Math Confidence ACT Scores and Studying Methodology

The exam was just like the ACT books and the content and sample test online http://www.act.org/.  I started to study about 4 months before June 9.  I would do one section a week then two, then three.  In the last month before the exam, I completed 1 section each day.  During the last week, I completed 2 sections each day.  I would rotate the sections to get a variety of practice and always timed myself to make sure I was being efficient.   The Math topics were all familiar to me and the Reading Comp was just fine.  The two sections that I had to study the most were English (Grammar) and Science.


I used the Real ACT books -- the one with 3 tests and then the newer one with 5 tests (the same 3 tests from the first book and 2 new tests).  The other book I used was Barron's ACT 36.   The content online included questions for each section and a full-length practice test which I completed the day before the exam.

All those months of studying paid off.    The day of the exam, Science was the most challenging section as I was not confident about 2 of the 7 sections.  I was not surprised to see that Science was my lowest score. National Ranks for Test Scores and Composite Scores Score Composite Score : 34 This is not an official ACT score report and is intended only for your informational use. Does your score meet the ACT College Readiness Benchmark? English 33   Yes. Usage/Mechanics 18 Rhetorical Skills 16 Mathematics 36   Yes. Pre-Algebra/Elem. Algebra 18 Algebra/Coord. Geometry 18 Plane Geometry/Trig. 18 Reading 34   Yes. Social Studies/Sciences 18 Arts/Literature 17 Science 31 & Yes. But you canfurther improve your science skills. Combined English/Writing Not yet available Writing (score range 2 to 12) 10

June 2012 Brain Teaser Solution

Q: My daughter has many sisters. She has as many sisters as she has brothers. Each of her brothers has twice as many sisters as brothers. How many sons and daughters do I have? 


A: Four daughters and three sons.

May 2012 Brain Teaser Solution

Q:A clock is observed. The hour hand is exactly at the minute mark, and the minute hand is six minutes ahead of it. Later, the clock is observed again. This time, the hour hand is exactly on a different minute mark, and the minute hand is seven minutes ahead of it. How much time elapsed between the first and second observations?


A:  The hour hand moves as the minute hand moves.  For example, at 2:30, the hour hand is halfway between 2 and 3.  The hour hand is exactly at the minute mark five times an hour on the 12, 24, 36, 48 and on the hour.
So 1:12 would make the hour hand pointing at 6 minutes and the minute hand pointing at 12 minutes.
24 minutes after on the minute hand would mean that the hour hand would have to be pointing at the 18 minute mark which would be more than halfway between 3 and 4.
36 minutes after the hour would mean that the hour hand would point exactly at the 6 and the hour hand only points exactly at the 6 when it is exactly at 6 o'clock.
48  minutes after the hour would mean that the hour hand would point at the 42 which would be a bit after 8 and not closer to nine.
So for the first observation the clock must be at 1:12.
The second observation cannot be 1:12 (since that it a 6minute difference) so we will now check the 24 minute after -- this would put the hour hand at the 17 which would work perfect with 3 as at 3:24, the hour hand has moved 2/5 between the 2 and 3.
Therefore the time elapsed between the first observation, 1:12, and the second observation, 3:24 is 2 hours and 12 minutes.

April 2012 Brain Teaser Solution

Q:You are in a pitch dark room selecting socks from a drawer that has only six socks, a mixture of black and white. If you choose two socks, the chances that you draw out a white pair is 2/3. What are the chances that you draw out a black pair?


A: 0 (it is impossible)


The good wrong answer is 1/3 because that is the remainder of the probability. This assumes that the pair of black socks has the same probability as a pair of white socks but there is also a chance that you pull out one black and one white.  The combinations with 4 white (W1, W2, W3 and W4) and 2 black (B1 and B2) are: 

W1W2, W1W3, W1W4, W2W3, W2W4, W3W4, W1B1, W2B1, W3B1, W4B1, W1B2, W2B2, W3B2, W4B2, and (last but not least) B1B2.  Far less than two-thirds of the aforementioned are a white pair.


So let's try 5 white and 1 black:  W1W2, W1W3, W1W4, W1W5, W2W3, W2W4, W2W5, W3W4, W3W5, W4W5, W1B1, W2B1, W3B1, W4B1, W5B1makes a total of 15 combination of which 10 are white pairs.  This simplifies to 2/3 so there must be 5 white and 1 black.  But with only one black sock, you can never get a black PAIR!

March 2012 Brain Teaser Solution

Which two whole numbers, neither containing any zeros, when multiplied together equal exactly 1,000,000,000? 
A: 512 and 1953125

The factorization of this number is 10^9
which can be broken down into its prime factors of (2 x 5)^9
so 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 2 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5 x 5
If we mix any 2 with any 5 we will get 10 and the number will end in a zero due to multiplication by 10
Therefore we must keep the 2s and the 5s separate from one another
2^9 is 512 and 5^9 is 1953125
These two factors (each with no zeros) will multiply out to 1 billion!

February 2012 Brain Teaser Solution

Q: Find three-digit numbers that the number itself, its double and its triple contain each digit from 1 to 9 exactly once. For example, 192 works because 192, 384, 576 contain 1 to 9 each once. 273 also works because 273, 546, 819 contain 1 to 9 each once.

What are the other two numbers that also have this property?

A: 219 and 327.
We need to keep the numbers low because if we are going to double and triple and maintain only 3 digits, then that means that the number must be less than 334 (334 tripled makes 1002 -- 4 digits).
We can rule out any numbers where the digits repeats as we have to use all 9 digits (1 through 9).
I used a guess and check to solve this problem.  I kept the first digit 1, 2 or 3 and kept out any repeated digits.  This led me to 219 and 327 but those are also the same digits as those in the given (192 and 273)!!! 


It was then that I realized that if you swap 19 and 2, and swap 27 and 3, you'll get them.

January 2012 Brain Teaser Solution

Andy and Sandy had to take a make-up class in math over the summer, a two-month, self-paced course with a test at the end of each of 12 chapters. The course requires a 70% grade to pass.  In the first month, they both had difficulties with the concepts. Andy averaged 60% on his exams; Sandy averaged 50%. In the second month, Andy averaged 90% on his exams; Sandy averaged 80%.  While Sandy got a passing grade of 75% in the class;  Andy failed with 65%. How did Sandy pass while Andy flunked? 

Their grades were weighted averages.

This can be solved using guess-and-check or through using algebra.

Let x = # of tests in the first month, therefore 12 - x would be the # of tests in the second month.

so Andy's tests can be modeled with algebra using: .6x + .9(12 - x) = .65 (12)

.6x + 10.8 -.9x = 7.8

-.3x + 10.8 = 7.8

-.3x = -3

x = 10 and 12 - x = 2

so Andy took 10 tests in the first month and 2 in the second month.

10(60) + 2(90) = 780 divided by 12 = 65

Sandy's tests can be modeled using .5x + .8(12 - x) = .75(12)

.5x + 9.6 - .8x = 9

9.6 - .3x = 9

-.3x = -.6

x = 2

so Sandy took 2 tests in the first month and 10 in the second month

2(50) + 10(80) = 900 divided by 12 = 75

December 2011 Brain Teaser Solution

Q:  In a hallway lined with 100 closed lockers, you begin by opening every locker. Then you close every second locker. Then you go to every third locker and open it (if it's closed) or close it (if it's open). Then you reverse every fourth locker, fifth, sixth , etc.  Continue reversing every nth locker on pass number n. After 100 passes, where you toggle only locker #100, how many lockers are open?

A: There are 10 lockers open -- #1, 4, 9, 16, 25, 36, 49, 64, 81, and 100 -- all perfect squares.
This problem is based on the factors of the locker number.

Think about factor pairs.  A number like 10 has two factor pairs, 1 and 10 and 2 and 5.  But a number like 9 has only 1 unique factor pair of 1 and 9 because the other factor pair is 3 and 3.  So 10 has four factors while 9 has only three.

Each locker is toggled by each factor; for example, locker #10 is toggled four times -- on pass numbers 1, 2, 5,and 10.  Locker #40 is toggled on pass numbers 1, 2, 4, 5, 8, 10, 20, and 40. That's eight toggles: open-closed-open-closed-open-closed-open-closed.

The only way a locker could be left open is if it is toggled an odd number of times. The only numbers with an odd number of factors are the perfect squares. Thus, the perfect squares are left open.
For example, locker #9 is toggled on pass number 1, 3, and 9 (three toggles): open-closed-open.

November 2011 Brain Teaser Solution

Q: You are driving at 50 miles per hour. If you decrease the time it takes you to travel 1 mile by 8 seconds, what is your new speed?


A: 56.25 mph (miles/hour)


At 50 miles per hour...one hour is 3600 seconds
3600 seconds/50 miles = 72 seconds per miles


If you decrease the time by 8 seconds to 64 seconds per mile
Then in 3600 seconds / 64 seconds per mile = 56.25 mph

October 2011 Brain Teaser Solution

If you use only a 5-liter bowl and a 3-liter bowl and have unlimited access to water, how would you get exactly 4 liters of water in the 5-liter bowl? 

Fill the 5-liter bowl and spill out 3 liters of water into the 3-liter bowl.  
Then empty the 3 liter bowl so you can put the 2 liters that remains in the 5 liter bowl into the 3 -liter bowl.  
Refill the 5-liter bowl and spill out 1 liter into the 3 -liter bowl (that had 2 liters in it and will now be full).  

4 liters of water will remain in the 5-liter bowl.

The Power of Zero!!

Anything raised to the zero power is 1.


Here is a way to remember this:


2^3 divided by 2^3 = 8/8 = 1
If you write this as (2)(2)(2)
                                (2)(2)(2)
All of the 2's will cancel and you will be left with no 2s
therefore 2 to the zero power :)

September 2011 Brain Teaser Solution

Q: A chicken farmer has figured out that a hen and a half can lay an egg and a half in a day and a half. How many hens does the farmer need to produce one dozen eggs in six days? A: 3 hens

If a hen and a half can lay an egg and a half in a day and a half, then  in three days (twice the amount of time), the same hen and a half can lay three eggs and in six days (doubling again), the same hen and a half can lay six eggs. We want twelve eggs in six days so we need to double the number of hens from 1.5 to 3. Another solution comes from Tim at Riverdale City Grill Bronx, NY:

You need 3 hens b/c each hen lays an egg every 36 hours. There are 144 hours in 6 days. That means each hen lays 4 eggs every 6 days. 3x4=12. 

New York Times Letter to the Editor re:Improving SAT Scores

Published on 9/26/11 in print and online
To the Editor:

Early childhood education can certainly be improved, but there are additional measures that will increase verbal scores on the SAT:

¶Subscribe to and complete daily the free SAT Question of the Day.

¶Take the 10 timed exams in The College Board’s Official SAT Study Guide.

Even adults should consider studying for and taking the SAT as a cognitive challenge. When I studied to retake the SAT in 2009, 29 years after I first took it, I filled in gaps in my education.

My verbal score increased over 200 points from high school — partly from living a few more decades but mostly from cracking the books. Jack LaLanne got us to exercise our bodies; now it is time to exercise our minds.

ROBIN SCHWARTZ
Bronx, Sept. 20, 2011

The writer is a math educator and author of the Build Math Confidence newsletter.

Response to NYT on Math Ed

This proposed sequence would help students prepare for life especially with respect to Finance. Another important consideration is how to prepare students for SAT, ACT and college placement tests. Until the Common Core Standards are implemented, we can use the SAT and ACT as standards. The College Board's SAT book has extremely accurate sample exams while the free SAT Question of the Day can help students, parents, teachers and society gauge what students should know. By studying this content, students can fill in their gaps and expand their opportunities.

http://www.amazon.com/Official-SAT-Study-Guide-2nd/dp/0874478529/ref=sr_1_1?ie=UTF8&qid=1314246854&sr=8-1

http://sat.collegeboard.org/practice/sat-question-of-the-day

Robin A. Schwartz, MBA, BSEE

www.mathconfidence.com

August 2011 Brain Teaser Solution

How many integers less than or equal to 1,000,000 are both perfect squares and perfect cubes?

Answer: 10
Anything that is both a perfect square and a perfect cube will be a number to the 6th power.
The lowest one is 1 = 1^2 and 1^3
2^6 = 64 = 8^2 and 4^3
3^6 = 729 = 27^2 and 9^3
4^6 = 4096 = 64^2 and 16^3
5^6 = 15625 = 125^2 and 25^3
6^6 = 46656 = 216^2 and 36^3
7^6 = 117649 = 343^2 and 49^3
8^6 = 262144 = 512^2 and 64^3
9^6 = 531441 = 729^2 and 81^3
10^6 = 1,000,000 = 1000^2 and 100^3

For an example of why this works, let's examine 2^6  which is 2 x 2 x 2 x 2 x 2 x 2 then these 2s can be split into 2 groups of 3 as (2 x 2 x 2)^2 which can be rewritten as 8^2
or 3 groups of 2 (2x2)(2x2)(2x2) = 4^3

How many people ran the NYC marathon in 2010?

About 45,000

http://www.nycmarathon.org/Results.htm

Formula Answers

1. What is the equation of a line?  y = mx + b  m = slope and b = y-intercept

2. The slope formula? m = change in y / change in x or (y2 - y1)/(x2-x1)

3. Circumference of a circle?  C = 2 pi r

4. Area of a circle? A = pi r squared

July 2011 Brain Teaser Solution

How many degrees are in the acute angle formed by the hands of a clock at 2:20PM?
A: 50 degrees


A circle has 360 degrees, so each of the 12 hours on the clock is 30 degrees (360/12)
Exactly 2:00 is 60 degrees from the twelve while 4:00 is 120 degrees.  So the difference between 2 and 4 would be 60 degrees EXCEPT that as the minute hand goes around the clock, the hour hand moves between the hours.


So while the long hand is exactly pointing at the 4, the little hand is in between the 2 and 3. As it is 20 minutes past the hour, 1/3 of the hour has elapsed. 


Since there is 30 degrees in between the 2 and the 3 and it has only transversed 1/3 of 30, it is 10 degrees closer than it was at exactly 2.


BTW, an acute angle is less than 90 so 50 fits the bill.  The angle going counterclockwise is 360 - 50 = 310.

Treasury Yields Not LInear!

http://blogs.wsj.com/marketbeat/2011/06/30/treasury-yields-go-parabolic/#

Maybe parabolic or even exponential!!

SAT II Scores

An 800 is only 88th percentile for Level 2 as so many students who take it are very good Math students and are planning on pre-engineering, pre-med, etc types of majors.
780 is 98th percentile.

TEST DATE	TEST	SCORE
06/2011	SAT Subject Test
	Mathematics Level 1	780
	Mathematics Level 2	800

Check back on June 28 for your full score report, with detailed analysis! If you took the SAT Reasoning Test you'll also be able to view a copy of the actual essay you wrote.

June 2011 Brain Teaser Solution

When I divide it by 2, the remainder is 1. When I divide it by 3, the remainder is 2. When I divide it by 4, the remainder is 3. When I divide it by 5, the remainder is 4. When I divide it by 6, the remainder is 5. When I divide it by 7, the remainder is 6. When I divide it by 8, the remainder is 7. When I divide it by 9, the remainder is 8. When I divide it by 10, the remainder is 9. Find the smallest number that satisfies these conditions.

Since the remainder is always so close to the next multiple, the main idea is lowest common multiple so as George Polya, suggested solve a simpler problem.
Polya's Main Ideas in How to Solve It

Let's use just the first three criteria

When I divide it by 2, the remainder is 1.When I divide it by 3, the remainder is 2.When I divide it by 4, the remainder is 3. The number will be one less than the lowest common multiple of 2, 3 and 4 Counting by 2s 3s and 4s will yield 2 4 6 8 10 12 3 6 9 12 4 8 12  So 12 is the LCM -- notice that if we were to multiply 2 x 3 x 4, we would get 24 which is not the lowest common multiple.  The number is one less than this so 11 would be the number that satisfies all three criteria. With a longer list of numbers we might not want to write out all of the multiples.  In fact, for 2, 3 and 4, if we break them down to prime factors 2 is 2 and 3 is 3 but 4 is 2 x 2.  If we already have 2 and 3 we only need one more 2 to make a 4. So 2 x 3 x 2 will cover 2, 3, and 4 (2 x 2). So for all the way up to 10, we need the LCM of 2, 3, 4, 5, 6, 7, 8, 9, 10 For 2, 3, 4, 5 we need to add a 5 so 2 x 3 x 2 x 5 = 60 For 2, 3, 4, 5, 6, we already have a 6 because 2 x 3 = 6 so the LCM of 2, 3, 4, 5 ,6 is 60. For 2, 3, 4, 5, 6, 7 we need to add a 7 so 2 x 3 x 2 x 5 x 7 = 420 For 2, 3, 4, 5, 6, 7, 8, we need to add a 2 (8 is 2 x 2 x 2 and so far we only have two 2s) = 840 For 2, 3, 4, 5, 6, 7, 8, 9 we need to add a 3 (9 is 3 x 3 and so far we only have one 3) = 2520 For 2, 3, 4, 5, 6, 7, 8, 9 , 10 we already have a 10 because 2 x 5 = 10 so the LCM = 2520.

So the answer is 1 less than 2520: 2519

May 2011 Brain Teaser Solution

Q: Six men have 6 bags each. In every bag there are 6 cats, each cat has 6 kittens. How many legs in all? A: 6060 Each man has 6 bags with 6 cats each -- that's 36 cats. 36 cats x 6 kittens = 216 kittens Each bag has 252 cats (36 cats and 216 kittens): 252 x 4 legs = 1008 cat legs per bag 6 men each have 1008 cat legs = 6048 Plus the 6 men have 12 legs so the total is 6060.

Three Steps to Shoo Away Math Anxiety

http://www.good.is/post/could-math-anxiety-become-a-thing-of-the-past/

MATHCONFIDENCE 
Three Steps to Shoo Away Math Anxiety as a Thing of the Past:

First, call it “Increasing Math Confidence”

Second, solve Math problems (Math opportunities) including multiple choice as compare/contrast with “good wrong answers” can increase knowledge, skills, attitude and scores!

Third, embrace Math mistakes! Learning from errors can be challenging emotionally but will improve critical thinking, build confidence and expand educational and career options.

Robin Schwartz aka Robin the Math Lady
www.mathconfidence.com
Author, Build Math Confidence e-newsletter

Response to Darren Hardy's SUCCESS Blog

Jim Rohn’s Challenge to Succeed along with Darren’s Living Your Best Year Ever are cutting edge tools for all ages. Their messages of accountability and discipline are essential inputs for planning and achieving one’s goals. As an educator and a parent, I champion these ideas and principles and am delighted to see SUCCESS magazine in the mainstream!

Many college students do not have financial independence as a goal and a college education may not pay back for quite a long time (especially with student loans). As Jim Rohn says “If they’d offered Wealth 1 and Wealth 2, I would have taken both classes”. Math provides the foundation for processing and understanding personal finance and economic terms to increase savviness and savings while reducing debt.

Math teachers often hear “When I am ever going to use this Math?” which is not really a question but a complaint posed as a question. I have prepared my response with an acronym — MATH teaches Mental Fitness, Accountability, Teamwork and Horizon. And these principles learned in Math class (or on the baseball field or at church or on a job) are life skills that can be applied to the entire Wheel of Life.

http://darrenhardy.success.com/2011/05/helping-grads/

Response to Change the Equation's on Learning to Love Math

 Thank you for featuring this book -- "Learning to Love Math" by Dr. Judy Willis has an awesome subtitle "Teaching Strategies That Change Student Attitudes and Get Results".  How can Math teachers share their enthusiasm so that students will embrace the challenges and enjoyment of Math?  While this book emphasizes the utility of Math, Dr. Willis also points out ways and reasons for students to learn including convincing people that they can change their intelligence, supporting students in setting short-term and long-term goals and reducing mistake anxiety.  She also recommends teachers/parents sharing their own stories of learning including tales of endurance and fortitude.  

These ideas are especially important for girls as it is still not very cool to be good at Math.  Having a stronger Math background creates more career choices and boosts  confidence. – for example, engineering is a great career and solid foundation and is still only about 20% women (about the same as the 1980's when I was in engineering school!).  Best wishes to your daughter, Barbara!

This book was reviewed in Math Confidence's e-newsletter in December 2010:

 http://archive.constantcontact.com/fs086/1102574170134/archive/1103856670244.html

April 2011 Brain Teaser Solution

Joe buys a 5 foot long fishing pole but cannot take the bus home because the bus driver will not let him board the bus with anything over 4 feet long.  Joe goes to a hardware store and buys one thing then returns and boards the bus.  The pole can not be cut, bent, broken, or taken apart. What did Joe buy to allow him to board the bus with the fishing pole?


Joe bought a BOX!!  Since the maximum dimension can be 4 feet maximum, use the Pythagorean Theorem!!

a^2 + b^2 = c^2

Joe bought a box that was at least 3 feet by 3 feet by 4 feet -- the diagonal of the box will be 5 feet (the sides of the box and the fishing pole will form a 3,4,5 right triangle).

March 2011 Brain Teaser Solution

Julie travels from A to B at 2 minutes per mile and returns over the same route at 2 miles per minute.  Find her average speed, in miles per hour, for the entire trip.


She travels from A to B at 30 miles per hour (60 minutes for 30 miles = 2 minutes per mile).
She travels from B to A at 120 miles per hour (2 miles per minute for 60 minutes).


We can pick a distance that works well with both 30 and 120 such as 120 miles.


From A to B at 30 mph, it will take her 4 hours to go 120 miles.
From B to A at 120 mph, it will take her 1 hour to go 120 miles.


Total distance = 120 + 120 = 240 miles
Total time = 4 + 1 = 5 hours


Average speed = (Total distance)/ Total time = 240 miles / 5 hours = 48 mph

Get Smart! Take the SAT! Power Point

Get Smart! Take the SAT! Power Point

February 2011 Brain Teaser Solution

There are 8 similar balls. Seven of them weigh the same and the eighth is a bit heavier. How would you identify the heavier ball if you could use a two-pan balance scale only twice?

1. Put three balls on each side of the balance scale.  If they balance with one another, then all of these six are the same weight.
2. Take the last two balls and put them on the balance scale to find the heavier one.

OR


1. Put three balls on each side of the balance scale.  If they do not balance, take the three balls from the heavier side for the next step.

2. Pick two out of these three balls and put one on each side of the balance scale.  If they are different weights, you will find the heavier ball.  If they balance, then the heavier ball is the third ball.

What is the right order for high school Math classes? Washington Post

This is a response to Valerie Strauss' blog:
http://voices.washingtonpost.com/answer-sheet/math/high-school-math-whats-the-rig.html

Thanks for this article on order of Math courses. I am not sure what order they should be in -- it may depend on how the topics are divided up. A2 is usually more rigorous than the others but it can depend on the school/class/state.


It would benefit students to learn and know the Math on the ACT/SAT/GED/ACCUPLACER (placement tests used by colleges). Many students have not seen the topics enough times, or have had the topics slivered (and are unused to multiple topics on the same exam), or have not developed the speed that will help them problem solve 20 questions in 25 minutes.

While the Common Core are under development, we already have these standards at the high school and college level.

Studying multiple choice items can improve metacognition due to compare/contrast and by studying "good wrong answers" (for example, exponent rules questions always have "good wrong answers"!!).

Students, teachers and parents can use the free SAT Question of the Day (and other free or reasonably priced resources) to better scores and knowledge and skills!
http://sat.collegeboard.com/practice/sat-question-of-the-day

Perhaps, we can bring academic and cognitive abilities up to the level of respect that athletics commands.

Robin Schwartz
Author, Build Math Confidence e-newsletter
http://www.mathconfidence.com/

Posted by: mathconfidence
January 14, 2011 12:37 PM

January 2011 Brain Teaser Solution

Q: What is the largest number of consecutive integers that will add up to 2011?

A:  A good way to think about this problem is to do what Polya said "Solve a simpler problem"
So first think about  -- What is the largest number of consecutive integers that will add up to 11?

The least number of numbers would be one -- 11
You could use two consecutive numbers -- 5 and 6.
If you think about negative integers, -4, 3, -2, -1 would cancel out 1, 2, 3, and 4
so - 4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6 would give a total of 11 numbers

But we can do even better

-10 would cancel out with 10
-9 would cancel out with 9 and so on

-10, -9, -8 , -7, -6, -5, - 4, -3, -2, -1, 0, 1, 2, 3, 4, 5, 6, 7, 8 , 9, 10 would make zero if you sum them.
so just now add an 11
-10 thru -1 is 10 integers
0 is 1 integer
1 thru 10 is 10 integers and
11 is 1 integer for a total of 22.
(22 = 2 x 11)

so the number of integers is always 2 times the number itself.

For 2011,
The lowest number would be -2010 (it would cancel out with +2010)
The next number would be -2009 (it would cancel out with +2009)
The next number would be -2008 (it would cancel out with +2008)
and so on...
until -2 cancels out with +2
and -1 cancels out with +1

and then there is 0

so there would be 2010 negative numbers -2010 through -1
2010 postiive numbers 1 through 2010
plus 0
and also 2011
2010 negative integerss + 2010 positive integers + 2 more (for 0 and 2011)
for a total of 4022
4022 is the answer.

December 2010 Brain Teaser and Solution

Q: At a hardware store, I can buy 1 for $0.75 and I can buy 2761 for $3.00. What am I buying?

A: House numbers

November 2010 Brain Teaser Solution

Which is larger -- 8^98 or (8^99 - 8^98)?  (8^98 means "8 to the 98th power")

You can put this into a calculator and it will give you back scientific notation because both these numbers are REALLY BIG.

so 8^98 is about 3.12 x 10^88 (10 to the 88th power) but 8^99 - 8^98 is about 2.22 x 10^89 (10 to the 89th power) therefore 8^99 - 8^98 is larger.  But by how much?
Here is where the cool factoring comes in:
8^99 - 8^98 can be rewritten as: 8^98(8 - 1) making it 7 times bigger than 8^98.

Math is a Gatekeeper

October 2010 Brain Teaser Solution

Q: What is the greatest possible product of two positive whole numbers whose sum is 100?
A: 2500 (50 x 50)

Let x = one number then 100 - x = other number
So we want to maximize the product of this algebra, x(100 - x) = 100x - x^2
see the graph below:

This result can also be achieved through Calculus.  If we take the derivative of the algebra and set that equal to 0, then solve for x.
The derivative of 100x - x^2 is 100 - 2x, when 100 - 2x = 0 is solved x = 50.
When we substitute 50 into 100x - x^2, we get
100(50) - (50)^2

5000 - 2500 = 2500
So the maximum point is at x = 50 -- at the (x,y) point (50, 2500).
The list below shows:
Column 1 first number
Column 2 100 - first number
Column 3 product of Column 1 and Column 2

1 99 99

2 98 196

3 97 291

4 96 384

5 95 475

6 94 564

7 93 651

8 92 736

9 91 819

10 90 900

11 89 979

12 88 1056

13 87 1131

14 86 1204

15 85 1275

16 84 1344

17 83 1411

18 82 1476

19 81 1539

20 80 1600

21 79 1659

22 78 1716

23 77 1771

24 76 1824

25 75 1875

26 74 1924

27 73 1971

28 72 2016

29 71 2059

30 70 2100

31 69 2139

32 68 2176

33 67 2211

34 66 2244

35 65 2275

36 64 2304

37 63 2331

38 62 2356

39 61 2379

40 60 2400

41 59 2419

42 58 2436

43 57 2451

44 56 2464

45 55 2475

46 54 2484

47 53 2491

48 52 2496

49 51 2499

50 50 2500

51 49 2499

52 48 2496

53 47 2491

54 46 2484

55 45 2475

56 44 2464

57 43 2451

58 42 2436

59 41 2419

60 40 2400

61 39 2379

62 38 2356

63 37 2331

64 36 2304

65 35 2275

66 34 2244

67 33 2211

68 32 2176

69 31 2139

70 30 2100

71 29 2059

72 28 2016

73 27 1971

74 26 1924

75 25 1875

76 24 1824

77 23 1771

78 22 1716

79 21 1659

80 20 1600

81 19 1539

82 18 1476

83 17 1411

84 16 1344

85 15 1275

86 14 1204

87 13 1131

88 12 1056

89 11 979

90 10 900

91 9 819

92 8 736

93 7 651

94 6 564

95 5 475

96 4 384

97 3 291

98 2 196

99 1 99

100 0 0

24 x 12

The huge Excel handbook had been my intense focus as a potential deskside banker support person.
 
On a third round at a big investment bank, a banker asked "What's 24 x 12?"
I said "288" He asked me how -- "12 x 12 doubled is 24 12s" = 288
He responded 24 x 10 + 24 x 2 = 240 + 48 = 288

I then said 24 x 24 = 576 -- take half of this (12 24s)which is 288

Both of us were at least 30, so I then said:
 "If you remember your high school algebra -- 24 and 12 are both 6 from 18 so you can write 24 x 12 as (18 + 6)(18-6) and when you FOIL that  it becomes 18^2 + 6(18) - 6(18) - 6^2. which is 324 - 36 = 288!"

I got the job!

September 2010 Brain Teaser Solution

When writing the whole numbers from 1 to 20, there are 12 1s (one in 1, 10, 12, 13, 14, 15, 16, 17, 18 and 19 and two in 11).  When writing the whole numbers from 1 to 1000, how many 1s will you write?

1- 99           20
100 - 199  120
200 - 299    20
300 - 399    20
400 - 499    20
500 - 599    20
600 - 699    20
700 - 799    20
800 - 899    20
900 - 999    20
1000              1

Total          301    

There are 12 1's from 1- 20

then 21, 31, 41, 51, 61, 71, 81, and 91

From 1 - 99, you will write a total of 20 ones

The same is true for 200 - 299, 300 - 399, 400 - 499, 500-599, 600 - 699, 700 - 799, 800 - 899, and 900-999. So from 200 - 999, there are a total of 160 ones (8 x 20)

From 100 - 199,

there are 20 ones in the second and/or third digit:

101, 110, 111, 112, 113, 114, 115, 116, 117, 118, 119, 121, 131, 141, 151, 161, 171, 181, 191

plus all the ones that begin every number from 100 - 199 inclusive: 100

"H" is for Horizon

August 2010 Brainteaser Solution

Determine the smallest integral (whole number) value of x that will make the product 4840x a perfect cube.

Thefactors of 4840 = 484 x 10 = 22 x 22 x 10 = 2 x 11 x 2 x 11 x 2 x 5 (the prime factors)

To make a perfect cube, you need three of each factor:

you have enough 2s, so you need 2 5's and one 11.

5 x 5 x 11 = 275

T is for Teamwork

July 2010 Newsletter Brain Teaser Solution

Which would you rather have? a) $1,000,000

or b) Doubling Pennies for 30 days
1 penny on day 1, 2 pennies on day 2, 4 pennies on day 3, 8 pennies on day 4, etc.
 
The doubling pennies become $5368709.12 on Day 30.



Day # of Pennies Dollars


1 1 0.01

2 2 0.02

3 4 0.04

4 8 0.08

5 16 0.16

6 32 0.32

7 64 0.64

8 128 1.28

9 256 2.56

10 512 5.12

11 1024 10.24

12 2048 20.48

13 4096 40.96

14 8192 81.92

15 16384 163.84

16 32768 327.68

17 65536 655.36

18 131072 1310.72

19 262144 2621.44

20 524288 5242.88

21 1048576 10485.76

22 2097152 20971.52

23 4194304 41943.04

24 8388608 83886.08

25 16777216 167772.16

26 33554432 335544.32

27 67108864 671088.64

28 134217728 1342177.28

29 268435456 2684354.56

30 536870912 5368709.12

A is for Accountability

Viewing the SAT as a Challenge and National Indicator

Viewing the SAT as a Challenge and National Indicator
in response to Washington Post article Your SAT Score Has Little to Do


As a Math peak performance coach, I have found that studying for the SAT can be challenging and entertaining while promoting brain fitness at any age. Although I graduated from college in 19XX, I took the SAT in 2009 to gain perspective, to have fun, and to boost mental fitness. I learned much reading and grammar while studying last summer.

Some fields of study use the SAT as a measure of being able to ‘keep up with the Joneses’. Engineering schools want high Math scores (to follow along with profs who write a dozen equations on the board); likewise, journalism schools want high verbal scores.

The SAT is important because there is no national standard of high school curriculum or a national exit exam. While the Core Standards have been in the works, the existing standard could be the SAT (or ACT or GED) as a unifier for a reasonable body of knowledge. http://www.corestandards.org/.

The GED is a formidable exam -- only 60% of high school graduates could pass the GED http://www.acenet.edu/Content/NavigationMenu/ged/etp/score.htm

If people could view the SAT like a marathon, it would give mental fitness a boost!! Test taking/studying just makes you smart just like working out makes you physically fit.

Try the free SAT Question of the Day!!
http://sat.collegeboard.com/practice/sat-question-of-the-day

Robin Schwartz
Author, Build Math Confidence e-newsletter
http://www.mathconfidence.com/

Math Would Help People Land Jobs

Workers Need Better Skills
American workers need 9th grade level Math to gain jobs in the manufacturing sector.  One of the slides shows 8th graders on a tour of a factory.

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.