notes for structures i session 18 11 8
TRANSCRIPT
STRUCTURES IThursday, 11/8/2012
Methods of MultiplicationView this presentation as a slide show so you hear the narration as well.
You will need to click to advance the slides. On some slides, you will need to click to bring up parts of the presentation on that slide.
Remember The Array ModelRemember The Array Model
Use an array model to multiply 17X53Use an array model to multiply 17X5350 3
The product is the
10500
30
The product is the sum of the pieces
500+350+30+2130850+30+21880+21901
7350 21
Traditional MethodTraditional Method
Multiply 17 X 53 using the traditional methodMultiply 17 X 53 using the traditional method.
53 17
7X3=21 7X5=35 35+2=37371 7X3=21, 7X5=35, 35+2=37
10X53=530
371530901
Connecting the Traditional Method to h d lthe Array Model
Note the sum of the rows50 3
Note the sum of the rows.
10500
30 53030
7350 21 371
Try It Again Do Both Array and Traditional Before Clicking Forwardy g
19X28
20 8
Traditional
28X19
10200
80
X19252280532
Sum of Rows
80280
9180 72 252
Partial ProductsPartial Products
Multiply 17 X 53 using the partial productsMultiply 17 X 53 using the partial products method.
5353x17500 10x50500 10x5030 10x3
350 7x5021 7x3
901Note: It is the array method without the array!
Using the Partial Products MethodUsing the Partial Products Method
Try 19x28 using the partial products methodTry 19x28 using the partial products method. Click to see the process when you have finished.
28X 192008018072532
Partial Product ConnectionsPartial Product Connections
• Note that the partial product method is anNote that the partial product method is an extension of the distributive property!– 17x53=(10+7)x(50+3)=10x50+10x3+7x50+7x3– 17x53=(10+7)x(50+3)=10x50+10x3+7x50+7x3– 19x28=(10+9)x(20+8)=10x20+10x8+9x20+9x8
Lattice MethodNamed for the lattice look to the modelNamed for the lattice look to the model
17x531. 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 lattice3. Multiply the digits putting the tens above the line and the units below the line4. Add down the diagonals5 The answer is read from top left to bottom right5. The answer is read from top left to bottom right
5 35 3
0
5
0
3
3 2
11 10
3
5
2
1
77
0
9
0 1
Using LatticeUsing LatticeTry 19x28 using the lattice method. Click to see the process when you have finished.
0 0
2 8
20
2
0
8
1 7
12
0
1
8
7
2
9
23
5
23
Try The Following Using Array, Partial Product and Lattice. Check using yourProduct and Lattice. Check using your
normal method.
1. 24 x 252 46 842. 46 x 843. 55 x 98
A Discovery ActivityA Discovery Activity• Use your calculator to complete the table
Number 1 Number 2 Product of the Two Numbers
Number 1 Number 2 Product of the Two Numbers
245 126
24.5 1.26
24.5 12.6
245 126 30870
24.5 1.26 30.870
24.5 12.6 308.70
2.45 1.26
.245 126
24 5 126
2.45 1.26 3.0870
.245 126 30.870
24 5 126 3 087024.5 .12624.5 .126 3.0870
• What do you notice about the digits in the answers?
Placing the DecimalPlacing 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 answerthree places in the answer.
• Looking at it mathematically, 2.45=245x10‐2 and 1.26=126x10‐2. 245x10‐2x126x10‐2=30870x10‐4.
Placing the Decimal by EstimationPlacing the Decimal by Estimation
• Compare the Estimate and Where the DecimalCompare the Estimate and Where the Decimal is Placed
Number 1 Number 2 Estimate ProductNumber 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
PracticePractice
• Given the information, place the decimal by , p yestimation.
• If 12x55=660, what estimation would you use to place the decimal for 1 2x5 5place 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
Using the Array for Multiplying Fractions
• Consider32
Consider 43Start with a 1x1 rectangleDivide one side into thirdsDivide the other side into fourthsDivide the other side into fourthsTake two‐thirds and three‐quarters and surround them with a rectangleThe rectangle has 6 pieces out of a total of twelve, 6/12 or ½.
41
41
41
41
31
4444
131
31
PracticePractice
• Use an array to illustrate the followingUse an array to illustrate the following products
3253
52
83
52
32
34
53 85
Looking at your arrays and the answers, what rule could you give so you don’t need to draw arrays all the time.
A ExplorationA Exploration
• Complete each and look for a relationshipComplete each and look for a relationship
83
52
82
53
85 85
53
32
52
33
53 53
34
43
85 85
What relationship do you see?How might it help you?How might it help you?
Multiplying FractionsMultiplying Fractions
• The arrays should have illustrated that the total ynumber 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 thenumerators. So, to multiply fractions, you multiply the numerators and multiply the denominators.
• In the exploration, you should have seen that the p ynumerators (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 beforeable 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.
Using an Array to Multiply Mixed bNumbers
• Consider 32 58 Consider 45 58
8 2/5
540
2
Answer=48 3/10
2
3/46 3/10
PracticePractice
• Use an array to find the following products:Use an array to find the following products:
53
32 94
93
92 67
31 1512 86 1512