The author's comments:

someone I knew was interested in getting better at math.

The first trick I will talk about is the 2 by 1 multiplications, one of the easier principles we can use. Some of the hardest, yet easiest problems we may run across are problems such as 47 times 7 and 89 times 8. Let’s look at the first problem 47 times 7. In case my reader didn’t know, multiplication is distributive so (10 times 3) + (5 times 3) is the same as 15 times 3. So 30 and 15 is 45. For the first problem, I mentally see 47 times 7 as (40 times 7) + (7 times 7). Thus 280 + 49 is 329, my answer. If the problem were 407 times 7, I would see (400 times 7) + (00 times 7) + (7 times 7). Let’s try the next problem. I break apart 89 times 8 to (80 times 8) + (9 times 8). 640 + 72 equals 712, the answer. Some other examples include; 76 times 7 for an answer of 532, 94 times 3 for an answer of 282, 89 times 2 for 178, et.al.

Let’s now go one step further with 3 by 1 multiplications. Now a difficult problem may be 738 times 4. However the 3 by 1’s follow the same principle as the 2 by 1’s except now we add the hundreds value. For example, I mentally break up 738 times 4 to get (700 times 4) + (30 times 4) + (8 times 4). Then I add the sums, 2,800, 120, and 32 and reach 2,952, the answer.

Let’s go to the 4 by 1 multiplications. One problem that might come up is 9,743 times 6. This may seem like a hard problem but it actually isn’t because it follows the same principles as the 3 by 1’s. I mentally broke apart the problem to get (9,000 + 700 + 40 + 3) and then multiplied the whole thing by 6. I ended up with 54,000+ 4,200 + 240 + 18 or 58,458, the answer. If the reader doesn’t understand how I arrived at this answer, keep at it. It will come with practice.

Let’s advance now to squaring numbers, the hardest yet easiest problems to solve. To square a number I round to an easy multiple, take the opposite, multiply, and add the squared difference. Let’s start with an easy problem, say 12 squared. If I didn’t know this, I would just use this principle. The number 12 is close to 10, an easy multiple, so I round down 2 to get 10, round up two for 14, multiply to get 140 and add the square of 2 or 4. Another way to solve this would be to round down to 10, add 4 times the original number and subtract 4. 100+ 4(12)-4 is 12 squared or 144. For numbers like 21, round down, double that number you started with and subtract 1. These 2 rules can be used to solve for any square since any square is 1 or 2 closest to 0 or 5. With the knowledge the reader has gained, try 86 squared. If the reader answered 7,396, they have got it. If not practice and I guarantee the reader will find squaring 2 digits numbers easy. Let’s try 92 squared. Don’t panic, numbers close to 100 are the easiest because we can round up to 100. I rounded up eight for 100, and subtracted 8 for 84. I then multiply 84 times 100 and added the square of eight to the total for an answer of 8,464, which in fact, is the answer.

Let’s try 3 digit numbers squared. For 972 squared I would have rounded up to 1,000, rounded down to 944(972-28) and added the square of 28, 784, for a total of 944,784. Now, what is 595 squared? I would have rounded up to 600 and rounded down to 590, then multiply them (this is a 2 by 1 in disguise). Using the 2 by 1 multiplication I multiply to get the product of 354 and add 3 zeros for 354,000 and added the square of 5 for the total.

Now, how about 4 digit numbers squared. 1,036 squared is what? If the reader correctly answered 1,073,296, they get my teaching. If not, go down 36 for 1000 and up for 1072. Multiply for 1,072,000. Now we are left with 36 squared. 36 squared we find is 1,296. When the 2 sums are added, we get 1,073,296, the answer. That was rather easy. One of the harder problems to compute would be 5,268 squared. However, did we get 27,751,824? If not, the same principle applies because this follows the same rule as every other square. Using the method I went up to 5,300 and down to 5,236 resulting in a 2 by 4 problem which I broke apart to get the answer. I mentally saw 53(00) times 5,236 as (50 times 5,236) + (3 times 5,236). We have now two 4 by 1’s. I added for 277,508 (00) and added 1,024(32 squared) for 27,751,824, the answer. The zeros in parentheses are the zero I dropped off and then added back on when I arrived at the product.

For 5 digit numbers squared try let’s try 24,974 squared and 16,517 squared. I got 623,700,676 and 272,811,289 respectively. I will not tell the reader how I arrived at this answer because I think based on the knowledge they have learned already, they know how I arrived at this answer. With lots of practice, the reader will soon become a mathemagician, if they choose so, and be able to square 5 digit numbers faster than it takes people to solve the same problem using pen and paper.

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