December 2013 Brain Teaser Solution

Q: How can you make the following equation true by drawing only one straight line: 5+5+5=550 ?
A: Draw a line on the first plus sign. This turns it into a 4! The equation then becomes true: 545+5=550.

Solution 2: Use the line to make the = into an unequal sign:

To think differently about the SAT...

Here is my letter published by the New York Times in response to E.D. Hirsch's How To Stop the Drop in Verbal Scores
http://www.nytimes.com/2011/09/19/opinion/how-to-stop-the-drop-in-verbal-scores.html

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.

Personal Development and Education

Success literature such as John Maxwell re: leadership or Jim Rohn re: personal development is often highly prized by salespeople because they need motivation and persistence to keep at their game.  I propose that students and teachers could highly benefit from these messages as they are always there when you want them!  Audio is especially helpful see below for Jim Rohn The Challenge to Succeed
By mixing personal development with education, we can help students, teachers and parents buy in to the idea of hard work and persistence that can vastly improve Math education outcomes and enjoyment!
Some of my favorites:  

The Math Classroom Management and Leadership Tool Kit Part II

Other ideas for classroom management and leadership:
Environmental changes: Lights up, lights down Open door, close door
Walk around with a pen so you can mark up student work
Popsicle sticks with student names for cold calling
Hands on activities
Guest speakers
Skeleton for notes

In today's fast paced society, where in person meetings must compete with online games and social media, we can help create opportunities for students to be actively learning.  This is a challenge as students may not see the importance of Math and/or education.  If we are willing to change modalities and keep up the pace, we may be able to lead our students and  engage them for a longer period of time thereby creating more learning opportunities.

Ideally, we would like students to be self-motivated but more on that in the next post.

The Math Classroom Management and Leadership Tool Kit Part I

Students can be a tough audience.  Most students are in Math class because it is a requirement so what can teachers do to keep students focused and engaged?  I know I cannot always delight and engage my Precalculus students for 75 minutes at a time.  It is hard to remember the "tool kit" in the moment.  Please send suggestions via email or post comments if you have any other suggestions to add or modify this "tool kit"

This partial list includes modalities of learning and teaching as well as physical items that may help in the Math classroom.

Using a timer (displaying it sometimes)
Switching between using technology and not
Calculator use vs not
Voting systems like remote clickers or if cell phones are allowed www.polleverywhere.com (this worked well for one section but the other section posted up inappropriate comments that displayed in real time on the screen in front of the classroom -- yikes!)
Independent work including exit slips/tickets, checkpoints and quizzes
Group work with groups they choose or groups you choose

Part II will include more Tool Kit items so please stay tuned

Math is Interesting and Exciting (as opposed to boring)

This week, Math Confidence's blog will contain posts in response to today's (Sunday 12/8/13) NYT Editorial "Who Says Math is Boring?".

The article focused on four ideas:

A More Flexible Curriculum
Very Early Exposure to Numbers
Better Teacher Preparation
Experience in the Real World

I will be tweeting a link to this blog for each new idea each weekday this week as I would like to add a few of my own ideas
Here is the Original OpEd link: http://www.nytimes.com/2013/12/08/opinion/sunday/who-says-math-has-to-be-boring.html

November 2013 Brain Teaser Solution

Q: On a 12-hour digital clock, what is the smallest interval between two times that are palindromic (can be read forwards and backwards as the same number)?

A:  2 minutes. the time from 9:59 to 10:01

NYT When Advanced Math Placement Is a Struggle

Is the student learning? Is she putting in effort?
These are the two top questions to ask the student and the family.

Too often, ppl rate themselves on Math by their grades.
Is she getting a C- because of exam scores? Very likely...how can she improve her scores?
She needs to evaluate the artifacts (old exams and quizzes) and figure out if she made conceptual errors (not understanding something) vs fuzzy errors (like 2 to the 3 power is 6 rather than 8!)

To prepare for an exam, she needs to practice problems over and over and over.
I have been a Math tutor for 13 years and work with ppl to get them to think on their own so they can do it when I am not there :)
Robin Schwartz BSEE MBA
www.mathconfidence.com

http://parenting.blogs.nytimes.com/2013/11/07/when-advanced-math-placement-is-a-struggle/#postComment

October 2013 Brain Teaser Solution

Q: Two babies born on the same day in the same year with the same mother and father are not twins. Can you explain how this can be? 
A: They are part of a set of triplets! Or Quadruplets! Quintuplets!

Don't Teach to a Test?? The SAT Can Make You College Ready and Smart too!!!

To the Editor:

If students learn the content on the SAT, they will be college ready so viewing test prep as an opportunity to learn new skills or strengthen existing knowledge can expand college major choices, career options and self-image!!
I always thought of myself as "only a Math person" which was reflected in my disparate SAT scores in high school (Math exceeded Verbal by over 200 points). Upon retaking the SAT at age 46, Reading was higher than Math!! This increase was due to three factors: more years of reading experience, lots of studying and an ability to "stay awake" to concentrate during the reading passages.  
Creativity and imagination in Math can be built through problem-solving and getting the Eureka!  Using multiple choice items can help students think about why the "good wrong answers" are wrong leading to better focus and self-awareness while learning more Math and perhaps even improving their scores!

Sincerely,

Robin A. Schwartz

Founder, www.mathconfidence.com

The New York Times did not publish my letter but this was in response to http://www.nytimes.com/2013/10/13/opinion/sunday/sunday-dialogue-too-tethered-to-tests.html?ref=opinion&_r=0

September 2013 Brain Teaser Solution

Q:
   SEND
+MORE
MONEY

  9567
+1085
10652
Figure out what number each letter stands for in the sum above.  Each letter represents its own digit (0-9) and multiple occurrences of the same letter represent the same digit.

M must be 1 as the two four digit numbers add up to a five digit number.  The highest four digit number is 9999 and if you add it twice you would get 19998.So M = 1

The rest of the puzzle is explained by Dr. Math on Math Forum here:
http://mathforum.org/library/drmath/view/60417.html

Math mistakes are a learning opp!

I already incorporate this idea of "mistakes are a learning opp" into my teaching having read Dweck's book which confirmed my instinct on my teaching style.  This was formed in engineering school in college as mistakes are so prevalent.  You get used to getting low scores and working hard for them.  When the average on a test is an 18 and that is a C you can take the mindset approach although we did not know it was called that and just work harder next time.  It was a bit crazy staying in Electrical Eng as my GPA was a 2.138 but it made me persistence and a good problem solver and it totally helps me to relate to my students!!!
Open and challenging problems are not necessarily the cure-all
You can learn a lot by studying and taking multiple choice tests
especially compare/contrast and also good wrong answers and why they are wrong!!!
Persistence is key -- that is how I got thru engineering school and I share this with many students as it was not easy nor GPA friendly but ppl love to hire engineers b/c we have been trained to be thinkers and persist!!
LOVE some of Prof Dweck's statements like:
"We should make kids feel cheated if the work is too easy for them"
"I can grow my Math brain"
"Math makes you work hard"
"Challenge is the new comfort zone"
but I disagreed with reminding teachers to keep track of progress b/c they probably already do this :)
3.7
Faster is not necessarily better but it sure helps with a lot of things
Obviously Laurent Schwartz did not know his own strengths
It would be great to interview someone who either has had a great experience with Math
and/or "the man/woman on the street" to get away from the academia perspective
Perhaps these interviews are meant to help the viewer feel empathy and connection but what about the student who has studied enough for the GRE to make it into the grad school of their dreams
How can you practice skills and knowledge if tests are too far apart?
Students learn while they take tests
It is not the testing culture -- ppl have felt this way abt Math for generations

August 2013 Brain Teaser Solution

Q: How many 5-digit numbers do not contain the numbers 3 and 5, and are multiples of 4?

(It cannot start with 0, ie. 01234 is not a 5 digit number) 

A: 9408
Multiples of 4 without 3s and 5s have the following last two digits: 00, 04, 08, 12, 16, 20, 24, 28, 40, 44, 48, 60, 64, 68, 72, 76, 80, 84, 88, 92, 96 this makes 21 possibilities for the last two digits together

The first digit has 7 possibilities: 1, 2, 4, 6, 7, 8, 9 

The second and third digits have 8 possibilities: 0, 1, 2, 4, 6, 7, 8, 9 

By using permutations, we multiply the different possibilities:

1st*2nd*3rd*(4th and 5th)=

7 x 8 x 8 x 21=9408

How to Learn Math: Schools and Mindset

If schools took on the mindset evidence seriously, what would they need to change?

"Mindset" refers to Carol Dweck's book


Schools would need to change parental attitudes and teacher attitudes and frankly societal attitude.
They can easily and inexpensively do this by listening to self-improvement experts like Jim Rohn, Tony Robbins, Darren Hardy and Robin Sharma.  These MP3s and books can be found at www.success.com
I do not know why the education sector has not embraced these philosophies!
Jim Rohn "The Challenge to Succeed" is the classic and has advice like "If you keep up the repetitive process, you can go from 5 pushups to 50!" and "How many years do you want your kid to be in 4th grade?  about 1!!"  and many many many other gems!!  These CDs help me with teaching but also with the rest of my life in parenting, relationships, fitness and finance.
The Challenge to Succeed CD set by Jim Rohn


I do incorporate these ideas into my teaching in the college classroom but not sure many of my colleagues are thinking about teaching and learning from this perspective.  I emailed our Academic Resource Center and Ed chair about spreading the mindset evidence and self-improvement idea that could change so many lives!


I think there should be a national campaign for adults to improve their cognitive ability.  Patricia Marx just wrote an article in the New Yorker on how she used brain training software which is cutting edge but we can use books from the SAT and especially ACT (since it is Common Core aligned!) and take the tests.  I took the ACT last year at age 48 -- I learned a lot of analytical science skills.  I highly recommend taking these exams as it puts us in the shoes of the student and makes us smarter -- I learned grammar in 2009 when I took the SAT ;)

How to Learn Math Do you think you can get smarter?

Question for students (and parents):
Do you think you can get smarter?

I ask this of my college students and at the end of the semester about 1/2 of them say "yes"
We think IQ is fixed when it is actually malleable.  This attitude may not take a learning disabled student to an Ivy League university just like some people are not likely to play professional basketball. But if you throw 100 free throws every day you will improve and if you do Math every day, you will also improve!

https://class.stanford.edu/courses/Education/EDUC115N/How_to_Learn_Math/about

Daily and Weekly Disciplines as of Summer 2013

Daily: A power walk with arms swinging above my head to two songs:  "Life" by Haddaway and "get Ready for This" by 2 Unlimited.   Then the cooldown is "Chariots of Fire" by Vangelis.

About 3 - 5 times/week: Workout for 30 mins either on the bike, elliptical or rower at the Y

Daily:  Stretch 10 minutes

Daily:  Green drink

About 3 - 5 times/week: at least 1 hour of fresh air

About 3-5 times/week: an average of about 2 hour of writing my book "Raising Math Confident Kids"

Daily: Clean at least 10 minutes

About 3 - 5 times/week: 10 minutes of project cleaning

About 3 - 5 times/week: listen to Jim Rohn or someone like him

Daily: Tweet, Facebook and LinkedIn sometimes set up previously on Hootsuite

July 2013 Brain Teaser Solution

Q: This is from the game show "Let's Make a Deal": Suppose you're on a game show, and you're given the choice of three doors: Behind one door is a car; behind the others, goats. You pick a door, say No. 1, and the host, who knows what's behind the doors, opens another door, say No. 3, which has a goat. He then says to you, "Do you want to pick door No. 2?" Is it to your advantage to switch your choice?

A: This famous problem from the early 1990's was first discussed and debated in Marilyn Vos Savant's (the person with the highest IQ!) Parade column.  She received many letters saying she was wrong but most people agree once they read through this:

Yes; you should switch. The first door has a 1/3 chance of winning, but the second door has a 2/3 chance. Here’s a good way to visualize what happened. Suppose there are a million doors, and you pick door #1. Then the host, who knows what’s behind the doors and will always avoid the one with the prize, opens them all except door #777,777. You’d switch to that door pretty fast, wouldn’t you?

and this :  http://marilynvossavant.com/game-show-problem/

June 2013 Brain Teaser Solution

Q: Between 1000 and 2000 you can get each integer as the sum of nonnegative consecutive integers. For example,
147+148+149+150+151+152+153 = 1050

There is only one number that you cannot get.
What is this number?

A: 1024

You cannot get 1024 because it is a power of 2
it is 2 to the 10th power
It is the only power of 2 between 1000 and 2000
For example 15 = 1+2+3+4+5
14 = 2+3+4+5
17 = 8 + 9
but 16 cannot be the sum of consecutive integers
neither can 2, 4, 8, 32, 64, 128, 256, 512, 1024, 2048 etc

Drawings of Number of Degrees in a Polygon

There are two less triangles in each shape than the number of sides
A triangle, is well, a triangle so 180 degrees
A square has two triangles in it so 360
A pentagon has three triangles in it so 540
The hexagon above was cut into two trapezoids so 2 x 360 = 720
See  the other post "Number of Degrees in a Polygon"

Number of Degrees in a Polygon

More Factoring with Algebra Tiles x^2 + 4x + 4

These algebra tiles form a square..hence the algebra that the tiles represent is a perfect square.
Look at each side of the square and you will see a length of x and then two little yellow squares making each side x + 2.
(x + 2)(x + 2) when FOILed is x^2 + 2x + 2x + 4 which when combining like terms becomes x^2 + 4x + 4.  If you look at the algebra tiles, there is the large blue square that represents x^2, four long green rectangles that are each x making 4x and 4 small yellow squares that represent 4.
see the post: http://mathconfidence.blogspot.com/2013/06/learning-factoring-with-algebra-tiles.html for more info and links on factoring polynomials with algebra tiles!

Learning Factoring with Algebra Tiles x^2 + 4x + 3

Algebra Tiles are a way of learning and teaching factoring skills.
The big blue square represents x^2 (x squared) and each green rectangle represents an x while the little yellow squares represent numbers (constants).
So if you look at both of these squares they are both composed of x^2 + 4x + 3
Factoring x^2 + 4x + 3 would be (x + 3)(x + 1) which if we FOILed would become x^2 + x + 3x + 3 and if we combined like terms would become x^2 + 4x + 3.
The rectangle on the left has dimensions x + 3 on the bottom and x + 1 at the top.
The rectangle on the right has the same total dimensions of x + 3 on the bottom and x + 1 at the top but is configured differently.
This is where Math is evidently a creative endeavor as two different students made these representations.
Intro to Algebra Factoring from Regents Prep
Another Factoring Example from Regents Prep
Student Set of Algebra Tiles from Amazon

SAT Book Wear and Tear in a PreEbook World

Do you think you studied enough?  Sometimes the amount of studying is evident by looking at the book.
This is what a used book that has been used can look like.
So far, the Official SAT Study Guide is not available as an e-book.
I will miss the dogeared pages and the falling off cover that come with repeated use of old-fashioned books!

Weekly Rhythm Register late May 2013

This shows my daily and weekly disciplines including stretching 10 mins a day so I can pick up the chalk!  A power walk with arms swinging above my head to two songs:  "Life" by Haddaway and "get Ready for This" by 2 Unlimited.   Then the cooldown is "Chariots of Fire" by Vangelis.

May 2013 Brain Teaser Solution

A function machine is when a number goes into this machine, a different number comes out. A basic formula is used to determine what comes out. If:
3 --> 90
4 --> 272
5 --> 650
6 --> 1332
7 --> 2450
What will come out if 10 goes in?
A:  10,100

3^4 + 3^2 = 81 + 9 = 90
4^4 + 4^2 = 256 + 16 = 272
5^4 + 5^2 = 625 + 25 = 650
6^4 + 6^2 = 1296 + 36 = 1332
7^4 + 7^2 = 2401 + 49 = 2450
...
10^4 + 10^2 = 10000 + 100 = 10100

Please Promote Math Positively!!

Inspired by a New York Times reporter who wrote in the Home section that "trying tiling was sort of like choosing to spend your weekend at an algebra slam!"

If we can promote Math in a positive way, it will help teachers and parents and most importantly students!

Please consider the way we speak of Math and how it can influence our students and society.  

Tiling 102 read the first paragraph

The article had photos of tiling tools so below please find some Math tools

How many ways can you arrange 10 books?

If you have 3 books, how many ways can you arrange them? 6 ways...there are 3 choices for  the first book, two choices for the second book leaving the last book to be the third one.  3 x 2 x 1= 6..this is also known as 3!  pronounced "3 factorial" as opposed to THREE!!
ABC
ACB
BAC
BCA
CAB
CBA

If you have 4 books, there are now 4! or 24 different ways to arrange these books.

Whats amazing is if you have 10 books, the number of possible arrangements is 10! or
10 x 9 x 8 x 7 x 6 x 5 x 4 x 3 x 2 x  1 which is 3,628,800

April 2013 Brain Teaser Solution

Q:Which unit fractions 1/2, 1/3, 1/4 etc have terminating decimals and which repeat? For example, 1/2 = 0.5 terminates but 1/3 = .3333etc and repeats.

How many unit fractions up to 1/1000 have terminating decimals?  What is the key to figuring this out?

A:  28
The first handful of fractions that terminate starting with 1/2 are:

1/2 = .5, 1/4 = .25, 1/8 = .125, 1/10 = .1

Notice the denominators are all composed of 2s and 5s

For example 4 = 2 x 2 and 8 = 2 x 2 x 2 and 10 = 2 x 5

This works becuase our number system is base 10 and base 10 is easy to express fractions as decimals if the factors of the denominator are 2s and/or 5s.

Here is a list of numbers that have 2 and/or 5 as their only factors:

2 4 5 8 10 16 

20 25 32 40 50 

64 80 100 125 128

160 200 250 256 320

400 500 512 625 640
800 1000

Is Utility the Only Reason for Math? Math vs The Odyssey

http://www.theatlantic.com/business/archive/2013/04/heres-how-little-math-americans-actually-use-at-work/275260/

Here's How Little Math Americans Actually Use at Work

Less than a quarter of employees do any calculations more complicated than basic fractions, and blue-collar workers generally do more advanced math than their white-collar friends.

and how many people use "The Odyssey" at work?

March 2013 Brain Teaser Solution

Q: A ship anchored in a port has a ladder which hangs over the side. The length of the ladder is 10 feet, the distance between each rung is 1 foot and the bottom rung touches the water.   The tide rises 18 inches per hour.  How long will it take for the water to reach the fifth rung from the top?

A: If the tide is rising, then the ship is also rising on the water. So water will still reach the bottom rung.

February 2013 Brain Teaser Solution

You must buy 100 pens for exactly $100 and purchase at least one pen from each of three stores. The first store charges 5 cents per pen, the second charges $1 per pen and the third charges $5 per pen. How many pens should you buy from each store?


Well I know it has to be an exact dollar amount. So the first store needs to be 20, 40 or 60 pens. I know the first 2 stores have to be a multipler of 5 for $ and the third store needs to be between 5 and 19 pens. And the second and third store has to be a multipler of 20 for pens.

So

80 Pens
1 Pen
19 Pens

The National Speakers' Association Challenge

The National Speakers Association is an amazing group of people who want to share their experise to help others reach their potential.

Math Confidence's Robin Schwartz loves sharing about Math and learning and would like to be the first NSA professional member to speak exclusively about Math and success.

Please support Robin's goal by thinking of organizations, groups of students or parents, teachers that would like a guest speaker who will inspire and motivate while encouraging mental fitness and fun!

MASTER YOUR MARKET AS A PROFESSIONAL MEMBER

Whether you want to grow or sustain your speaking business, reach a wider audience or generate revenue, a professional NSA membership will help you become a master of your market.

To be eligible, you only need to meet ONE of the following professional membership criteria:

• Option 1 – You have received compensation for 20 or more presentations within the 12 months prior to application. Supporting documentation can be any of the following: contracts, paid invoices, check copies, speaker agreements, tax returns or documents showing revenue has been generated as a result of your speaking engagements.

http://www.nsaspeaker.org/join/types-of-membership/professional#