by Josh Rappaport
So you’re taking that long summer car trip, and being dutiful homeschoolers, you’ve taken some time to quiz your kids on the U.S. presidents, the spelling of “australopithecus,” and the answers to the good old fashioned times tables.
But they’re getting a bit bored with the process. So why not add a little pizazz — at least with the math part of the summer quizzing process —by tossing in some “wild and crazy” math tricks.
I myself still remember my dad reading from a book and teaching me at age 10 my first math trick: a trick for multiplying 11 by two-digit numbers. I’m not talking about the fact that 11 x 3 = 33, and 11 x 9 = 99. I’m talking about a trick that showed me how to multiply 11 x 24 and get 264 — in less than three seconds and all in my head!
The trick is that the answer has three digits. The first digit of the answer is the same as the 10s digit of 24, namely 2. The last digit of the answer is the 1s digit of 24, namely 4. And the middle digit of the answer— and here’s the neat part — turns out to be the sum of 2 and 4, namely 6. So the answer is 2-6-4, meaning 264.
Since my dad had not known the trick either, the two of us shared the same sense of delight and wonder. And the pleasure in learning that trick still resounds for me, so my dad and I still share a sense of fun when it comes to numbers and number tricks.
So today I’d like to encourage you to take some time teaching your children a few math tricks. The benefits are many: 1) Children learn that there are somewhat magical tricks that make math easier, 2) Children see that math has many interesting patterns, 3) If mental math tricks are taught well, children develop an appetite to learn more tricks, not to mention the ability to develop tricks of their own, and 4) As children learn more tricks, they boost their “number sense,” that indefinable yet extremely important knack for working with numbers which, once obtained, can last a lifetime.
Trick #2) Multiplying by 5. Since 5 is half of 10, it stands to reason that the product obtained when multiplying by 5 must be half of the product obtained when multiplying by 10. Note: if this idea is hard for your kids to grasp, bring it down to earth by likening 10 x 4 to 10 separate groups of 4 beans. Using that model, 5 x 4 would mean 5 groups of 4 beans. Children can see that 5 groups of 4 beans is half as much as 10 groups of 4 beans. So they can understand that 5 x 4 must be half as much as 10 x 4. Especially with younger children, it’s a good idea to demonstrate this visually with beans (or coins or whatever is at hand).
In any case, this principle leads to a simple trick for multiplying by 5, which we’ll learn by thinking about the problem 5 x 48. First, though, examine the simpler problem, 10 x 48. This is easy because multiplying by 10 means you just tack a zero onto the number. Tacking a zero onto 48 you get 480, so 10 x 48 = 480. Then, to get 5 x 48, follow these steps: 1st) Take half of 48 = 24, 2nd) Now that you’ve got half, tack on the zero, to get 240. And that’s it: 5 x 48 = 240. It’s that simple.
Trick #3) Multiplying by 25. Most children who’ve worked with money know that four quarters make a dollar. Thinking about that, children can see this means that 4 x 25 = 100, since 4 coins worth 25 cents are worth 100 cents. Taking that a step further, children can see that 8 quarters would make $2, so 8 x 25 = 200; 12 quarters would make $3, so 12 x 25 = 300. With a bit of time, children can see that this pattern will extend, so that every additional 4 quarters leads to another 100. And this means that to work a problem like 25 x 28, for example, all children need to do is figure out how many groups of four there are in 28. From division, they know that there are 7 groups of 4 in 28. So that means that 25 x 28 is like $7, so 25 x 28 = 700.
Then, you can help children streamline the process into two steps for multiplying numbers by 25: 1st) Divide the number (other than 25) by 4 and get the quotient, 2nd) Tack on two zeros at the end of the quotient to get the answer.
You might notice that this trick, as stated, works only when the other number is a perfect multiple of 4, such as 8, 12, 16, 20, etc. That is true, but you can easily fix this shortcoming. To see how, let’s try to use the procedure just outlined for the problem 14 x 25. In the first, dividing stage, we’d get 14 ÷ 4 = 3 with a remainder of 2. Going back to money, this means that 14 quarters gives us 3 whole dollars, with 2 quarters to boot (the remainder of 2). Interpreting the remainder of 2 as 2 quarters, we see that it is worth 50 cents. So in all you get $3.00 + 50 cents = $3.50. And this is equivalent, in pure numbers, to 350.
So for problems in which the other number is not a multiple of 4, we can streamline the process like this: 1st) Divide the number (other than 25) by 4 to get the quotient and remainder, 2nd) Tack on two zeros to the main part of the quotient,
3rd) Multiply the remainder by 25 and get the product. 2, 4th) Add the results from steps 2 and 3 to get the answer.
Example: 25 x 19. 1st) 19 ÷ 4 = 4 remainder 3, 2nd) The 4 stands for 400,
3rd) 3 x 25 = 75, 4th) 400 + 75 = 475, so 19 x 25 = 475.
One last point for today’s lesson. As long as we’re noticing shortcomings, you probably also noticed that the first trick — for multiplying two-digit numbers by 11 — works in some situations but not others. The trick requires you to add the digits of the two-digit number and to write down that sum as the middle digit of the answer. But what would you do if the sum of the digits is NOT a one-digit number? For example, what you would you do for the problem: 11 x 48? The sum of 4 and 8 is 12, which is not a one-digit number. And you obviously can’t write down 12 as the single, middle digit of the answer!
The solution to this problem relates to the idea of “carrying” when adding. Using the trick, the answer to 11 x 48 would seem to be: 4 – 12 – 8, but that makes no sense. However, by using the idea of “carrying,” you “carry” the 1 in the front of 12 by adding it to the 4. So the 4 gets boosted up to a 5. And then you keep the 2 from 12, and that 2 becomes the middle digit of the answer. So the answer actually ends up being: 5 – 2 – 8, which is 528, meaning that 11 x 48 = 528.
There are many more tricks in basic arithmetic. These represent the “tip of the iceberg,” so to speak. You can find more tricks online or in books. But I’ll be bringing more up in future articles as well. In the meantime, enjoy the tricks — and your car trips too.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
So you’re taking that long summer car trip, and being dutiful homeschoolers, you’ve taken some time to quiz your kids on the U.S. presidents, the spelling of “australopithecus,” and the answers to the good old fashioned times tables.
But they’re getting a bit bored with the process. So why not add a little pizazz — at least with the math part of the summer quizzing process —by tossing in some “wild and crazy” math tricks.
I myself still remember my dad reading from a book and teaching me at age 10 my first math trick: a trick for multiplying 11 by two-digit numbers. I’m not talking about the fact that 11 x 3 = 33, and 11 x 9 = 99. I’m talking about a trick that showed me how to multiply 11 x 24 and get 264 — in less than three seconds and all in my head!
The trick is that the answer has three digits. The first digit of the answer is the same as the 10s digit of 24, namely 2. The last digit of the answer is the 1s digit of 24, namely 4. And the middle digit of the answer— and here’s the neat part — turns out to be the sum of 2 and 4, namely 6. So the answer is 2-6-4, meaning 264.
Since my dad had not known the trick either, the two of us shared the same sense of delight and wonder. And the pleasure in learning that trick still resounds for me, so my dad and I still share a sense of fun when it comes to numbers and number tricks.
So today I’d like to encourage you to take some time teaching your children a few math tricks. The benefits are many: 1) Children learn that there are somewhat magical tricks that make math easier, 2) Children see that math has many interesting patterns, 3) If mental math tricks are taught well, children develop an appetite to learn more tricks, not to mention the ability to develop tricks of their own, and 4) As children learn more tricks, they boost their “number sense,” that indefinable yet extremely important knack for working with numbers which, once obtained, can last a lifetime.
Trick #2) Multiplying by 5. Since 5 is half of 10, it stands to reason that the product obtained when multiplying by 5 must be half of the product obtained when multiplying by 10. Note: if this idea is hard for your kids to grasp, bring it down to earth by likening 10 x 4 to 10 separate groups of 4 beans. Using that model, 5 x 4 would mean 5 groups of 4 beans. Children can see that 5 groups of 4 beans is half as much as 10 groups of 4 beans. So they can understand that 5 x 4 must be half as much as 10 x 4. Especially with younger children, it’s a good idea to demonstrate this visually with beans (or coins or whatever is at hand).
In any case, this principle leads to a simple trick for multiplying by 5, which we’ll learn by thinking about the problem 5 x 48. First, though, examine the simpler problem, 10 x 48. This is easy because multiplying by 10 means you just tack a zero onto the number. Tacking a zero onto 48 you get 480, so 10 x 48 = 480. Then, to get 5 x 48, follow these steps: 1st) Take half of 48 = 24, 2nd) Now that you’ve got half, tack on the zero, to get 240. And that’s it: 5 x 48 = 240. It’s that simple.
Trick #3) Multiplying by 25. Most children who’ve worked with money know that four quarters make a dollar. Thinking about that, children can see this means that 4 x 25 = 100, since 4 coins worth 25 cents are worth 100 cents. Taking that a step further, children can see that 8 quarters would make $2, so 8 x 25 = 200; 12 quarters would make $3, so 12 x 25 = 300. With a bit of time, children can see that this pattern will extend, so that every additional 4 quarters leads to another 100. And this means that to work a problem like 25 x 28, for example, all children need to do is figure out how many groups of four there are in 28. From division, they know that there are 7 groups of 4 in 28. So that means that 25 x 28 is like $7, so 25 x 28 = 700.
Then, you can help children streamline the process into two steps for multiplying numbers by 25: 1st) Divide the number (other than 25) by 4 and get the quotient, 2nd) Tack on two zeros at the end of the quotient to get the answer.
You might notice that this trick, as stated, works only when the other number is a perfect multiple of 4, such as 8, 12, 16, 20, etc. That is true, but you can easily fix this shortcoming. To see how, let’s try to use the procedure just outlined for the problem 14 x 25. In the first, dividing stage, we’d get 14 ÷ 4 = 3 with a remainder of 2. Going back to money, this means that 14 quarters gives us 3 whole dollars, with 2 quarters to boot (the remainder of 2). Interpreting the remainder of 2 as 2 quarters, we see that it is worth 50 cents. So in all you get $3.00 + 50 cents = $3.50. And this is equivalent, in pure numbers, to 350.
So for problems in which the other number is not a multiple of 4, we can streamline the process like this: 1st) Divide the number (other than 25) by 4 to get the quotient and remainder, 2nd) Tack on two zeros to the main part of the quotient,
3rd) Multiply the remainder by 25 and get the product. 2, 4th) Add the results from steps 2 and 3 to get the answer.
Example: 25 x 19. 1st) 19 ÷ 4 = 4 remainder 3, 2nd) The 4 stands for 400,
3rd) 3 x 25 = 75, 4th) 400 + 75 = 475, so 19 x 25 = 475.
One last point for today’s lesson. As long as we’re noticing shortcomings, you probably also noticed that the first trick — for multiplying two-digit numbers by 11 — works in some situations but not others. The trick requires you to add the digits of the two-digit number and to write down that sum as the middle digit of the answer. But what would you do if the sum of the digits is NOT a one-digit number? For example, what you would you do for the problem: 11 x 48? The sum of 4 and 8 is 12, which is not a one-digit number. And you obviously can’t write down 12 as the single, middle digit of the answer!
The solution to this problem relates to the idea of “carrying” when adding. Using the trick, the answer to 11 x 48 would seem to be: 4 – 12 – 8, but that makes no sense. However, by using the idea of “carrying,” you “carry” the 1 in the front of 12 by adding it to the 4. So the 4 gets boosted up to a 5. And then you keep the 2 from 12, and that 2 becomes the middle digit of the answer. So the answer actually ends up being: 5 – 2 – 8, which is 528, meaning that 11 x 48 = 528.
There are many more tricks in basic arithmetic. These represent the “tip of the iceberg,” so to speak. You can find more tricks online or in books. But I’ll be bringing more up in future articles as well. In the meantime, enjoy the tricks — and your car trips too.
Josh Rappaport lives and works in Santa Fe, New Mexico, along with his wife and two children, now teens. Josh is the author of the briskly-selling Algebra Survival Guide, and companion Algebra Survival Guide Workbook. Josh is also co-author of the Card Game Roundup books, and author of PreAlgebra Blastoff!, a playful approach to positive and negative numbers. Josh is currently working on the Geometry Survival Flash Cards, a colorful approach to learning the key facts of geometry.
At his blog, Josh writes about the “nuts-and-bolts” of teaching math. Josh also leads workshops on math education at school and homeschooling conferences., and he tutors homeschoolers nationwide using SKYPE. You can reach Josh by email at: josh@SingingTurtle.com
