Solution for June 2010 Brain Teaser

Find the average of all multiples of 7 between 7 and 777, inclusive.   Answer: 392

This is like the Gaussian problems (adding consecutive numbers) as it is the pairs of numbers that allow us to more easily solve these types of problems.
For example, adding the numbers 1 - 10 can be done by grouping the smallest and largest 1 and 10 to make 11, (then the next smallest and largest) 2 and 9 to make 11, 3 and 8, 4 and 7 and 5 and 6 to get 5 pairs of 11.  So the sum of the numbers form 1 to 10 is 5 x 11 = 55.

From 7 to 777 inclusive is just like 1 to 111 inclusive (divide each by 7).
One of Math educator and writer Polya's methods:  Solve a Simpler Problem.  Step 2: Devise a Plan

Choosing easier numbers can often make the solution easier and simpler to understand.
To find the average of 1 to 111, find the average of each pair -- the average of 1 and 111 is  56 (1+111)/2.
The average of 2 and 110 is 56 and so on.

So the 392 is the average of 7 and 777 and also the average of 14 and 770 (the next two multiples) is also 392.

Please visit the Excel spreadsheet on docstoc and scroll down to see that the answer is 392!!
http://www.docstoc.com/docs/45498081/Math-Confidence-Brain-Teaser-June-2010

M is for Mental Fitness

On-Line Learning of 100 Pairs by Math Confidence

Quick, what number plus 43 adds up to 100?  Studying 100 pairs (for example, 60 and 40 are 100 pair as they sum to 100) can help people with mental Math and cash register Math.
Math Confidence 100 Pairs Activity on Quia

If you can't divide 300 by 2, should you qualify for a loan?

Weak Math Skills Linked to Default -- borrowers with poor Math skills were three times more likely to go into foreclosure.

This article starts off "If you can't divide 300 by 2, should you qualify for a loan?"
The survey led by a Columbia University prof, Stephan Meier, had five questions -- only two were in the article -- the one listed above and "How much is 10% of 1000?"
16% of the respondents got one of these 2 questions incorrect.

Financial education is heavily based on Math education and personal finance is an excellent application of understanding numbers and how they affect life.  In addition, practicing problem-solving can build the mental skills and perseverance that can help people to read the fine print on a mortgage.

Researchers study cues to predict students’ math performance - The Boston Globe

Researchers study cues to predict students’ math performance - The Boston Globe

The SAT vs the Marathon

When I took the SAT in October 2009, the students thought I was the proctor!  It was helpful to relive the test-taking experience plus I learned a lot of grammar.  A marathon is an interesting comparison as preparation for an athletic event is very similar to an academic event. 
There is a peak performance aspect -- the preparation is the key -- actual timed practice as well as self-care (sleep, food, etc) help with mental and physical  performance.  At the end of the day regardless of the results, the payoff is in the energy and effort expended.

I was inspired by a 2009 Wall Street Journal article written by a reporter whose teenage son dared her to take the SATs What I Learned from the SATs.

SAT                      vs.                           Marathon

4.5 hours                                              Variable

Indoors                                                 Outdoors

Teens                                                    Various age levels

Increase cognitive abilities                      Improve physical fitness

Final Exam Success Tip 1: Don't Write on Your Review Sheet

A review sheet is a gift ;)

Most teachers are giving away much of the exam on their review sheets!!

Don't write the answers on your review sheet!

It's the independent practice of reworking the problems that will boost learning, confidence and scores.

Review Sheet Tip from Math Confidence Blog

May 2010 Newsletter Brain Teaser Solution

Tom can beat Dick by one-tenth of a mile in a five-mile race. Dick can beat Harry by one-fifth of a mile in a five-mile race. By how much can Tom beat Harry in a five-mile race?

Tom runs 5 miles in the time it takes Dick to run 4.9 miles.  This is a rate of 98% or .98.
Dick runs 5 miles in the time it takes Harry to run 4.8 miles.  This is a rate of 96% or .96.

So the rate of Harry to Dick is (0.98)(0.96)5 so Harry was 4.704 miles along when Tom finished.
So Tom beat Harry by 0.296 miles.

Grades 3-8 Sample Math Questions...Man/Woman on the Street

log 10 + log 10 = log 100

April 2010 Brain Teaser Answer

Using all the digits from 0,1,2...9, form two 5-digit numbers so that their sum is the

a)smallest sum:   34047


One example:  20468 + 13579
Another example is: 10468 + 23579
 
You cannot have a leading zero (the number cannot start with a zero) so for the least sum, start with 1 for the first number and 2 for the second number. Then make 0 the second digit of one of them and 3 the other second digit.  Then continue with 4 and 5 and so on.

b)greatest sum: 183951

One example is: 97531 + 86420
Another example is: 87531 + 96420

For the greatest sum: First digits 8 or 9, Second digits 6 or 7 and so on.

Millions and Billions

New York Hall of Science

Career and College Ready

According to this Washington Post article: http://voices.washingtonpost.com/answer-sheet/no-child-left-behind/what-is-being-college-and-care.html, only 23% of college students do not remediation.  The article quotes an ACT spokeswoman "Readiness for college means not needing to take remedial courses." 

High schools and society would do well to focus on college and career readiness in addition to graduation rates.  While there is no high school exit exam, only 60% of high school graduates could pass the GED (according to the GED).

Preparing Math Teachers

Financial Literacy as a School Subject

NYT article on Financial Literacy in School
Often students and parents will ask "When I am ever going to use this Math?".  Studying trig and algebra will make you smarter and therefore you will have the capacity to understand Finance and Economics.  An excellent topic mentioned in the article is Time Value of Money.  A good example is:
How much will you have if you invest $100 for 3 years at 10% compounded annually?
This question inspires dialogue and learning on interest rates, banking, decimals, percents, exponents as well as opening a window into understanding the way the world works.

Identifying Problems...ok Math Opportunities

Metacognition (thinking about thinking) is the feedback loop or the "little voice in your head that you hear when you do Math -- such as add 4 to each side and divide both sides by 2".

Self-awareness of knowledge and skills can be achieved by categorizing Math problems into types of problems.  The ability to identify problem types can refine learning and bring clarity while increasing the enjoyment and flow of Math opportunities.  This is especially important in Math as gauging the level of challenge can help students know what to study and how to invest their time and energy under peak perfromance coniditions (tests).

It is a problem that they call them problems --  so how about Math opportunities?

Math can Build Non-Cognitive Skills too!

Studying Math may seem like a purely academic endeavor but it improves focus, persistence and self-advocacy which are takeaways that are "life skills".  This article quotes Dan Pink author of Drive on parents focusing on academics at the expense of "life skills" but school can also fuel these skills and talents that can then be applied to any endeavor whether academic, athletic or otherwise.
New York Times on Determination and Deliberate Practice

Planning for Success in Math and Life

A planner is vastly different than a calendar.  A calendar keeps track of appointments and scheduling but a planner focuses on prioritization and short- and long-term planning!

The Personal Digital Assistant (PDA) has become a common tool in school and business and many people use their phone to store their calendar and appointments.  While the Treo phone has a very basic calendar --I went back to the low-tech paper version of Franklin Covey Planners -- the Weekly pocket format.

The planner inspires vision and goals which leads to thinking about the Important but Not Urgent tasks such as studying for next week's exam, filing your taxes or scheduling an annual checkup.  In addition to feeling more inspired and organized, perusing your paper planner is more socially correct than glancing at your phone.  A zipperable planner protects your plans from the elements (like a water bottle!) and offers a place to store a pen or pencil. Zippered Planner Binders

Franklin Covey offers Plan Plus Online which can be reached from any web browser -- planning on paper or on-line can raise productivity and motivation to stay focused and reach goals.

March 2010 Brain Teaser Solution

Amanda went to the mall to buy supplies. She spent 1/2 of what she had plus $2.00 in the first store. Then she went to the second store and spent 1/2 of what she had left plus $1.00. In the third store, she spent 1/2 of what she had left plus $1.00. In the fourth store and last, she spent 1/2 of what she had left. After all of her purchases, she was left with only $3.00. How much money did she start with and how much did she spend in each store?

Amanda started with $64
she spent $34 in store 1
$16 in store 2
$8 is store 3
$3 in store 4

she spent a total of 34 + 16 + 8 + 3 = 61
therefore she has $3 left


After store 1 she has x - (1/2x + 2) which becomes 1/2x -2

then store 2
1/2 of answer 1 - 1

so 1/2(1/2x - 2) - 1

1/4x - 1- 1
so after store 2: 1/4x - 2


then store 3
1/2(1/4x-2) - 1

1/8x - 1- 1

so after store 3:1/8x - 2

then store 4:

1/2(1/8x - 2)

1/16x - 1

this is = 3

1/16x - 1 = 3

add 1 to both sides

1/16x = 4

x = 64

After store 1 she has 1/2(64)-2 = 30
so after store 2: 1/4x - 2 = 1/4(64) - 2 = 14
so after store 3:1/8x - 2 = 1/8(64) -2 = 6
then store 4: 6/2 = 3

How Far Can Math and Patience Take Kids?

Patience is a virtue!!  Math builds space telescopes and also life skills such as persistence and endurance.

Washington Post article on Math and Hubble -- the sky is the limit! has some great quotes from Neil deGrasse Tyson, director of the Hayden Planetarium in New York, "who gets to peer into the starry great beyond for clues to the origin of the universe. "

"Tyson earned a bachelor's degree in physics from Harvard and a doctorate in astrophysics from Columbia, but it was at the Bronx High School of Science in New York where he learned to not to be deterred when the going gets rough. " 
"I remember the transition from algebra to calculus and thinking: 'I will never in my life figure this out. I better try something else.' How many people do we lose because they don't think they are getting it as fast as they should? What I did was take a little bit at a time. Over the months, it was [as] though a misty fog began to dissipate and I was slowly emerging with a fluency in the new and powerful symbols of calculus."

Author Courtland Milloy encourages schools and parents to inspire students by showing an interest in Math and Science ;)