and B = {3, 2, 1} EQUAL SIGNThe equal sign shows that two numerals or expressions name the same number.Example: 10 + 9 = 19In a true sentence, the equal sign shows that the numerals on each side of the sign name the same number.

  33. Equation/Equivalent Sets/Estimate EQUATIONA number sentence in which the equal sign = is used in an equation.Example: 6 + = 10 and 8 - 3 = are equations EQUIVALENT SETSIf the members of two sets can be matched one to one, the sets are equivalent. Equivalent sets have the same number of members/elements. ESTIMATEAn estimate is an approximate answer found by rounding numbers.Example: 22 + 39 = , 22 may be rounded to 20, 39 may be rounded to 40. The estimated sum is 20 + 40 or 60

  34. Even Number/Expanded Numeral/Exponent EVEN NUMBERAn integer that is divisible by 2 without a remainder.Example: 0, 2, 4, 6. . . Are even numbers EXPANDED NUMERALAn expanded numeral is a name for a number which shows the value of the digits.Example: An expanded number for 35 is 30 + 5 or ( 3 x 10) + (5 x 1) EXPONENTA number which tells how many times a base number issued as a factor. In the example below the base numbers are 10, 3, and 9.Example: 10 = 10 x 10 3 = 3 x 3 x 3 10 = 10 x 10 x 10 x 10 x 10 x 10 9 = 9 x 9 x 9 x 9

  35. Factors/Factor Tree/Fahrenheit FACTORSNumbers to be multiplied. In 2 x 4 = 8, the factor are 2 and 4.FACTOR TREEA diagram used to show the prime factors of a number. Example: 24 6 x 4 2 x 3 2 x 2 24 = 2 x 3 x 2 x 2 or 2 x 3 FAHRENHEITOf or according to the temperature scale of which 32 degrees (32º) is the freezing point of water and 212 degrees is the boiling point of water.

  36. Fraction-Fractional Numbers/Greater Than/Greatest Common Factor FRACTION FRACTIONAL NUMBEREqual parts of a whole thing, group or set. A number named by a numeral such as 1/2, 2/3, 6/2, 8/4. GREATER THANLarger than or bigger than something else. In greater than the symbol >, means that the number named at the left is greater than the number named at the right.Example: 8 > 3 is a true sentence GREATEST COMMON FACTORThe greatest common factor (GCF) of two or more counting numbers is the largest counting which is a factor of each of the counting numbers.Example: 10 = {1, 2, 5} 12 = {1, 2, 3, 4, 6, 12} 2 is the G.C.F. for 10 and 12

  37. Graph GRAPHA graph shows two sets of related information by the use of pictures, bars, lines or a circle. Graphs may be constructed using horizontal or vertical positions. BOYS’ PERFECT ATTENDANCETEMPERATURE RECORD MonthGirls Present 20 April 10 May June 0 Each symbol represents 3 girls 10 11 12 1 2 3 Graphs continued on next page             

  38. Graph/Hindu Arabic Numeration System GRAPHS(continued) 10,000 9,000 8,000 7,000 6,000 5,000 4,000 3,000 2,000 1,000 0 Caribbean Red North Japan HINDU ARABIC NUMERATION SYSTEM(Base Ten Decimal Numeration System)There are 10 digits; namely, 0, 1, 2, 3, 4, 5, 6, 7, 8 and 9. All whole numbers may be represented by using the digits and Base Ten place value (one, tens, hundreds. . .)Example: 96,5200 = (9 x 10,000) + (6 x 1,000) + (5 x 100) + (2 x 10) + (0 x 1)or (9 x 10) + (6 x 10) + (5 x 10) + (2 x 10) + (0 x 1)

  39. Horizontal/Identity Element of (Addition)(Multiplication)/Inequality/Integer HORIZONTALStraight across. Travels from west to east and east to west.Example: 965 x 4 = 3,860 IDENTITY ELEMENT OF ADDITIONThe sum of any number and zero is the other number.Example: 6 + 0 = 6 IDENTITY ELEMENT OF MULTIPLICATIONThe sum of any number and one is that number.Example: 6 x 1 = 6 INEQUALITYA mathematical sentence which states that two expressions de not name the same number. The signs < and > are usually used. INTEGERThe integers consist of the counting numbers, zero and the negatives of the counting numbers.Example: -5, -4, -3, -2, -1, 0, 1, 2, 3, 4, 5. . .

  40. Intersection of Sets/Joining Sets/Kilometer INTERSECTION OF SETSThe set consisting of all members which are common to two or more sets.Example: 12 14 3 1 7 4 2 6 12 14 JOINING SETSForming one set which contains all the members of two or more sets.Example: If Set A = {a, b} and Set B = {3, 4}, Sets A and B may be joined to form the set C = {a, b, 3, 4} KILOMETERA unit of length in the metric system. A kilometer (KM) is equal to 1000 meters, or about .62 of a mile.

  41. Least Common Multiple/Length LEAST COMMON MULTIPLEThe least common multiple of two or more counting numbers is the smallest counting numbers which is a multiple of each of the counting numbers.Example: What are some multiples of both 4 and 6? Set of multiples of 4 = {4, 8, 12, 16, 20, 24, 28, 32,. . .} Set of multiples of 6 = {6, 12, 18, 24, 30, 36,. . .} 12 is multiple of both 4 and 6. Another multiple of both 4 and 6 is 24. Therefore, 12 and 24 are called common multiples of 4 and 6. 12 is the Least Common Multiple (LCM). LENGTHThe distance from one end to the other end. Long represents how long something is from the beginning to the end. Endpoint to endpoint.

  42. Less Than/Lowest Terms/Measure/Measure of a Set LESS THANSmaller than something else. In less than the symbol “<“ means that the number to the left of the symbol is smaller than the number to the right of the symbol.Example: 104 < 140; 5 + 6 < 6 + 6; 1/6 < 1/4 LOWEST TERMSA fraction is in the lowest or simplest form if the numerator and denominator have no other common factors besides 1.Example: The lowest terms of 8/32 is 1/4 MEASURETo find or show the size, weight or amount of something. MEASURE OF A SETEach thing belonging to a set is a member of the set. It is also called an element of the set.Example: In a set, A = {R, S, T}, R, S, and T are members/elements of set A.

  43. Meter/Metric System/Minuend/Minus METERThe basic unit of measure is the metric system. The meter is about 39 inches long. METRIC SYSTEMA decimal system used for practically all scientific measurement. The standard unit of length is the meter. MINUENDThe number of things, members or elements in all (whole set) before subtracting.Example: 904 is the minuend of 904 - 756 = 148 The number from which another number is taken away (subtracted). MINUSDecreased by. Lower or less than.Example: 12 - 5 = 7 The numeral 12 is decreased by 5 or minus 5.

  44. Mixed Numeral/Multiple/Multiplicand/Multiplication MIXED NUMERALA numeral which consists of numerals for a whole number and a fractional number.Example: 3 MULTIPLE A number that is multiplied a certain number of times.Example: Multiples of 10 are 10, 20, 30, 40, 50. . . Multiples of 3 are 6, 9, 12, 15, 18. . . MULTIPLICAND A number that is to be multiplied by another number.Example: 36 x 14, 36 is the multiplicand MULTIPLICATIONThe operation of taking a number and adding it to itself a certain number of times.Example: 4 x 3 = 4 + 4 +4 25 x 6 = 25 + 25 + 25 + 25 + 25 + 25

  45. Multiplier/Multiply/Natural Numbers/Negative Numbers/Number Sentence MULTIPLIERA number that tells how many times to multiply anotherExample: 7 x 4 means that 7 will be multiplied 4 times. MULTIPLYTo add a number to itself a certain number of times. Shortcut to addition. NATURAL NUMBERSCounting numbers. NEGATIVE NUMBERSNumbers less than 0.Example: -5, -6, -7, -4, -3, -2. . . NUMBER SENTENCEA sentence of numerical relationship.Example: 2 + 5 = 1 + 6 3 + 8 > 6 1 x 3 < 9 - 2

  46. Numeral/Numeration/Numerator NUMERALA symbol for a number.Example: The number word six may be denoted by the symbol 6; thus, 6 is a numeral. NOTE: The fundamental operations(addition, subtraction, multiplication, division) are performed with numbers, not with numerals. The word “numeral” is used only when referring to the whether to use the word “number” or “numeral,” use the word NUMERATION A system to name numbers in various ways. NUMERATOR In 3/5, the numerator is 3. The numerator tells the number of equal parts, groups or sets that is being used.

  47. Odd Number/One-to-One Correspondence ODD NUMBERAn integer which is divisible by 2 with a remainder.Example: /// ONE-TO-ONE CORRESPONDENCE A one -to-one matching relationship. If to every member in one set there corresponds one and only one member in a second set, and to every member in the second set there corresponds one and only member in the first set, the sets are said to be in one-to-one correspondence.Example: If every seat in a room is occupied by a person, and no person is standing, there is a one-to-one correspondence between the number of persons and the number of seats.

  48. Open Sentence/Operation/Order OPEN SENTENCEA mathematical sentence which contains a variable such as n, x, , or.Example: 3 + = 8An open sentence cannot be judged true or false. When the variable is replaced by a numeral, the open sentence becomes a statement. OPERATION A specific process for combining quantities.Example: Addition, subtraction, multiplication, division ORDER The way in which something is arranged.Example: 1, 2, 3, 4. . . A, B, C, D. . . 9, 8, 7, 6. . . 3, 6, 9, 12. . . Z, Y, X, W. . . First, Second, Third, Fourth. . .

  49. Ordinal Number/Pair/Per/Percent ORDINAL NUMBERA number which indicates the order place of a member of a set in relation to other members of the same set. Example: 1st, 2nd, 3rd. . . PAIR Two persons, animals, or things that are alike/ that go together.Example: A pair of gloves PER For each. Similar and are matched to go together.Example: eggs per dozen PERCENTRatio with 100 as its second number. Percent means per hundred.Example: % = /100

  50. Picture Graph/Place Value/Prime Number PICTURE GRAPHA graph which uses picture symbols to show number information.Example: The pictograph shows how much money 4 children earned last week. Each means 10 cent.Cierra  Alex      Paul     Calin     PLACE VALUE Place value is the value of each place in a plan for naming numbers. The value of the first place on the right, in our system of naming whole numbers is one. The value of the place to the left of ones place is then. . . [Tens/Ones] PRIME NUMBER A number greater than one which has factors of only itself and one. 2, 3, 5, 7, 11 and 13 are just a few of the prime numbers.

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