Near(ly) Perfect Squares

Perfect squares are great landmarks for Mental Math. Math has changed since 19XX as calculators have changed the way that Math is taught and learned and tested. Increasing Mental Math ability increases confidence, empowerment and self-reliance.

Nearly Perfect Squares
Perfect squares (like 4^2 = 16 and 5^2 = 25) can help people find close facts like 3 x5 and 4 x 6 (answers are both one less than the perfect squares) .
This will work for:
9 x 11 (99) = 1 less than 10^2 (100)
19 x 21 (399) = 1 less than 20^2 (400)
29 x 31 (899) = 1 less than 30^2 (900)
and for any number!!
(x-1)(x+1) = x^2 - 1 always one less than x^2

Somewhat Nearly Perfect Squares
Perfect squares (like 4^2 = 16 and 5^2 = 25) can help people find somewhat close facts like 2 x 6 and 3 x 7 (answers are both four less than the perfect squares) .
This will work for:
8 x 12 (96) = 4 less than 10^2 (100)
18 x 22 (396) = 4 less than 20^2 (400)
28 x 32 (899) = 4 less than 30^2 (900)
and for any number!!
(x-2)(x+3) = x^2 - 4 always four less than x^2

February 2010 Brain Teaser answer and solution

Solve for a, b, c, d, and e.

a (b + c + d + e) = 128
b (a + c + d + e) = 155
c (a + b + d + e) = 203
d (a + b + c + e) = 243
e (a + b + c + d) = 275

Check out the prime factors of each of the numbers.
155 = 5 x 31...therefore b = 5
203 = 7 x 29...therefore c = 7
243 = 3 x 81...therefore d = 3
275 = 11 x 25...therefore e = 11

a (b + c + d + e) = 128
a(5 + 7 + 3 + 11) = hey this doesn't work

ok let's try 243 = 9 x 27 so d = 9
a(5 + 7 + 9 + 11) = a ( 32) therefore a = 4
4(5 + 7 + 9 + 11) = 128

Check:
5(4 + 7 + 9 + 11) = 5(31) = 155
7(4 + 5 + 9 + 11) = 7(29) = 203

9(4 + 5 + 7 + 11) = 9(27) = 243

11(4 + 5 + 7 + 9) = 11(25) = 275

so: a = 4,b = 5, c = 7, d = 3, e = 11

GED as a Possible Standard

http://voices.washingtonpost.com/class-struggle/2010/02/crawling_toward_national_tests.html

How about using the GED as a high school standard?According to the ACE that administers the GED, "Only 60% of graduating high school seniors would pass the GED Tests on their first attempt". http://www.acenet.edu/Content/NavigationMenu/ged/pubs/GED_Testing_Program_Fact_Sheet_v1_2010(3).pdf

Since institutions already accept the GED as an established standard, it would be interesting to consider the GED as an exit exam rather than defining new ones.
For lower grades, SAT or ACT (or placement tests like Accuplacer or COMPASS) could be used as goals (whether or not these exams cover what is needed, they are already an accepted standard).

Math Helps People Think Better

Seth Godin writes:
Can you factor this?
If you're like most people, you get a little queasy at the thought. And when you were in tenth grade, you surely wondered why they were bothering you.
(the answer is (x-2) times (x-2), in case you were curious.)
It turns out that the real reason you needed to do this work was to be able to play with numbers in your head. Abstract numerical thought is an important skill among educated people.

My response:
Great analogy...except no queasiness ;)

Why Take Math comment on about.com

http://math.about.com/b/2010/02/08/why-take-math.htm#gB3

Instilling Core Values with the Life Skill and Discipline of Math

Info for Parents on Math Ed

http://www.wheresthemath.com/Pages/informationforparents.aspx

This advocacy group based in Seattle has been a strong proponent of parental involvement and curriculum choice.

Checking Up on Student Understanding

http://www.denverpost.com/education/ci_14350149

In this the article, the reference is to POP (Proof of Purchase) -- some call them exit slips. Sometimes I call them checkpoints. A short quiz at the end of class or at the end of a lesson (within a class) can help both students and teachers assess what they know and what they still need to learn.

The GED is a Common Standard

http://www.acenet.edu/Content/NavigationMenu/ged/test/math.htm

A Common Core accepted standard is the GED.

Kids Need Work to Do

http://voices.washingtonpost.com/answer-sheet/homework/give-kids-homework-while-schoo.html?referrer=emaillinkpg
With beforeschooling and afterschooling, just add snowschooling ;)

Letters on Playing to Learn

http://www.nytimes.com/2010/02/08/opinion/l08teach.html?ref=opinion

Two of the letter writers would like to see elementary Math education to go beyond the four operations.
"In addition, in any mathematics curriculum, including early childhood, children are capable of learning much more than the four basic operations. Where are geometry and early algebra? What about logic, measurement and estimation?"

The same letter writer as quoted above also points out the lack of social studies in Engel's ideal early schooling.

What should a 12 year old know?

http://www.nytimes.com/2010/02/02/opinion/02engel.html?ref=opinion

Reading out loud is a terrific skill to develop!

However, by age 12, other Math concepts can be added to the four operations -- especially the middle school merry-go-round of fraction/decimal/percent.

Division with Fractions 1 and 3/4 divided by 1/2

This is an example from Liping Ma's book:

Using Decimals
How much is 1.75 divided by .5?
1.75/.5 = 3.5


Using Fractions
1 and 3/4 divided by 1/2
1 and 3/4 = 7/4
7/4 divided by 1/2
How many half cups are in 7/4 cups?
1 half cup = 1/2
2 half cups = 2/2 = 1
3 half cups = 3/2
4 half cups = 4/2 = 2 -- this is too much

7/4 divided by 1/2
multiply 7/4 by the reciprocal of 1/2
7/4 x 2/1 = 14/4 = 7/2 = 3 and 1/2

Division with Fractions is "a topic at the summit of arithmetic". In the introduction, she asks a division with fractions question: 13/4 divided by 1/2 and offers three additional problem-solving methodologies (besides Keep, Change, Flip) -- Dividing Using Decimals, Applying the Distributive Law and You Don't Have to Multiply.

Checking Multiplication with Casting Out Nines

Multiplication can be checked using Casting Out Nines. (rather than long division)

By studying the digits in the problem and comparing them to the digits in the answer, we can gain confidence that we have done the problem correctly.

Here are some examples:
12 x 12 = 144
The digits in 12 add up to 3 (1+2)
The digits in the other 12 add up to 3 (1+ 2)
3 x 3 = 9 (144 adds up to 9)

13 x 13 = 169
The digits in 13 add up to 4 (1+3)
The digits in the other 13 add up to 4 (1+ 3)
4 x 4 = 16 (169 adds up to 16)
We could also add up the digits in the 16 of 4 x 4 and get 7
and get 7 in the answer by casting out the 9 in the 169 so we are left with a 16 that add up to 7.

125 x 4 = 500
The digits in 125 add up to 8 (1+2+5)
The 4 is just a 4
8 x 4 = 32 and the digits in 32 add up to 5
The answer 500 also adds up to 5.

What You Know About Math?

Great and enthusiastic fun video about the TI-84
http://www.youtube.com/watch?v=Ooa8nHKPZ5k

Learning Math Takes Patience

Feature on an award-winning Calc teacher in North Carolina who emphasizes practice for improvement and likens the process to bettering athletic ability. The article also talks about how Math attitude affects students and performance.

Personal Finance Classes in High School

This is such a needed app for Math and Life!!

Economics is also terrific but can be theoretical rather than practical.

Math provides the foundation for processing and understanding personal finance and economic terms to increase savviness and savings while reducing debt.

Singapore Math Success

Singapore Math in Focus is hailed by students and teachers.
http://www.kentucky.com/142/story/1109176.html

ACCUPLACER: College Placement Exam

The ACCUPLACER is used by many colleges for placement into Math and English.
Passing these tests is important in order to avoid non-credit remediation classes.
The title link goes to a page with a pdf of sample questions for both Math and English.

Tips for the ACCUPLACER:
http://www.collegeboard.com/student/testing/accuplacer/accuplacer-tips.html

Where is the GED?

While states are spending money and resources on exit exams, it would be interesting to think about using the GED as a standard. According to this document, only 60% of graduating high school seniors would pass the GED Tests on their first attempt:
http://www.acenet.edu/Content/NavigationMenu/ged/pubs/GED_Testing_Program_FactSheet_20092.pdf

GED sample questions:
http://www.acenet.edu/Content/NavigationMenu/ged/test/prep/sample_questions.htm

Math Show for Preschoolers

This new Nickelodeon preschool show will debut on January 25th.
Web site of show
http://www.nickjr.com/team-umizoomi/


http://www.nickjr.com/family-entertainment/kids-books/nickjr-quick-picks/best-math-books-for-kids.html