October 2014 Brain Teaser Solution

Q: Can you transform the word THINK into BRAIN while changing only one letter at a time in a manner that each of the word in the process is a real word?


A:THINK

Replacing N with C
THICK

Replacing H with R
TRICK

Replacing I with A
TRACK

Replacing K with T
TRACT

Replacing C with I
TRAIT

Replacing T with N
TRAIN

Replacing T with B
BRAIN

September 2014 Brain Teaser Solution

Q: I traveled home in an old car that was leaking gas and could only be driven at a steady 30 miles per hour while getting gas mileage of 20 miles per gallon.   At the start of the trip, there were exactly 10 gallons of fuel but the fuel tank lost fuel at the rate of half a gallon per hour.   Just as I arrived home, the car stopped because it had run out of fuel and I had only just made it.  How far did I drive?

A: 150 miles

Solution: 30 mph with 20 miles per gallon means that 20 miles would use 1 gallon so 30 miles uses 1.5 gallons/hour and we must also include the 1/2 gallon an hour due to the leak. 

So the gas level is decreasing by a rate of 2 gallons per hour which means that the 10 gallons will last for 5 hours.

5 hours at the steady 30 mph = 150 miles

August 2014 Brain Teaser Solution

Q:

What is the next number in this series?

1, 11, 21, 1211, 111221, 312211 

 

A: 13112221 
as the last number listed above is 312211
one 3 one 1 two 2 two 1 

1, 11, 21, 1211 , 111221, 312211, 13112221, ...

The pattern is that the next term comes from counting how many of each numbers are in the previous term. For example, the second term is 11 because there is ONE "1" in the previous term. The third term, likewise, is 21 because there are TWO "1" in the previous term. Next term is ONE "2", ONE "1", and then ONE "1", ONE "2", TWO "1", and so on and so forth.

 

July 2014 Brain Teaser Solution

Q: 8, 5, 4, 9, 1, 7, 6, 10, 3, 2, 0...what is special about this sequence?

     Where would 11 fit into this sequence?

A: The numbers are arranged alphabetically so 11 would fit in between 8 and 5

Add Placement Test Examples: Comment to Higher Ed for Higher Standards

It is wonderful to see attention on this important national issue
It would be great to have samples of the placement tests on your web site:
http://media.collegeboard.com/digitalServices/pdf/accuplacer/accuplacer-sample-questions-for-students.pdf
http://www.act.org/compass/sample/index.html
These contain the content that students need to know especially at the community college level.
Please consider adding links on your web site to these important documents  so that stakeholders can familiarize themselves with the content.  Access to these examples could benefit students, teachers, parents and society at large.
Respectfully submitted,
Robin Schwartz
Founder www.mathconfidence.com
Adjunct Prof, College of Mt St Vincent

June 2014 Brain Teaser Solution

Q: Gauss said: "The cube and the fourth power of my age contain each digit from 0 to 9 exactly once."  The professor said: "It's a good coincidence because the square and the cube of my age contain each digit from 0 to 9 exactly once."  Find their ages.

A: 18 and 69
Looking at the last digit can give us many clues in Math solutions  and BC (before calculators) was how Math was often done!
In this case, we do not want to repeat a digit in both the cube and the fourth power because all the digits have to be different.
Here is a list of perfect cubes and perfect fourths from 1 to 10
Check out how the last digits are the same for 0 1 5 6
                 Cube                   To the fourth
1                  1                                 1
2                  8                               16
3                27                               81
4                64                             256
5              125                             625
6              216                           1296
7              343                           2401
8              512                           4096
9              729                           6561
10          1000                         10000

Their ages can't end with 0, 1, 5 or 6.  

See this spreadsheet: https://docs.google.com/spreadsheets/d/1rEkYk8uShyQ8gPfwTC4Losp-RPzNIYgvMo2nu0NMV4M/edit?usp=sharing

Gauss's age is easy. Since 17^3=4913, 17^4=83521, 22^3=10648, 22^4=234256, Gauss's age is between 18 and 21. The only possibility is 18. 18^3=5832, 18^4=104976.

The professor's age is a little harder. It can be 47 to 99, and the possibilities are: 48, 53, 54, 57, 62, 63, 69, 72, 78, 84, 87, 89, 93, 98, 99. By trial and error, 69 satisfies this condition: 69^2=4761, 69^3=328509.

Therefore, Gauss was 18 and the professor was 69.

 

May 2014 Brain Teaser Solution

Q: What is the largest number possible using only three digits?  

A:   9 to the 9th to the 9th9^9 does not equal 81. 
9^9=387420489 
387420489^9=?  a big number!!!!!!

April 2014 Brain Teaser Solution

Q: Find the two numbers whose digit sum is half of the number itself.

A: 0 and 18

March 2014 Brain Teaser Solution

Q: Find the two right triangles (with whole number sides) whose perimeter and area are 24.

Meant to ask 

Find the two right triangles (with whole number sides) whose perimeter and area are the same.

A: 6 8 10 and 5 12 13

February 2014 Brain Teaser Solution

Q; Jack and Jill  have two children and one of the children is a boy. Assuming that the probability of each gender is 50/50.  What is the probability that the other child is also a boy? 

A: 1/3

The following are possible combinations of two children:

Boy - Girl
Girl - Boy
Boy - Boy
Girl - Girl

Out of the above options 3 contain at least one boy and only one of them has two boys.  So once you know that one of them is a boy, you can just focus on these 3 possible outcomes and since only one of the outcomes has two boys, the probability would be 1 out of 3 or 1/3

The Pleasures of Teaching to the Test: SAT and ACT

SAT tutoring is teaching to the test as the College Board book has old tests in it! as does the ACT.

I wish the Common Core testing would be as straightforward...Grades 3- 8 Math and ELA have samples of last year's test (but not the whole test).  Grades 3-8 Sample Questions
Spoiler alert: The answers are on the same page as the questions

Here is an article from the Wall Street Journal that shares my view that if kids know what is on the test, it can make them smarter and help us to focus on what are important skills and knowledge.

January 2014 Brain Teaser Solution

Q: Can you circle exactly four of these numbers such that the total is twelve?

1 6 1

6 1 6

1 6 1

6 1 6

A: This brain teaser is about perspective as if you turn the page (ahem, ipad, tablet, PC screen) upside down the 6 will become a 9 and then it will be easy ;)

Positive Spin on Academics Please!!! SAT is an Opportunity

To the Editor:

We need good PR in Education and even more so in Math Education.  The front page article with "The Loser Has to Do Math" could be retitled "Loser Gets to Learn Math and Loves the Challenge" to inspire people to learn rather than normalize negative feelings about Math.

We, as a nation, can raise our SAT scores along with knowledge, skills and attitude by developing a positive outlook on testing.  This could prepare students for college and greatly reduce remediation while helping people realize that studying improves one's mind just as marathon training boosts fitness. 

A 2009 WSJ article on the SAT inspired me to retake the exam but it also had a glum view of testing with the phrase "the terrifying experience of the SAT" (http://online.barrons.com/article/SB124337711110856009.html).   I prepared by doing actual old exams from the College Board book and was able to raise my old 1980 verbal score by over 200 points.  This did away with 3 decades of thinking "I am just a Math person" and furthered my belief in the power of self-improvement and the ability to influence one's own future.  

Please consider how the Wall Street Journal which often publishes caring and insightful pieces on education can help students, parents, teachers and society view test taking as an opportunity rather than a threat.

Sincerely,

Robin A. Schwartz

Founder www.mathconfidence.com

Bronx NY

In reference to http://online.wsj.com/news/articles/SB10001424052702304887104579302523403171080

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