Continuing finger arithmetic from Part Eight. This one is called Part Nine. It is the tenth part in the series. It introduces subtraction.
Subtraction is the opposite of addition. It's sort of addition backwards. You might be able to figure it out, using what you've learned from addition. Indeed, i was unable to teach my eight year old subtraction. I announced that i would teach him, and he just went ahead and did it. Now, if he'd done it wrong, i'd have corrected him. And when he makes mistakes, i walk him through the steps. But he really seemed to go through all nine lessons in reverse, taking about ten seconds. I don't hold him back. The correct direction is forward to the goal.
In lesson zero, we had counting. Count from nine to zero. It was then pointed out that counting was adding one. Counting down is the same as subtracting one. Remember to say the answers out loud. Try counting from twenty to zero.
Now we'll subtract small numbers. Consider 4 - 3. Set four fingers and subtract three fingers. Read the result: One. This covers 1 - 1, 2 - 1, 3 - 1, 4 - 1, 2 - 2, 3 - 2, 4 - 2, 3 - 3, 4 - 3, 4 - 4. Practice these.
Some big numbers with small numbers are easy. Consider 9 - 3. Set nine. Take away three fingers. Read the answer: six. This covers 9 - 1, 9 - 2, 9 - 3, 9 - 4, 8 - 1, 8 - 2, 8 - 3, 7 - 1, 7 - 2, 6 - 1.
Next is a half carry, using Five's Compliments. Recall that the Five's Compliment of 1 is 4, and 4 is 1, and 2 is 3 and 3 is 2. Consider 7 - 3. Set seven. Note that there aren't three fingers to subtract. We're going perform this by observing that 7 - 3 is the same as 7 + 2 - 5. Remember that the Five's Compliment of three is two, so you add two and subtract five. Add two fingers. Subtract the thumb. Read the answer: four.
Subtracting five is easy. Consider 7 - 5. Set seven. Take away the thumb. Read the answer: two. This covers 9 - 5, 8 - 5, 7 - 5, 6 - 5, and 5 - 5.
Subtracting bigger numbers. Consider 9 - 7. Set nine. Seven is five plus two - thumb and two fingers. You have two fingers to take away, so do it. You have a thumb to take away, so do it. Read the answer: two. This covers 9 - 9, 9 - 8, 9 - 7, 9 - 6, 8 - 8, 8 - 7, 8 - 6, 7 - 7, 7 - 6, 6 -6. Need you be reminded to practice?
Next lesson: two digit subtraction. Bring your thinking caps.
Again, you are referred to the example generator, which can provide you with an infinite number of examples. Each time you click here, you get a new page. Use your back button to get back to the lesson. This link provides two digit examples. As this lesson hasn't taught two digit subtraction, feel free to ignore any problems involving two digits anywhere.
You might just want to go to the web site, so you can ask the generator yourself. Addition, subtraction or mixed problems, how many digits, more than just two numbers per problem? Answers on the same page or another page? Maybe you want one hundred pages to get you through vacation.
Tuesday, July 25, 2006
Thursday, July 13, 2006
All Wet
When Oxygen Tech played Hydrogen U.
The Game had just begun,
when Hydrogen scored two fast points
And Oxygen still had none.
Then Oxygen scored a single goal
And thus it did remain,
At Hydrogen 2 and Oxygen 1
Called because of rain.
The Game had just begun,
when Hydrogen scored two fast points
And Oxygen still had none.
Then Oxygen scored a single goal
And thus it did remain,
At Hydrogen 2 and Oxygen 1
Called because of rain.
Wednesday, July 12, 2006
Finger Arithmetic Part Eight
Continuing finger arithmetic from Part Seven. This one is called Part Eight. It is the ninth part in the series.
Back in Part Seven, we finished up addition of single digit numbers. We took care of carries with Ten's Compliments and Five's Compliments. In this lesson, we continue forward with two digit addition.
For example, how does one add 13 + 21? Set thirteen by making two fists, set three fingers on the right hand and one finger on the left hand. The twenty-one can be added in either order. Add twenty, then add one, or add one, then add twenty. These lessons always proceed from right to left. To add one to the three on the right hand, just add one more finger. To add twenty, just add two fingers to the left hand. Read the answer: 34.
The last example was easy. There were no carries. Consider adding 34 + 35. Set thirty four by making two fists, setting three fingers on the left hand and four fingers on the right hand. Add five (of 35) to the right hand by adding the thumb. Still no carry. Then consider adding thirty. You would add three fingers to the left hand, but there aren't three fingers to add. So, subtract the Five's Compliment of three, which is two fingers, and add five, which is the thumb. Read the answer: 69. This example added five, and also added using a Five's Compliment.
Consider 26 + 48. Set twenty six by starting with two fists. Set two fingers on the left and a finger and thumb on the right. We need to add eight. Eight is five plus three. You can add the three by adding three fingers. You can't add the five, so you must subtract five by subtracting the right thumb, and adding ten, which is adding one finger on the left hand. To add forty, you'd want to add four fingers to the left hand. However, there aren't enough fingers to add. So, you subtract the Five's Compliment of four, which is one, and add five, which is the thumb, to the left hand. Read the answer: 74. This example has a carry from the unit's digit (hand) to the ten's digit (hand), and also has a Five's Compliment add.
Consider 19 + 27. Set nineteen by making two fists, setting one finger on the left hand and all fingers and thumb on the right hand. To add seven, note that seven is five plus two. There aren't two fingers to add, so you use Ten's Compliment. Subtract the Ten's Compliment of seven, which is three, and add ten, which is one finger on the left hand. To add twenty, add two more fingers on the left hand. Read the answer: 46. Use Ten's Compliments for numbers over five, and Five's Compliments for numbers less than five.
Consider 46 + 14. You might still have 46 on your hands, or set 46. To add four, recognize that there aren't four fingers to add. Subtract the Five's Compliment of four, which is one, and add five. You already have the thumb up, so you subtract five (the thumb) and add ten. You don't have a finger to add to the left hand, so you subtract the Five's Compliment of one, which is four, and add five, which is the thumb. That was just to add the four of the fourteen. To add the ten of the fourteen, add one finger to the left hand. Read the answer: sixty. You would think that this is as complicated as carries get for finger addition. However, there is one more case.
Consider 37 + 63. Set thirty seven on your hands. You want to add three, but can't, as there are only two available fingers. So, you subtract two and add five. You can subtract two, but can't add five, as there is already a thumb set. You add five by subtracting five and adding ten, which is one finger on the left hand. Now you need to add sixty. Considering that sixty is 1 + 5, you note that you can't add one finger. So, you subtract the Ten's Compliment of six, which is four, and add ten - which is one hundred. Well, most people don't have another hand, so you'll have to remember it. Perhaps your nose. Read the answer. Remembering the one hundred, you find that the answer is: 100.
In the previous example, we used the Five's Compliment of three to add it to the existing seven. There were a number of confusing steps to get through adding just the one digit. While this method works, you could instead use the Ten's Compliment of three, which is seven. Here's how that works. Set thirty seven on your hands. To add three, subtract the Ten's Compliment of three, which is seven, and add ten. You can add the ten, so you just do it. I find it conceptually easier to use Five's Compliments for digits less than five, and Ten's Compliments for digits more than five. This always works, though it leads to extra steps now and then. As your proficiency improves, feel free to experiment with Ten's Compliments where they make sense.
You might spend a few weeks on this before moving on to subtraction.
Again, you are referred to the example generator, which can provide you with an infinite number of examples. Each time you click here, you get a new page. Use your back button to get back to the lesson. This isn't the same link as in the Part Six. This link provides two digit examples.
The example generator is the same, with different parameters. This is the form that lets you select the parameters you might want. For example, if you have a Japanese Abacus, called a Soroban, you'll have more than two hands, as each hand is basically one rod of this device. Let's say that you have a child in elementary school. If they're reading for enjoyment, then handing them some books to read over the summer will allow them to advance, or at least not backslide too much. What about arithmetic? It is estimated that children backslide an average of about two and a half months over the summer. This problem generator can help. I try to work in five minutes with my son every day. There is also a problem generator for multiplication on my site. Feel free to use it for division problems as well.
Back in Part Seven, we finished up addition of single digit numbers. We took care of carries with Ten's Compliments and Five's Compliments. In this lesson, we continue forward with two digit addition.
For example, how does one add 13 + 21? Set thirteen by making two fists, set three fingers on the right hand and one finger on the left hand. The twenty-one can be added in either order. Add twenty, then add one, or add one, then add twenty. These lessons always proceed from right to left. To add one to the three on the right hand, just add one more finger. To add twenty, just add two fingers to the left hand. Read the answer: 34.
The last example was easy. There were no carries. Consider adding 34 + 35. Set thirty four by making two fists, setting three fingers on the left hand and four fingers on the right hand. Add five (of 35) to the right hand by adding the thumb. Still no carry. Then consider adding thirty. You would add three fingers to the left hand, but there aren't three fingers to add. So, subtract the Five's Compliment of three, which is two fingers, and add five, which is the thumb. Read the answer: 69. This example added five, and also added using a Five's Compliment.
Consider 26 + 48. Set twenty six by starting with two fists. Set two fingers on the left and a finger and thumb on the right. We need to add eight. Eight is five plus three. You can add the three by adding three fingers. You can't add the five, so you must subtract five by subtracting the right thumb, and adding ten, which is adding one finger on the left hand. To add forty, you'd want to add four fingers to the left hand. However, there aren't enough fingers to add. So, you subtract the Five's Compliment of four, which is one, and add five, which is the thumb, to the left hand. Read the answer: 74. This example has a carry from the unit's digit (hand) to the ten's digit (hand), and also has a Five's Compliment add.
Consider 19 + 27. Set nineteen by making two fists, setting one finger on the left hand and all fingers and thumb on the right hand. To add seven, note that seven is five plus two. There aren't two fingers to add, so you use Ten's Compliment. Subtract the Ten's Compliment of seven, which is three, and add ten, which is one finger on the left hand. To add twenty, add two more fingers on the left hand. Read the answer: 46. Use Ten's Compliments for numbers over five, and Five's Compliments for numbers less than five.
Consider 46 + 14. You might still have 46 on your hands, or set 46. To add four, recognize that there aren't four fingers to add. Subtract the Five's Compliment of four, which is one, and add five. You already have the thumb up, so you subtract five (the thumb) and add ten. You don't have a finger to add to the left hand, so you subtract the Five's Compliment of one, which is four, and add five, which is the thumb. That was just to add the four of the fourteen. To add the ten of the fourteen, add one finger to the left hand. Read the answer: sixty. You would think that this is as complicated as carries get for finger addition. However, there is one more case.
Consider 37 + 63. Set thirty seven on your hands. You want to add three, but can't, as there are only two available fingers. So, you subtract two and add five. You can subtract two, but can't add five, as there is already a thumb set. You add five by subtracting five and adding ten, which is one finger on the left hand. Now you need to add sixty. Considering that sixty is 1 + 5, you note that you can't add one finger. So, you subtract the Ten's Compliment of six, which is four, and add ten - which is one hundred. Well, most people don't have another hand, so you'll have to remember it. Perhaps your nose. Read the answer. Remembering the one hundred, you find that the answer is: 100.
In the previous example, we used the Five's Compliment of three to add it to the existing seven. There were a number of confusing steps to get through adding just the one digit. While this method works, you could instead use the Ten's Compliment of three, which is seven. Here's how that works. Set thirty seven on your hands. To add three, subtract the Ten's Compliment of three, which is seven, and add ten. You can add the ten, so you just do it. I find it conceptually easier to use Five's Compliments for digits less than five, and Ten's Compliments for digits more than five. This always works, though it leads to extra steps now and then. As your proficiency improves, feel free to experiment with Ten's Compliments where they make sense.
You might spend a few weeks on this before moving on to subtraction.
Again, you are referred to the example generator, which can provide you with an infinite number of examples. Each time you click here, you get a new page. Use your back button to get back to the lesson. This isn't the same link as in the Part Six. This link provides two digit examples.
The example generator is the same, with different parameters. This is the form that lets you select the parameters you might want. For example, if you have a Japanese Abacus, called a Soroban, you'll have more than two hands, as each hand is basically one rod of this device. Let's say that you have a child in elementary school. If they're reading for enjoyment, then handing them some books to read over the summer will allow them to advance, or at least not backslide too much. What about arithmetic? It is estimated that children backslide an average of about two and a half months over the summer. This problem generator can help. I try to work in five minutes with my son every day. There is also a problem generator for multiplication on my site. Feel free to use it for division problems as well.
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