Download Multiplication Notes

Multiplication Unit Definitions:
Factor: The numbers used in a multiplication problem OR A factor of a
given number is any number that divides evenly into a given number with no
remainder.
Multiple: A multiple of a given number is the product of that number and
any natural number (counting number).
Product: The answer to a multiplication problem
Composite Number: A number with more than two factors.
Prime Number: A number with two factors: the number 1 and itself.
15 prime numbers in all from 1-50
1s (4) 2, 3, 5, 7
10s (4)11, 13, 17, 19
20s (2) 23, 29
30s (2) 31, 37
40s (3) 41, 43, 47
peRIMeter: The distance around the RIM of a figure.
Area: The measure of covering inside a figure. It is measured in square units
(units²).
Divisibility: If a number “a” is divisible by a number “b”, “b” can divide
evenly into “a” with no remainder.
Multiplication Unit:
Multiplication is expressed in 3 ways. Take 3 x 5
1. As a “group of” objects/things.
(3 x 4) 3 groups of 4 pencils
2. As “repeated addition”
3 x 4 = 4 + 4 + 4
4 x 3 = 3 + 3 + 3 + 3
3. As a “model for area” in the form of an array (modeled on
next page).
Multiplication Fact Strategies:
1. Memorize the facts!
7 x 8
7 x 8 = 56
2. Use a fact that you know, then add or subtract.
7 x 7 = 49
49 + 7 = 56
3. Combine two facts that you know.
5 x 7= 35
3 x 7= 21 35+21 = 56
4. Half one of the factors, then double the product.
4 x 7= 28
28 x 2 = 56
5. Skip count the multiples.
8, 16, 24, 32, 40, 48, 56
Factor Pairs and Arrays- Finding the Area and Perimeter:
Every factor pair has a matching array.
** Remember: R C cola Rows x Columns**
Given the factor pair 3 x 6, draw the array. Make sure to label it.
6
3
The formula for PERIMETER of a rectangle:
L= length and W = width
P = 2L + 2W
= (2 x 3) + (2 x 6)
= 6 + 12
= 18 units
P =
=
=
=
(L + W) x 2
(3 + 6) x 2
9 x 2
18 units
The total’s label for PERIMETER is whatever measurement you
are using. If there is no given measurement, you can just use the
term “units”
The formula for AREA of a rectangle:
A= L x W
=3x6
= 18 units² or squared units
The total’s label for AREA is squared units or units
Properties of Multiplication
1) Identity Property: (Multiplicative Identity)
Any number multiplied by 1 equals itself.
Example:
5 x 1 = 5
2) Zero Property: Any number times 0 equals 0.
Example:
7 x 0 = 0
3) Associative Property: Changing the position of the parenthesis
(grouping symbol) does NOT affect the product.
Example: (a x b) x c = a x (b x c)
(2 x 3) x 5 = 2 x (3 x 5)
(6) x 5 = 2 (15)
30 = 30
**Think of an association, like the girl scouts association; It
is a group.**
4) Commutative Property: changing the position of the factors
does NOT change the product.
Example: a x b = b x a
10 x 11 = 11 x 10
110 = 110
** just think your parents COMMUTE to and from work **
The Rules of Divisibility
Divisibility: a number “a” is divisible by a number “b” if “b” divides
evenly into “a” with no remainder.
Example: 30 is divisible by 6 because 6 divides evenly into 30 with
no remainder.
30 ÷ 6 = 5
30 is NOT divisible by 7 because 7 divides into 30 4 times with a
remainder of 2.
Divisibility Rules for:
 2,5,10
o Look at the LAST DIGIT
 2 – 0,2,4,6,8
 5 – 0,5
 10 - 0
 3 and 9
o Sum of the digits - has to be a multiple of 3 or 9
-below they are recorded in () by the number
 3 {3,6,9,12,15,18,21,24…}
 9 {9,18,27,36,45,54,63…}
 6 (2/3)
o Product of two factors – number has to be divisible by
2 and 3 for it to be divisible by 6.
32,730 (15)
471 (12)
4,023 (9)
68,202 (18)
53 (8)
8,291 (20)
2
X
X
3
x
X
x
x
5
x
6 (2/3)
X
x
9
X
x
10
x
Create a 5 Digit Number that Meets Certain
Divisibility Requirements
**draw out 5 dash marks for each place value of the number**
1. Create a 5 digit number divisible by 2 and 9
(9)
2 0, 3 0 4
0,2,4,6,8
2. Create a 5 digit number divisible by 2, 5, and 10
5 1, 8 9 0
0
3. Create a 5 digit number divisible by 3 and 5, but not 10.
(21)
4 3, 7 2 5
5, 0
Multiply Numbers by Multiples of 10, 100, 1000, etc.
1. Box out the digits (natural/counting numbers). Multiply them
together.
2. Count the zeroes in both FACTORS using a . Place that many zeroes
in the PRODUCT.
 The TOTAL number of zeroes in both factors should equal the
number of zeroes in the product.
**be careful when dealing with products that are multiples of 10- make
sure you are not short a zero!**
3,000 x 70 = 210, 000
600 x 50 = 30,000
4,000 x 80 = 320,000
90 x 600,000 = 54,000,000
800 x 100 = 80,000
200 x 150 = 30,000
400 x 5,000 = 2,000,000
7,000 x 800 = 5,600,000
Partial Product Method of Multiplication
 ESTIMATE to find out approximately what your answer will be.
 Write numbers in EXPANDED FORM to show the VALUE of each digit.
2 digit x 1 digit
39
x4
Estimate: 40 x 4 = 160
30 + 9
x 4
36 (4 x 9)
+ 120 (4 x 30)
156
3 digit x 1 digit
537
x 8
500 + 30 + 7
x 8
56 (8 x 7)
240 (8 x 30)
+ 4,000 (8 x 500)
4,296
Estimate: 500 x 8 = 4,000
2 digit x 2 digit
(lock it and block it)
47
x 84
Estimate: 50 x 80= 4,000
40 + 7
x 80 + 4
28 (4 x 7)
80 ( 4 x 40)
560 (80 x 7)
+ 3200 (80 x 40)
3,868
Traditional Method of Multiplication
 ESTIMATE to find out approximately what your answer will be.
 use REGROUPING as you multiply
2 digit x 1 digit
3
37
x5
185
Estimate:
40 x 5 - 200
1. Begin by multiplying 7 x 5 = 35
2. 5 remains in the ones PV.
3. REGROUP the 3 into the tens PV.
4. Multiply 5 x 3 = 15. And ADD the 3 you
regrouped.
5 x 3 = 15 + 3 = 18
3 digit x 1 digit
2 5
537
x 8
4,296
Estimate:
500 x 8 = 4,000
1. 7 x 8 = 56. Regroup the 5.
2. 8 x 3 = 24 + 5 = 29. Regroup the 2.
3. 8 x 5 = 40 + 2 = 42
2 digit x 2 digit
(lock it and block it)
2
3
68
x 34
272
+ 2040
2,31 2
Estimate:
70 x 30 = 2,100
1. 4 x 8 = 32. Leave the 2 in the ones PV and
regroup the 3.
2. 4 x 6 = 24 + 3 = 27.
3. LOCK IT AND BLOCK IT! Put a 0 below the 2 to
represent the 10s PV (LOCK IT) that you are
multiplying by. BLOCK OUT any number that you
have already regrouped by putting a line through
it.
a. In 34, the value of the 3 is 30.
4. 3 x 8 = 24. Leave the 4 and regroup the 2.
5. 3 x 6 = 18 + 2 = 20.
6. Add the two products together.
3 digit x 2 digit
(lock it and block it)
2
1 4
327
x 36
1962
+ 9810
11,772
Estimate:
300 x 40 = 12,000
Follow the same directions as above; just go one
extra step as you multiply across by the 6 and the
3 in the bottom factor.
Four Step Problem Solving Method
Mark lives on a farm where he grows fruits and vegetables.
He planted 25 rows of corn with 30 plants in each row. How
many corn plants did Mark plant?
1 – FIND OUT- what does the problem mean? What question
must you answer?
X = number of corn plants Mark planted
2 – CHOOSE A STRATEGY – write an equation.
X = (number of rows) x (number of plants in each row)
X = 25 x 30
3 – SOLVE IT – work it out; find the answer; record your work.
1
25
X 30
750
4 – LOOK BACK – check the problem; check your work; rewrite
the equation with the answer.
X = 25 x 30 = 750 corn plants
Greatest Common Factor
The greatest common factor, GCF, is the largest factor common
to or shared by 2 or more numbers.
Example: Find the GCF for 16 and 24.
I.
List the factors for each number
16 {1, 2, 4, 6, 8, 16}
GCF = 8
24 { 1, 2, 3, 4, 6, 8, 12, 24}
II. “Prime” Factorization / “Monkey Division”
*Start with the smallest PRIME NUMBER and work your way up*
2
2x2x2 =
2³
GCF = 8
16
2
8
2 4
2
24
12
6
3