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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

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

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.

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

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!

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

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.

#####################################################

ATTENTION READERS:

We are always looking for more readers. If you like what the Algebra Times

has to offer, tell your friends and colleagues to subscribe by using our

easy newsletter sign-up form on the website: www.mathkits.com Or they

could subscribe by writing SUBSCRIBE as the subject when writing

Subscriptions are free!

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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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Having trouble with math? Check out my free

Algebra Times newsletter. Each issue offers a

lesson plan, teaching & learning tips and a fun

Problem of the Month. Also, if you're having

trouble w/algebra, learn about my Algebra

Survival Kit. All at: