Math Explorer's Club


Introduction Lesson 1: Addition Lesson 2: Subtraction Lesson 3: Multiplication Lesson 4: Division Lesson 5: Calendar computations Lesson 6: Guessing a number and other tricks 
Addition tipsStart from left to right, not from right to leftAddition is our first contact with mathematics. Most people feel comfortable adding numbers to their heart's content. Here we provide some tips that can make adding numbers even easier, specially if your tryint to improve your skills to add numbers in your head.We normally start adding numbers from right to left, and sometimes we "carry" some digit. This is all well and good, but when doing mental computations, starting from the left may prove more useful. For instance, consider What about adding numbers with three (or more) digits? Let's say First notice that it's easier to think of 1932 + 415 (add the smaller number to the bigger one). Now, 1932 + 400 + 10 + 5 = 2332 + 10 + 5 (we added 4 to 19) = 2342 + 5 (we added 1 to 3) = 2347 (we added 5 to 2). With practice, you'll learn to appreciate the advantages of starting from the left, and hopefully will find it more useful to do mental computations. Combine it with subtractionIf you want to add 9 to a number, you can just add then and then subtract 1. Similarly if you want to add 8, you can add then and then subtract 2. This can be particularly helpful when we are adding numbers that end with a "large" digit, such as 7, 8, or 9.For instance, Similarly, if we want to add 37, we can add 40 and then subtract 3. In general, to add a number ab, we can just add (a + 1)0 = 10 × (a + 1), and then subtract 10 &minus b. For instance, Activity: Adding numbers in your head
Apply the techniques discussed above to do the following sums in your head.
Nocarry additionThere exists a way to avoid some carrying in addition by simply writing the total for each column, and then adding these. For instance, considerNotice that the space is needed, since the 18 in the second row actually represents 18 × 10, since it was obtained by adding up the decimal units. Activity: More sums
Casting out nines to check your answerHere is a nice fact: If we divide an integer n by 9, the remainder obtained is the same as the sum of the digits of n, module 9. For instance, take n = 67. Since 67 = 9 × 7 + 4, the remainder is 4. Now 6 + 7 = 13 &equiv 4 (mod 9). Casting out nines to check addition is simply the process of checking if the remainder module 9 of the numbers in the sum add up to the remainder module 9 of our answer. For instance, considerwhere the numbers in red represent the remainders mod 9. Notice that 8 + 5 &equiv 4 (mod 9). So we have some evidence that our answer is correct. Notice that casting out nines is not a proof that our answer is correct. However, it is a method to determine if the answer is wrong: If the sum of remainders doesn't add up to the remainder of our answer, we know for sure we made a mistake somewhere. For instance, consider the sum Notice that the sum of the remainders mod 9 is 8 + 3 + 1 &equiv 3 (mod 9). So if we obtained, say, 1322 &equiv 8 (mod 9), we can be certain that we made a mistake. We remark that casting out nines can be used to check subtraction and multiplication. Activity: Checking your answer

