Number Sense
Techniques to make your mental math faster
Number Sense
The tips and strategies are used to help make
solving math problems easier and faster. It
requires memorization and practice to become
accurate every time and recognize the rules
quickly.
Always try to practice using a time limit. Speed
and accuracy are important.
Number Sense Scoring
-Numbers are written in pen and must be legible.
-Showing your work, mark outs, or overwrites will result in
an error.
-Each correct answer is 5 points.
-Each skipped or missed costs 4 points.
-Illegible numbers are errors. Numbers written under the
line are errors.
Number Sense Rules
-Starred problems are approximate answers
and must be within 5% of the correct answer.
-Exponentials must be written out into their full
number form. 83 must be written 512.
-Fractions must be reduced to their lowest
terms.
Squaring Numbers
One of the first skills to learn and memorize is
your squares. You should know your first 12
from your multiplication facts.
12 = 1 42 = 16
72 = 49
102 =
100
22= 4
52 = 25
82 = 64
112
= 121
32= 9
62 = 36
92 = 81
122
Tip for Solving for a 2 Digit Square
-5
40
Take 40 x 50 and multiply.
45
X
+5
50
Then add the square of the
number added/subtracted.
5 squared = 25.
Now let’s look at the next set of squares
132 =
192 =
252 =
142 =
202 =
262 =
152 =
212 =
272 =
162 =
222 =
282 =
Learning Perfect Squares and Cubes
After you’ve learned your perfect squares, it is
time to learn your perfect cubes from 13 = 1 to
252= 15625.
You should learn these in both directions:
172 = 289 and the
289
Squaring Numbers Ending in 5
752
1. First two digits - take the first digit and take the digit +1
2. Add the last two digits of 25. (You are squaring the 5.)
7(7 + 1)
25 = 5625
Multiplying Consecutive Tens
35 x 45
1. First two digits = the small ten’s digit times one
more than the large tens digit.
2. Last two digits are always 75.
3(4+1) 75
=1575
Ending in 5….Ten’s Digits are Even
45 x 85
1. First two digits = the product of the ten’s digits plus ½
the sum of the ten’s digit.
2. Last two digits are always 25.
4(8) + ½ (4+8)
25 =
3825
Ending in 5….Ten’s Digits are Odd
35 x 75
1. First two digits = the product of the ten’s digits plus ½
the sum of the ten’s digit.
2. Last two digits are always 25.
3(7) + ½ (3+7)
25 =
2625
Ending in 5….Ten’s Digits are Odd & Even
35 x 85
1. First two digits = the product of the ten’s digits plus ½
the sum of the ten’s digit. Drop the remainder.
2. Last two digits are always 75.
3(8) + ½ (3+8)
75 =
2975
The Rainbow Method
FOIL - First outside inside last
23 x 12
2(1)
2(2) + 3(1)
276
3(2)
Multiplying by 25
32 x 25
1. Divide by non-25 number by 4.
2. Add two zeroes.
32 divided by 4 = 800
Multiplying by 50
32 x 50
1. Divide by non-50 number by 2.
2. Add two zeroes.
32 divided by 2 = 1600
Multiplying by 75
32 x 75
1. Divide by non-75 number by 4.
2. Multiply by 3.
3. Add two zeroes.
32 divided by 4 = 8 x 3 = 2400
Multiplying by 125
32 x 125
1. Divide by non-125 number by 8.
2. Add three zeroes.
32 divided by 8 = 4000
Multiplying Two Numbers in 90s
97 x 94
1. 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) 3 (6) = 91 18 = 9118
Multiplying Two Numbers in 100s
109 x 106
1. The first number will always be 1.
2. The middle two numbers equal the sum of the units
digits.
3. The last two digits equals the product of the units digits.
1 9+6 9(6) = 1 15 54 = 11,554
Multiplying a 2-Digit Number by 11
92 x 11
1. Last digit is the units digit.
2. The middle digit is the sum of the tens and
the units digit.
3. The first digit is the tens digit + any carry.
9+1 9+2 2 = 10 1 2 = 1012
Multiplying a 3-Digit Number by 11
192 x 11
1. 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 hundreds digit + any carry.
1+1 1+9+1 9+2 2 = 2 1 1 2 = 2112
Multiplying a 2-Digit Number by 111
41 x 111
1. 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 hundreds + carry.
4. The next digit is the tens digit + carry.
4 4+1 4+1 1 = 4 5 5 1 = 4551
Multiplying a 3-Digit Number by 111
192 x 111
1. Last digit is the units digit.
2. 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.
3. The next digit is the sum of the tens and hundreds + carry.
4. The next digit is the hundreds digit + carry.
1+1 1+9+1 1+9+2+1 9+2 2= 2 1 3 1 2
= 21312
Multiplying a 2 Digit Number by 101
93 x 101
1. The first two digits are the 2 digit number x1.
2. The last two digits are the 2-digit number x1.
93 (1) 93 (1) = 93 93 = 9393
Multiplying a 3 Digit Number by 101
934 x 101
1. 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.
934 + 9 34 = 943 34 = 94334
Halving and Doubling
52 x 13
1. Take half of one number.
2. Double the other number.
3. Multiply together.
52 divided by 2 13(2) = 26(26) = 676
Fraction Foil
8½ x6¼
1. Multiply the fractions together.
2. Multiply the outside two numbers.
3. Multiply the inside two numbers.
4. Add the results and then add tothe product of the whole
numbers.
(8)(6)
+ ½(6) + ¼(8) (½ x ¼) = 53 ⅛
Adding Reciprocals
⅞ + 8/7
1. Keep the denominators.
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 (8-7)2 = 2 1/56
7x8
Percent Missing the 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 00 = 400
Converting Base N to Base 10
426 = _____10
1. Multiply the left digit times the base.
2. Add the number in the units column.
4(6) + 2 = 26 10