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NUMBER SENSE AT A FLIPNUMBER SENSE AT A FLIP
NUMBER SENSE AT A FLIPNUMBER SENSE AT A FLIP
Number Sense Number Sense is memorization and
practice. The secret to getting good at number sense is to learn how to recognize and then do the rules accurately . Then learn how to do them quickly. Every practice should be under a time limit.
The First StepThe first step in learning number sense should be to memorize the
PERFECT SQUARES from 12 = 1 to 402 = 1600 and the PERFECT CUBES
from 13 = 1 to 253 = 15625. These squares and cubes should be learned
in both directions. ie. 172 = 289 and the 289 is 17.
2x2 Foil (LIOF)
23 x 1223 x 121. The last number is the units digit of the
product of the unit’s digits2. Multiply the outside, multiply the inside
3. Add the outside and the inside together plus any carry and write down the units digit
4. Multiply the first digits together and add and carry. Write down the number
Work Backwards
The Rainbow Method
3(2)2(2)+3(1)2(1)672
276
Used when you forget a rule about 2x2 multiplication
2x2 Foil (LIOF)
23 x 1223 x 12Work Backwards
The Rainbow Method
Used when you forget a rule about 2x2 multiplication
1.45 x 31=
2.31 x 62=
3.64 x 73=
4.62 x 87=
5.96 x74=
Squaring Numbers Ending In 5
757522
1. First two digits = the ten’s digit times one more than the ten’s digit.
2. Last two digits are always 25
7(7+1) 25 =56 25
Squaring Numbers Ending In 5
757522
1.45 x 45=
2.952=
3.652=
4.352=
5.15 x 15=
Consecutive Decades
35 x 4535 x 451. First two digits = the small ten’s digit times
one more than the large ten’s digit.2. Last two digits are always 75
3(4+1) 75 =15 75
Consecutive Decades
35 x 4535 x 451.45 x 55=
2.65 x 55=
3.25 x 35=
4.95 x 85=
5.85 x75=
Ending in 5…Ten’s Digits Both Even
45 x 8545 x 851. First two digits = the product of the ten’s
digits plus ½ the sum of the ten’s digits.2. Last two digits are always 25
4(8) + ½ (4+8) 25 =38 25
Ending in 5…Ten’s Digits Both Even
45 x 8545 x 851.45 x 65=
2.65 x 25=
3.85 x 65=
4.85 x 25=
5.65 x65=
Ending in 5…Ten’s Digits Both Odd
35 x 7535 x 751. First two digits = the product of the ten’s
digits plus ½ the sum of the ten’s digits.2. Last two digits are always 25
3(7) + ½ (3+7) 25 =26 25
Ending in 5…Ten’s Digits Both Odd
35 x 7535 x 751.35 x 75=
2.55 x 15=
3.15 x 95=
4.95 x 55=
5.35 x 95=
Ending in 5…Ten’s Digits Odd&Even
35 x 8535 x 851. First two digits = the product of the ten’s digits
plus ½ the sum of the ten’s digits. Always drop the remainder.
2. Last two digits are always 75
3(8) + ½ (3+8) 75 =29 75
Ending in 5…Ten’s Digits Odd&Even
35 x 8535 x 851.45 x 75=
2.35 x 65=
3.65 x 15=
4.15 x 85=
5.55 x 85=
Multiplying By 12 ½
32 x 12 ½32 x 12 ½ 1. Divide the non-12 ½ number by 8.
2. Add two zeroes.
8 = 4+00
=4 00
(1/8 rule)
32
Multiplying By 12 ½
32 x 12 ½32 x 12 ½
(1/8 rule)
1. 12 ½ x 48=
2. 12 ½ x 88 =
3. 888 x 12 ½ =
4. 12 ½ x 24 =
5. 12 ½ x 16=
Multiplying By 16 2/3
42 x 16 2/342 x 16 2/3 1. Divide the non-16 2/3 number by 6.
2. Add two zeroes.
6 = 7+00
=7 00
(1/6 rule)
42
Multiplying By 16 2/3
42 x 16 2/342 x 16 2/3
(1/6 rule)
1. 16 2/3 x 42 =
2. 16 2/3 x 66 =
3. 78 x 16 2/3 =
4. 16 2/3 x 48=
5. 16 2/3 x 120=
Multiplying By 33 1/3
24 x 33 1/324 x 33 1/3
243 = 8+00
=8 00
(1/3 rule)
1. Divide the non-33 1/3 number by 3.
2. Add two zeroes.
Multiplying By 33 1/3
24 x 33 1/324 x 33 1/3
(1/3 rule)
1. 33 1/3 x 45=
2. 33 1/3 x 66=
3. 33 1/3 x 123=
4. 33 1/3 x 48=
5. 243 x 33 1/3=
Multiplying By 25
32 x 2532 x 251. Divide the non-25 number by 4.
2. Add two zeroes.
4 = 8+00
=8 00
(1/4 rule)
32
Multiplying By 25
32 x 2532 x 25
(1/4 rule)
1. 25 x 44=
2. 444 x 25=
3. 25 x 88=
4. 25 x 36=
5. 25 x 12=
Multiplying By 50
32 x 5032 x 501. Divide the non-50 number by 2.
2. Add two zeroes.
2 = 16+00
=16 00
(1/2 rule)
32
Multiplying By 50
32 x 5032 x 50
(1/2 rule)
1. 50 x 44=
2. 50 x 126=
3. 50 x 424=
4. 50 x 78=
5. 50 x 14=
Multiplying By 75
32 x 7532 x 751. Divide the non-75 number by 4.
2. Multiply by 3.
3. Add two zeroes.
4= 8x3=24+00
=24 00
(3/4 rule)
32
Multiplying By 75
32 x 7532 x 75
(3/4 rule)
1. 75 x 44=
2. 75 x 120=
3. 75 x 24=
4. 48 x 75=
5. 84 x 75=
Multiplying By 37 1/2
37 1/2 x 2437 1/2 x 24
00=9 00
(3/8 rule)
(3/8)24
Multiplying By 62 1/2
62 1/2 x 5662 1/2 x 56
00=35 00
(5/8 rule)
(5/8)56
Multiplying By 87 1/2
87 1/2 x 4887 1/2 x 48
00=42 00
(7/8 rule)
(7/8)48
Multiplying By 83 1/3
83 1/3 x 3683 1/3 x 36
00=30 00
(5/6 rule)
(5/6)36
Multiplying By 66 2/3
66 2/3 x 6666 2/3 x 66
00=44 00
(2/3 rule)
(2/3)66
Multiplying By 125
32 x 12532 x 1251. Divide the non-125 number by 8.
2. Add three zeroes.
328 = 4+000
=4 000
(1/8 rule)
Multiplying By 125
32 x 12532 x 125
(1/8 rule)
1. 125 x 48=
2. 125 x 88=
3. 125 x 408=
4. 125 x 24=
5. 125 x 160=
Multiplying When Tens Digits Are Equal And The Unit Digits Add To 10
32 x 3832 x 381. First two digits are the tens digit
times one more than the tens digit
2. Last two digits are the product of the units digits.
3(3+1)=12 16
2(8)
Multiplying When Tens Digits Are Equal And The Unit Digits Add To 10
32 x 3832 x 381. 34 x 36=
2. 73 x 77=
3. 28 x 22=
4. 47 x 43=
5. 83 x 87=
Multiplying When Tens Digits Add To 10 And The Units Digits Are Equal
67 x 4767 x 471. First two digits are the product of the tens
digit plus the units digit
2. Last two digits are the product of the units digits.
6(4)+7=31 49
7(7)
Multiplying When Tens Digits Add To 10 And The Units Digits Are Equal
67 x 4767 x 471. 45 x 65=
2. 38 x 78=
3. 51 x 51=
4. 93 x 13=
5. 24 x 84=
Multiplying Two Numbers in the 90’s
97 x 9497 x 941. Find out how far each number is from 100
2. The 1st two numbers equal the sum of the differences subtracted from 100
3. The last two numbers equal the product of the differences
100-(3+6)=91 18
3(6)
Multiplying Two Numbers in the 90’s
97 x 9497 x 941. 98 x 93=
2. 92 x 94=
3. 91 x 96=
4. 96 x 99=
5. 98 x 98=
Multiplying Two Numbers Near 100
109 x 106109 x 1061. First Number is always 1
2. The middle two numbers = the sum on the units digits
3. The last two digits = the product of the units digits
1= 1 15 54
9+6 9(6)
Multiplying Two Numbers Near 100
109 x 106109 x 1061. 106 x109=
2. 103 x 105=
3. 108 x 101=
4. 107 x 106=
5. 108 x 109=
Multiplying Two Numbers With 1st Numbers = And A 0 In The Middle
402 x 405402 x 4051. The 1st two numbers = the product of the hundreds digits
2. The middle two numbers = the sum of the units x the hundreds digit
3. The last two digits = the product of the units digits
4(4)= 16 28 10
4(2+5) 2(5)
Multiplying Two Numbers With 1st Numbers = And A “0” In The Middle
402 x 405402 x 4051. 405 x 405=
2. 205 x 206=
3. 703 x 706=
4. 603 x 607=
5. 801 x 805=
Multiplying By 3367
18 x 336718 x 33671. Divide the non-3367 # by 3
2. Multiply by 10101
18/3 = 6 x 10101== 60606
10101 Rule
Multiplying By 3367
18 x 336718 x 3367
10101 Rule
1. 3367 x 33=
2. 3367 x123=
3. 3367 x 66=
4. 3367 x 93=
5. 3367 x 24=
Multiplying A 2-Digit # By 11
92 x 1192 x 111. Last digit is the units digit
2. The middle digit is the sum of the tens and the units digits
3. The first digit is the tens digit + any carry
9+1= 10 1 2
9+2 2
121 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
Multiplying A 2-Digit # By 11
92 x 1192 x 11
121 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1. 11 x 34=
2. 11 x 98=
3. 65 x 11=
4. 11 x 69=
5. 27 x 11=
Multiplying A 3-Digit # By 11
192 x 11192 x 111. Last digit is the units digit
2. The next digit is the sum of the tens and the units digits
3. The next digit is the sum of the tens and the hundreds digit + carry
4. The first digit is the hundreds digit + any carry
1+9+1=2 1 1 2
9+2 2
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1+1
Multiplying A 3-Digit # By 11
192 x 11192 x 11
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1. 11 x 231=
2. 11 x 687=
3. 265 x 11=
4. 879x 11=
5. 11 x 912=
Multiplying A 3-Digit # By 111
192 x 111192 x 1111. Last digit is the units digit
2. The next digit is the sum of the tens and the units digits
3. The next digit is the sum of the units, tens and the hundreds digit + carry
4. The next digit is the sum of the tens and hundreds digits + carry
5. The next digit is the hundreds digit + carry
1+9+1= 2 1 3 1 2
9+2 2
12321 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1+1 1+9+2+1
Multiplying A 3-Digit # By 111
192 x 111192 x 111
12321 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1. 111 x 213=
2. 111 x 548=
3. 111 x825=
4. 936 x 111=
5. 903 x 111=
Multiplying A 2-Digit # By 111
41 x 11141 x 1111. Last digit is the units digit
2. The next digit is the sum of the tens and the units digits
3. The next digit is the sum of the tens and the units digits + carry
4. The next digit is the tens digit + carry
4= 4 5 5 1
4+1 1
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
4+1
Multiplying A 2-Digit # By 111
41 x 11141 x 111
1221 Pattern
(ALWAYS WORK FROM RIGHT TO LEFT)
1. 45 x 111=
2. 111 x 57=
3. 111 x93=
4. 78 x 111=
5. 83 x 111=
Multiplying A 2-Digit # By 101
93 x 10193 x 1011. The first two digits are the 2-digit number x1
2. The last two digits are the 2-digit number x1
= 93 93 93(1) 93(1)
Multiplying A 2-Digit # By 101
93 x 10193 x 1011. 45 x 101=
2. 62 x 101=
3. 101 x 72=
4. 101 x 69=
5. 101 x 94=
Multiplying A 3-Digit # By 101
934 x 101934 x 1011. The last two digits are the last two digits
of the 3-digit number
2. The first three numbers are the 3-digit number plus the hundreds digit
= 943 34 934+9 34
Multiplying A 3-Digit # By 101
934 x 101934 x 1011. 101 x 658=
2. 963 x 101=
3. 101 x 584=
4. 381 x 101=
5. 101 x 369=
Multiplying A 2-Digit # By 1001
87 x 100187 x 10011. The first 2 digits are the 2-digit number x 1
2. The middle digit is always 0
3. The last two digits are the 2-digit number x 1
= 87 0 87 87(1) 0 87(1)
Multiplying A 2-Digit # By 1001
87 x 100187 x 10011. 1001 x 66=
2. 91 x 1001=
3. 1001 x 53=
4. 1001 x 76=
5. 5.2 x 1001=
Halving And Doubling
52 x 1352 x 131. Take half of one number
2. Double the other number
3. Multiply together
= 26(26)= 676 52/2 13(2)
Halving And Doubling
52 x 1352 x 131. 14 x 56=
2. 16 x 64=
3. 8 x 32=
4. 17 x 68=
5. 19 x 76=
One Number in the Hundreds And One Number In The 90’s
95 x 10895 x 1081. Find how far each number is from 100
2. The last two numbers are the product of the differences subtracted from 100
3. The first numbers = the difference (from the 90’s) from 100 increased by 1 and subtracted from the larger number
= 102 60 108-(5+1) 100-(5x8)
One Number in the Hundreds And One Number In The 90’s
95 x 10895 x 1081. 105 x 96=
2. 98 x 104=
3. 109 x 97=
4. 98 x 105=
5. 97 x 107=
Fraction Foil (Type 1)
8 ½ x 6 ¼ 8 ½ x 6 ¼ 1. Multiply the fractions together
2. Multiply the outside two number3. Multiply the inside two numbers
4. Add the results and then add to the product of the whole numbers
= 53 1/8 (8)(6)+1/2(6)+1/4(8) (1/2x1/4)
Fraction Foil (Type 1)
8 ½ x 6 ¼ 8 ½ x 6 ¼ 1. 9 1/2 x 8 1/3
2. 5 1/5 x 10 2/5
3. 10 1/7 x 14 1/2
4. 3 1/4 x 8 1/3
5. 6 1/4 x 8 1/2
Fraction Foil (same fraction)
7 ½ x 5 ½ 7 ½ x 5 ½ 1. Multiply the fractions together
2. Add the whole numbers and divide by the denominator
3. Multiply the whole numbers and add to previous step
= 41 1/4 (7x5)+6 (1/2x1/2)
Fraction Foil (Type 2)
7 ½ x 5 ½ 7 ½ x 5 ½ 1. 9 1/2 x 7 1/2
2. 4 1/5 x 11 1/5
3. 10 1/6 x 14 1/6
4. 2 1/3 x 10 1/3
5. 6 1/7 x 8 1/7
Fraction Foil (fraction adds to 1)
7 ¼ x 7 ¾ 7 ¼ x 7 ¾ 1. Multiply the fractions together
2. Multiply the whole number by one more than the whole number
= 56 3/16 (7)(7+1) (1/4x3/4)
Fraction Foil (Type 3)
7 ¼ x 7 ¾ 7 ¼ x 7 ¾ 1. 8 1/2 x 8 1/2
2. 10 1/5 x 10 4/5
3. 9 1/7 x 9 6/7
4. 5 3/4 x 5 1/4
5. 2 1/4 x 2 3/4
Adding Reciprocals
7/8 + 8/77/8 + 8/71. Keep the common denominator
2. The numerator is the difference of the two numbers squared
3. The whole number is always two plus any carry from the fraction.
2
=2 1/56
(8-7)2
7x8
Adding Reciprocals
7/8 + 8/77/8 + 8/71. 5/6 + 6/5
2. 11/13 + 13/11
3. 7/2 + 2/7
4. 7/10 + 10/7
5. 11/15 +15/11
Percent Missing the Of
36 is 9% of __36 is 9% of __1. Divide the first number by
the percent number
2. Add 2 zeros or move the decimal two places to the right
36/9= 400
00
Percent Missing the Of
36 is 9% of __36 is 9% of __1. 40 is 3% of ______=
2. 27 is 9% of ______=
3. 800 is 25% of ____=
4. 70 is 4% of ______=
5. 10 is 2 1/2 % of _____=
Percent Missing the Of
36 is 9% of __36 is 9% of __1. 40 is 3% of ______=
2. 27 is 9% of ______=
3. 800 is 25% of ____=
4. 70 is 4% of ______=
5. 10 is 2 1/2 % of _____=
Base N to Base 10
424266 =____ =____10101. Multiply the left digit times the base
2. Add the number in the units column
4(6)+2
= 2610
Base N to Base 10 Of
424266 =____ =____1010
1. 546=_____10
2. 347=_____10
3. 769=_____10
4. 1245=_____10
5. 2346=_____10
Multiplying in Bases
4 x 534 x 5366=___=___661. Multiply the units digit by the multiplier
2. If number cannot be written in base n subtract base n until the digit can be written
3. Continue until you have the answer
= 3406
= 4x3=12 subtract 12 Write 0
= 4x5=20+2=22 subtract 18 Write 4= Write 3
Multiplying in Bases
4 x 534 x 5366=___=___66
1. 2 x 426= _____6
2. 3 x 547=_____7
3. 4 x 678=_____8
4. 5 x 345=_____5
5. 3 x 278=_____8
N/40 to a % or Decimal
21/40___21/40___decimaldecimal1. Mentally take off the zero
2. Divide the numerator by the denominator and write down the digit
3. Put the remainder over the 4 and write the decimal without the decimal point
4. Put the decimal point in front of the numbers
5 25.21/4 1/4
N/40 to a % or Decimal
21/40___21/40___decimaldecimal
1. 31/40=
2. 27/40=
3. 51/40=
4. 3/40=
5. 129/40=
Remainder When Dividing By 9
867867//9=___9=___remainderremainder
1. Add the digits until you get a single digit
2. Write the remainder
8+6+7=21=2+1=3
= 3
Remainder When Dividing By 9
867867//9=___9=___remainderremainder
1. 3251/9=
2. 264/9=
3. 6235/9=
4. 456/9=
5. 6935/9=
Base 8 to Base 2
73273288 =____ =____221. Mentally put 421 over each number
2. Figure out how each base number can be written with a 4, 2 and 1
3. Write the three digit number down
7
421 Method
3 2421 421 421
111 011 010
Base 8 to Base 2
73273288 =____ =____22
421 Method
1. 3548= _____2
2. 3258=_____2
3. 1568=_____2
4. 3548=_____2
5. 5748=_____2
Base 2 to Base 8 Of
11101101011101101022 =___ =___88
1. Separate the number into groups of 3 from the right.
2. Mentally put 421 over each group
3. Add the digits together and write the sum
7
421 Method
3 2
421 421 421
111 011 010
Base 2 to Base 8 Of
11101101011101101022 =___ =___88
421 Method
1. 1100012= _____8
2. 1111002=_____8
3. 1010012=_____8
4. 110112=_____8
5. 10001102=_____8
Cubic Feet to Cubic Yards
33ftft xx 6 6ftft x x 1212ftft=__yds=__yds33
1. Try to eliminate three 3s by division
2. Multiply out the remaining numbers
3. Place them over any remaining 3s
3 1263 3 3
1 x 2 x 4 = 8 Cubic yards
Cubic Feet to Cubic Yards
33ftft xx 6 6ftft x x 1212ftft=__yds=__yds33
1. 6ft x 3ft x 2ft=
2. 9ft x 2ft x 11ft=
3. 2ft x 5ft x 7ft=
4. 27ft x 2ft x5ft=
5. 10ft x 12ft x 3ft=
Ft/sec to MPH
44 44 ft/sec ft/sec ____mphmph1. Use 15 mph = 22 ft/sec
2. Find the correct multiple
3. Multiply the other number
22x2=4415x2=30 mph
Ft/sec to mph
44 44 ft/sec ft/sec ____mphmph
1. 88 ft/sec=_____mph
2. 120 mph=_____ft/sec
3. 90 mph =______ft/sec
4. 132 ft/sec = _____mph
5. 45 mph= ____ft/sec
Subset Problems
{F,R,O,N,T}{F,R,O,N,T}=______=______
1. Subsets=2n
2. Improper subsets always = 1
3. Proper subsets = 2n - 1
4. Power sets = subsets
SUBSETS
25=32 subsets
Subset Problems
{F,R,O,N,T}{F,R,O,N,T}=______=______SUBSETS
1. {A,B,C}=
2. {D,G,H,J,U,N}=
3. {!!, $, ^^^, *}=
4. {AB, FC,GH,DE,BM}=
5. {M,A,T,H}=
Repeating Decimals to Fractions
.18=___.18=___fractionfraction
1. The numerator is the number
2. Read the number backwards. If a number has a line over it then there is a 9 in the denominator
3. Write the fraction and reduce
1899 = 2
11
______
Repeating Decimals to Fractions
.18=___.18=___fractionfraction
______
1. .25
2. .123
3. .74
4. .031
5. .8
Repeating Decimals to Fractions
.18=___.18=___fractionfraction
1. The numerator is the number minus the part that does not repeat
2. For the denominator read the number backwards. If it has a line over it,
it is a 9. if not it is a o.
18-190 = 17
90
__
Repeating Decimals to Fractions
.18=___.18=___fractionfraction
__
1. .16
2. .583
3. .123
4. .45
5. .92
Gallons Cubic Inches
2 gallons=__in2 gallons=__in33
1. Use the fact: 1 gal= 231 in3
2. Find the multiple or the factor and adjust the other number. (This is a direct variation)
2(231)= 462 in3
(Factors of 231 are 3, 7, 11)
Gallons Cubic Inches
2 gallons=__in2 gallons=__in33
1. 3 gallons =_____in3
2. ½ gallon =______in3
3. 77 in3=_______gallons
4. 33 in3=_______gallons
5. 1/5 gallon=______in3
Finding Pentagonal Numbers
55thth Pentagonal Pentagonal # =# =____1. Use the house method)
2. Find the square #, find the triangular #, then add them together
25+10=355
5
25
1+2+3+4= 10
Finding Pentagonal Numbers
55thth Pentagonal Pentagonal # =# =____1. 3rd pentagonal number=
2. 6th pentagonal number=
3. 10th pentagonal number=
4. 4th pentagonal number=
5. 6th pentagonal number=
Finding Triangular Numbers
66thth Triangular Triangular # =# =____1. Use the n(n+1)/2 method
2. Take the number of the term that you are looking for and multiply it by one more than that term.
3. Divide by 2 (Divide before multiplying)
6(6+1)=4242/2=21
Finding Triangular Numbers
66thth Triangular Triangular # =# =____1. 3rd triangular number=
2. 10th triangular number=
3. 5th triangular number=
4. 8th triangular number=
5. 40th triangular number=
Pi To An Odd Power
1313=____=____approximationapproximation
1. Pi to the 1st = 3 (approx) Write a 3
2. Add a zero for each odd power of Pi after the first
3000000
Pi To An Odd Power
1313=____=____approximationapproximation
1. Pi11
2. Pi7
3. Pi9
4. Pi5
5. Pi3
Pi To An Even Power
1212=____=____approximationapproximation
1. Pi to the 2nd = 95 (approx) Write a 95
2. Add a zero for each even power of Pi after the 4th
950000
Pi To An Even Power
1212=____=____approximationapproximation
1. Pi10
2. Pi8
3. Pi6
4. Pi14
5. Pi16
The “More” Problem
17/15 x 1717/15 x 171. The answer has to be more than the whole number.
2. The denominator remains the same.
3. The numerator is the difference in the two numbers squared.
4. The whole number is the original whole number plus the difference
=19 4/15
(17-15)2
1517+2
The More Problem
17/15 x 1717/15 x 171. 19/17 x 19=
2. 15/13 x 15=
3. 21/17 x 21=
4. 15/12 x 15=
5. 31/27 x 31=
The “Less” Problem
15/17 x 1515/17 x 151. The answer has to be less than the whole number.
2. The denominator remains the same.
3. The numerator is the difference in the two numbers squared.
4. The whole number is the original whole number minus the difference
=13 4/17
(17-15)2
1715-2
The Less Problem
15/17 x 1515/17 x 151. 13/17 x 13=
2. 21/23 x 21=
3. 5/7 x 5=
4. 4/7 x4=
5. 49/53 x49=
Multiplying Two Numbers Near 1000
994 x 998994 x 9981. Find out how far each number is from 1000
2. The 1st two numbers equal the sum of the differences subtracted from 1000
3. The last two numbers equal the product of the differences written as a 3-digit number
1000-(6+2)=992 012
6(2)
Multiplying Two Numbers Near 1000
994 x 998994 x 9981. 996 x 991 =
2. 993 x 997 =
3. 995 x 989 =
4. 997 x 992 =
5. 985 x 994 =
The (Reciprocal) Work Problem
1/6 + 1/5 = 1/X1/6 + 1/5 = 1/X1. Use the formula ab/a+b.
2. The numerator is the product of the two numbers.
3. The deniminator is the sum of the two numbers.
4. Reduce if necessary
=30/11
=6(5)=6+5
Two Things Helping
The (Reciprocal) Work Problem
1/6 + 1/5 = 1/X1/6 + 1/5 = 1/X
Two Things Helping
1. 1/3 + 1/5 = 1/x
2. 1/2 + 1/6 =1/x
3. 1/4 + 1/7 = 1/x
4. 1/8 + 1/6 =1/x
5. 1/10 + 1/4 = 1/x
The (Reciprocal) Work Problem
1/6 - 1/8 = 1/X1/6 - 1/8 = 1/X1. Use the formula ab/b-a.
2. The numerator is the product of the two numbers.
3. The denominator is the difference of the two numbers from right to left.
4. Reduce if necessary
=24
=6(8)=8-6
Two Things working Against Each Other
The (Reciprocal) Work Problem
1/6 - 1/8 = 1/X1/6 - 1/8 = 1/X
Two Things working Against Each Other
1. 1/8 – 1/5 = 1/x
2. 1/11 – 1/3 = 1/x
3. 1/8 – 1/10 = 1/x
4. 1/7 – 1/8 = 1/x
5. 1/30 – 1/12 = 1/x
The Inverse Variation % Problem
30% of 12 = 20% of ___30% of 12 = 20% of ___1. Compare the similar terms as a reduced ratio
2. Multiply the other term by the reduced ratio.
3. Write the answer
30/20=3/2
=18 3/2(12)=18
The Inverse Variation % Problem
30% of 12 = 20% of ___30% of 12 = 20% of ___
1. 27% of 50= 54% of _____
2. 15% of 24 = 20% of _____
3. 90% of 70 = 30% of _____
4. 75% of 48 = 50% of _____
5. 14% of 27 = 21% of _____
6. 26% of 39 = 78% of _____
Sum of Consecutive Integers
1+2+3+…..+201+2+3+…..+201. Use formula n(n+1)/2
2. Divide even number by 2
3. Multiply by the other number
10(21)= 210 (20)(21)/2
Sum of Consecutive Integers
1+2+3+…..+201+2+3+…..+20
1. 1+2+3+….+30=
2. 1+2+3+….+16=
3. 1+2+3+….+19=
4. 1+2+3+…+49=
5. 1+2+3+….100=
Sum of Consecutive Even Integers
2+4+6+…..+202+4+6+…..+201. Use formula n(n+2)/4
2. Divide the multiple of 4 by 4
3. Multiply by the other number
5(22)= 110 (20)(22)/4
Sum of Consecutive Even Integers
2+4+6+…..+202+4+6+…..+20
1. 2+4+6+….+16=
2. 2+4+6+….+40=
3. 2+4+6+….+28=
4. 2+4+6+….+48=
5. 2+4+6+….+398=
Sum of Consecutive Odd Integers
1+3+5+…..+191+3+5+…..+191. Use formula ((n+1)/2)2
2. Add the last number and the first number
3. Divide by 2
4. Square the result
102 = 100(19+1)/2=
Sum of Consecutive Odd Integers
1+3+5+…..+191+3+5+…..+19
1. 1+3+5+….+33=
2. 1+3+5+….+49=
3. 1+3+5+….+67=
4. 1+3+5+….+27=
5. 1+3+5+….+47=
Finding Hexagonal Numbers
Find the 5Find the 5thth Hexagonal NumberHexagonal Number
1. Use formula 2n2-n
2. Square the number and multiply by2
3. Subtract the number wanted from the previous answer
50-5=2(5)2= 50
45
Finding Hexagonal Numbers
Find the 5Find the 5thth Hexagonal NumberHexagonal Number
1. Find the 3rd hexagonal number=
2. Find the 10th hexagonal number=
3. Find the 4th hexagonal number=
4. Find the 2nd hexagonal number=
5. Find the 6th hexagonal number=
Cube Properties
Find the Surface Area of a Cube Find the Surface Area of a Cube Given the Space Diagonal = 12Given the Space Diagonal = 12
1. Use formula Area = 2D2
2. Square the diagonal
3. Multiply the product by 2
2(144)=2(12)(12)
288
Cube Properties
Find the Surface Area of a Cube Find the Surface Area of a Cube Given the Space Diagonal of 12Given the Space Diagonal of 12
1. Space diagonal = 24
2. Space diagonal = 10
3. Space diagonal = 50
4. Space diagonal = 21
5. Space diagonal = 8
Cube Properties
2
S
3S
S
Find S, Then Use It To Find Find S, Then Use It To Find Volume or Surface AreaVolume or Surface Area
Cube Properties
2
S
3S
S
Find S, Then Use It To Find Find S, Then Use It To Find Volume or Surface AreaVolume or Surface Area
Finding Slope From An Equation
3X+2Y=103X+2Y=101. Solve the equation for Y
2. The number in front of X is the Slope
3X+2Y=10Y = -3X +5
2Slope = -3/2
Finding Slope From An Equation
3X+2Y=103X+2Y=101. Y = 2X + 8
2. Y = -7X + 6
3. 2Y = 8X - 12
4. 2X + 3Y = 12
5. 10X – 4Y = 13
Hidden Pythagorean Theorem Find The Distance Between These PointsFind The Distance Between These Points
(6,2) (6,2) andand (9,6) (9,6)1. Find the distance between the X’s
2. Find the distance between the Y’s
3. Look for a Pythagorean triple
4. If not there, use the Pythagorean Theorem
9-6=3 6-2=43 4 5
The distance is 5
3 4 55 12 137 24 258 15 17
Common Pythagorean triples
Hidden Pythagorean Theorem Find The Distance Between These PointsFind The Distance Between These Points
(6,2) (6,2) andand (9,6) (9,6)1. (4,3) and (7,7)
2. (8,3) and (13,15)
3. (1,2) and (3,4)
4. (12,29) and (5,5)
5. (3,4) and (2,4)
Finding Diagonals
Find The Number Of Find The Number Of Diagonals In An OctagonDiagonals In An Octagon
1. Use the formula n(n-3)/2
2. N is the number of vertices in the polygon
8(8-3)/2=20
Finding Diagonals
Find The Number Of Find The Number Of Diagonals In An OctagonDiagonals In An Octagon
1. # of diagonals in a pentagon
2. # of diagonals of a hexagon
3. # of diagonals of a decagon
4. # of diagonals of a dodecagon
5. # of diagonals of a heptagon
Finding the total number of factors
24= ________24= ________1. Put the number into prime factorization
2. Add 1 to each exponent
3. Multiply the numbers together
31 x 23=1+1=2 3+1=4
2x4=8
Finding the total number of factors
24= ________24= ________
1. 12=
2. 30=
3. 120=
4. 50=
5. 36=
Estimating a 4-digit square root
802=6400
7549 = _______7549 = _______1. The answer is between 802 and 902
2. Find 852
3. The answer is between 85 and 90
4. Guess any number in that range
902=8100852=7225 87
Estimating a 4-digit square root 7549 = _______7549 = _______
3165
6189
1796
9268
5396
1.
2.
3.
4.
5.
Estimating a 5-digit square root
192=361
37437485 = _______85 = _______1. Use only the first three numbers
2. Find perfect squares on either side
3. Add a zero to each number
4. Guess any number in that range
202=400195190-200
Estimating a 5-digit square root 37437485 = _______85 = _______
31651
61893
17964
92682
53966
1.
2.
3.
4.
5.
C F
9/5(55) + 32
55C = _______F55C = _______F1. Use the formula F= 9/5 C + 32
2. Plug in the F number
3. Solve for the answer
= 13199+32
C F 59C = _______F59C = _______F1. 4500C=______F
2. 400C =_____F
3. 650C =_____F
4. 250C=_____F
5. 900C=_____F
C F
5/9(50-32)
50F = _______C50F = _______C1. Use the formula C = 5/9 (F-32)
2. Plug in the C number
3. Solve for the answer
= 105/9(18)
C F 50F = _______C50F = _______C1. 680F=
2. 590F=
3. 1130F=
4. 410F=
5. 950F=
Finding The Product of the Roots
6 / 4 = 3/2
4X4X22 + 5X + 6 + 5X + 61. Use the formula c/a
2. Substitute in the coefficients
3. Find answer
a b c
Finding The Product of the Roots 4X4X22 + 5X + 6 + 5X + 6a b c
1. 5x2 + 6x + 2
2. 2x2 + -7x +1
3. 3x2 + 4x -1
4. -3x2 +2x -4
5. -8x2 -6x +1
Finding The Sum of the Roots
-5 / 4
4X4X22 + 5X + 6 + 5X + 61. Use the formula -b/a
2. Substitute in the coefficients
3. Find answer
a b c
Finding The Sum of the Roots 4X4X22 + 5X + 6 + 5X + 6a b c
1. 5x2 + 6x + 2
2. 2x2 + -7x +1
3. 3x2 + 4x -1
4. -3x2 +2x -4
5. -8x2 -6x +1
Estimation
26/7 =3r5
142857 x 26 = 142857 x 26 = 1. Divide 26 by 7 to get the first digit
2. Take the remainder and add a zero
3. Divide by 7 again to get the next number
4. Find the number in 142857 and copy in a circle
999999 Rule
5+0=50/7=73 714285
Estimation 142857 x 26 = 142857 x 26 = 999999 Rule
1. 142857 x 38
2. 142857 x 54
3. 142857 x 17
4. 142857 x 31
5. 142857 x 64
Area of a Square Given the Diagonal
½ D1 D2
Find the area of a square Find the area of a square with a diagonal of 12 with a diagonal of 12
1. Use the formula Area = ½ D1D2
2. Since both diagonals are equal
3. Area = ½ 12 x 12
4. Find answer
72½ x 12 x 12
Area of a Square Given the Diagonal Find the area of a square Find the area of a square with a diagonal of 12 with a diagonal of 12
1. Diagonal = 14
2. Diagonal = 8
3. Diagonal = 20
4. Diagonal = 26
5. Diagonal = 17
Estimation of a 3 x 3 Multiplication
35 x 30
346 x 291 = 346 x 291 = 1. Take off the last digit for each number
2. Round to multiply easier
3. Add two zeroes
4. Write answer
1050001050 + 00
Estimation of a 3 x 3 Multiplication
346 x 291 = 346 x 291 = 1. 316 x 935
2. 248 x 603
3. 132 x 129
4. 531 x 528
5. 248 x 439
Dividing by 11 and finding the remainder 7258 / 11=_____7258 / 11=_____1. Start with the units digit and add up every other number
2. Do the same with the other numbers
3. Subtract the two numbers
4. If the answer is a negative or a number greater than 11 add or subtract 11 until you get a number from 0-10
10-12= -2 +11= 998+2=10 7+5= 12
RemainderRemainder
Dividing by 11 and finding the remainder 7258 / 11=_____7258 / 11=_____ RemainderRemainder
1. 16235 / 11
2. 326510 / 11
3. 6152412 / 11
4. 26543 / 11
5. 123456 / 11
Multiply By Rounding 2994 x 6 = 2994 x 6 = 1. Round 2994 up to 3000
2. Think 3000 x 6
3. Write 179. then find the last two numbers by multiplying what you added by 6 and subtracting it from 100.
6(6)=36 100-36=643000(6)=179_ _
=17964
Multiply By Rounding 2994 x 6 = 2994 x 6 = 1. 3994 x 7
2. 5991 x 6
3. 4997 x 8
4. 6994 x 4
5. 1998 x 6
The Sum of Squares 121222 + 24 + 2422= = 1. Since 12 goes into 24 twice…
2. Square 12 and multiply by 10
3. Divide by 2
144x10=122=144
=1440/2
(factor of 2)
=720
The Sum of Squares 121222 + 24 + 2422= = (factor of 2)
1. 142 + 282
2. 172 + 342
3. 112 + 222
4. 252 + 502
5. 182 + 362
The Sum of Squares 121222 + 36 + 3622= = 1. Since 12 goes into 36 three times…
2. Square 12 and multiply by 10
144x10=122=144
=1440
(factor of 3)
The Sum of Squares 121222 + 36 + 3622= = (factor of 3)
1. 142 + 422
2. 172 + 512
3. 112 + 332
4. 252 + 752
5. 182 + 542
The Difference of Squares 323222 - 30 - 3022= = 1. Find the sum of the bases
2. Find the difference of the bases
3. Multiply them together
32+30=62
32-30=2
(Sum x the Difference)
62 x 2 =124
The Difference of Squares 323222 - 30 - 3022= = (Sum x the Difference)
1. 222 - 322
2. 732 - 272
3. 312 - 192
4. 622 - 422
5. 992 - 982
Addition by Rounding 2989 + 456= 2989 + 456= 1. Round 2989 to 3000
2. Subtract the same amount to 456, 456-11= 445
3. Add them together
2989+11= 3000
456-11=4453000+445=3445
Addition by Rounding 2989 + 456= 2989 + 456= 1. 2994 + 658
2. 3899 + 310
3. 294 + 498 + 28
4. 6499 + 621
5. 2938 +64
123…x9 + A Constant 123 x 9 + 4 123 x 9 + 4 1. The answer should be all 1s. There should be 1 more 1
than the length of the 123… pattern.
2. You must check the last number. Multiply the last number in the 123… pattern and add the constant.
3x9 + 4 =31
1111
(1111…Problem)
123…x9 + A Constant 123 x 9 + 4 123 x 9 + 4 (1111…Problem)
1. 1234 x 9 + 5
2. 12345 x 9 + 6
3. 1234 x 9 + 7
4. 123456 x 9 + 6
5. 12 x 9 + 3
Supplement and Complement
1. The answer is always 90
=90
Find The Difference Of The Find The Difference Of The Supplement And The Supplement And The
Complement Of An Angle Of Complement Of An Angle Of 40. 40.
Supplement and Complement Find The Difference Of The Find The Difference Of The Supplement And The Supplement And The
Complement Of An Angle Of 40. Complement Of An Angle Of 40. 1. angle of 70
2. angle of 30
3. angle of 13.8
4. angle of 63
5. angle of 71 ½
Supplement and Complement
1. Use the formula 270-twice the angle
2. Multiple the angle by 2
3. Subtract from 270
270-80=
Find The Sum Of The Find The Sum Of The Supplement And The Supplement And The
Complement Of An Angle Of Complement Of An Angle Of 40. 40.
=190
Supplement and Complement Find The Sum Of The Find The Sum Of The Supplement And The Supplement And The
Complement Of An Angle Of 40. Complement Of An Angle Of 40. 1. angle of 70
2. angle of 30
3. angle of 13.8
4. angle of 63
5. angle of 71 ½
Larger or Smaller
1. Find the cross products
2. The larger fraction is below the larger number
3. The smaller number is below the smaller number
Larger = 5/4
55 13134 114 11++
55 52
Smaller = 13/11
Larger or Smaller
55 13134 114 11++
55 52
Two Step Equations(Christmas Present Problem)
1. Start with the answer and undo the operations using reverse order of operations
11+1=12
12 x3 = 36
AA33
-- = 11= 1111
Two Step Equations(Christmas Present Problem) AA33
-- = 11= 11111. 2x -1 =8
2. x/3 - 4 =6
3. 5x -12 = 33
4. x/2 + 5 =8
5. x/12 +5 = 3
Relatively Prime(No common Factors Problem)
* One is relatively prime to all numbers
1. Put the number into prime factorization
2. Subtract 1 from each exponent and multiply out all parts separately
3. Subtract 1 from each base
4. Multiply all parts together
How Many #s less than 20 How Many #s less than 20 are relatively prime to 20?are relatively prime to 20?
22 x 51 =21 x 50 =2 x 12 x 1 x 1 x 4 = 8
Relatively Prime(No common Factors Problem)
* One is relatively prime to all numbers How Many #s less than 20 are How Many #s less than 20 are relatively prime to 20?relatively prime to 20?1. less than 18
2. less than 50
3. less than 12
4. less than 22
5. less than 100
Product of LCM and GCF
1. Multiple the two numbers together
Find the Product of the GCF Find the Product of the GCF and the LCM of 6 and 15and the LCM of 6 and 15
6 x 15 = 90
Product of LCM and GCF
Find the Product of the GCF Find the Product of the GCF and the LCM of 6 and 15and the LCM of 6 and 15
1. 21 and 40
2. 38 and 50
3. 25 and 44
4. 12 and 48
5. 29 and 31
Estimation
1. Take the number in the middle and cube it
15 x 17 x 1915 x 17 x 19
173=4913
Estimation
15 x 17 x 1915 x 17 x 191. 7 x 8 x 9
2. 11 x 13 x 15
3. 19 x 20 x 21
4. 38 x 40 x 42
5. 9 x 11 x 13
Sequences-Finding the Pattern
1. If the pattern is not obvious try looking at every other number. There may be two patterns put together
7, 2, 5, 8, 3, 147, 2, 5, 8, 3, 14
7, 2, 5, 8, 3, 14
Find the next number in this patternFind the next number in this pattern
1
Sequences-Finding the Pattern
7, 2, 5, 8, 3, 147, 2, 5, 8, 3, 14Find the next number in this patternFind the next number in this pattern
1. 5,10,15,20,25…..
2. 11, 12, 14, 17,…..
3. 8,9,7,8,6……
4. 7,13,14,10,21,7…..
5. 2,8,5,4,6,10,6,4,15…
Sequences-Finding the Pattern
1. If nothing else works look for a Fibonacci Sequence where the next term is the sum of the previous two
1, 4, 5, 9, 14, 231, 4, 5, 9, 14, 23
1, 4, 5, 9, 14, 23
Find the next number in this patternFind the next number in this pattern
14+23=37
Sequences-Finding the Pattern
1, 4, 5, 9, 14, 231, 4, 5, 9, 14, 23Find the next number in this patternFind the next number in this pattern
1. 1,4,5,9,14,23……
2. 2,3,5,10,18,33,……
3. 1,4,9,16,25…….
4. 8, 27,64,125….
5. 10,8,6,4,….
Degrees Radians
1. If you want radians use π X/180
2. If you want degrees use 180 x/ π
909000= _____= _____
90(π)/180
RadiansRadians
= π/2
Degrees Radians
909000= _____= _____RadiansRadians
1. 1800=
2. 450=
3. 2700=
4. 1800=
5. 1350=
