1. STRUCTURES I
Thursday, 11/8/2012
Methods of Multiplication
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2. Remember The Array Model
Use an array model to multiply 17X53
50 3
The product is the
sum of the pieces
10
500 500+350+30+21
30
850+30+21
880+21
901
7
350 21
4. Connecting the Traditional Method to
the Array Model
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
19X28
Traditional
20 8
28
X19
Sum of 252
10 Rows 280
200
80 532
280
9 252
180 72
6. Partial Products
Multiply 17 X 53 using the partial products
method.
53
x17
500 10x50
30 10x3
350 7x50
21 7x3
901
Note: It is the array method without the array!
7. 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
• 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
17x53
1. Draw an array based on the number of digits in the numbers (2 by 2 in this case)
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. 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 1
10. Using Lattice
Try 19x28 using the lattice method. Click to see the process when you have
finished.
2 8
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
normal method.
1. 24 x 25
2. 46 x 84
3. 55 x 98
12. 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 .126 3.0870
• What do you notice about the digits in the
answers?
13. Placing the Decimal
• We probably all remember what we were taught;
count the total number of decimal places and
ensure that number of places are in the answer.
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.
• Looking at it mathematically, 2.45=245x10-2 and
1.26=126x10-2. 245x10-2x126x10-2=30870x10-4.
14. Placing the Decimal by Estimation
• Compare the Estimate and Where the Decimal
is Placed
Number 1 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
estimation.
• If 12x55=660, what estimation would you use to
place the decimal for 1.2x5.5.
• If 26x37=962, what estimation would you use to
place the decimal for 26x3.7.
• If 87x932=81084, what estimations would you
use for
– 8.7x93.2
– 8.7x9.32
– .87x93.2
16. Using the Array for Multiplying
Fractions
2 3
• Consider 3 4
Start with a 1x1 rectangle
Divide one side into thirds
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
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
• 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?
19. Multiplying Fractions
• The arrays should have illustrated that the total
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 and multiply the denominators.
• 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
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
• Consider 8 25 5 43
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:
3
42 95
3
2 3
79 69
1 3
12 6 15 8
