My apologies for not being able to update my blog the last month. I have been a little busy at home, but everything is working out. Thanks for your wonderful kids, they were stupendous on Monday for the substitute teacher. If you don't hear from me again, Merry Christmas!
I have been worried sick about my brother, he will be living with my family for the next month or so.
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Wednesday, December 16, 2009
Tuesday, December 15, 2009
Eight Times Tables
Eight Times Tables put into a Table
8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
8 x 11 = 88
8 x 12 = 96
If you have a look at the tens digits here, you'll see a very interesting in the first five rows :
8 x 1 = 08
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
So up to five, eight times something starts with one less than the something.
• Eight times 1 starts with 0,
• eight times 2 starts with 1,
• eight times 3 starts with 2,
• eight times 4 starts with 3, and
• eight times 5 starts with 4.
The next few rows have a similar pattern
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
For numbers in the range 6 to 10, eight times the number starts with two less than the number.
• Eight times 6 starts with 4,
• eight times 7 starts with 5,
• eight times 8 starts with 6,
• eight times 9 starts with 7, and
• eight times 10 starts with 8.
The next block of 5 rows has, again, a similar pattern.
• Eight times 11 starts with 8,
• eight times 12 starts with 9,
• eight times 13 starts with 10,
• eight times 14 starts with 11, and
• eight times 15 starts with 12.
Well, that might help remember the tens digit, but what about the ones digit?
Note that within each block of five rows, the ones digit follows a very simple pattern
8, 6, 4, 2, 0
And this repeats over and over and over!
If your child has a good handle on the two and ten times tables, then it is not too hard perhaps to mentally multiply by eight. To multiply something by eight, remember that eight times a number is ten times the number minus two times the number. This is because 8 = 10 - 2. For example
• To work out 8 times 7 :
• 10 times 7 is 70
• 2 times 7 is 14
• 70 - 14 is 60 - 4, or 56.
Alternatively, if their four times table is strong, eight times a number is four times the number, added to four times the number. That is, eight times is twice four times. For example
• To find 8 times 7
• 4 times 7 is 28, and
• 28 + 28 is 56.
Another nice pattern appears in the digits used in certain rows of the table. Check out the following two sums.
8 x 8 = 6 4, and
8 x 6 = 4 8
Notice that you can get one from the other just by shifting all the digits sideways. Another pair of sums like this is
4 x 8 = 3 2, and
8 x 3 = 2 4
I used one of these pairs as the inspiration for one of the times table mazes on this site.
Before I close, let me mention this... Did you know that to test if a number is divisible by 8, you only need to look at the last 3 digits? Here's how the test works.
• Firstly, the last digit should be 8, 6, 4, 2 or 0.
This makes sure that the number is an even number. Odd numbers ae not divisible by two, so they can't be divisible by eight!
• If the last digit is 0, 4 or 8 (a multiple of 4), the second last digit should be even.
• If the last digit is 2 or 6 (a multiple of 2, but not 4), the second last digit should be odd.
These two conditions make sure the number is divisible by 4. If a number is not divisble by 4, it can't be divisble by 8! For example, 3098 is not divisble by 4 - the last digit is 8, but the second last is not even. On the other hand, 3932 is divisible by 4.
• If the last two digits make up a multiple of 8, then the third last digit must be even.
• If the last two digits don't make up a multiple of 8, then the third last digit must be odd.
Of course, it helps to be familar with the eight times tables (up to 8 x 12) for this last test.
It's worth giving a few examples of applying these rules...
• 314159 is not a multiple of 8. It fails the first test, so it's not even a multiple of 2. The last digit is 9, not 8,6,4,2 or 0.
• 314158 passes the first test. The last digit is 8. The next test requires that the second last digit be even, but it's not. So 314158 is not a multiple of 4, let alone 8.
• 314162 passes the first test also. The last digit is 2. If the last digit is 2 ot 6, the next test requires the second last digit to be odd. Again, the number fails to be a multiple of 4!
• 314156 passes the first and second tests! So it's a multiple of 4. The last two digits make up 56, which is a multiple of 8. The third and final test tells us that the third last digit should be even. But it's odd! So 314156 is a multiple of 4, but not of 8. (On the other hand, 314256 is a multiple of 8, by this very same test)
• 314152 passes the first and second tests also! So it's a multiple of 4. The last two digits make up 52, and since I know my 8 times table, I know that 52 is not eight times anything. The third and final test tells me that the third last digit should be odd - and it is! So 314152 is, indeed, a multiple of 8 - and so would be any number at all ending in 152.
Note that 152 = 8 x 19, and 314152 = 8 x 39269.
8 x 1 = 8
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
8 x 11 = 88
8 x 12 = 96
If you have a look at the tens digits here, you'll see a very interesting in the first five rows :
8 x 1 = 08
8 x 2 = 16
8 x 3 = 24
8 x 4 = 32
8 x 5 = 40
So up to five, eight times something starts with one less than the something.
• Eight times 1 starts with 0,
• eight times 2 starts with 1,
• eight times 3 starts with 2,
• eight times 4 starts with 3, and
• eight times 5 starts with 4.
The next few rows have a similar pattern
8 x 6 = 48
8 x 7 = 56
8 x 8 = 64
8 x 9 = 72
8 x 10 = 80
For numbers in the range 6 to 10, eight times the number starts with two less than the number.
• Eight times 6 starts with 4,
• eight times 7 starts with 5,
• eight times 8 starts with 6,
• eight times 9 starts with 7, and
• eight times 10 starts with 8.
The next block of 5 rows has, again, a similar pattern.
• Eight times 11 starts with 8,
• eight times 12 starts with 9,
• eight times 13 starts with 10,
• eight times 14 starts with 11, and
• eight times 15 starts with 12.
Well, that might help remember the tens digit, but what about the ones digit?
Note that within each block of five rows, the ones digit follows a very simple pattern
8, 6, 4, 2, 0
And this repeats over and over and over!
If your child has a good handle on the two and ten times tables, then it is not too hard perhaps to mentally multiply by eight. To multiply something by eight, remember that eight times a number is ten times the number minus two times the number. This is because 8 = 10 - 2. For example
• To work out 8 times 7 :
• 10 times 7 is 70
• 2 times 7 is 14
• 70 - 14 is 60 - 4, or 56.
Alternatively, if their four times table is strong, eight times a number is four times the number, added to four times the number. That is, eight times is twice four times. For example
• To find 8 times 7
• 4 times 7 is 28, and
• 28 + 28 is 56.
Another nice pattern appears in the digits used in certain rows of the table. Check out the following two sums.
8 x 8 = 6 4, and
8 x 6 = 4 8
Notice that you can get one from the other just by shifting all the digits sideways. Another pair of sums like this is
4 x 8 = 3 2, and
8 x 3 = 2 4
I used one of these pairs as the inspiration for one of the times table mazes on this site.
Before I close, let me mention this... Did you know that to test if a number is divisible by 8, you only need to look at the last 3 digits? Here's how the test works.
• Firstly, the last digit should be 8, 6, 4, 2 or 0.
This makes sure that the number is an even number. Odd numbers ae not divisible by two, so they can't be divisible by eight!
• If the last digit is 0, 4 or 8 (a multiple of 4), the second last digit should be even.
• If the last digit is 2 or 6 (a multiple of 2, but not 4), the second last digit should be odd.
These two conditions make sure the number is divisible by 4. If a number is not divisble by 4, it can't be divisble by 8! For example, 3098 is not divisble by 4 - the last digit is 8, but the second last is not even. On the other hand, 3932 is divisible by 4.
• If the last two digits make up a multiple of 8, then the third last digit must be even.
• If the last two digits don't make up a multiple of 8, then the third last digit must be odd.
Of course, it helps to be familar with the eight times tables (up to 8 x 12) for this last test.
It's worth giving a few examples of applying these rules...
• 314159 is not a multiple of 8. It fails the first test, so it's not even a multiple of 2. The last digit is 9, not 8,6,4,2 or 0.
• 314158 passes the first test. The last digit is 8. The next test requires that the second last digit be even, but it's not. So 314158 is not a multiple of 4, let alone 8.
• 314162 passes the first test also. The last digit is 2. If the last digit is 2 ot 6, the next test requires the second last digit to be odd. Again, the number fails to be a multiple of 4!
• 314156 passes the first and second tests! So it's a multiple of 4. The last two digits make up 56, which is a multiple of 8. The third and final test tells us that the third last digit should be even. But it's odd! So 314156 is a multiple of 4, but not of 8. (On the other hand, 314256 is a multiple of 8, by this very same test)
• 314152 passes the first and second tests also! So it's a multiple of 4. The last two digits make up 52, and since I know my 8 times table, I know that 52 is not eight times anything. The third and final test tells me that the third last digit should be odd - and it is! So 314152 is, indeed, a multiple of 8 - and so would be any number at all ending in 152.
Note that 152 = 8 x 19, and 314152 = 8 x 39269.
Times Tables – Multiplication Tricks
I don't recall where I obtained these tricks, but I have some more tricks that I will post in the coming weeks. Here are some Times Table Tricks for 0, 1, 2, 4, 5, 9, 10, and 11 Times Tables.
Zero and One Times Tables : Could This Get Any Easier?
Check this out: if you multiply ANYTHING by zero, the answer is zero. Anything. 4 x 0 = 0 and 1,000,000 x 0 = 0. One times tables are almost as easy. Any number multiplied by one is always itself. "Huh," you say? Well, check it out: 1 x 10 = 10, 1 x 42 = 42, 8,726 x 1 = 8,726. Simple, right?
Two Times Tables : Double Your Pleasure
When you multiply a number by two, you just double that number. That's some pretty simple math, even for the most numerically challenged of us. So, if you want to figure out what 2 x 7 is, you just add 7 + 7 (the answer is 14, by the way). Any number times two is the same as that number PLUS itself. Here's one more example: 2 x 5 is the same as 5 + 5, which equals 10. Practice a couple -- it's really easy
Four Times Tables : Double, Double Trouble
OK, so now that you have the twos figured out, it's time to get doubling again. This is not as confusing as it may sound. When you multiply four with anything, you have to use the doubling-up trick (that's the one you used for the two times table) twice. Here's an example: 4 x 7 is the same as 7 + 7 = 14 and then 14 + 14 = 28. So 4 x 7 = 28. Here's another double, double example: 4 x 10 is the same as 10 + 10 = 20, so then 20 + 20 = 40. So the answer is 4 x 10 = 40.
Five Times Tables : It's Why You Have Fingers
First things first - can you count to five? Yeah? Then you can figure out your five times tables. So, when you want to multiply a number by five you just count up by fives that may times. Let's review how to count by fives: 5, 10, 15, 20, 25... and so on. Got it? So if you want to multiply 5 x 7, you just count by fives, seven times. 5, 10, 15, 20, 25, 30, 35. So 7 x 5 = 35. If you have trouble keeping track, just use you fingers
Nine Times Tables - One Seriously Handy Tip
Here is a fun (OK, "fun" is a relative term) way to remember your nine times tables. For this method you are going to need to have two hands. Put your hands in front of you with your palms towards you. Your fingers represent the numbers one to ten (one is your left thumb; ten is your right thumb). Now you're ready to do your nine times tables. Let's say the question is 9 x 4. Count to the fourth finger (if you've counted right, it will be the ring finger on your left hand) and curl that finger under. Now you have three fingers up before that finger and 6 up after it. So the answer is 36. Let's try 9 x 8. Put down the 8th finger (middle finger on the right hand) so that you have seven fingers up before the finger you curled under, and two fingers afterward. So the answer is 72. If you're confused, check out the diagram at the top of this page on the right. It'll clear things up
Ten Times Tables : Just Add Zero and Stir
Here's another no brainer. If you want to multiply something by 10, just add a zero on the end. Here's an example: 10 x 8 = 80 or 10 x 100 = 1,000. Try it with any number - from one to a billion. It's just too easy.
Eleven Times Tables :: Seeing Double
Here's another one that's as easy as pie. Any number - up to nine - multiplied by 11 is itself written out twice. Confused? Just check it out: 9 x 11 = 99, 4 x 11 = 44, 3 x 11 = 33 and so on.
* Hint *
• Times tables work both ways (5 x 10 is the same as 10 x 5) so if you have the question 3 x 11, don’t stress that there isn't a trick for the three times tables -- use the trick for 11 instead!
Zero and One Times Tables : Could This Get Any Easier?
Check this out: if you multiply ANYTHING by zero, the answer is zero. Anything. 4 x 0 = 0 and 1,000,000 x 0 = 0. One times tables are almost as easy. Any number multiplied by one is always itself. "Huh," you say? Well, check it out: 1 x 10 = 10, 1 x 42 = 42, 8,726 x 1 = 8,726. Simple, right?
Two Times Tables : Double Your Pleasure
When you multiply a number by two, you just double that number. That's some pretty simple math, even for the most numerically challenged of us. So, if you want to figure out what 2 x 7 is, you just add 7 + 7 (the answer is 14, by the way). Any number times two is the same as that number PLUS itself. Here's one more example: 2 x 5 is the same as 5 + 5, which equals 10. Practice a couple -- it's really easy
Four Times Tables : Double, Double Trouble
OK, so now that you have the twos figured out, it's time to get doubling again. This is not as confusing as it may sound. When you multiply four with anything, you have to use the doubling-up trick (that's the one you used for the two times table) twice. Here's an example: 4 x 7 is the same as 7 + 7 = 14 and then 14 + 14 = 28. So 4 x 7 = 28. Here's another double, double example: 4 x 10 is the same as 10 + 10 = 20, so then 20 + 20 = 40. So the answer is 4 x 10 = 40.
Five Times Tables : It's Why You Have Fingers
First things first - can you count to five? Yeah? Then you can figure out your five times tables. So, when you want to multiply a number by five you just count up by fives that may times. Let's review how to count by fives: 5, 10, 15, 20, 25... and so on. Got it? So if you want to multiply 5 x 7, you just count by fives, seven times. 5, 10, 15, 20, 25, 30, 35. So 7 x 5 = 35. If you have trouble keeping track, just use you fingers
Nine Times Tables - One Seriously Handy Tip
Here is a fun (OK, "fun" is a relative term) way to remember your nine times tables. For this method you are going to need to have two hands. Put your hands in front of you with your palms towards you. Your fingers represent the numbers one to ten (one is your left thumb; ten is your right thumb). Now you're ready to do your nine times tables. Let's say the question is 9 x 4. Count to the fourth finger (if you've counted right, it will be the ring finger on your left hand) and curl that finger under. Now you have three fingers up before that finger and 6 up after it. So the answer is 36. Let's try 9 x 8. Put down the 8th finger (middle finger on the right hand) so that you have seven fingers up before the finger you curled under, and two fingers afterward. So the answer is 72. If you're confused, check out the diagram at the top of this page on the right. It'll clear things up
Ten Times Tables : Just Add Zero and Stir
Here's another no brainer. If you want to multiply something by 10, just add a zero on the end. Here's an example: 10 x 8 = 80 or 10 x 100 = 1,000. Try it with any number - from one to a billion. It's just too easy.
Eleven Times Tables :: Seeing Double
Here's another one that's as easy as pie. Any number - up to nine - multiplied by 11 is itself written out twice. Confused? Just check it out: 9 x 11 = 99, 4 x 11 = 44, 3 x 11 = 33 and so on.
* Hint *
• Times tables work both ways (5 x 10 is the same as 10 x 5) so if you have the question 3 x 11, don’t stress that there isn't a trick for the three times tables -- use the trick for 11 instead!
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