## Monday, April 28, 2014

When adding fractions with unlike denominators, we must change each fraction so they have the same denominator. Here's a simple example to start with.

1/2 + 1/4

To add these two fractions, we must get a denominator that is the same. This is called the common denominator. We can get a common denominator by multiplying the two denominators, in this problem it would be 2 x 4 = 8. But that is not the lowest common denominator, which is preferred when adding and subtracting fractions. To get the lowest common denominator, let's look at all the multiples of 2 and then the multiples of 4.

Multiples of 2 are

2, 4, 6, 8, 10, 12, …...

Multiples of 4 are

4, 8, 12, 16, 20, …..

The lowest common denominator is the smallest number that is the same in both sets of multiples. Notice the 4 is bold in each set. That is the lowest common denominator.

Now that we have the common denominator, we have to make ½ into an equivalent fraction with 4 as the denominator. . If we multiply the numerator and denominator by 2, we get 2/4.. Notice ¼ already has a denominator of 4, so we don't have to change this fraction in order to add. We now get...1/4 + 2/4 = 3/4

## Thursday, April 24, 2014

Check out this site for many interesting photos and tidbits of information about math and science. Here's one from the page of interest. Enjoy!

http://www.sciencedump.com/subcategory/math

## Monday, April 21, 2014

To my mathematically minded friends...this book sounds interesting. Check it out and let me know what you think. I will have to read this and I'm sure much of it will be fascinating!

## Wednesday, April 16, 2014

Over the course of many years, I have come across mathematical puzzles and tricks that can baffle even the best mathematical minds. But what most of these tricks come down to is simple algebra. Here's some math tricks that you may have encountered. but couldn't figure out. Enjoy the art behind the mathematical magic presented in the next several paragraphs.

Here's some mathematical "magic" you can perform with a single person or many people in an audience. Pick a person and ask that person to pick any single or double digit number. Then ask the person to do the following, double the number, add 14, divide by two and subtract the original number. You will astound most people when you ask, "Is the number you are now thinking of seven?" Try this yourself and you'll see that the answer will always be 7.

There reason is the simple algebra behind this. The number chosen is x, double the number to give you 2x. Add 14 to get 2x + 14. Divide by 2 to get x + 7. Now subtract the original number x, to get x + 7 - x = 7. This will work no matter what number is used. The answer will always be half the number you tell the person to add in the second step of the problem.

Here's another trick in which the answer will always be 1,089. Write down any three-digit number in decreasing order of digits. For example, 875 and 942 would both work. Let's use 875. Reverse the number and subtract from the original number. Therefore we have 875 - 578 = 297. Add this result to the reverse of itself. So we add 297 to 792 and we get 1,089. Try this with any three-digit number in which the numbers are from largest to smallest.

For the next trick, give someone a piece of paper and a pencil or pen. Have the person write the numbers 1 through 10 down the left hand side. Ask the person to pick two numbers between 1 and 20 and write them on lines 1 and 2. Next have the person write the sum of lines 1 and 2 and write in line 3, the sum of lines 2 and 3 and write in line 4 and so on. After the card is filled, ask the person to show the card. You should quickly be able to tell the person the sum of all 10 numbers quicker than he or she could add them on a calculator.

For example, if the person starts with the numbers 9 and 2, you can quickly tell that the sum of the numbers is 671. If the numbers chosen are 3 and 19, you can quickly say the sum is 1,837. How can this be done so fast? SImply multiply the number in line 7 by 11. If you want to check this out to show that it works, Get a card and go through the entire process. You can also astound by asking the person to divide the number in line 10 by the number in line 9. You will instantly be able to tell the person that the first three digits in that result is 1.61.

How does this work? The first result is simply based on the fact that you assign the value in line 1 the variable x and the value in line 2 the variable y. Line 3 is then x + y, line 4 is x + 2y, line 5 is 2x + 3y, line 6 is 3x + 5y, line 7 is 5x + 8y. When you get through and add all 10 lines you get 55x + 88y. That is 11 times 5x + 8y, which is line 7.

Knowing that line 10 divided by line 9 is around 1.61 is a bit more complex, involving four variables and adding fractions badly (adding numerators and denominators), showing that the value falls between two original fractions. It's not important to know how every result is arrived, but you can amaze friends by having this answer ahead of time by writing 1.61 on the back of the card.

Some of these tricks I have used over the years with my students, while the last method I obtained form the book, Secrets of Mental Math, by Arthur Benjamin and Michael Shermer.

Look for a few more mental math tricks in upcoming articles.

## Sunday, April 13, 2014

In previous articles, I've given you techniques needed to figure out exact answers to math problems mentally. Sometimes, we don't need an exact answer and an estimation or "best guess" is satisfactory. Suppose you are getting quotes from different banks on a personal loan. All that is needed is a close approximation to the monthly payments. Another example where an estimation is satisfactory is settling a restaurant bill with some friends, where it's not important to calculate to the exact penny. This article will explain techniques to help you master mathematical estimation.

We'll first examine addition estimation. The trick is to round the original numbers up or down. For example, 4,561 + 2,233 = 6,794. If we round to the nearest hundred, rounding up at 50 or above and rounding down below 50, we get 4,600 + 2,200 is approximately 6,800. If you always round off to the nearest hundred, the estimate will always been within 100 of the correct answer. This is within one percent of the correct answer when the answer is 10,000 or more.

We can use this technique when shopping in a supermarket. Suppose you want to approximate the total bill before the cashier rings up your order. If you round all items to the nearest 50 cents, you'd be surprised how accurate your estimation will be. Let's try it with these prices: \$1.69, \$2.43, \$0.79, \$1.57, \$0.40, \$4.23, \$1.75, \$1.35, \$2.65, \$0.89. The actual cost for these items is \$17.75. When rounding off the prices become \$1.50, \$2.50, \$1.00, \$1.50, \$0.50, \$4.00, \$2.00, \$1.50, \$2.50, \$1.00. Adding these gives the estimation of \$18.00, amazingly close to the total cost.

Estimation involving subtraction is done the same way. The most accurate estimation is when numbers are rounded to the nearest hundred. For example, 9,251 - 3,771 = 5,480. The estimated answer is found by taking 9,300 - 3,800 = 5,500.

For estimation involving multiplication problems, the process is similar. We still want to round numbers. If we have a multiplication problem with two, two-digit numbers, we can round each to the nearest 10. While that is quickest and easiest, it's not the most accurate. For example, take 56 times 89. We round 56 to 60 and 89 to 90 to get 60 times 90 equals 5,400. The actual answer is 4,984.

A more accurate way to estimate the problem above is to round one number up to the next 10 and the other down by the same amount the other number was round up. For example, we have 56 times 89. If we round 89 up by one to 90, we round 56 down by one to 55. Now we have 90 times 55 equals 4,950. The estimation is only 34 off from the actual answer and much easier to multiply 90 and 55 mentally than 89 and 56.

The same technique can be used when multiplying a three digit to a two digit number and two three digit numbers. Note that while the estimation technique will work, the multiplication is more difficult to perform mentally. Master the mental multiplication techniques before attempting problems above two digits.
There are other techniques for estimating answers to division problems and square roots, which will be covered in a later article. I have explained these techniques to my students over the past 14 years and hope you find some use out of these as well.

## Wednesday, April 9, 2014

### An old post on how to simplfy square roots, an important concept

In a first year algebra class, students will encounter problems involving simplifying square roots. But many times the teacher simply doesn't explain the process by which this is accomplished in a manner that students can understand. I will explain a way that will make simplifying square roots easier for students of any ability level.

The first method I use when teaching students how to calculate square root is to look to see if the number is a perfect square first. I suggest students memorize the perfect squares from 1 to 25 as follows: 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. So if you are asked to calculate the square root of 441, you automatically know the answer is 21 or -21, since a negative number times another negative number yields a positive number.

For non-perfect squares or larger numbers that you are unsure of, if the number is a perfect square, I suggest using a factor tree. For example, suppose you want to simplify a square root of 48. It is not a perfect square since 6 times 6 equals 36 and 7 times 7 equals 49. No whole number times itself equals 48. So break it down into factors. I always suggest trying to find a perfect square as one of the factors and in this case, 16 times 3 equals 48 and 16 is a perfect square. Remember: Since you are dealing with square root, the factors are also square root. So the square root of 48 equals the square root of 16 times the square root of 3. Three is a prime number so you cannot break down 3 any farther using a factor tree. We know that the square root of 16 is 4, so the answer is 4 times the square root of 3.

Another more difficult example---say we need to find the square root of 2025. With such a large number, most people won't know if this is a perfect square, so use the factor tree. Know that any number ending in 5 is divisible by 5. So 5 times 405 is 2025. But 405 can be broken down into factors, using the same rule, therefore 5 times 81 is 405. Now we have the square root of 5 times the square root of 5 times the square root of 81. Notice then that the square root of 5 times the square root of 5 equals the square root of 25. Now this problem becomes simple because you notice we have two perfect squares here, 25 and 81. The square root of 25 is 5 and square root of 81 is 9. So the answer is 5 times 9, which is 45 and -45, since -45 times -45 equals 2025. This problem is actually a perfect square, but if you do not recognize it as such, you can use the factor tree method I just described.

When dealing with the square root of a negative number, imaginary numbers come into play. The square root of -1 equals an imaginary number denoted as "i". So in the above problems if we had the square root of -48, you have square root of -1 times square root of 3 times square root of 16. The answer is 4i times the square root of 3, and -4i times square root of 3. If we have the square root of -2025, the answer is simply 45i and -45i.

Hope my method helps you when trying to figure out the square root of both positive and negative numbers.

## Friday, April 4, 2014

### A review of a previous topic I posted

Simplifying square roots

In a first year algebra class, students will encounter problems involving simplifying square roots. But many times the teacher simply doesn't explain the process by which this is accomplished in a manner that students can understand. I will explain a way that will make simplifying square roots easier for students of any ability level.

The first method I use when teaching students how to calculate square root is to look to see if the number is a perfect square first. I suggest students memorize the perfect squares from 1 to 25 as follows: 1, 4, 9, 16, 25, 49, 64, 81, 100, 121, 144, 169, 196, 225, 256, 289, 324, 361, 400, 441, 484, 529, 576, 625. So if you are asked to calculate the square root of 441, you automatically know the answer is 21 or -21, since a negative number times another negative number yields a positive number.
For non-perfect squares or larger numbers that you are unsure of, if the number is a perfect square, I suggest using a factor tree. For example, suppose you want to simplify a square root of 48. It is not a perfect square since 6 times 6 equals 36 and 7 times 7 equals 49. No whole number times itself equals 48. So break it down into factors. I always suggest trying to find a perfect square as one of the factors and in this case, 16 times 3 equals 48 and 16 is a perfect square. Remember: Since you are dealing with square root, the factors are also square root. So the square root of 48 equals the square root of 16 times the square root of 3. Three is a prime number so you cannot break down 3 any farther using a factor tree. We know that the square root of 16 is 4, so the answer is 4 times the square root of 3.
Another more difficult example---say we need to find the square root of 2025. With such a large number, most people won't know if this is a perfect square, so use the factor tree. Know that any number ending in 5 is divisible by 5. So 5 times 405 is 2025. But 405 can be broken down into factors, using the same rule, therefore 5 times 81 is 405. Now we have the square root of 5 times the square root of 5 times the square root of 81. Notice then that the square root of 5 times the square root of 5 equals the square root of 25. Now this problem becomes simple because you notice we have two perfect squares here, 25 and 81. The square root of 25 is 5 and square root of 81 is 9. So the answer is 5 times 9, which is 45 and -45, since -45 times -45 equals 2025. This problem is actually a perfect square, but if you do not recognize it as such, you can use the factor tree method I just described.
When dealing with the square root of a negative number, imaginary numbers come into play. The square root of -1 equals an imaginary number denoted as "i". So in the above problems if we had the square root of -48, you have square root of -1 times square root of 3 times square root of 16. The answer is 4i times the square root of 3, and -4i times square root of 3. If we have the square root of -2025, the answer is simply 45i and -45i.
Hope my method helps you when trying to figure out the square root of both positive and negative numbers.

## Tuesday, April 1, 2014

Remember to check my page for my algebra books. Working on a books for early to middle elementary school children as well.

http://www.lulu.com/spotlight/KKauffman1969

Read more on mental math tricks at tipes from the book, "Secrets of Mental Math" by Benjamin and Shermer. It's sure to increase your mental math ability.