Download Math90 Day02 Math Notes

Math90 Day02 Math Notes - for Rounding:
I sometimes require notations and techniques that are not covered completely in the text.
To get full credit, these should always be used in your homework, tests and quizzes.
Showing work when ROUNDING a number
Problems: 45. Round 4973 to Hundreds
45. 4 9 7 3
Show work:
1
 5 0 00
51. Round 253,678 to Ten-thousands
51. 2 5 3, 6 7 8
0
 2 5 0, 0 0 0
Summary: Write down the number to be rounded (leave space between digits)
Underline the digit in the rounding position and all digits to its left
Circle the digits immediately following the ones you underlined
(the first circled digit is called the “check digit”).
Rounding Up: If the check digit is 5 or greater,
Write +1 directly under the right-most underlined digit, circle the digit 7
Since 7 ≥ 5, Add 1 to the underlined digits above (49 + 1 = 50 in the first example)
Replace the check digit and all digits to its right with 0’s (73 becomes 00)
Rounding Off: If the check digit is 4 or less,
Write +0 under the right-most underlined digit, circle the check digit 3
Since 3 ≤ 4, Add 0 to the underlines digits above it (25 + 0 = 25 in the second example)
Replace the check digit and all digits to its right with 0’s (3,678 becomes 0,000)
Some students question this:
“MrV, I already know how to do rounding in my head. Why do I have to use this technique?”
My reply:
“First, I need to grade you accurately. Showing work this way proves you properly identified the rounding
position and the check digit before determining the answer. For 4973 in the 1st example, you might have
looked at the 9 in the 100ths position as the check digit, so you would wind up with 5000 as the same answer.”
“Second, you get the opportunity to improve your math habits. People who don’t like math have a
tendency to write down as little intermediate work as possible. As math gets more complex, this often leads
to lower grades.”
“Third, it reduces the temptation to scam homework. Some students have actually written down the
answers from the back of the book … and then expected magic to occur on tests!”
More Examples: Round 389,702 to Hundreds
3 8 9, 7 0 3
Show work:
0
 3 8 9, 7 0 0
Round 236,753,678 to Hundred-thousands
2 3 6, 7 5 3, 6 7 8
1
 2 3 6, 8 0 0, 0 0 0
Math Notes for Whole Number Subtraction:
MrV sometimes requires notations and techniques that are not covered completely in the text.
These should always be used in your homework, tests and quizzes.
Subtraction always involves two numbers, with the procedure of borrowing
The text does a good with how to do subtraction, including how to do borrowing one digit at a time.
Borrowing into 0 could use a better step-by-step explanation.
Step-by-step Examples of borrowing into a 0 digit:
46.
Original
Borrow 10
problem
from the 4
Borrow 10
from the 8
No borrow
for 7 – 4
Borrow 10
from the 2
1 10
Borrow 10
from the 4
3 10
7 13 10
3 1110
840
8 4/ 0/
8/ 4/ 0/
76. 4 2 0 7
4 2/ 0/ 7
4/ 2/ 0/ 7
 7 43
 7 43
 7 43
 16 2 4
 16 2 4
 16 2 4
7
97
3
83
2 583
The text doesn’t give you a nice shortcut when you have to borrow from the digit 0.
What is one less than 40? 39, of course.
What is one less than 700? 699, of course.
What is one less than 6000? 5999, of course.
Trickier: What is one less than 100? Use 099 when placing the borrowing digits.
Examples, showing multiple steps of the process.
Original
Borrow 10
problem
from the 70
Borrrow 10
from the 3
6 9 16
37 0 6
3---7 0 6/
2 16 9 16
 2957
 2957
3/---7 0 6/
 2957
58.
49
7 49
Original
problem
Borrow 10
from the 8000
7 9 9 9 14
80. 8 0,0 0 4
 8,2 3 7
8 0, 0 0 4/
-------- 8, 2 3 7
7 1,7 6 7
Sometimes a subtraction results in a negative number (the bottom # is bigger that the top #).
Example: 17 – 89 = ?
Let’s look at how to set your work area up in a special MrV way. (not in the text)
Normally, you would set it up as:
17
but switch the #’s, put – in the answer line,
–89
then do the subtraction:
can’t subtract
89
–17
–72
Math Notes for Multiplication:
MrV sometimes requires notations and techniques that are not covered completely in the text.
These should always be used in your homework, tests and quizzes.
Proof of multiplication involves showing all carries and adding single digit products
This chapter section only does a fair job showing how to do multiplication; it fails to give you any procedure
for showing multiplication carries.
Showing multiplication carries, step-by-step explanations.
Single digit multiplication example:
Original
4x9=
problem
3
4 x 5 +3 =
2
3
6
3
28.
559

4
559

2 3
2 3
559
559

4
6
4 x 5 +2 =
22
4
36

(little # is a carry digit, big #’s go on product line)
4
2236
Two digit multiplication example; two rows of products will be added; and usually two little rows of carries:
I recommend using – ‘s (dashes) rather than zeros to insure that the second and subsequent product rows
are lined up correctly.
Original
problem
56. 9 5 7 7

35
5 x 9577
showing carries
Cross off carries,
3 x 9577
Add 2 product rows,
set up 2nd row show new carries
show carries
2 3 3
-----2 3 3
1 2 2
2 3 3
------
957 7
957 7
957 7

35

35

35
1 2 2
2 3 3
------
957 7

35
1 1 1
4 7 885
4 7 885
4 7 885
4 7 885
–
2 8 7 31 –
 2 8 7 31 –
0
0
3 3 5,19 5
The text doesn’t give you an Easy Way to multiply numbers with trailing 0’s.
Let’s use 6,000 x 700 as an example.
1. Set up the multiplication as though the 0’s were not there. ( 6 x 7)
2. Do the multiplication. (6 x 7 = 42)
3. How many 0’s were ignored? Place them after the product. ( 4200000 )
4. Insert commas from the right. ( 4,200,000 )
Examples, showing multiple steps of the process.
66.
Original
Set it up for
problem
the easy way
700
 274
Do 7 x 274,
but ignore 0’s
274

7 00
Count 0’s, place them
after the product, add comma(s)
5 2
5 2
274
274

7 00
1 9 18

7 00
191, 8 0 0
The text shows an uncommon way of handling inside 0 multiplication. Use this instead:
I recommend using –’s (dashes)rather than zeros to insure that the second and subsequent product rows
are lined up correctly. You don’t need a product line for interior 0’s; just use an additional –.
Cross off carries,
Original
9 x 758
problem
Show carries
5 7
74.
se to skip one pos, 2 x 758
Add 2 product rows,
Use a 2nd to skip x0 show new carries show carries, add comma
1 1
1 1
5 7
----
5 7
-----
7 58
5 7
----
7 58
7 58
7 58
7 58
 209
 209
 209
 209
 209
68 2 2
68 2 2
68 2 2
68 2 2
– –
15 1 6 – –
 15 1 6 – –
1
15 8, 4 2 2
Math Notes for Long Division:
Proof of division involves showing all work for each digit in the quotient.
This chapter section does a good job showing how to do division, but it doesn’t show you how to simplify you
division when both numbers have trailing 0’s.
Simplifying divisions that have matching right-hand zeroes; step-by-step explanations.
Some problems sometimes have lots of matching trailing zeros in both divisor and dividend:
Note that the divisions below all have the same quotient. If you can remove the same number of trailing
zeros from both divisor and dividend; it makes the division simpler.
9
9
9
9
6 54 60 540 600 5400 6000 54000
How would you simplify the divisions below?.
80 220  8 22
700 6480  70 648
Using MrV’s Times Table to do Division – selected pictures from the Powerpoint.
Math Notes for Shorthand Division:
Proof of division involves showing all work for each digit in the quotient.
The textbook does not address shorthand division, which can save you a lot of time when you have to do
Prime Factorization (the next page). Mostly, this technique is used for dividing by 2, 3, 5 or 7.
Here are 2 examples that compare both ways. The notation is different: carries above, quotient below.


Class Notes for Prime Factorization:
I sometimes require notations and techniques that are not covered completely in the text.
These should always be used in your homework, tests and quizzes.
Prime Factorization is the key to finding GCF’s (Greatest Common Factor) and LCM’s
(Least Common Multiple).
The textbook doesn’t emphasize the importance of doing prime factorization, which is the key skill needed for
all operations involving fractions. The Factor Ladder technique used in the text also works, but it is not very
flexible and it doesn’t use your ability to use shorthand division. The Factor Tree technique exposes the primes
to be multiplied with a minimum of side calculations.
Factor Tree technique for doing Prime Factorization (PF):
1. Write down the number to be PF’ed.
2. Find one prime factor at a time (using tests below), going down one level for each new one.
a. Find all factors of 2 (if it’s an even number).
b. Find all factors of 5 (if it ends on 0 or 5).
c. Find all factors of 3 (if sum of digits divides by 3).
d. Check for other prime factors by using long division.
3. When all lower numbers are prime, write the PF from the “dangling” primes.
Example: Find the PF of 630: (shown as step-by step – circle the primes as you find them)
630
Factor out 2
10
10
10
10
630
630
630
630
ᴧ
15
2 31 5
Factor out 5
ᴧ
2 31 5
ᴧ
15
5
63
ᴧ
ᴧ
2 31 5
2 31 5
ᴧ
ᴧ
5 63
5 63
ᴧ
ᴧ
3 21
3 21
ᴧ
15
15
Factor out 3
Factor out 3 again
Write the PF as a product of the primes,
Put them in order, Use exponents
630  2  3  3  5  7
 2  32  5  7
3 7
More examples, showing final result.
Find the PF of 144:
Find the PF of 78:
1 44
ᴧ
2 72
ᴧ
2 36
ᴧ
2 18
ᴧ
2 9
ᴧ
3 3

Factor out 2
12
18
78
ᴧ
2 39
ᴧ
3 13
Factor out 2


Factor out 2
16
Factor out 2

Factor out 2
78  2  3  13

Factor out 3
the other 3 is prime
All are primes

Factor out 3
13 is also prime

144  2  2  2  2  3  3
 2 4  32
Find the PF of 225:
Find the PF of 46:
25
Factor out 5
Factor out 5
Factor out 3
and other 3
22 5
ᴧ
5 45
ᴧ
5 9
ᴧ
3 3



225  3  3  5  5
 32  5 2
Factor out 2
Can’t be 5 or 3,
7 = 49, can’t be 7
2
46
ᴧ
2 23

46  2  23