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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 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.
The Rainbow Method
Work Backwards
2x2 Foil (LIOF)23 x 12
Used when you forget a rule about 2x2 multiplication
The last number is the units digit of the product of the unit’s digits
Multiply the outside, multiply the inside
Add the outside and the inside together plus any carry and write down the units digit
Multiply the first digits together and add and carry. Write down the number
2(1)
2(2)+3(1)
3(2)
2
7
6
276
Squaring Numbers Ending In 5752
First two digits = the ten’s digit times one more than the ten’s digit.
Last two digits are always 25
7(7+1)25
=5625
Consecutive Decades35 x 45
First two digits = the small ten’s digit times one more than the large ten’s digit.
Last two digits are always 75
3(4+1)75
=1575
Ending in 5…Ten’s Digits Both Even45 x 85
First two digits = the product of the ten’s digits plus ½ the sum of the ten’s digits.
Last two digits are always 25
4(8) + ½ (4+8)25
=3825
Ending in 5…Ten’s Digits Both Odd35 x 75
First two digits = the product of the ten’s digits plus ½ the sum of the ten’s digits.
Last two digits are always 25
3(7) + ½ (3+7)25
=2625
Ending in 5…Ten’s Digits Odd&Even35 x 85
First two digits = the product of the ten’s digits plus ½ the sum of the ten’s digits. Always drop the remainder.
Last two digits are always 75
3(8) + ½ (3+8)75
=2975
(1/8 rule)
Multiplying By 12 ½ 32 x 12 ½
32
Divide the non12 ½ number by 8.
Add two zeroes.
4+00
=
8
=400
(1/6 rule)
Multiplying By 16 2/3 42 x 16 2/3
42
Divide the non16 2/3 number by 6.
Add two zeroes.
7+00
=
6
=700
(1/3 rule)
Multiplying By 33 1/3 24 x 33 1/3
24
8+00
=
3
=800
(1/4 rule)
Multiplying By 2532 x 25
32
Divide the non25 number by 4.
Add two zeroes.
8
+00
=
4
=800
(1/2 rule)
Multiplying By 5032 x 50
32
Divide the non50 number by 2.
Add two zeroes.
16
=
+00
2
=1600
(3/4 rule)
Multiplying By 7532 x 75
Divide the non75 number by 4.
Multiply by 3.
Add two zeroes.
32
=
8x3=24+00
4
=2400
(3/8 rule)
Multiplying By 37 1/237 1/2 x 24
=900
00
(3/8)24
(5/8 rule)
Multiplying By 62 1/262 1/2 x 56
=3500
00
(5/8)56
(7/8 rule)
Multiplying By 87 1/287 1/2 x 48
=4200
00
(7/8)48
(5/6 rule)
Multiplying By 83 1/383 1/3 x 36
=3000
00
(5/6)36
(2/3 rule)
Multiplying By 66 2/366 2/3 x 66
=4400
00
(2/3)66
(1/8 rule)
Multiplying By 12532 x 125
Divide the non125 number by 8.
Add three zeroes.
32
4+000
=
8
=4000
Multiplying When Tens Digits Are Equal And The Unit Digits Add To 1032 x 38
First two digits are the tens digit times one more than the tens digit
Last two digits are the product of the units digits.
3(3+1)
2(8)
=1216
Multiplying When Tens Digits Add To 10 And The Units Digits Are Equal67 x 47
First two digits are the product of the tens digit plus the units digit
Last two digits are the product of the units digits.
6(4)+7
7(7)
=3149
Multiplying Two Numbers in the 90’s97 x 94
Find out how far each number is from 100
The 1st two numbers equal the sum of the differences subtracted from 100
The last two numbers equal the product of the differences
100(3+6)
3(6)
=9118
Multiplying Two Numbers Near 100109 x 106
First Number is always 1
The middle two numbers = the sum on the units digits
The last two digits = the product of the units digits
1
9+6
9(6)
= 11554
Multiplying Two Numbers With 1st Numbers = And A 0 In The Middle402 x 405
The 1st two numbers = the product of the hundreds digits
The middle two numbers = the sum of the units x the hundreds digit
The last two digits = the product of the units digits
4(4)
4(2+5)
2(5)
= 162810
10101 Rule
Multiplying By 336718 x 3367
Divide the non3367 # by 3
Multiply by 10101
18/3 = 6 x 10101=
= 60606
121 Pattern
Multiplying A 2Digit # By 1192 x 11
(ALWAYS WORK FROM RIGHT TO LEFT)
Last digit is the units digit
The middle digit is the sum of the tens and the units digits
The first digit is the tens digit + any carry
9+1
9+2
2
= 1012
1221 Pattern
Multiplying A 3Digit # By 11192 x 11
(ALWAYS WORK FROM RIGHT TO LEFT)
Last digit is the units digit
The next digit is the sum of the tens and the units digits
The next digit is the sum of the tens and the hundreds digit + carry
The first digit is the hundreds digit + any carry
1+1
1+9+1
9+2
2
=2112
12321 Pattern
Multiplying A 3Digit # By 111192 x 111
(ALWAYS WORK FROM RIGHT TO LEFT)
Last digit is the units digit
The next digit is the sum of the tens and the units digits
The next digit is the sum of the units, tens and the hundreds digit + carry
The next digit is the sum of the tens and hundreds digits + carry
The next digit is the hundreds digit + carry
1+1
1+9+1
1+9+2+1
9+2
2
= 21312
1221 Pattern
Multiplying A 2Digit # By 11141 x 111
(ALWAYS WORK FROM RIGHT TO LEFT)
Last digit is the units digit
The next digit is the sum of the tens and the units digits
The next digit is the sum of the tens and the units digits + carry
The next digit is the tens digit + carry
4
4+1
4+1
1
= 4551
Multiplying A 2Digit # By 10193 x 101
The first two digits are the 2digit number x1
The last two digits are the 2digit number x1
93(1)
93(1)
= 9393
Multiplying A 3Digit # By 101934 x 101
The last two digits are the last two digits of the 3digit number
The first three numbers are the 3digit number plus the hundreds digit
934+9
34
= 94334
Multiplying A 2Digit # By 100187 x 1001
The first 2 digits are the 2digit number x 1
The middle digit is always 0
The last two digits are the 2digit number x 1
87(1)
0
87(1)
= 87087
Halving And Doubling52 x 13
Take half of one number
Double the other number
Multiply together
52/2
13(2)
= 26(26)= 676
One Number in the Hundreds And One Number In The 90’s95 x 108
Find how far each number is from 100
The last two numbers are the product of the differences subtracted from 100
The first numbers = the small number difference from 100 increased by 1 and subtracted from the larger number
108(5+1)
100(5x8)
= 10260
Fraction Foil (Type 1)8 ½ x 6 ¼
Multiply the fractions together
Multiply the outside two number
Multiply the inside two numbers
Add the results and then add to the product of the whole numbers
(8)(6)+1/2(6)+1/4(8)
(1/2x1/4)
= 531/8
Fraction Foil (Type 2)7 ½ x 5 ½
Multiply the fractions together
Add the whole numbers and divide by the denominator
Multiply the whole numbers and add to previous step
(7x5)+6
(1/2x1/2)
= 411/4
Fraction Foil (Type 3)7 ¼ x 7 ¾
Multiply the fractions together
Multiply the whole number by one more than the whole number
(7)(7+1)
(1/4x3/4)
= 563/16
(87)2
2
7x8
=21/56
Adding Reciprocals7/8 + 8/7
Keep the denominator
The numerator is the difference of the two numbers squared
The whole number is always two plus any carry from the fraction.
Percent Missing the Of36 is 9% of __
Divide the first number by the percent number
Add 2 zeros or move the decimal two places to the right
36/9
00
= 400
Base N to Base 10 426 =____10
Multiply the left digit times the base
Add the number in the units column
4(6)+2
= 2610
Multiplying in Bases4 x 536=___6
Multiply the units digit by the multiplier
If number cannot be written in base n subtract base n until the digit can be written
Continue until you have the answer
= 4x3=12subtract 12Write 0
= 4x5=20+2=22subtract 18Write 4
= Write 3
= 3406
N/40 to a % or Decimal21/40___decimal
Mentally take off the zero
Divide the numerator by the denominator and write down the digit
Put the remainder over the 4 and write the decimal without the decimal point
Put the decimal point in front of the numbers
.
5
25
21/4
1/4
Remainder When Dividing By 9867/9=___remainder
Add the digits until you get a single digit
Write the remainder
8+6+7=21=2+1=3
= 3
421 Method
Base 8 to Base 27328 =____2
Mentally put 421 over each number
Figure out how each base number can be written with a 4, 2 and 1
Write the three digit number down
421
421
421
7
3
2
111
011
010
421 Method
Base 2 to Base 8 Of1110110102 =___8
Separate the number into groups of 3 from the right.
Mentally put 421 over each group
Add the digits together and write the sum
421
421
421
011
010
111
7
3
2
Cubic Feet to Cubic Yards3ftx 6ftx 12ft=__yds3
Try to eliminate three 3s by division
Multiply out the remaining numbers
Place them over any remaining 3s
3
6
12
3
3
3
1 x 2 x 4 = 8
Cubic yards
Ft/sec to MPH44 ft/sec __mph
Use 15 mph = 22 ft/sec
Find the correct multiple
Multiply the other number
22x2=44
15x2=30 mph
25=32
subsets
Subset Problems{F,R,O,N,T}=______
SUBSETS
Subsets=2n
Improper subsets always = 1
Proper subsets = 2n  1
Power sets = subsets
___
Repeating Decimals to Fractions.18=___fraction
The numerator is the number
Read the number backwards. If a number has a line over it then there is a 9 in the denominator
Write the fraction and reduce
18
2
=
99
11
_
Repeating Decimals to Fractions.18=___fraction
The numerator is the number minus the part that does not repeat
For the denominator read the number backwards. If it has a line over it, it is a 9. if not it is a o.
181
17
=
90
90
Gallons Cubic Inches2 gallons=__in3
(Factors of 231 are 3, 7, 11)
Use the fact: 1 gal= 231 in3
Find the multiple or the factor and adjust the other number. (This is a direct variation)
2(231)= 462 in3
Finding Pentagonal Numbers5th Pentagonal# =__
Use the house method)
Find the square #, find the triangular #, then add them together
1+2+3+4= 10
25+10=35
25
5
5
Finding Triangular Numbers6th Triangular# =__
Use the n(n+1)/2 method
Take the number of the term that you are looking for and multiply it by one more than that term.
Divide by 2 (Divide before multiplying)
6(6+1)=42
42/2=21
Pi To An Odd Power13=____approximation
Pi to the 1st = 3 (approx) Write a 3
Add a zero for each odd power of Pi after the first
3000000
Pi To An Even Power12=____approximation
Pi to the 4th = 95 (approx) Write a 95
Add a zero for each even power of Pi after the 4th
950000
The “More” Problem17/15 x 17
The answer has to be more than the whole number.
The denominator remains the same.
The numerator is the difference in the two numbers squared.
The whole number is the original whole number plus the difference
(1715)2
17+2
15
=194/15
The “Less” Problem15/17 x 15
The answer has to be less than the whole number.
The denominator remains the same.
The numerator is the difference in the two numbers squared.
The whole number is the original whole number minus the difference
(1715)2
152
17
=134/17
Multiplying Two Numbers Near 1000994 x 998
Find out how far each number is from 1000
The 1st two numbers equal the sum of the differences subtracted from 1000
The last two numbers equal the product of the differences written as a 3digit number
1000(6+2)
6(2)
=992012
Two Things Helping
The (Reciprocal) Work Problem1/6 + 1/5 = 1/X
=6(5)
Use the formula ab/a+b.
The numerator is the product of the two numbers.
The deniminator is the sum of the two numbers.
Reduce if necessary
=6+5
=30/11
Two Things working Against Each Other
The (Reciprocal) Work Problem1/6  1/8 = 1/X
Use the formula ab/ba.
The numerator is the product of the two numbers.
The denominator is the difference of the two numbers from right to left.
Reduce if necessary
=6(8)
=86
=24
30/20=3/2
3/2(12)=18
=18
The Inverse Variation % Problem30% of 12 = 20% of ___
Compare the similar terms as a reduced ratio
Multiple the other term by the reduced ratio.
Write the answer
Sum of Consecutive Integers1+2+3+…..+20
(20)(21)/2
Use formula n(n+1)/2
Divide even number by 2
Multiply by the other number
10(21)= 210
Sum of Consecutive Even Integers2+4+6+…..+20
(20)(22)/4
Use formula n(n+2)/4
Divide the multiple of 4 by 4
Multiply by the other number
5(22)= 110
Sum of Consecutive Odd Integers1+3+5+…..+19
Use formula ((n+1)/2)2
Add the last number and the first number
Divide by 2
Square the result
(19+1)/2=
102 = 100
Finding Hexagonal NumbersFind the 5th Hexagonal Number
Use formula 2n2n
Square the number and multiply by2
Subtract the number wanted from the previous answer
2(5)2= 50
505=
45
Cube PropertiesFind the Surface Area of a Cube Given the Space Diagonal of 12
Use formula Area = 2D2
Square the diagonal
Multiply the product by 2
2(12)(12)
2(144)=
288
S
S
S
2
3
Find S, Then Use It To Find Volume or Surface Area
Cube Properties
Y = 3X +5
2
Finding Slope From An Equation3X+2Y=10
3X+2Y=10
Solve the equation for Y
The number in fron of X is the Slope
Slope = 3/2
Hidden Pythagorean Theorem Find The Distance Between These Points(6,2) and (9,6)
Find the distance between the X’s
Find the distance between the Y’s
Look for a Pythagorean triple
If not there, use the Pythagorean Theorem
96=3
62=4
3 4 5
5 12 13
3
4
5
7 24 25
8 15 17
The distance is 5
Common Pythagorean triples
Finding Diagonals Find The Number Of Diagonals In An Octagon
Use the formula n(n3)/2
N is the number of vertices in the polygon
8(83)/2=
20
Finding the total number of factors 24= ________
Put the number into prime factorization
Add 1 to each exponent
Multiply the numbers together
31 x 23=
1+1=2 3+1=4
2x4=8
7549 = _______
Estimating a 4digit square root
802=6400
852=7225
87
902=8100
37485 = _______
Estimating a 5digit square root
192=361
190200
195
202=400
55C = _______F
C F
9/5(55) + 32
99+32
= 131
50F = _______C
C F
5/9(5032)
5/9(18)
= 10
4X2 + 5X + 6
a
b
c
Finding The Product of the Roots
6 / 4 = 3/2
4X2 + 5X + 6
a
b
c
Finding The Sum of the Roots
5 / 4
999999 Rule
142857 x 26 =
Estimation
26/7 =3r5
5+0=50/7=7
3 714285
Find the area of a square with a diagonal of 12
Area of a Square Given the Diagonal
½ D1 D2
½ x 12 x 12
72
346 x 291 =
Estimation of a 3 x 3 Multiplication
35 x 30
1050 + 00
105000
7258 / 11=_____
Dividing by 11 and finding the remainder
Remainder
8+2=10 7+5= 12
1012= 2 +11= 9
2994 x 6 =
Multiply By Rounding
3000(6)=179_ _
6(6)=36 10036=64
=17964
(factor of 2)
122 + 242=
The Sum of Squares
122=144
144x10=
=1440/2
=720
(factor of 3)
122 + 362=
The Sum of Squares
122=144
144x10=
=1440
(Sum x the Difference)
322  302=
The Difference of Squares
32+30=62
3230=2
62 x 2 =124
(Sum x the Difference)
2989 + 456=
Addition by Rounding
2989+11= 3000
45611=445
3000+445=3445
(1111…Problem)
123 x 9 + 4
123…x9 + A Constant
3x9 + 4 =31
1111
Find The Difference Of The Supplement And The Complement Of An Angle Of 40.
Supplement and Complement
=90
Find The Sum Of The Supplement And The Complement Of An Angle Of 40.
Supplement and Complement
27080=
=190
513
+
4 11
55
52
Larger or Smaller
Larger = 5/4
Smaller = 13/11
A

1
= 11
3
Two Step Equations(Christmas Present Problem)
11+1=12
12 x3 = 36
22 x 51
=21 x 50
=2 x 1
How Many #s less than 20 are relatively prime to 20?
Relatively Prime(No common Factors Problem)* One is relatively prime to all numbers
2 x 1 x 1 x 4 = 8
Find the Product of the GCF and the LCM of 6 and 15
Product of LCM and GCF
6 x 15 = 90
15 x 17 x 19
Estimation
173=4913
7, 2, 5, 8, 3, 14
SequencesFinding the Pattern
Find the next number in this pattern
7, 2, 5, 8, 3, 14
1
1, 4, 5, 9, 14, 23
SequencesFinding the Pattern
Find the next number in this pattern
1, 4, 5, 9, 14, 23
14+23=37
900= _____
Degrees Radians
Radians
90(Pi)/180
=Pi/2