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ALGEBRA> Algebra Times -- June 1998

NEW NEWSLETTER ADDITION:
From: josh rappaport [mailto:]
Sent: Tuesday, June 23, 1998 2:52 PM
Subject: Algebra Times -- June 1998

Hi,

Hope you're all enjoying the onset of summer!

A thousand apologies that this issue is arriving later than
usual. After getting listed in the American Homeschool Association
electronic newsletter, I received so many new requests that I had
to set up an automated system. Note that there are revised directions
for subscribing and unsubscribing at the bottom of the newsletter.

In this issue you'll find:

-- some tips on how to keep the math spirit alive during
the hot summer months
-- a lesson that uses fun activities to teach the concept
of proportion
-- a math riddle involving a cowboy and his horses
-- a Problem of the Month for people who enjoy playing
around with circles and squares.
-- and More!

If you like what you read in the Algebra Times, feel free to
forward this newsletter to any friends or colleagues who may
like to read it..

And if you experienced any technical problems in receiving this
newsletter, please let me know by sending an e-mail briefly describing
the problem to: Finally, if you received this
issue twice, it's just a fluke due to this being the first issue that
is going out through the automated service.

Thanks for your interest in the Algebra Times. And feel free to
give me your feedback.

Sincerely,
Josh

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The Algebra Times

-- a newsletter --

Vol. 1, Issue 8

June 1998

QUOTE OF THE MONTH --

"Remember that what is hard to endure will be sweet to recall."

-- Tote Yamada

ALGEBRA SURVIVAL KIT UPDATE!

The Kit has been printed and is now being assembled. It will be
available in July. To reserve your copy now, send email to my
publisher, Kathy Gayle at: In the subject line
of your email, write: "Reserve A.S.K."

More info on the Kit -- and now photos of it too! -- are found at:
www.mathkits.com

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BEATING THE SUMMER MATH-AMNESIA PROBLEM

For those who worry their children may forget much of what they learned
during the school year, here are some tips on how you can keep their math
skills sharp during the summer.

My suggestion is that you do PRACTICAL MATH . Here are some examples:

CAR TRIP MATH ...

a) Get out the map and show your kids how to find the distance from town
to town. When you're traveling, say, from Dayton to Detroit, have them add
up all the little legs of the trip and tell you how far the trip is.

b) Once your child has figured out the mileage, let them use the formula:
(rate) x (time) = (distance) to figure out how long it should take you to
reach your destination.

c) Miles per gallon. Fill up the car when you leave, and have your child
note the mileage on the odometer. Drive for a while, and then fill it up
again, and note the new odometer reading. By calculating the total amount
of gas used and the total amount of miles driven, and by dividing the miles
by the amount of gas, kids can figure out how many miles per gallon your
car really gets. Let them do this for city/town driving, and then let them
do it for highway driving. Let them see whether the car's mileage is what
the dealer says it should be. If not, maybe it's time for a tune-up. Let
them do this for more than one car. Doing this trains your child to be a
good consumer as well as a good mathematician.

d) If you eat at any restaurants during your road trip, kids can help you
figure out the "tip." If you do a 15% tip, you can even teach them this
nice trick for calculating 15%.

1st) Round off the amount of the bill. [i.e.: $23.87 rounds to $24]

2nd) Take half that amount. [$24 divided by 2 equals $12]

3rd) Multiply that by 3 [$12 x 3 = $36]

4th) Move decimal point one place to the left. That's the "tip."
[$36 becomes $3.60, and that's the "tip."]

KITCHEN / SHOPPING MATH ...

a) Find a recipe that your child will like to help out with (dare I
suggest chocolate chip cookies). If it's a good recipe, double it. But
assign your child the task of doubling all the fractions correctly.

b) When shopping, take along a calculator and ask your child to figure
out which of two purchases is the better deal. (See proportion lesson below
for details.)

c) Shopping also gives your child the opportunity to practice the skill of
rounding off. Take a pad with you, and have your child round off each
purchase to the nearest dollar. Before you check out, have him or her tally
up the rounded-off values of all the purchases. When the cashier gives you
your exact total, ask your child if this is close to his/her estimate. In
addition to reinforcing the rounding off skill, you are teaching your child
to be a careful shopper. (Recent studies have shown that even those
electric scanners often make mistakes!)

SPORTS MATH ...

Some new math twists on some old games

a) Prime Basketball: Play one-on-one basketball with your child. But
instead of counting buckets the normal way -- 1,2,3, etc. -- count by
prime numbers: 2,3,5,7,11,13,17,19,23, 29 First one to reach 29 wins.
And anyone who accidentally says a composite (non-prime) loses a bucket
and goes down one prime. Knowing the first 10 primes is not just an idle
exercise. As I will show in future newsletters, knowing the primes can save
considerable amounts of time when reducing fractions.

b) "Power" Toss. Throw a ball back and forth with your child. But each
time you toss the ball, you have to call out a power (exponential value) of
a number. For example, let's say you start with the powers of 2. The first
person to throw the ball says: 2 (for 2^1). The second person who tosses
the ball says 4 (for 2^2) [By the way, ^ means that the number following
it is an exponent: that is, 3^2 means 3 to the 2nd power, or 9] . The next
person to toss calls out: 8 (for 2^3). The next person says: 16 (for 2^4).
See how high you can go. The person who gives the highest power wins.
Then try this with different numbers as bases: 3, 4, 5, etc.

c) Times-Table Catch or Times-Table Soccer -- Either toss a ball back and
forth -- or kick a ball back and forth -- with your child. But when you
throw or kick the ball, call out some times-table fact. For example, while
throwing or kicking, shout: "Seven times eight." Your child must throw or
kick the ball back with the right answer.

d) Algebra Catch or Algebra Soccer -- With this variation, you call out
algebra formula facts, and your child must complete the formula. For
example, throw the ball and call out "a to the x times a to the y." Your
child must throw it back saying, "a to the x plus y." If you're re-learning
algebra yourself, you might want to switch roles and let your child quiz
you too. They LOVE that!

HOME IMPROVEMENT MATH ...

Summer is time for home improvements, so why not let your children help
out by doing a few calculations.

a) PYTHAGOREAN THEOREM -- If you are building something square or
rectangular ( a garden border, a sandbox, etc) first tell your child the
lengths of the sides, and have her/him use the Pythagorean theorem to
calculate the length of the diagonal. Then have her/him actually measure
it out with a tape measure to make sure that you have a proper right angle.

b) PAINTING MATH -- Are you painting or repainting a room? If so, your
child can calculate how manuy gallons of paint you need. Most paint
containers tell you how many square feet they will cover. Have your child
first calculate the square foot area of the walls and/or ceiling you need
painted. Then multiply that by the number of coats you need to apply.
Finally take that figure and divide it by the square foot area on the
gallon of paint. The answer will be the number of gallons you need
to buy.

c) ENGLISH-METRIC EQUIVALENTS Doing any sorts of measurements
in the English system (inches, feet, yards)? Why not have your child
convert those to the metric system using metric equivalents like 2.54
centimeters equals 1 inch.

WHAT ARE YOUR IDEAS FOR PRACTICAL MATH?

Send what works best for you, and I'll print some of the more interesting
suggestions for everyone to see in the July issue.

Include your full name and hometown. And tell whether you are a
teacher, tutor, homeschooling parent, homeschooling student or
school student.

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"TRICK OR TREAT"

Last month I offered a calculation "trick," so this month I offer a "treat."

Here's a fun little riddle to ask someone:

Before he dies, an old rancher wills away all
his horses to his three sons according to this formula:
-- the oldest son shall receive half of the horses.
-- the second son shall receive one third of the horses.
-- the youngest son shall receive one-ninth of the horses.

But when the rancher dies, he has exactly 17 horses.

So his sons begin quarreling because 17 cannot
be divided evenly by 2, 3 or 9.

Then you come galloping along and see the three sons
arguing. You want to settle the argument. How can you
possibly settle the argument fairly?

Answer at bottom of newsletter.

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QUICK & EASY LESSON PLAN #5

(Note: since many of my readers are teachers -- either school teachers or
homeschooling parents, I've decided to make the second item in the
newsletter a lesson plan that you can actually take and use. This month I
offer the third "Quick-and-Easy" Lesson Plan. Send me your feedback on
how well a lesson plan works for you! Also, even if you are a student, you
should be able to follow the lesson plan and learn from it.)

-- PLAYING WITH PROPORTIONS --

Last month I described a lesson for teaching ratios through manipulatives
-- using different colored beans.

Now I offer thoughts on how you can extend this lesson to teach the concept
of proportion, and then captivate students with some real-world activities
involving ratio and proportion.

When we left off, we had helped students gain a solid grasp of the concept
of ratio by working with red and white beans.

Next it's time to introduce the concept of proportion.

Point back to the blackboard, specifically to the equation: red/white = 3/5
Point out that while "3 to 5" is a ratio, the entire equation is a
proportion. For a proportion is a statement that two ratios are equal.

Then ask students to help you make another proportion.

"In our first way of groupng the beans," you can remind the students, "we
had 3 red beans and 5 white beans. How many of each did we have in our
second group?" (Someone notes that we had 6 reds and 10 whites.)

Then write up: 3/5 = 6/10

"This," you say, "is also a proportion because it shows that two ratios are
equal." Also ask the class how they know this is true, using what they know
about fractions. (Someone usually points out that 6/10 reduces to 3/5.)

"Now," you continue, "what if you have a proportion, but one of the numbers
is missing. Would you be able to find the missing number? Let's give it a
shot."

Ask students if they can work with their beans to figure out what the
missing number would be in the proportion: 3/5 = x/25

Usually some group figures out that x = 15. Ask this group to show the rest
of the class how they figured it out using the beans.

Then ask students if anyone can figure out how you can do this same
operation without manipulatives, using the basic rules for solving
equations. Again put up the problem:

3/5 = x/25

Often someone recalls that you can multiply both sides by the denominator,
to get:

25 (3/5) = (x/25) 25

x = 15

Then ask students if they can use the rules for solving equations to solve
a proportion in which the variable is in the numerator. For example:

3/5 = 15/x

Note: I myself teach this by showing that with a proportion, you can flip
numerators and denominators on both sides, to get:

5/3 = x/15

And from here you just multiply both sides by the denominator under the
variable (here 15). Some educators prefer to use cross-multiplying. Use
your preferred approach.

Follow this up by putting up some simple proportion problems of both types
and letting students solve them using algebra. Here are a few simple
proportions you might want to start with:

a) 2/3 = x/9

b) 3/5 = 12/x

c) 1/4 = x/20

d) 5/6 = 15/x

Then give students some tougher ones. Here are some examples:

a) 7/11 = x/55

b) 9/13 = 36/x

c) 12/19 = x/57

d) 17/23 = 85/x

After students show they can solve proportions like those above, I ask them
to make up their own proportion problems and exchange them with other
groups. Students must do each others' problems. Then they check each
others' work.

FOLLOW-UP ACTIVITIES --

After all of this has been mastered, students understand the basic concept
of ratio and proportion. Now comes the "dessert." For now it's time to
let students have some fun applying these concepts with follow-up
activities. The first two activities strengthen the concept of ratio; the
third activity helps students compare two ratios.

1) PEER POLL: Have students poll their classmates or friends about
questions of interest to teens, but questions that can have only one of two
answers. For example, questions like:
Which do you like more: rap music or alternative music?
Which sport do you like more: baseball or basketball?
Yes or no: should the voting age be dropped to 16?
Yes or no: should girls be able to ask guys out on dates?

Or, for a math/social studies interdisciplinary project, do a yes or no
question, and do it about any interesting controversy.
The scope can be local, state, national or international.

First have students guess what the ratios will be for the answers. Then
have them actually go out and poll their peers and keep track of the
results.

When they bring their results back, teach students how to find round number
approximations for their findings. For example, if after interviewing 41
students, they find that 10 preferred rap while 31 preferred rock, the
ratio of 10/31 can be rounded off to a 1/3 ratio.

You might also ask students if they think that their ratio findings would
hold up for teenagers all over the nation. As a crititcal thinking
exercise, ask them to offer reasons why or why not.

If you're at a school with a school paper, see if the editors will publish
the results in an upcoming issue.

2) LOOK, COUNT & LEARN: Same idea as the PEER POLL, but instead
of asking questions of people, students merely go somewhere and make
observations of interest to them. For example, a student might want to know
the ratio of domestic to foreign cars in his hometown. S/he could go to a
few different shopping store parking lot (preferably in different areas of
town) and simply record the numbers of each kind of car. Then s/he would
round the figures off to a simple ratio. Note that here too students must
take some topic that resolves into a pure dichotomy: i.e.: the observed
thing must be either this or that. You can't have more than two options.

Other ideas for ratios to check out:
-- ratio of boys to girls in video arcades.
-- ratio of cats to dogs in a child's neighborhood, or at a park.
-- ratio of local to out-of-state license plates on cars parked on a
major street.
-- ratio of locally owned restaurants to chain restaurants in your town.
-- ratio of locally owned vs. nationally owned stores along five blocks
of your downtown.

3) SHOPPING CART: Have students go to a grocery store and check out the
prices of items that can be bought in smaller and larger containers. Then
let them figure out which is the better deal. For example, a student might
find an 8-oz can of mayonnaise for $1.59 and a 12-oz can for $2.19. Ask
them to use ratios and proportions to determine which is the better deal.
Here's how:

Divide the cost of the item by the quantity of the item to find the "price
per unit."

For example, if an 8-oz can of mayonnaise costs $1.59, you divide $1.59
by 8 to find out that each ounce of mayonnaise costs roughly 20 cents.
Similarly, if a 12-oz can of the same mayonnaise costs $2.19, you divide
$2.19 by 12 to find out that each ounce in this container costs roughly 18
cents. Since the mayonnaise in the larger container is cheaper, ounce for
ounce, it offers the better deal.

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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: www.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.

May's Problem --

Hooray!

A total of ten students put on their "thinking caps" and got May's Problem
of the Month right. Of course, only one of them could send it to me first,
and this month that speedy student was Courtney Whitworth, of Dallas,
Texas. Courtney is a 16-year-old homeschooler who lists her hobbies as:
music, computers, school, astrology, figure skating for fun and magic the
gathering (the card game). Congratulations, Courtney. We'll send that
Algebra Survival Kit to you as soon as we get it back from the printer.

In order of my receiving the answer, here are the other students who got
this problem right:
Shannon O'Connor, a 12-year-old homeschooling student in London, Ohio.
Nika Strzelecka, a 14-year-old student whose favorite subjects are math and
science.
Chris Fortson, a 14-year-old student in Tampa, Florida.
G.W. Wilson and Katy Wilson, 9th and 6th grade homeschoolers
Christopher Kagehiro, 12, from Dale City, Virginia.
Elcie Potts, 14, of Boulder, Colorado.
Caitlin Letts, 13, of Beaverton, Oregon.
Karen Nishikawa, homeschooler, age 13, from Grand Junction, Colorado.

Congratulation to all!

Here again was May's problem:
The ratio of sharks to marlins is 2 to 3.
The ratio of marlins to tuna is 6 to 11.
And the ratio of tuna to guppies is 33 to 48.
What is the ratio of sharks to guppies?

Using ratios, Courtney and the others discoverd that the ratio of
sharks to guppies is 1 to 4.

And now, here's JUNE'S PROBLEM OF THE MONTH --

Last month, I limited the responses to the problem of the month to
students, as I hadn't heard much from students. This month the problem
is open to all.

First 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.
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QUESTIONS FROM READERS --

I received a rather interesting question this month from Phil Shapiro in
Washington, D.C. Phil works with the Community Technology Centers'
Network (CTCNet) as an "out of school" educator, helping to set up
"community technology centers" in the Washington DC metro area. The
national nonprofit organization he works for is funded primarily by the
National Science Foundation. Its website, if you care to check it out, is

Phil writes: "One of my favorite teachers in high school was a math teacher
with a great sense of humor. He almost always had a glimmer in his eye,
and had a great zest for life. In what ways can humor help enliven
presentations in the classroom?

Here are some thoughts I had in reply:

Certainly humor can always help make good teaching better.

And I'd like to draw a distinction between two forms of humor that
you can use in teaching, for I think it's helpful to note the difference.

One kind of humor is the humor of puns and little jokes. Jokes like I
told a few months ago when I gave the old algebra question:
"If 2a + 3a = 5a, what would be 8q + 2 q?" Answer: "10q"
Response: "You're welcome!"
This type of humor can help relax students and frankly, make them feel
a little more comfortable about what may be a very uncomfortable subject
-- math.

But there's another type of humor that you can use in a math class. And
this is a type of humor that is more content-oriented, less "ha-ha" and
truly even more helpful in helping students understand math. It's the
type of humor that makes mathematical concepts come alive.

Here's an example. In working with negative exponents, students must
learn that when an exponential term moves from the numerator to the
denominator -- or vice versa -- the sign of its exponent changes. If its
sign was positive, it becomes negative. If it's sign was negative, it
becomes positive. In my book, the Algebra Survival Kit, I had my
wonderful illustrator, Sally Blakemore, make a cartoon for this concept.
She drew a little muscle man who was trying to push an exponential term
from the numerator to the denominator. In the first frame, the muscle man
is gathering his strength. By the second frame, the muscle man has pushed
the term halfway across, and the term is saying: "What's happening here?"
By the third frame, the term has been pushed up to the numerator, and the
term exclaims: "My sign, it changed!"

While this is not meant to be "ha-ha" funny, it does have a bit of humor,
and even more importantly, it helps students get the idea. I've found,
through my teaching and tutoring, that when we the educators can personify
some aspect of math -- make it come alive -- students are much more likely
to remember it. It's the same idea that good teachers use in social studies
classes. If a social studies educator can have students actually act out the
process of how a bill becomes a law, for example, students are likely to
remember that process much better than if they simply memorize it.

So in response to this question about humor in the math classroom, my basic
response is: go out there and look for analogies and metaphors that make
math come alive. Your students will appreciate that you made the effort.

IF YOU HAVE A QUESTION ABOUT ALGEBRA OR EDUCATION
that you would like answered, send it to:

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 "Treat" Riddle

Here's what you do.

Since you ride up on a horse, you get off of your mount and tell the
sons that you are adding the horse to the other 17 so that there is
a total of 18 horses.

Now the first son receives 9 horse as his half of the total.
The second son receives 6 horses as his third of the total.
And the youngest son receives 2 horses as his ninth of the total.

Together the three sons take 17 horses (9 + 6 + 2), so there is one
left over. You mount that horse and ride away!
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