Structures I session 18 11 8
- 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. 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
- 3. Traditional Method Multiply 17 X 53 using the traditional method. 53 17 371 7X3=21, 7X5=35, 35+2=37 530 10X53=530 901
- 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