Over the 20 or so years I've been learning, teaching, and using math, I've put together a kind of mental toolbox. In it are the tips, tricks, and shortcuts I've found the most useful. The first group are some general tips and the second are hints for mastering arithmetic.

1. Knowing something about what the answer should look like

If I could go back in time and give one piece of advice to my younger self, I would say that before you try to work a problem, you should try to get an idea of what the answer should look like. If you know what the answer should look like (what units, bigger or smaller than the starting number, positive or negative) you have very good clues as to how to work the problem. For example, 345 + 678 = ? Well, I can round 345 to 300 and 678 to 700. 300 plus 700 is 1000, so the actual answer should be close to 1000.

In word problems, you should look for the question mark. That will tell you the units of the answer.

2. Break down the problem into steps

Just about any math problem more complicated than 1 +1 = 2 requires more than one step to solve. Just like you can't bake a cake in one step, neither can you try to solve a math problem all at once.

3. Try to check your answer

In most math problems, you will have an equation you can use to check your answer. Put your answer into that equation and see if you get the same thing on both sides of the equal sign.

4. Write down each step

It's far easier to find mistakes by writing down each step in your solution. At first, it's best to write every single change on a new line. As you get better, you can do some easy steps in your head.

Here are some tips for mastering arithmetic

1. Learn to add and subtract single numbers by heart

There are ten single numbers: 0 to 9. This means there are 100 sums and 100 differences so 200 things to memorize. You can chop that number for addition from 100 to 50 by realizing that in addition, order does not matter- 1 + 2 is the same as 2 +1. You can reduce that 50 to 30 by realizing that any number plus zero is the other number and that a number plus one just goes to the next higher number. Once you have the sums memorized, subtraction is a piece of cake. If your know that 2+3=5., then 5-2 must be 3 because that's the part leftover. So really, you only need to remember 30 sums.

2. Learn the multiplication tables by heart

Like addition, you need to learn all the products of one digit numbers. Again, this makes 100 products to memorize. Like addition, you can cut that in half by knowing that order does not matter in multiplication.

2 x 3 = 3 x 2 .= 6. You can cut this down to 30 by remembering that a number times one is the other number and that a number times zero is zero. It is impossible to learn how to do division and fractions without mastering multiplication.

3. Move the decimal point when multiplying or dividing a number by a number like 10, 100, etc.

Multiplying or dividing by a multiple of 10 is a piece of cake. If you're multiplying, write what the other number is and then add on the total number of zeroes. For example, 30 x 60 = ? Well, 3 x 6 = 18 and there are two zeroes. Write the 18 and then two zeroes and you get the answer: 1800. For division, for every zero on the top, you can cross out a zero on the bottom. For example, 30 / 60 = 3 / 6. 3 / 6 simplifies to 1 /2 or one half. With decimals, count the zeros on the multiple of 10. Move the decimal point to the left if dividing and to the right if multiplying. Examples:

3.4 x 10 = 34 (one zero; moved the decimal point one space to the right)

3.4 / 10 = 0.34 (one zero, move the decimal point one space to the left.)

There are other tricks, but these are the one that have been the most useful to me.

1. Knowing something about what the answer should look like

If I could go back in time and give one piece of advice to my younger self, I would say that before you try to work a problem, you should try to get an idea of what the answer should look like. If you know what the answer should look like (what units, bigger or smaller than the starting number, positive or negative) you have very good clues as to how to work the problem. For example, 345 + 678 = ? Well, I can round 345 to 300 and 678 to 700. 300 plus 700 is 1000, so the actual answer should be close to 1000.

In word problems, you should look for the question mark. That will tell you the units of the answer.

2. Break down the problem into steps

Just about any math problem more complicated than 1 +1 = 2 requires more than one step to solve. Just like you can't bake a cake in one step, neither can you try to solve a math problem all at once.

3. Try to check your answer

In most math problems, you will have an equation you can use to check your answer. Put your answer into that equation and see if you get the same thing on both sides of the equal sign.

4. Write down each step

It's far easier to find mistakes by writing down each step in your solution. At first, it's best to write every single change on a new line. As you get better, you can do some easy steps in your head.

Here are some tips for mastering arithmetic

1. Learn to add and subtract single numbers by heart

There are ten single numbers: 0 to 9. This means there are 100 sums and 100 differences so 200 things to memorize. You can chop that number for addition from 100 to 50 by realizing that in addition, order does not matter- 1 + 2 is the same as 2 +1. You can reduce that 50 to 30 by realizing that any number plus zero is the other number and that a number plus one just goes to the next higher number. Once you have the sums memorized, subtraction is a piece of cake. If your know that 2+3=5., then 5-2 must be 3 because that's the part leftover. So really, you only need to remember 30 sums.

2. Learn the multiplication tables by heart

Like addition, you need to learn all the products of one digit numbers. Again, this makes 100 products to memorize. Like addition, you can cut that in half by knowing that order does not matter in multiplication.

2 x 3 = 3 x 2 .= 6. You can cut this down to 30 by remembering that a number times one is the other number and that a number times zero is zero. It is impossible to learn how to do division and fractions without mastering multiplication.

3. Move the decimal point when multiplying or dividing a number by a number like 10, 100, etc.

Multiplying or dividing by a multiple of 10 is a piece of cake. If you're multiplying, write what the other number is and then add on the total number of zeroes. For example, 30 x 60 = ? Well, 3 x 6 = 18 and there are two zeroes. Write the 18 and then two zeroes and you get the answer: 1800. For division, for every zero on the top, you can cross out a zero on the bottom. For example, 30 / 60 = 3 / 6. 3 / 6 simplifies to 1 /2 or one half. With decimals, count the zeros on the multiple of 10. Move the decimal point to the left if dividing and to the right if multiplying. Examples:

3.4 x 10 = 34 (one zero; moved the decimal point one space to the right)

3.4 / 10 = 0.34 (one zero, move the decimal point one space to the left.)

There are other tricks, but these are the one that have been the most useful to me.

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