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In
the
first addition lesson, you learned a little about
adding with "one-to-one" correspondence, and
how to tell how many numbers are represented by fingers.
This
lesson covers addition to more than ten.
Adding two numbers which total more than 10
is the first traumatic
arithmetical operation which everyone faces.
Why is that? Because they usually try to do
an addition like that before they learn to recognize
groups of fingers as numbers (instead of individual,
one-to-one correspondences), So they still try
to do things like 6 + 8 on their fingers! How
frustrating is THAT to a kid?
Of
course it seems simple to you, now, at your age, but
a little child agonizes over that until s/he learns
to recognize groups of fingers as numbers.
There
is a way to deal with that, which you can use to help
children get over their first math-related trauma. It
is also a powerful method for you to explore the way
numbers work. The method will also help you to
get very good at Math Mojo with the other operations
in arithmetic.
The
method allows a child to use one-to-one correspondence
with his/her fingers to add to over ten, although
the child (probably) doesn't have more than
ten fingers. How can they do that? Read on.
The
method is sometimes called addition-by-subtraction
(although, like many learning-methods, there
is no one official name for it). It uses tens-complements.
A tens-complement of a number is simply that
number, subtracted from ten. Or, in other words,
the tens-complement of a number is the amount
you would need to make that number a ten. See
the box below.
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A
Short Lesson on Tens-complements
The
tens-complement of any digit is
the number you would need to make
the number a ten.
For instance, if you had the number
7, you would need 3 more to make
it a 10.
So 3 is the tens-complement of 7,
and 7 is the tens-compement of 3.
- The
tens complement of 1 is 9, and
vice-versa,
- The
tens complement of 2 is 8, and
vice-versa,
- The
tens complement of 3 is 7, and
vice-versa,
- The
tens complement of 4 is 6, and
vice-versa,
- The
tens complement of 5 is 5.
Those
are all the tens-complements. It
is worth it to learn them, because
there are hundreds of shortcuts
in math which use them. I hope to
cover most of them in this site,
eventually.
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How
to use Tens-complements to add
numbers which total above 10
When
you look at two numbers, you can tell right
away if they area going to add up to over
10 if you need more than 10 fingers to add
them. That is kind of obvious, isn't it?
So,
as long as you know that you are
going to have a total over 10, you
might as well put a 1 in the tens-column
of the answer before you go any
further. It can't be anything but
a 1, because there is no pair of
1-digit numbers which add to higher
than 18 (that would be 9 + 9), so
there can't be a 2 or anything in
the tens column.
Now
look at the addend instead
of the augend. Take the tens-complement
of the addend
and subtract it (that's right -
subtract!)
it from the augend. This will never
be a problem (as long as they original
digits will add to 10 or more),
because you will always be subtracting
a very small number from a larger
one-digit number. You will see how
to do that on your fingers in the
little interactive lesson below.
When
you have done that very simple subtraction,
write the answer to it in the ones-column
of your answer, and you will be
done! That is because you already
wrote the 1 in the tens column.
That makes up for way they usually
teach you to "carry" (which
you won't have to do anymore).
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The
interactive lesson below shows how to use the ten-complements
to add above 10 on your fingers.
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Why
this works
The
reason you do subtract the tens complement
of the addend and put a 1 in the tens column:
What
we are really doing, is adding 10
to the augend. That is what putting
the 1 in the tens-column is about.
We add ten, because it so easy -
you just put a 1 in front of the
augend, and you have added 10! What
could be simpler than that?
But,
wait, we only wanted to add the addend,
didn't we? So if we added 10, we really
added too much. How much too much? Well,
the difference between 10 and the addend,
of course! And that is what a tens-complement
of the addend is, isn't it?
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One
more thing...
You
don't have to always have to subtract
the tens-complement of the addend
from the augend. You could do it
the other way around.
For
example, in the problem above (7
+ 6) you could have written a 1
in the tens column, then subtracted
the tens complement of 7 (which
is 3) from 6 (which would had given
you the same answer as the other
way around). Actually, if you think
about it, taking 3 from 6 is even
easier than taking 4 from 7.
I
taught it to you the first way,
because it is easier to teach
it and have it make sense to
you that way. Like most things
in math, though, it is flexible.
Does that surprise you? It turns
out that there are usually many
more ways to do things than
they teach you in school. Teachers
sometime lose sight of that,
because they have to answer
to unenlightened administrators
and politicians.
It's
not terrible for them to teach
you only one way, but it is
terrible that they make you
think that there is only one way.
They should tell you that the
way they teach is the way that
is easiest for them, but that
there are other ways, too. Math
(real math) is much more
about freedom and imagination
than about grades and tests.
I
think the best rule of thumb in
this case, is to take the tens-complement
of the larger number and subtract
it from the smaller. Experiment.
See what you like best, and get
really good at doing it. |
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If,
for any reason, the above lesson confuses you,
you can check out the lesson on adding
with the abax.
It covers the same material in a different way.
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There
will be more addition lessons up as soon as I can. If you want me to hurry up with these lessons, please send me an e-mail. You can use the "contact Math Mojo" link at the bottom right on this page.
The next lesson (third addition) will be a short one
about how examples can be written or spoken, and how
to handle them when you experience them different ways.
The
one after that will be on adding two 2-digit numbers
which total over 10 and less than 100 (like 38 + 29).
It will use the same method which was used here, so
it pays to learn this lesson well.
Until
the next lesson is up, you can practice your newly-acquired
addition skills by clicking the link in the box below.
Don't forget to use the method we learned above (the
addition-by-subtraction method).
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Test
your basic addition skill. This link
is just a little interactive chance to test
how well you can add two digits that total
10 or more. Like 7+8. It
will eventually have a timer, and you will
be able to record your scores, and test
for improvement after you learn speed-math
methods, which will also be up on this page
soon.
Actually,
if you use this to practice getting
used to adding by subtracting with tens-complements,
you will do better than testing what you
haven't worked on, yet.
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