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# NUMBER SENSE AT A FLIP - PowerPoint PPT Presentation

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### Number Sense me know at

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 me know at

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 me know at

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 5 me know at 752

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 Decades me know at 35 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 Even me know at 45 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 Odd me know at 35 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&Even me know at 35 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) me know at

### Multiplying By 12 ½ 32 x 12 ½

32

Divide the non-12 ½ number by 8.

4+00

=

8

=400

(1/6 rule) me know at

### Multiplying By 16 2/3 42 x 16 2/3

42

Divide the non-16 2/3 number by 6.

7+00

=

6

=700

(1/3 rule) me know at

### Multiplying By 33 1/3 24 x 33 1/3

• Divide the non-33 1/3 number by 3.

24

8+00

=

3

=800

(1/4 rule) me know at

### Multiplying By 2532 x 25

32

Divide the non-25 number by 4.

8

+00

=

4

=800

(1/2 rule) me know at

### Multiplying By 5032 x 50

32

Divide the non-50 number by 2.

16

=

+00

2

=1600

(3/4 rule) me know at

### Multiplying By 7532 x 75

Divide the non-75 number by 4.

Multiply by 3.

32

=

8x3=24+00

4

=2400

(3/8 rule) me know at

### Multiplying By 37 1/237 1/2 x 24

=900

00

(3/8)24

(5/8 rule) me know at

### Multiplying By 62 1/262 1/2 x 56

=3500

00

(5/8)56

(7/8 rule) me know at

### Multiplying By 87 1/287 1/2 x 48

=4200

00

(7/8)48

(5/6 rule) me know at

### Multiplying By 83 1/383 1/3 x 36

=3000

00

(5/6)36

(2/3 rule) me know at

### Multiplying By 66 2/366 2/3 x 66

=4400

00

(2/3)66

(1/8 rule) me know at

### Multiplying By 12532 x 125

Divide the non-125 number by 8.

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’s Digits Are Equal97 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 100 Digits Are Equal109 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 1 Digits Are Equalst 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 Digits Are Equal

### Multiplying By 336718 x 3367

Divide the non-3367 # by 3

Multiply by 10101

18/3 = 6 x 10101=

= 60606

121 Pattern Digits Are Equal

### Multiplying A 2-Digit # 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 Digits Are Equal

### Multiplying A 3-Digit # 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 Digits Are Equal

### Multiplying A 3-Digit # 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 Digits Are Equal

### Multiplying A 2-Digit # 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 2-Digit # By 101 Digits Are Equal93 x 101

The first two digits are the 2-digit number x1

The last two digits are the 2-digit number x1

93(1)

93(1)

= 9393

### Multiplying A 3-Digit # By 101 Digits Are Equal934 x 101

The last two digits are the last two digits of the 3-digit number

The first three numbers are the 3-digit number plus the hundreds digit

934+9

34

= 94334

### Multiplying A 2-Digit # By 1001 Digits Are Equal87 x 1001

The first 2 digits are the 2-digit number x 1

The middle digit is always 0

The last two digits are the 2-digit number x 1

87(1)

0

87(1)

= 87087

### Halving And Doubling Digits Are Equal52 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’s Digits Are Equal95 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) Digits Are Equal8 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) Digits Are Equal7 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) Digits Are Equal7 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

(8-7) Digits Are Equal2

2

7x8

=21/56

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 Of Digits Are Equal36 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 Digits Are Equal426 =____10

Multiply the left digit times the base

Add the number in the units column

4(6)+2

= 2610

### Multiplying in Bases Digits Are Equal4 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 Decimal Digits Are Equal21/40___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 9 Digits Are Equal867/9=___remainder

Add the digits until you get a single digit

Write the remainder

8+6+7=21=2+1=3

= 3

421 Method Digits Are Equal

### 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 Digits Are Equal

### 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 Yards Digits Are Equal3ftx 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 MPH Digits Are Equal44 ft/sec __mph

Use 15 mph = 22 ft/sec

Find the correct multiple

Multiply the other number

22x2=44

15x2=30 mph

2 Digits Are Equal5=32

subsets

### Subset Problems{F,R,O,N,T}=______

SUBSETS

Subsets=2n

Improper subsets always = 1

Proper subsets = 2n - 1

Power sets = subsets

___ Digits Are Equal

### 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

_ Digits Are Equal

### 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.

18-1

17

=

90

90

### Gallons Cubic Inches Digits Are Equal2 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 Numbers Digits Are Equal5th 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 Numbers Digits Are Equal6th 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 Power Digits Are Equal13=____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 Power Digits Are Equal12=____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” Problem Digits Are Equal17/15 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

(17-15)2

17+2

15

=194/15

### The “Less” Problem Digits Are Equal15/17 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

(17-15)2

15-2

17

=134/17

### Multiplying Two Numbers Near 1000 Digits Are Equal994 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 3-digit number

1000-(6+2)

6(2)

=992012

Two Things Helping Digits Are Equal

### 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 Digits Are Equal

### The (Reciprocal) Work Problem1/6 - 1/8 = 1/X

Use the formula ab/b-a.

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)

=8-6

=24

30/20=3/2 Digits Are Equal

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.

### Sum of Consecutive Integers Digits Are Equal1+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 Integers Digits Are Equal2+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 Integers Digits Are Equal1+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 Numbers Digits Are EqualFind the 5th Hexagonal Number

Use formula 2n2-n

Square the number and multiply by2

Subtract the number wanted from the previous answer

2(5)2= 50

50-5=

45

### Cube Properties Digits Are EqualFind 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 Digits Are Equal

S

S

2

3

Find S, Then Use It To Find Volume or Surface Area

### Cube Properties

Y = - Digits Are Equal3X +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 Digits Are EqualFind 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

9-6=3

6-2=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 Digits Are EqualFind The Number Of Diagonals In An Octagon

Use the formula n(n-3)/2

N is the number of vertices in the polygon

8(8-3)/2=

20

### Finding the total number of factors Digits Are Equal24= ________

Put the number into prime factorization

Multiply the numbers together

31 x 23=

1+1=2 3+1=4

2x4=8

7549 = _______ Digits Are Equal

### Estimating a 4-digit square root

• The answer is between 802 and 902

• Find 852

• The answer is between 85 and 90

• Guess any number in that range

802=6400

852=7225

87

902=8100

374 Digits Are Equal85 = _______

### Estimating a 5-digit square root

• Use only the first three numbers

• Find perfect squares on either side

• Add a zero to each number

• Guess any number in that range

192=361

190-200

195

202=400

55C = _______F Digits Are Equal

### C F

• Use the formula F= 9/5 C + 32

• Plug in the F number

9/5(55) + 32

99+32

= 131

50F = _______C Digits Are Equal

### C F

• Use the formula C = 5/9 (F-32)

• Plug in the C number

5/9(50-32)

5/9(18)

= 10

4X Digits Are Equal2 + 5X + 6

a

b

c

### Finding The Product of the Roots

• Use the formula c/a

• Substitute in the coefficients

6 / 4 = 3/2

4X Digits Are Equal2 + 5X + 6

a

b

c

### Finding The Sum of the Roots

• Use the formula -b/a

• Substitute in the coefficients

-5 / 4

999999 Rule Digits Are Equal

142857 x 26 =

### Estimation

• Divide 26 by 7 to get the first digit

• Take the remainder and add a zero

• Divide by 7 again to get the next number

• Find the number in 142857 and copy in a circle

26/7 =3r5

5+0=50/7=7

3 714285

Find the area of a square with a diagonal of 12 Digits Are Equal

### Area of a Square Given the Diagonal

• Use the formula Area = ½ D1D2

• Since both diagonals are equal

• Area = ½ 12 x 12

½ D1 D2

½ x 12 x 12

72

346 x 291 = Digits Are Equal

### Estimation of a 3 x 3 Multiplication

• Take off the last digit for each number

• Round to multiply easier

35 x 30

1050 + 00

105000

7258 / 11=_____ Digits Are Equal

### Dividing by 11 and finding the remainder

Remainder

• Do the same with the other numbers

• Subtract the two numbers

• If the answer is a negative or a number greater than 11 add or subtract 11 until you get a number from 0-10

8+2=10 7+5= 12

10-12= -2 +11= 9

2994 x 6 = Digits Are Equal

### Multiply By Rounding

• Round 2994 up to 3000

• Think 3000 x 6

• Write 179. then find the last two numbers by multiplying what you added by 6 and subtracting it from 100.

3000(6)=179_ _

6(6)=36 100-36=64

=17964

(factor of 2) Digits Are Equal

122 + 242=

### The Sum of Squares

• Since 12 goes into 24 twice…

• Square 12 and multiply by 10

• Divide by 2

122=144

144x10=

=1440/2

=720

(factor of 3) Digits Are Equal

122 + 362=

### The Sum of Squares

• Since 12 goes into 36 three times…

• Square 12 and multiply by 10

122=144

144x10=

=1440

(Sum x the Difference) Digits Are Equal

322 - 302=

### The Difference of Squares

• Find the sum of the bases

• Find the difference of the bases

• Multiply them together

32+30=62

32-30=2

62 x 2 =124

(Sum x the Difference) Digits Are Equal

2989 + 456=

• Round 2989 to 3000

• Subtract the same amount to 456, 456-11= 445

2989+11= 3000

456-11=445

3000+445=3445

(1111…Problem) Digits Are Equal

123 x 9 + 4

### 123…x9 + A Constant

• The answer should be all 1s. There should be 1 more 1 than the length of the 123… pattern.

• You must check the last number. Multiply the last number in the 123… pattern and add the constant.

3x9 + 4 =31

1111

Find The Difference Of The Supplement And The Complement Of An Angle Of 40.

### Supplement and Complement

• The answer is always 90

=90

### Supplement and Complement

• Use the formula 270-twice the angle

• Multiple the angle by 2

• Subtract from 270

270-80=

=190

5 Angle Of 40. 13

+

4 11

55

52

### Larger or Smaller

• Find the cross products

• The larger fraction is below the larger number

• The smaller number is below the smaller number

Larger = 5/4

Smaller = 13/11

A Angle Of 40.

-

1

= 11

3

### Two Step Equations(Christmas Present Problem)

11+1=12

12 x3 = 36

2 Angle Of 40. 2 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

• Put the number into prime factorization

• Subtract 1 from each exponent and multiply out all parts separately

• Subtract 1 from each base

• Multiply all parts together

2 x 1 x 1 x 4 = 8

### Product of LCM and GCF

• Multiple the two numbers together

6 x 15 = 90

15 x 17 x 19 Angle Of 40.

### Estimation

• Take the number in the middle and cube it

173=4913

7, 2, 5, 8, 3, 14 Angle Of 40.

### Sequences-Finding the Pattern

Find the next number in this pattern

• If the pattern is not obvious try looking at every other number. There may be two patterns put together

7, 2, 5, 8, 3, 14

1

1, 4, 5, 9, 14, 23 Angle Of 40.

### Sequences-Finding the Pattern

Find the next number in this pattern

• 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, 23

14+23=37

90 Angle Of 40. 0= _____