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# Notes for structures I session 18 11 8

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Presentation on methods of multiplication for 11-8-2012 class.

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### Notes for structures I session 18 11 8

1. 1. STRUCTURES I Thursday, 11/8/2012 Methods of Multiplication View 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.
2. 2. Remember The Array Model Remember The Array ModelUse an array model to multiply 17X53Use 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. 3. Traditional Method Traditional MethodMultiply 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. 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. 5. Try It Again  Do Both Array and Traditional Before Clicking Forward y g 19X28 Traditional 20                                                               8 28 X19 Sum of  25210 Rows 280 200 80 532 2809 252 180 72
6. 6. Partial Products Partial ProductsMultiply 17 X 53 using the partial products Multiply 17 X 53 using the partial productsmethod. 53 x17 500     10x50 500 10x50 30      10x3 350      7x50 21      7x3 901Note: It is the array method without the array!
7. 7. Using the Partial Products MethodUsing the Partial Products MethodTry 19x28 using the partial products method.  Try 19x28 using the partial products methodClick to see the process when you have finished. 28 X 19 200 80 180 72 532
8. 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. 9. Lattice Method Named for the lattice look to the model Named 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 diagonals5.5 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. 10. Using Lattice Using LatticeTry 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. 11. Try The Following Using Array, Partial Product and Lattice.  Check using your Product and Lattice. Check using your normal method.1. 24 x 252. 46 x 842 46 843. 55 x 98
12. 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. 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. 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. 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. 16. Using the Array for Multiplying  Fractions 2 3• Consider 3  4 Consider Start 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. 1 1 1 1 4 4 4 4 1 3 1 3 1 3
17. 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. 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. 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. 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. 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