This document provides a summary of different methods for multiplying multi-digit numbers, including the array model, traditional method, partial products method, and lattice method. It includes examples of using each method to multiply numbers like 17x53 and 19x28. The document also discusses placing decimals in products and using arrays to multiply fractions.

1. STRUCTURES I
Thursday, 11/8/2012
Methods of Multiplication
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2. Remember The Array Model
Remember The Array Model
Use an array model to multiply 17X53
Use an array model to multiply 17X53
50 3
The product is the
The product is the
sum of the pieces
10
500 500+350+30+21
30
850+30+21
880+21
901
7
350 21

3. Traditional Method
Traditional Method
Multiply 17 X 53 using the traditional method.
Multiply 17 X 53 using the traditional method
53
 17
371 7X3=21, 7X5=35, 35+2=37
7X3=21 7X5=35 35+2=37
530 10X53=530
901

4. Connecting the Traditional Method to
the Array Model
h d l
Note the sum of the rows.
Note the sum of the rows
50 3
10
500
30 530
7
350 21 371

5. Try It Again
Do Both Array and Traditional Before Clicking Forward
y g
19X28
Traditional
20 8
28
X19
Sum of 252
10 Rows 280
200
80 532
280
9 252
180 72

6. Partial Products
Partial Products
Multiply 17 X 53 using the partial products
Multiply 17 X 53 using the partial products
method.
53
x17
500 10x50
500 10x50
30 10x3
350 7x50
21 7x3
901
Note: It is the array method without the array!

7. Using the Partial Products Method
Using the Partial Products Method
Try 19x28 using the partial products method.
Try 19x28 using the partial products method
Click to see the process when you have finished.
28
X 19
200
80
180
72
532

8. Partial Product Connections
Partial Product Connections
• Note that the partial product method is an
Note that the partial product method is an
extension of the distributive property!
– 17x53=(10+7)x(50+3)=10x50+10x3+7x50+7x3
– 19x28=(10+9)x(20+8)=10x20+10x8+9x20+9x8

9. Lattice Method
Named for the lattice look to the model
Named for the lattice look to the model
17x53
1. Draw an array based on the number of digits in the numbers (2 by 2 in this case)
y g ( y )
2. Draw diagonal lines to create the lattice
3. Multiply the digits putting the tens above the line and the units below the line
4. Add down the diagonals
5.
5 The answer is read from top left to bottom right
The answer is read from top left to bottom right
5 3 5 3
0 0 1
1 0 1
5 3
7 3 2
7
9
5 1
0
0 1

10. Using Lattice
Using Lattice
Try 19x28 using the lattice method. Click to see the process when you have
finished.
2 8
0
0 0 2
0 1
2 8
1 7
9
5
8 2
3 2

11. Try The Following Using Array, Partial
Product and Lattice. Check using your
Product and Lattice. Check using your
normal method.
1. 24 x 25
2. 46 x 84
2 46 84
3. 55 x 98

12. A Discovery Activity
A Discovery Activity
• Use your calculator to complete the table
Number 1 Number 2 Product of the Two
Numbers
245 126 30870
24.5 1.26 30.870
24.5 12.6 308.70
2.45 1.26 3.0870
.245 126 30.870
24.5
24 5 .126
126 3.0870
3 0870
• What do you notice about the digits in the
answers?

13. Placing the Decimal
Placing the Decimal
• We probably all remember what we were taught;
p y g ;
count the total number of decimal places and
ensure that number of places are in the answer.
But why does it work?
But why does it work?
• Start with 245x126=30870. 2.45x1.26 moves
each number two places to the left, so move four
places to the left in the answer. 24.5x1.26 moves
one place in 245 and two places in 126, so move
three places in the answer.
three places in the answer
• Looking at it mathematically, 2.45=245x10‐2 and
1.26=126x10‐2. 245x10‐2x126x10‐2=30870x10‐4.

14. Placing the Decimal by Estimation
Placing the Decimal by Estimation
• Compare the Estimate and Where the Decimal
Compare the Estimate and Where the Decimal
is Placed
Number 1 Number 2
Number 2 Estimate Product
245 126 30870
24.5 1.26 24x1=24 30.870
24.5 12.6 25x12=300 308.70
2.45 1.26 2x1=2 3.0870
.245 126 .2x100=20 30.870
24.5 .126 24x.1=2.4 3.0870

15. Practice
• Given the information, place the decimal by
,p y
estimation.
• If 12x55=660, what estimation would you use to
place the decimal for 1.2x5.5.
place the decimal for 1 2x5 5
• If 26x37=962, what estimation would you use to
place the decimal for 26x3.7.
place the decimal for 26x3.7.
• If 87x932=81084, what estimations would you
use for
– 8.7x93.2
– 8.7x9.32
– .87x93.2
87x93 2

16. Using the Array for Multiplying
Fractions
2 3
• Consider 3  4
Consider
Start with a 1x1 rectangle
Divide one side into thirds
Divide the other side into fourths
Divide the other side into fourths
Take two‐thirds and three‐quarters and surround them with a rectangle
The rectangle has 6 pieces out of a total of twelve, 6/12 or ½.
1 1 1 1
4 4 4 4
1
3
1
3
1
3

17. Practice
• Use an array to illustrate the following
Use an array to illustrate the following
products
2 3 2 3
 
5 5 5 8
2 3 4 3
 
3 5 5 8
Looking at your arrays and the answers, what rule could you give so you
don’t need to draw arrays all the time.

18. A Exploration
A Exploration
• Complete each and look for a relationship
Complete each and look for a relationship
2 3 3 2
 
5 8 5 8
2 3 3 2
 
3 5 3 5
4 3 3 4
 
5 8 5 8
What relationship do you see?
How might it help you?
How might it help you?

19. Multiplying Fractions
Multiplying Fractions
• The arrays should have illustrated that the total
y
number of pieces is the product of the denominators
and the number in the rectangle is the product of the
numerators. So, to multiply fractions, you multiply the
numerators So to multiply fractions you multiply the
numerators and multiply the denominators.
p y
• In the exploration, you should have seen that the
numerators (or the denominators) could be switched
and still yield the same result. Therefore, you might be
able to use this concept to simplify the problem before
able to use this concept to simplify the problem before
multiplying. For example, seeing 2/3x3/5 was the
same as 3/3x2/5 makes it 1x2/5 or 2/5.

20. Using an Array to Multiply Mixed
Numbers
b
• Consider 82 54
Consider 5 3
8 2/5
Answer=48 3/10
5
40
2
3/4
6 3/10

21. Practice
• Use an array to find the following products:
Use an array to find the following products:
4 2 9 5
3
3
7 9 6 9
2 3
12 1
6  15 3
8