Kiss those Math Headaches GOODBYE!

Fun Math Problem #2

Here is the second in my series of “Fun Math Problems.”

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit! I will post the answer to the problems two days later, after people have had time to respond.

To provide your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
Please show how you worked the problem. Thanks. I will post the names of the first three people who get this right.

The Problem:  Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange. The minute and the hour hand have exchanged places from the positions they had just before you went to lunch.

The question is:  how long were you away?

Image by The Hidaway (Simon) via Flickr

How to Multiply by 25 in Your Head

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

Here are the answers:
a) 300
b) 700
c) 1,200
d) 1,500
e) 2,100
f) 2,400

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

And here are the answers:

A) 525
B) 650
C) 875
D) 1,050
E) 1,575
F) 2,025

Turtle Talk – January 2010

Note: Below is a copy of my January 2010 ezine, Turtle Talk.

If you’d like to subscribe to Turtle Talk to get it as soon as it gets published, just go to this site:

http://singingturtle.com/pages/turtle_talk.html

There is a cute animation of moving turtles.

Turtle Talk
— a newsletter —
January 2010
Vol. XIII, Issue #1

QUOTE OF THE MONTH —

“If you think dogs can’t count, try putting three dog biscuits
in your pocket and then giving Fido only two of them.”
— Phil Pastoret

MathChat Blog Update:

In case you did not know, I have been spending a fair amount of time
writing on my blog, and there are many articles and ideas over there.
I have articles on many topics. Here’s just a small sampler:

Multiplication tricks
Mental math shortcuts
Dividing a fraction by a fraction using the “bologna-cheese” sandwich
Using color to elucidate ideas in geometry
Using colors to clarify concepts in algebra
How to solve algebraic mixture problems
Using “master equations” to make word problems less scary

To check out the blog, just visit:
http://www.mathchat.wordpress.com

Feel free to leave comments, too.

Twitter presence too:

I send out tweets about blogs and other issues of interest
to math teachers and students. If you’d like to FOLLOW ME,
just go to twitter, type in joshmathguy and sign up to follow.

January Problem of the Month

Samantha gets 12 out of 14 problems correct on a test.
Then she gets half of the remaining problems correct.
If every problem is worth the same number of points,
and Samantha ends up getting a score of 60% on the test,
how many problems are on the test altogether?

To get credit you must show your work as well as get the problem right.

First person to send in the right answer gets a FREE COPY of any
Singing Turtle Press book.

Send your answers to: josh@SingingTurtle.com

In the next newsletter I’ll name the first 9 others who get this
right.

DON’T FORGET: when you submit answers for the Problem
of the Month, please write
POTM
on the Subject line of your email’s header and, knowing that
it is optional, feel free to share:

— your age (if a child)
— your state or hometown
— status: student, teacher, tutor, etc.
— whether in school or homeschooled
— anything else about you of interest

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Discussion: A Way to Help Students Notice
& Understand the Mistakes they make in Algebra

Have you ever noticed that students often have a hard time spotting and understanding what is actually wrong — when they make mistakes in algebra?
One reason is that — when students enter the “Land of Xs and Ys,” they have little intuition as to what should happen. It’s as if they know that “they’re not in Kansas anymore,” so as far as they’re concerned, there could be “munchkins” around the corner. With such a level of uncertainty, they have little idea what to expect.
And if students don’t know what SHOULD happen, they also don’t know what SHOULD NOT happen. However, it’s helpful to remember that there is still one way to reach such early algebra students — appeal to their understanding
of how numbers work.
Here’s an example of how you can use this to your advantage:
Suppose a student makes the following mistake when solving an equation:
3x – 4 – 2x = 12
+ 2x + 2x
5x – 4 = 12
What’s the student is doing wrong? Adding 2x to the SAME SIDE of the equation TWICE, instead of adding it to BOTH SIDES of the equation ONCE.
Of course, if you asked the student why s/he did this, the student would probably offer up some half-true, yet misguided phrase like, “you have to add 2x on both sides.”
What you do here is get a “change of venue.” Move the problem from the field of algebra to the field of arithmetic.
Simply substitute numbers for letters. Ask the student if the following steps would make sense:
10 – 3 = 7  + 3 + 3
13 = 7
Generally, students will see right away that the answer is wrong. And that will alert them that something is wrong up above. And usually students can see that it is weird to add 3 twice on the left side.
Once they get that idea — either on their own, or with a little prodding — go back to the algebraic situation and ask them if they can NOW see what’s wrong. Very often they can.
When I was studying to become a teacher long ago, I learned a key idea about how people learn: to learn anything we need to relate new information to something we already know and understand. In essence, that’s what I am suggesting we do when students make procedural mistakes in algebra. Students know arithmetic (hopefully!). So use that. Put the algebra in an arithmetic context. Usually students will see the problem in the arithmetic setting. Then they’ll realize that algebra, even though it looks different than arithmetic, still works a lot like arithmetic. Getting that idea across will help students greatly as they continue in their study of algebra.
In any case, feel free to give this a shot, and see how it works for you. And feel free to share any feedback on how this works for you.
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Online Algebra now Available from Author
of the Algebra Survival Guide

Josh is now offering services online through SKYPE:
algebra tutoring sessions, and entire Algebra 1 classes.

If interested, please send email to: josh@SingingTurtle.com,
telling what kind of help you are looking for.

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Singing Turtle PRODUCTS

To see the line of Singing Turtle products for math education,
visit: http://www.singingturtle.com/pages/PARENTS_new.html

You can buy all of our books on Amazon.com
If you do, please consider writing a review.

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Well, that’s all for this month, though I will continue to blog and send out tweets on items of interest.

Have a great month, and I’ll be back in February.

— Josh