Nancy from MI

An interestig forward from another home school list.

~Nancy


--- In HomeschoolSupport@y..., Marsha Ransom <mransom@c...> wrote:
Hi, Just had to send along this Algebra Times - it includes a great
article about grade-level appropriate math - if you want to subscribe
go to the end. Marsha

Hi Everybody,

Here's what's you can check out in our full-length
(and I do mean "full-length") springtime issue:

- Discussion on the concept of
"age appropriate" curriculae

- A "trick" on using prime numbers to help you
reduce fractions.

- A springtime lesson plan on ways children
can use math in setting up and developing a garden.

- A Problem of the Month on how to be
an efficient gardener.


Happy Springtime!
- Josh

P.S.: Note that if you want our
subscribe/unsubscribe instructions are
at the bottom of each section of the
NL.

-------------------------------------------------

The Algebra Times

- a newsletter -

Vol. 4, Issue 8

April 2001


QUOTE OF THE MONTH -

"Nature's great book is written in mathematical symbols."
- Galileo Galilei

***********************************************

"AGE-APPROPRIATE" CURRICULAE

For this month's discussion, I'd like to address the
general idea of "age appropriate"curriculae. There
are ways that what I'll say has relevance for both
classroom teachers and homeschoolers, so I'll
try to address both groups.

First of all, I want to say that I attended public school
from kindergarten through 12th grade, and I came out of
that experience with the expected prejudices about "age
appropriate" curriculae. Namely, that at certain ages, children
become ready for certain concepts; that they are not ready
beforehand; and that they would be bored if you were to
teach a topic too late. [i.e: you do in Algebra 1
in eighth grade, not before, not after.]

Lately, in my work with both schooled and
homeschooled children, it's as if a metaphorical wrecking
ball has come along and demolished this notion.

Three examples to show what I mean -

Example #1: I'm working with a public schooled kindergarten child
whose parents want me to do math enrichment.

A few weeks ago she learned quite a bit about negative numbers,
and I'd like to share how this happened.

She and I were playing around with subtraction
problems, and I had just asked her what
8 - 3 was. She said 5, and then she asked, "What's 3 - 8?"
I asked her what she thought it would be. She pondered
for a moment and said, "You can't do it." I said, "Well,
actually, in much more advanced math, you can do it,
and the answer would be negative 5."

At that, this girl started chuckling. She kept saying
"negative three" and laughing. I offered her a way
of understanding this by asking: "Imagine that you
have three pennies, but you owe someone 8. What
would happen?" She said, "I'd give them the three, but
then I'd still owe them more." I said, "Can you figure out
how many more you'd owe." She took a moment, and then
counted up from 3 to 8, using her fingers, and eventually
said five. I said, "Right, you'd owe five more. And when
you end up owing money, the amount is negative. Since you'd owe
five, 3 - 8 is negative 5.

For some reason, this girl loved this concept. It just
intrigued her. She asked me to give her more problems
with negative numbers. What was I going to do? Refuse?!

So since then, she and I have been doing a slew
of problems with negative numbers. Once she even
figured out 9 - 23. Wish you could have seen the look
on her face when I told her she was right!

Here's a kindergartner learning something that is ordinarily
taught in seventh grade.

Example #2: I'm working with a first-grade boy who is homeschooled
and his parents also want enrichment. One day he was
standing in the entrance to my office, which has saltillo
tiling, large tile whose grout lines form a square grid pattern. I
wondered
if we might use this floor design to introduce the notion
of the coordinate plane. I explain that where he was standing
we would call the center; I explained that distances to the right of
the center would be called x distances, and that distances
up from the center would be called y-distances. I sensed that he
liked the sound of those terms, so I talked a bit about sci-fi shows
like Star Trek where people need to give their coordinates so that
others can beam them up or locate them. He thought that was cool.
Then I asked him if thought he could locate me by my x- and
y-distances if I stood at a certain intersection point where the
grout lines met. He said he wasn't sure. I told him I'd help him
figure it out.

It really wasn't hard. Within 10 minutes, he was able to give my
x- and y-coordinates. Oh, I guess I must confess. Part of what
motivated him is this. I told him that if he stated my coordinates
correctly, then he would blow me up. The first time he exploded me,
he was hooked. (Boys are boys, right?).

Of course, in this case, the boy was learning about the first quadrant
of the coordinate plane. But I'm confident that with a little more
work,
he would be able to locate a point in any of the four quadrants.

Here's a first-grader learning something that is ordinarily
taught in 8th or 9th grade.

Example #3: The same kindergarten girl that I mentioned above
was working on counting with me one day. As we approached
the number 10, I had her take 10 popsicle sticks and bind
them with a rubber band. To help her grasp the idea
of place value, we divided my desk into two sections,
divided by a chopstick taped down with masking tape
(I love Asian food, what can I say!).

In the section to the right of the chopstick she placed single
popsicle
sticks; in the section to the left of the chopstick she placed bundles
of 10 popsicle sticks. With a little bit of work, she was able to
see that a number like 37 means three bundles and 7 singles,
that the number in the 10's place means number of bundles,
while the number in the 1's place means number of singles.

During this first session on the topic, this girl did problems like 23
+ 34
using the manipulatives. By the next session, she was doing problems
like 23 + 34 (no carrying) in her head. Four sessions later, she was
able to do
problems like 26 + 38 in her head. She would mutter to herself:
"O.K. 2 bundles and 3 bundles is 5 bundles. 6 and 8 is ....... 14.
You take 10 from the 14 and bundle them up. Now you have ... 6
bundles, and you have four singles left. So it's sixty-four."

Here is a kindergartner doing second grade math, but mentally,
and with command of place value.

You might be thinking ... these must be exceptional children.
Well, I've been tutoring/teaching for 11 years now, and I can
tell you that while these children are bright, they are
not the most mathematically intuitive youngsters
I've worked with. They are just curious and fun, like most kids.

You might also wonder if I'm advocating that we speed kids
along and try to make them all little math geniuses by third grade?
No. All I'm saying is that if we pay attention to clues kids give us
(like the girl asking what 3 - 8 would be, then giggling at the
answer),
we will find that children are often ready to explore math ideas that
are slated for later grades.

In this respect, hats off to the National Council
of Mathematics Teachers. This organization in 2000 put out
general standards stating that children should be taught in five
content areas
from kindergarten through 12th grade. These are:
- number & operation
- algebra
- geometry
- measurement
- data analysis and probability

If you'd like to read these standards and see how they
can help you do all kinds of interesting math with your
children, just go to:
http://standards.nctm.org/document/index.htm
to get the standards.

For more info about the NCTM in general, visit:
http://www.nctm.org

The point is: if we just keep our eyes and ears open to the curiosity
of children, we will see that they are often interested
in and ready to explore math concepts beyond what
the state education departments or local school districts
tell us they should be learning.

For me, the key has been listening to the questions of
children, and having faith that they can learn a lot more
than we might think.

++++++++++++++++++++++++++++++++++

PASS IT ALONG -

If you like what you read in the Algebra Times,
feel free to forward a copy to a friend - or
to your distribution list of fellow educators
or homeschoolers. Many homeschoolers
send it out along their email loop, and it's
fine with me if you do.

++++++++++++++++++++++++++++++++++


TRICK OR TREAT? TIME FOR A TRICK

The other day I was working on reducing fractions
with a tutee, and I saw something new and useful
about reducing fractions.

When we run into a fraction like 12/25, we know
that it can't be reduced further, and if you ask most
teachers (at least most I know), they'll say it's because
the two numbers, 12 and 25, have no common factors.

This is certainly true, and this is a perfectly valid way of
explaining why the fraction can't be reduced further.

But all of a sudden I realized another way of looking
at the situation. If you look at the gap (or difference)
between 12 and 25, you'll see that it's 13, a prime number.

If you just think about this for a moment, you'll see that
since this gap is a prime number, there are only two
numbers that will divide that gap evenly: 1 and 13.

1 has no power to reduce a fraction, and 13 does not
go into 25, so this leads to a shortcut:

If the difference between the numerator and denominator of a fraction
turns out to be a prime number, you need only check if that
prime number goes into the numerator or denominator.

If it goes into one, it will go into the other, and the fraction
can be reduced in one step. Examples of this would be:

14/21 (7 apart, and 7 goes into both),
reducing to 2/3

39/52 (13 apart, and 13 goes into both),
reducing to 3/4

If the gap number does not go into either, then the fraction
cannot be reduced. Examples of this would be:

38/45 (7 apart, but 7 goes into neither),
so it can't be reduced.

80/93 (13 apart, but 13 goes into neither),
so it can't be reduced.

And now you can get a little bit of practice with this idea:

For each of the fractions below, find the gap between
the fraction's numerator and denominator, and see if it
is a prime number. Then use the ideas above to see whether
or not the fraction can be reduced, and if it can,
what it reduces to. (Answers at end of newsletter.)

a) 32/39

b) 40/51

c) 25/38

d) 28/33

e) 52/65

f) 27/44

g) 88/95

h) 42/65

j) 22/25

k) 51/80


XYXYXYXYXYXYXYXYXYXYXYXYXYXXYX

ALGEBRA SURVIVAL GUIDE UPDATE

The Algebra Survival Guide is helping homeschoolers,
teachers, tutors and students. To get your copy of this
delightful book, just follow the links at my website:
http://algebrawizard.com/askinfo.html
or go to your nearest Barnes & Noble or Waldenbooks,
or any local bookstore, for that matter. The book is
available through INGRAM, the largest wholesaler
of books in the country.

XYXYXYXYXYXYXYXYXYXYXYXYXYXXYX

END OF PART I

Copyright 2001, by Josh Rappaport. All rights reserved.
May be redistributed if the entire newsletter, including signature,
is used.

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--
Josh Rappaport
Publisher, Singing Turtle Press
#770, 3530 Zafarano Drive #6
Santa Fe, New Mexico 87505

Voice: 1-505/438-3418
Toll-Free: 1-888/308-MATH
Fax: 1-505/438-7742
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Website: http://AlgebraWizard.com

{#} ----------------------------------------------------+[ Algebra
]+---
--- End forwarded message ---