My next posts will include the lost art of estimating division problems....so many people have used a calculator since the 5th grade that they have forgotten how to estimate each digit of the quotient in a division problem...I will review that process of estimating in each step of the division process.
I will also be talking about the most common mistakes that I have seen students make on the TABE test (the current test that is suppose to indicate a student's readiness in taking and passing the GED) As with all multiple choice tests, sometimes a student can make a pure guess at an answer and get it correct without having any idea of how to actually calculate to get the answer...and sometimes a student can have a good understanding of how to work a problem but miss it because of a careless error like miscopying a number...I can tell more about what a student knows about a mathematical process by looking at their scratch paper on which they did their calculations (which I do) than I can by looking at the wrong answer they picked...unfortunately the TABE test results do not even tell you the test takers actual response to a question, it only tells you what kind of problem they did not answer correctly. In mathematics, I am not a very big fan of multiple choice questions.
Sunday, December 13, 2009
Left to Right Multiplication and Estimation
Left to right multiplication gives its user a better sense of the true value of a number and is easier to calculate in your head than right to left multiplication.
Right to left multiplication starts the process by multiplying the smallest part of the numbers in the ones’ or units’ place. This gives you a poor estimate of the product of the whole numbers. When you multiply 435 times 8, the process of right to left multiplication starts with 8 times 5 whose product is 40...a poor estimate of the product of 435 times 8. Using this process, you also calculate the last number of the answer first so that you are obtaining the answer in reverse order. You have to keep track of the product of the two numbers you multiply…then you have to carry a number to add to the product of the next two numbers, and this process continues until you have the first and largest number of your answer. The process of multiplying 435 time 8 would go like this:
Step 1.…..multiply 8 times 5 to get 40
Step 2.…..remember the answer ends in 0
Step 3.…..multiply 8 times 3 to get 24 and add the 4 you carried over from the last step to arrive at 28.
Step 4.….remember the second to last number in the answer is 8 and the last number in the answer is 0.
Step 5.….multiply 8 times 4 to get 32 and add the 2 you carried over from the last step to arrive at 34.
Step 6.….you remembered the last number of the answer first and the first number of the answer last, so now you have to reverse the order of the numbers you calculated to arrive at the product of 435 times 8.
This is a difficult process to do in your head without writing the numbers down.
In left to right multiplication you multiply the largest part of the numbers together first arriving at a much closer estimate of the true value of the product of the two numbers. You look at the numbers at their true value…The 4 in 435 represents 400, not the single digit 4. The 3 represents its true value of 30. The process goes like this:
8 times 400 = 3200
8 times 30 = 240 thirty two hundred plus two forty is thirty four forty
Add these #s 3440 just remember thirty four forty in your head.
8 times 5 = 40 add 40 to 3440 to get 3480 thirty four eighty
The first step in the left to right multiplication process can be used to estimate the product of two numbers by finding the range for the solution. The range consists of two numbers…the actual product of the two numbers must fall between this range consisting of a number that is too small to be a possible solution and a number that is too large to be a possible solution. This process involves multiplying only the first digit of both numbers….all digits to the right of the first two digits will become zeros in the low and high range.
Look at the example 369 multiplied by 86...
369.… 3 (69) the two numbers to the right of 3 become 0’s in the estimate.
86.….8 (6) the one number to the right of 8 becomes a 0 in the estimate.
Total of 3 numbers to the right of 3 and 8 become 0’s
Low estimate…… 3 X 8 = 24 (000) or 24,000 …shortcut for 300 X 80
High estimate… 4 X 9 = 36 (000) or 36,000 …shortcut for 400 X 90
By rounding 369 down to 300 and 86 down to 80, you are multiplying two numbers that are smaller than the given numbers; so it follows that the product of 300 X 80 should be smaller than the product of 369 X 86.
By rounding 369 up to 400 and 86 up to 90, you are multiplying two numbers that are larger than the original numbers; so it follows that the
product of 400 X 90 should be larger than the product of 369 X 86.
Since 24,000 is too small to be the true product of 369 X 86, and 36,000 is too large to be the true product of these numbers, 24,000---36,000 is the range that the real product of 369 X 86 must fall between.
Also since 369 is closer to 400 than 300, and 86 is closer to 90 than 80, the product of 369 X 86 should be more than half-way between 24,000 and 36,000...a rough, but educated guess should be approx 32,000. The answer should also have a 4 as the last digit since the last two digits of 369 and 86 are 9 and 6.….9 X 6 = 54 . The actual answer is 31,734.
Finding the range for the product of larger numbers is just as easy as it is for multiplying smaller numbers. Look at 3,845,123 multiplied by 83,295. These numbers start with the digits 3 and 8, the same first digits of the previous example. The total number of digits that follow these first two digits total 10 (845125.…6 digits and 3295..4 digits).
Since the first digits are 3 and 8, the range for the product of these larger numbers is:
Low Range (3 X 8 = 24) 24 followed by 10 0’s is 240,000,000,000
High Range (4 X 9 =36) 36 followed by 10 0’s is 360,000,000,000
The time needed to calculate the range of the products of the larger numbers exceeded the time to calculate the estimate of the products of the smaller numbers only by the time it took to count how many additional digits were to the right to the first two digits of the larger number (10 digits) compared to the smaller number (4 digits). Counting an additional 6 digits may take a couple of seconds. So, estimating the range for 3,845,123 X 83,295 should only take a couple more seconds than estimating the range for 368 X 86.
If I wanted a closer estimate for the product of the larger numbers, I would consider that 3,845,000 is closer to 4,000,000 than 3,000,000 and 83,00 is closer to 80,000 than90,000, I would estimate the answer is about 315,000,000,000 and the last digit is 5 (3 X 5 = 15 ) The actual answer is 320,279,520,285.
These estimates take less than 15 seconds, but I quite frequently see students taking multiple choice tests picking answers to multiplication problems that are outside the range of possibility and picking answers that do not have the correct number of digits. For example, in the first multiplication problem above, 369 X 86, whose range is between 24,000 and 36,000.…both 5 digit numbers…..it would be illogical to pick an answer with fewer or more than 5 digits, or an answer whose last digit was not 4.
Here are a few examples of finding the range of answers in multiplication problems.
567 X 93 ( 3 digits follow the 5 and the 3) 5 X 9 = 45 low est. is 45,000
6 X10 = 60 high est is 60,000
Last digits 7 X 3 = 21 answer’s last digit must be 1.
Answer should about half-way between high and low ranges.…why?
Actual answer is 52,731.
98,743 X 923 6 digits follow the 9s’
9 X 9 = 81 low range is 81,000,000
10 X 10 = 100 high range 100,000,000
3 X 3 = 9 last digit of answer must be nine.
Answer should be about half-way between high and low range.
Actual answer is 91,139,789.
Practice left to right multiplication.
75 X 8 think 75 = 70 + 5 325 X 7 think 300 +20 + 5
X 8 X 7 560 + 40 = 6oo 2100 + 140 = 2240 +35 = 2275
Say numbers with fewest words… twenty-one hundred plus one-forty is twenty-two-forty plus thirty-five is twenty-two-seventy-five. Once you add 2100 to 140 to get 2240, you only have to keep the 2240 in your head to continue the process of multiplying and adding...you don’t have to remember how you got to that number…in other words, you don’t have to remember the 2100 and the 140.
With practice, you can multiply a two or three digit number by a one digit number faster in your head than you can punch the numbers into the keyboard of your calculator… you can also add a column of numbers in your head working from left to right, the same way an adding machine computes a sum.
There are many methods to multiply large numbers and make seemingly difficult mathematical calculations easier…I recommend a book written by Scott Flansburg called Math Magic to explore some of these calculating shortcuts. You can find more information about Flansburg aka The Human Calculator at www.humancalculator.com .
Watch the video at http://www.youtube.com/watch?v=1LyoeWLmclU&feature=related to find out how fast this guy can process complex calculations….
Right to left multiplication starts the process by multiplying the smallest part of the numbers in the ones’ or units’ place. This gives you a poor estimate of the product of the whole numbers. When you multiply 435 times 8, the process of right to left multiplication starts with 8 times 5 whose product is 40...a poor estimate of the product of 435 times 8. Using this process, you also calculate the last number of the answer first so that you are obtaining the answer in reverse order. You have to keep track of the product of the two numbers you multiply…then you have to carry a number to add to the product of the next two numbers, and this process continues until you have the first and largest number of your answer. The process of multiplying 435 time 8 would go like this:
Step 1.…..multiply 8 times 5 to get 40
Step 2.…..remember the answer ends in 0
Step 3.…..multiply 8 times 3 to get 24 and add the 4 you carried over from the last step to arrive at 28.
Step 4.….remember the second to last number in the answer is 8 and the last number in the answer is 0.
Step 5.….multiply 8 times 4 to get 32 and add the 2 you carried over from the last step to arrive at 34.
Step 6.….you remembered the last number of the answer first and the first number of the answer last, so now you have to reverse the order of the numbers you calculated to arrive at the product of 435 times 8.
This is a difficult process to do in your head without writing the numbers down.
In left to right multiplication you multiply the largest part of the numbers together first arriving at a much closer estimate of the true value of the product of the two numbers. You look at the numbers at their true value…The 4 in 435 represents 400, not the single digit 4. The 3 represents its true value of 30. The process goes like this:
8 times 400 = 3200
8 times 30 = 240 thirty two hundred plus two forty is thirty four forty
Add these #s 3440 just remember thirty four forty in your head.
8 times 5 = 40 add 40 to 3440 to get 3480 thirty four eighty
The first step in the left to right multiplication process can be used to estimate the product of two numbers by finding the range for the solution. The range consists of two numbers…the actual product of the two numbers must fall between this range consisting of a number that is too small to be a possible solution and a number that is too large to be a possible solution. This process involves multiplying only the first digit of both numbers….all digits to the right of the first two digits will become zeros in the low and high range.
Look at the example 369 multiplied by 86...
369.… 3 (69) the two numbers to the right of 3 become 0’s in the estimate.
86.….8 (6) the one number to the right of 8 becomes a 0 in the estimate.
Total of 3 numbers to the right of 3 and 8 become 0’s
Low estimate…… 3 X 8 = 24 (000) or 24,000 …shortcut for 300 X 80
High estimate… 4 X 9 = 36 (000) or 36,000 …shortcut for 400 X 90
By rounding 369 down to 300 and 86 down to 80, you are multiplying two numbers that are smaller than the given numbers; so it follows that the product of 300 X 80 should be smaller than the product of 369 X 86.
By rounding 369 up to 400 and 86 up to 90, you are multiplying two numbers that are larger than the original numbers; so it follows that the
product of 400 X 90 should be larger than the product of 369 X 86.
Since 24,000 is too small to be the true product of 369 X 86, and 36,000 is too large to be the true product of these numbers, 24,000---36,000 is the range that the real product of 369 X 86 must fall between.
Also since 369 is closer to 400 than 300, and 86 is closer to 90 than 80, the product of 369 X 86 should be more than half-way between 24,000 and 36,000...a rough, but educated guess should be approx 32,000. The answer should also have a 4 as the last digit since the last two digits of 369 and 86 are 9 and 6.….9 X 6 = 54 . The actual answer is 31,734.
Finding the range for the product of larger numbers is just as easy as it is for multiplying smaller numbers. Look at 3,845,123 multiplied by 83,295. These numbers start with the digits 3 and 8, the same first digits of the previous example. The total number of digits that follow these first two digits total 10 (845125.…6 digits and 3295..4 digits).
Since the first digits are 3 and 8, the range for the product of these larger numbers is:
Low Range (3 X 8 = 24) 24 followed by 10 0’s is 240,000,000,000
High Range (4 X 9 =36) 36 followed by 10 0’s is 360,000,000,000
The time needed to calculate the range of the products of the larger numbers exceeded the time to calculate the estimate of the products of the smaller numbers only by the time it took to count how many additional digits were to the right to the first two digits of the larger number (10 digits) compared to the smaller number (4 digits). Counting an additional 6 digits may take a couple of seconds. So, estimating the range for 3,845,123 X 83,295 should only take a couple more seconds than estimating the range for 368 X 86.
If I wanted a closer estimate for the product of the larger numbers, I would consider that 3,845,000 is closer to 4,000,000 than 3,000,000 and 83,00 is closer to 80,000 than90,000, I would estimate the answer is about 315,000,000,000 and the last digit is 5 (3 X 5 = 15 ) The actual answer is 320,279,520,285.
These estimates take less than 15 seconds, but I quite frequently see students taking multiple choice tests picking answers to multiplication problems that are outside the range of possibility and picking answers that do not have the correct number of digits. For example, in the first multiplication problem above, 369 X 86, whose range is between 24,000 and 36,000.…both 5 digit numbers…..it would be illogical to pick an answer with fewer or more than 5 digits, or an answer whose last digit was not 4.
Here are a few examples of finding the range of answers in multiplication problems.
567 X 93 ( 3 digits follow the 5 and the 3) 5 X 9 = 45 low est. is 45,000
6 X10 = 60 high est is 60,000
Last digits 7 X 3 = 21 answer’s last digit must be 1.
Answer should about half-way between high and low ranges.…why?
Actual answer is 52,731.
98,743 X 923 6 digits follow the 9s’
9 X 9 = 81 low range is 81,000,000
10 X 10 = 100 high range 100,000,000
3 X 3 = 9 last digit of answer must be nine.
Answer should be about half-way between high and low range.
Actual answer is 91,139,789.
Practice left to right multiplication.
75 X 8 think 75 = 70 + 5 325 X 7 think 300 +20 + 5
X 8 X 7 560 + 40 = 6oo 2100 + 140 = 2240 +35 = 2275
Say numbers with fewest words… twenty-one hundred plus one-forty is twenty-two-forty plus thirty-five is twenty-two-seventy-five. Once you add 2100 to 140 to get 2240, you only have to keep the 2240 in your head to continue the process of multiplying and adding...you don’t have to remember how you got to that number…in other words, you don’t have to remember the 2100 and the 140.
With practice, you can multiply a two or three digit number by a one digit number faster in your head than you can punch the numbers into the keyboard of your calculator… you can also add a column of numbers in your head working from left to right, the same way an adding machine computes a sum.
There are many methods to multiply large numbers and make seemingly difficult mathematical calculations easier…I recommend a book written by Scott Flansburg called Math Magic to explore some of these calculating shortcuts. You can find more information about Flansburg aka The Human Calculator at www.humancalculator.com .
Watch the video at http://www.youtube.com/watch?v=1LyoeWLmclU&feature=related to find out how fast this guy can process complex calculations….
Subscribe to:
Posts (Atom)
