In this issue you'll find:
-- Discussion of a recent Wall Street Journal article on
the importance of algebra
-- A lesson from one of our readers on shopping math
for the summer months (or anytime)
-- A trick for multiplying any number by 25
-- "Quick-and-Easy" Lesson Plan that introduces
younger children to algebraic thinking
-- A follow-up activity to this lesson using creative
writing!
-- Problem of the Month using a basket of goodies
-- and More!
If you know anyone who might enjoy the Algebra Times, feel free to
forward this newsletter to friends or colleagues.
And if you experience any technical problems in receiving this
newsletter, please let me know by sending an e-mail briefly describing
the problem to: josh@mathkits.com
Thanks for your interest in the Algebra Times. And feel free to
send me your feedback.
Sincerely,
Josh
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The Algebra Times
-- a newsletter --
Vol. 1, Issue 9
July 1998
QUOTE OF THE MONTH --
"Knowledge is the wing wherewith we fly to heaven."
-- Shakespeare (from "Henry VI")
ALGEBRA SURVIVAL KIT UPDATE!
The Kit is here, and we have begun shipping out copies!
If you would like to receive your copy for the special
"early bird" price, you may still do so as long as you place
your order by the end of this month -- July 31, 1998,
to be exact. All who order by this date will get the book,
which retails for $22.95, for the discount price of $18.95
plus shipping and handling.
To order your copy now, call toll free at 1/888-308-MATH
or visit the Singing Turtle website (http://mathkits.com)
and check out all your ordering options (phone, fax, email,
snail-mail).
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** WHY THE WALL STREET JOURNAL JUST MADE US FAMOUS **
Whether you love, hate -- or feel indifferent toward -- The Wall
Street Journal, it's hard to deny that it occupies a prominent place in
American journalism.
That makes it all the more remarkable that this paper, which specializes
in coverage of business news, just ran an in-depth article on nothing
less than algebra. And no, this article was not buried in a little-read
section of the paper; this was front-page news!
So why did The Wall Street Journal devote its daily feature on June 16,
1998, to algebra? If you read the article, you'll see it's because algebra
has make-or-break importance in determining whether or not secondary
students go on to higher education.
Here's an excerpt:
"Youngsters who take algebra tend to go to college, research shows,
and low-income youngsters who take it are almost as likely to go to
college as middle- and upper-income kids. The gap in test scores between
students in private school and those in public school largely disappears
if they take upper-level math courses, beginning with algebra."
In other words, algebra levels the playing field between the haves and the
have-nots. A child from a low-income family who takes and succeeds in
algebra stands virtually the same chance of going to college as a child
from an upper-income family who passes this course.
This has startling implications. First, it means that if we want our own
children to thrive in today and tomorrow's high-tech society, we must
ensure that they master algebra. There is no subject more important.
Another implication is that those committed to bettering the plight
of low-income children would do well to focus their energies on helping
these children master the higher maths, beginning with algebra.
The article also touches on why algebra, more than any other subject,
holds such great importance. Again, an excerpt:
"... algebra is what teachers call a gatekeeper course; you have to go
through it to reach the possibilities beyond. Algebra is the language
of math and science, 'the language of problem solving,' says University
of Chicago math professor Zalman Usiskin. It deals in abstractions --
using letters to generalize math operations -- that expand thinking skills.
In a technology-fueled society, says Mr. Usiskin, not knowing algebra
"limits what you can do."
What the reporter is getting at here is the well-known fact that algebra,
because it deals in abstractions, builds abstract thinking skills. And
abstract thinking skills are essential for any job or vocation that
requires analytical thought. In a society like ours that is information-
based, analytical thinking ability is the key that will unlock the door to
job and career opportunities. This, I believe, is why the business-oriented
Wall Street Journal zeroed in on this subject. In the newspaper's
editorial office, they probably looked at it this way:
Passing Algebra = Job Opportunities.
And Job Opportunites = Economic Growth.
So Passing Algebra = Economic Growth
(by the transitive property of equality).
But of course, there are other reasons students should focus in on algebra.
For one thing, learning algebra simply improves our ability to think
clearly; it provides an excellent mental exercise that strengthens the
mind. Also, since algebra requires creative problem-solving ability,
learning algebra bolsters our ability to solve all kinds of problems.
It gives us the mental ability to stick with tough problems from life and
work them through to solutions. Beyond that, learning algebra teaches us
about ourselves. Since algebra is a product of human thought, we are
learning about our own thought patterns when we study algebra.
By the way, the primary source for the information excerpted above in
this Wall Street Journal article comes from a 1990 study by the College
Board called "Changing the Odds; Factors Increasing Access to College." It
makes some interesting reading, and if you'd like to check it out, call the
College Board's publishing house at 800/323-7155 for ordering information.
(I bought it some time ago for about $10.)
In fairness to the research, I should note that the College Board
study stressed that for students to benefit from taking algebra, they should
certainly follow it with a year of geometry.
In the meanwhile, let's keep learning and teaching algebra. Now,
perhaps, we have a clearer sense of mission.
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PRACTICAL MATH / A LESSON FROM ONE OF OUR READERS ...
Last month I offered suggestions on how you can help your children
beat the summertime "math amnesia" problem.
I also invited readers to offer their suggestions on "practical math"
exercises they do with their children during the summer to keep those
math neurons firing.
One of our readers, Kris King, of Longmont, Colorado, kindly shared a
way of teaching shopping math that has evolved from her work with her
children.
For the record, Kris and her husband Rob have two children: Helen and
Emily, ages 9 and 11 and both homeschooled. I will simply let Kris
speak in her own words:
"We have been using a variety of your shopping math for a few years.
The kids take a clipboard, paper and pencil to the grocery store and add up
the bill as we shop. In the beginning they rounded to the nearest 10 cents.
As they got better at that, they started adding in the exact amount. For
produce they estimate the weight of things before we weigh them. In
the beginning I would add items to make the produce come out to an even
number of pounds -- to simplify the multiplication. But as they got better
at that, I left the weights as they were, to give them practice handling
multiplication of fractions.
"We talked about all sorts of strategies for adding prices that end in
.99, .89, etc. When we get to the check-out stand they take the
cashiers total, subtract the tax, add back in the discount for bags.
If they are within 1% (they have to figure out what 1% of the total
is), they "win", and get a token to put whatever they want on the
grocery list next time. Then they have to figure out what percent
they were off from the total. Sometimes they are way off, and we
suspected it wasn't in their calculations, but a difference in the
printed price of items and the scanned price.
"So now my kids keep track of the printed price of each item
in addition to the running total. At the end, if they are more than 1%
off, if they can show that taking into account the differences in the
printed and scanned price they are with 1% they still get a token. So the
game, originally to get them to add two-digit numbers, has expanded far
beyond that.
"Our shopping math has evolved over the two years we have been doing it.
We started out just adding up the items rounded to the nearest 10 cents.
The additions and changes in the game have come about mainly from the
kids. As their skills grew, we added different challenges: adding the
actual price, getting 1/2 and 1/4 pounds of produce, estimating the
weight of produce, getting withn 1% of the total, finding the percent their
total was from the actual total, finding discrepancies in the printed price
and scanned price."
Kris' exercise shows how -- by being both creative and attuned to the
development of your children's skill level -- you can take a single
exercise and develop it over time so that it becomes more and more
sophisticated.
Thank you, Kris!
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"TRICK OR TREAT"
Last month I offered a "treat" in the form of a math riddle,
so this month I offer a "trick."
This is a simple trick that anyone can easily learn. It is just a trick
for multiplying a number by 25.
If someone asked you what 25 times 36 equals, you'd probably be tempted
to reach for a calculator and start punching buttons. But remarkably,
you'd probably be able to work it out even faster in your head.
Since 25 is one-fourth of 100, multiplying by 25 is the same thing as
multiplying by 100 and dividing by 4. Or, even more simply:
first divide by 4,
then add two zeros.
Here's the example:
Problem: 36 x 25
First divide 36 by 4 to get 9.
Then add two zeros to get: 900.
That, amazingly enough, is the answer.
Another example: 88 x 25
First divide 88 by 4 to get 22.
Then add two zeros to get: 2,200.
Now try these problems in your head:
a) 25 x 12
b) 25 x 28
c) 25 x 48
d) 25 x 60
e) 25 x 84
f) 25 x 96
(Answers at end of newsletter.)
But, you say, what if the number you start with is not divisible by 4.
No problem. Just use this fact:
if the remainder is 1, that is the same as 1/4 or .25
if the remainder is 2, that is the same as 2/4 or .50
if the remainder is 3, that is the same as 3/4 or .75
So take a problem like this: 25 x 17
dividing 17 by 4, you get 4 remainder 1.
But that is the same as 4.25
Now just move the decimal right two places (same as multiplying by 100)
Answer is: 425
Another example: 25 x 18
dividing 18 by 4, you get 4 remainder 2.
But that is the same as 4.50
Now move the decimal right two places.
Answer: 450
Another example: 25 x 19
dividing 19 by 4, you get 4 remainder 3.
But that is the same as 4.75
Now move the decimal two places to the right.
Answer is: 475
Now try these in your head:
A) 25 x 21
B) 25 x 26
C) 25 x 35
D) 25 x 42
E) 25 x 63
F) 25 x 81
All answers at bottom of the newsletter!
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ALGEBRA HUMOR -- REVISITED
Way back in the April newsletter, I invited people to invent an answer
to this math joke:
Q: Why did the exponent cross the road?
Finally, Jeff LeMieux, a teacher in Washington state, did it. At
least, he created and came up with the answer to a similar riddle:
Q: Why did the radical cross the road?
A: To get to the root of the problem.
Thanks, Jeff.
Jeff, a veteran of the classroom for 25 years, teaches 7th and 8th
grade math at the Oak Harbor Middle School in Oak Harbor, Washington.
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QUICK & EASY LESSON PLAN #6
-- ALGEBRA FOR ELEMENTARY SCHOOL-AGE KIDS --
A number of parents have written to me recently asking what they
can do to introduce their elementary-age students to algebra.
Since this newsletter discusses the importance of algebra (see
discussion above on Wall Street Journal article), what better time, I
figured, to introduce younger children to algebra.
Here is a lesson plan that teaches elementary-age students a fun-
damental algebraic concept.
PURPOSES:
1) To introduce the idea that variables stand for numbers whose
values we do not immediately know.
2) To teach students how to figure out the value of variables for the
simple operations of addition, subtraction, multiplication and division.
3) To lay the foundation for the abstract reasoning skills necessary for
doing algebra.
MATERIALS NEEDED:
-- a stack of plain white 3 x 5 index cards,
-- a pair of scissors,
-- brightly colored magic markers
-- small post-it notes.
SET-UP:
1st) Cut the cards in half so each card is shaped 3 x 2 1/2 inches.
2nd) Using one color, take the cards, one-by-one, and write numbers
on them as follows:
-- on one card, write a large 1
-- on another write a large 2
-- keep going until you get to say, 20
3rd) Now take six more cards. Using a different colored marker,
-- on one make a big addition sign
-- on one make a big subtraction sign
-- on one make a big multiplication sign
-- on one make a big division sign
-- on one make a big equal sign
-- on one make the letter "y"
PROCEDURE:
1st) Clear off a large smooth, flat area (coffee tables or school desks
work well), and first just let your children/students play around with
the cards, making as many true statements as they can. For example, your
child/student might arrange the cards to make statements like:
4 + 3 = 7
12 - 8 = 4
6 x 3 = 18
12 (division symbol) 4 = 3
2nd) After they have become comfortable doing this, take a post-it
note and do something like this:
-- write a "4" on the post-it note and, without showing this to your
children/students, stick it on the back of the "y" card.
-- then create a statement with the "y" side showing, like this:
y + 5 = 9
3rd) Ask you children/students what "y" must be to make the statement
true.
[Helpful hint: If they have trouble with this idea, tell them that in
place of "y," they can use the word "what" so that the statement could
be read as a question: "What plus 5 equals 9?"]
4th) Once your child/student gives an answer for "y," have him/her look
at the back of the card and see if the answer is right. The child will
see the post-it note with "4" written on it.
5th) Explain to your child that in algebra, we let letters from the
alphabet stand for numbers. Sometimes we don't know the value of these
numbers, and in algebra we use our minds to figure them out. Tell them
that when they say their answer, they should say it in the form of:
" y equals 6" or "y equals 10"
6th) Here are some more sample problems you might make up for your
children/students:
7 + y = 16
4 - y = 2
y - 3 = 3
y x 2 = 6
5 x y = 15
y (division sign) 3 = 3
16 (division sign) y = 8
7th) Once your children/students have become comfortable looking at
equations with variables and figuring out the value of the variable,
youecan take the exercise a step further by having them make up problems
of their own. Children/students can take turns making up problems and
giving them either to you or to one another.
CREATIVE WRITING EXTENSION ON THIS LESSON!!
People seldom consider using creative writing to teach math, but you
certainly can use it. This is especially helpful for students who like
to write. Here's a creative writing extension on this lesson on
variables.
Essentially, this lesson teaches students that in algebra we use
letters to stand for unknown quantities and then try to figure out the value
of those quantities. In a creative writing extension, you have children
write about a real situation -- or make one up -- in which they faced and
solved some mystery.
Here are the guidelines:
a) Think of a time (or else make up a story) when you solved some
mystery.
Perhaps you were trying to figure out where you left something, or who
started a certain rumor, or even why the moon changes in appearance
from day to day.
b) Write this up as a short story.
c) At the end,
1) tell what the unknown thing was,
2) assign a letter to represent that unknown,
3) tell what the letter equals.
Here is an complete model showing how you do this:
One day I couldn't find my new kitten. I had seen it just last night.
In fact, it had slept with me in my bed. But when I woke up it was gone.
I looked all over the house for it, but I couldn't find it anywhere. I
thought maybe it got outside. So I went to my dresser to put some
socks on so I could go outside. And as soon as I opened up my sock drawer,
there it was, and it jumped right out and started meowing.
The mystery was the location of my kitten.
Let the letter l = location of kitten
Solution: l = inside my sock drawer
Students: if you write a story like this and send it to me, I will
publish those that follow the guidelines in the next issue. Send all
submissions
to: josh@mathkits.com
Make sure you follow the guidelines carefully. Also, please tell me:
-- your full name
-- your age
-- where you live (optional but nice)
Looking forward to reading your stories!
Next month I will offer another lesson plan that builds on this one.
Stay tuned.
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SEEN THE WEB SITE?
If you received this newsletter as a pass-along, you are warmly
invited to visit the web site from which it springs: http://mathkits.com
Check out the FAQ's about algebra, details about my tutoring and
consulting services, and information on the Algebra Survival Kit, a new and
exciting educational tool that will bring algebra to life for your children
or students!
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PROBLEM OF THE MONTH
Remember: a free copy of my Algebra Survival Kit to whoever solves it
and sends me the answer first.
JULY'S PROBLEM OF THE MONTH:
-- Five pounds of walnuts cost as much as 20 pounds of apples and 6
pounds of bananas.
-- Twelve pounds of bananas cost as much as one pound of walnuts and
5 pounds of apples.
-- How many pounds of apples could you get for the same price as 3
pounds of walnuts?
SOLUTION to JUNE'S PROBLEM OF THE MONTH --
Four people solved June's Problem of the Month.
The "early bird" who captured a free copy of the Algebra
Survival Kit is Paul Atherton, a teacher in Florida.
HERE AGAIN IS THE PROBLEM:
Think of a square. Now try to imagine a circle whose area is equal
to that of the square. Your goal is to figure out the relationship
between the diagonal of the square and the diameter of the circle.
More specifically ...
If a square has diagonal d, what is the diameter of the circle whose area
equals the area of the square? Express the length of the diameter
in terms of d.
Here is the answer:
Diameter of circle = diagonal of square times the square root of (2 x
pi)/pi
To get the answer, you must set the two equal areas equal to each other.
First step: get the area of the square in terms of its diagonal, d.
Here's how:
Since d = the diagonal of the square,
the side of the square must be diagonal divided by the square root of 2
(by the Pythagorean theorem)
But, since the area of the square is the side times itself,
the area of the square must be diagonal squared divided by 2, or d
squared/2
Now you turn to the circle.
The standard formula for the area of a circle is Pi times radius squared.
But in this problem, we want to work not with the circle's radius but
with its diameter. Since diameter is twice the radius, we can
re-write the formula for the area of a circle, so instead of
Pi times radius squared, we can call it:
Pi times (diameter/2) squared.
But Pi times (diameter/2) squared equals Pi times diameter squared/4
So putting the two areas together, we get:
Area of square equals area of circle
diagonal squared divided by 2 = Pi times diameter squared/4
If you solve this equation for diameter, you get:
Diameter of circle = diagonal of square times the square root of 2/Pi.
And using the rules for rationalizing denominators, this turns into:
Diameter of circle = diagonal of square times the square root of (2 x
pi)/pi
Paul Atherton, June's Problem of the Month winner, has been teaching
Applied Math and Algebra I in the Orange County Public Schools. This
past year he taught at Cypress Creek High School in Orlando, Florida.
He has been teaching for five years.
In addition to enjoying math, Paul likes to read Science Fiction and
Fantasy, he enjoys computer programming and tropical fish aquariums.
Paul's motto: Don't say you can't if you haven't tried and tried
again, and again and again......"
And here are the others who got this problem right:
-- Kathleen Reynolds, a mother of two children in Middletown, Rhode Island.
-- Caitlin Letts, a 13-year-old student in Beaverton, Oregon.
-- One person, who chose not to be named, also solved this problem.
Note on May's Problem --
We received one more correct solution to the problem.
It came in from Shanda Williams, age 9, who lives in Nanaimo, British
Columbia, Canada.
Congratulations Shanda!
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IF YOU HAVE A QUESTION ABOUT ALGEBRA OR EDUCATION
to which you would like my reply, send it to: josh@mathkits.com
If it is of general interest, I will print the answer in the next
newsletter.
If it's of a more particular nature, I'll try to answer it by e-mail.
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Answer to the "Trick" Problems
a) 300
b) 700
c) 1,200
d) 1,500
e) 2,100
f) 2,400
A) 525
B) 650
C) 875
D) 1,050
E) 1,575
F) 2,025
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Copyright 1998, by Josh Rappaport. All rights reserved.
May be redistributed if the entire newsletter, including signature,
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