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
Tuesday, April 30, 2013
Sunday, April 28, 2013
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:
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
800 1000
Friday, April 26, 2013
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?
Thursday, March 28, 2013
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.
Thursday, February 14, 2013
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
Thursday, January 31, 2013
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.
Wednesday, January 16, 2013
January 2013 Brain Teaser Solution
Q: 2 Pencil Problems:
1. I have some pencils and some jars. If I put 4 pencils into each jar I will have one jar left over. If I put 3 pencils into each jar I will have one pencil left over. How many pencils and how many jars?
2. Again I have some pencils and some jars. If I put 9 pencils into each jar I will have two jars left over. If I put 6 pencils into each jar I will have three pencils left over. How many pencils and how many jars?
A: 1. 5 jars 16 pencils
2. 7 jars 45 pencils
(9p*J)-18 = (6p*J)+3
2. 7 jars 45 pencils
(9p*J)-18 = (6p*J)+3
Sunday, December 30, 2012
Managing Time for Success: Math Confidence Workshop
Managing Time for Success
Using productivity and organizational tools, performance and learning can be maximized while leaving time for other important activities including down time. This talk includes Robin's sharing her own system of time management including setting goals, listening to powerful speakers such as Tony Robbins and Jim Rohn, implementing tools like Franklin Covey planner and Darren Hardy's Weekly Rhythm Register. This talk can be for students as well as people in the corporate world, parents and anyone who would like to take their life to the next level!
College of Mount Saint Vincent, Weds. 1/30/13 4PM Elizabeth Seton Library |
Friday, December 28, 2012
December 2012 Brain Teaser Solution
Q: Paula and Georgia play a game with dice that have colors instead of numbers. Some faces are pink and some are green. Paula wins when the two top faces are the same color. Georgia wins when the colors are different. Their chances are even.
The first die has 5 pink faces and 1 green face. On the second die, how many faces are pink and how many are green?
A: 3 pink faces and 3 green faces
At first it may seem as if the second die should have the reverse of the first (1 pink and 5 green) but that would make Georgia win too often as the chance of different colors would be too high.
This can be done with a guess and check.
5 pink and 1 green matched with 2 pink and 4 green would give
P P
P P
P G
P G
P G
G G
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PG, PG, PG, GG, PP, PP, PG, PG, PG, GG,PP, PP, PG, PG, PG, GG, GP, GP, GG, GG, GG, GG
still too many that match giving Paula an advantage
This can be done with a guess and check.
5 pink and 1 green matched with 3 pink and 3 green would give
P P
P P
P P
P G
P G
G G
(keep the second column steady while changing the first die)
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP,
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)
If you want to keep the first column steady and change the second:
PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG,, PP, PP, PP, PG, PG, PG, GP, GP, GP, GG, GG, GG
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)
Solution 2:
Bill Adams said that he solved it using probability fractions:
Pink 5/6 x Pink 3/6 = 15/36 Pink Pink
Pink 5/6 x Green 3/6
Green1/6 x Pink 3/6
Green 1/6 x Green3/6 = 3/36 Green Green
Add those together and 15/36 + 3/36 = 18/36 = 1/2!
The first die has 5 pink faces and 1 green face. On the second die, how many faces are pink and how many are green?
A: 3 pink faces and 3 green faces
At first it may seem as if the second die should have the reverse of the first (1 pink and 5 green) but that would make Georgia win too often as the chance of different colors would be too high.
This can be done with a guess and check.
5 pink and 1 green matched with 2 pink and 4 green would give
P P
P P
P G
P G
P G
G G
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PG, PG, PG, GG, PP, PP, PG, PG, PG, GG,PP, PP, PG, PG, PG, GG, GP, GP, GG, GG, GG, GG
still too many that match giving Paula an advantage
This can be done with a guess and check.
5 pink and 1 green matched with 3 pink and 3 green would give
P P
P P
P P
P G
P G
G G
(keep the second column steady while changing the first die)
PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP, PP, PP, PP, PP, PP, GP,
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)
If you want to keep the first column steady and change the second:
PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG, PP, PP, PP, PG, PG, PG,, PP, PP, PP, PG, PG, PG, GP, GP, GP, GG, GG, GG
PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG, PG,PG,PG, PG, PG, GG
This makes 15 PP and 3 GG making 18 that are the same (and 18 different)
Solution 2:
Bill Adams said that he solved it using probability fractions:
Pink 5/6 x Pink 3/6 = 15/36 Pink Pink
Pink 5/6 x Green 3/6
Green1/6 x Pink 3/6
Green 1/6 x Green3/6 = 3/36 Green Green
Add those together and 15/36 + 3/36 = 18/36 = 1/2!
Friday, December 07, 2012
The Math Confidence Challenge
Take The Math Confidence Challenge
Study for and take the SAT and/or ACT
Math Coaching by Math Confidence
Robin has worked with children, young adults and adults since 2000 building Math knowledge, confidence and study skills for academic, professional and personal success. Her refreshing and comforting approach engages the learner and enables them to accomplish their academic and personal goals.
Some students have needed a study partner while others have needed a turnaround.
Many learners have improved their knowledge, skills, and scores preparing with Robin for standardized exams (state tests, SHSAT, Regents, SAT, SAT II) and/or classroom tests while boosting their attitude and comfort with Math. Students and their families benefit from the Math Confidence advantage that helps students learn, retain and understand the material in a relaxed and secure manner.
College level programs of study, outside of math-specific fields still require challenging math courses. Achievement in these courses, such as calculus, teaches students perseverance and problem-solving skills essential in many professional careers (i.e. medicine). These skills provide professional candidates with a competitive edge in their chosen fields and give a sense of personal accomplishment.
Prof. Schwartz designs and delivers workshops both as a presenter and facilitator and attends many seminars and conferences for lifelong learning. Her memorable nicknames for Math concepts such as "the lonely girl at the dance" for the middle term in (x+3)^2 and "the Math police" who drive up when fractions are 'illegally' canceled make learners smile (and remember)!
FAQ about Math and Math Confidence
Q: Why take Math?
A: Even nontechnical positions like advertising are appreciating quantitative skills according to the New York Times Ad Companies Face a Widening Talent Gap. The actual Math one learns in school may or may not be used on the job, but the process of learning and doing Math teaches problem solving and critical thinking as well as persistence. These skills are useful in work and in life.
Q. How can people build their Math Confidence?
A: 1. Practice!! Solve Math problems (Math opportunities) including multiple choice as compare/contrast with "good wrong answers" can increase knowledge, skills, attitude and scores!
2. Treat Math exams like athletic events -- prepare by studying but also by eating breakfast, getting a good night's rest and packing up the night before.
3. Embrace Math mistakes! Learning from errors can be challenging emotionally but will improve critical thinking, build confidence and expand educational and career options.
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Q: What are the top factors for success in Math?
A: Here are some top tips for academic success and confidence:
Be prepared and proactive
A review sheet is a gift
Metacognition (raise self-awareness about your thinking)
Attention to Detail (reduce fuzzy errors)
Self-Care (sleep, food, stress management)
Effort expended (the feeling of a job well done)
Q. Are Math facts important?
Some of us from the 1900s remember standing up and reciting facts in class to learn times tables. Very few schools still do this but automaticity of Math facts is advantageous as it builds confidence and speed and makes Math more enjoyable. Although many types of calculators are available on cell phones, there are also Math challenges like
Android App Math Workout and other great tools available likeGreat Fact Practice at Dad's Worksheets , Developing Automaticity with Math Facts from Mrs Russ and FASTTMATH for Automaticity of Math Facts.
Q. Isn't Math just formulas?
1. What is the equation of a line? 2. The slope formula?
3. Circumference of a circle? 4. Area of a circle?
If Math were just formulas then open book exams would be easy!
Many professors of Math and other technical subjects along with the Professional Engineering licensing exam allow open book exams.
A: No, Math is much more than formulas. Applying the formulas is the challenge -- many formulas are tricky (like exponent rules) so examples are best to improve recall. Progress can be made by thinking about and understanding the formulas. For example, volume is three dimensional, therefore, the units will be units cubed such as ft cubed or cm^3. For area , the answer will contain square units and perimeter (or circumference) will be just units.
Q: What are "fuzzy errors"?
A: How much is 2^0? (2 to the zero power) Answer
People often get the incorrect answer due to a lack of focus rather than a lack of understanding. Some common topics that illustrate this idea are: Area vs Perimeter, Exponent Rules, the word "NOT"
and the ubiquitous minus sign mistake.
One excellent way to reduce fuzzy errors is to study multiple choice items that have tempting good wrong answers."Is that your final answer?" is a way to help raise awareness and critical thinking -- it would be great if "Who Wants to be a Millionaire?" had an Algebra version!
Monday, November 19, 2012
November 2012 Brain Teaser Solution
Q: What mathematical symbol can be placed between 5 and 9, to get a number greater than 5 and smaller than 9?
A: A decimal point
A: A decimal point
Wednesday, October 24, 2012
October 2012 Brain Teaser Solution
Q:What value of X makes the seven digit number 6X92114 divisible by 11? (caveat: no calculator or long division)
A: 6
To determine if a number divisible by 11, add up the odd digits and compare that to the sum of the even digits. For example, 242 is divisible by 11 because the 1st digit (2) and 3rd digit (2) add up to 4 and is equivalent to the (only) even digit.
Zip Code 10471: the odd digits (digits 1, 3, 5) are 1, 4, 1 and add up to 6 while the even digits (digits 2 and 4) add up to 7. Therefore 10471 is not divisible by 11.
Zip Code 10065: the wealthiest zip code in the United States IS divisible by 11 as the odd digits (digits 1, 3, 5) are 1, 0, 5 and add up to 6 while the even digits (digits 2 and 4) add up to 6. Therefore 10065 is divisible by 11.
Therefore:
In the number 6X92114. the odd digits are: 6 9 1 4 and add up to 20
The even digits are x 2 1 and add up to x + 3
If we set x + 3 = 20 then x would be 17 which cannot fit in one digit
So we have to think about what happens when the numbers add up to more than 11
For example 990 is divisible by 11 as is 979 and 968 -- notice that in 979 the sum of the odd digits is 11 more than the even digit (9 + 9 is 11 more than 7). This is also true in 968 as the sum of the odd digits is 17 which is 11 more than the even digit of 6.
For 6X92114, if we set x + 3 = 20 then x would be 17 which cannot fit in one digit but just like 968, we can use a 6 instead of a 17 (notice that 6 is 11 away from 17!)
A: 6
To determine if a number divisible by 11, add up the odd digits and compare that to the sum of the even digits. For example, 242 is divisible by 11 because the 1st digit (2) and 3rd digit (2) add up to 4 and is equivalent to the (only) even digit.
Zip Code 10471: the odd digits (digits 1, 3, 5) are 1, 4, 1 and add up to 6 while the even digits (digits 2 and 4) add up to 7. Therefore 10471 is not divisible by 11.
Zip Code 10065: the wealthiest zip code in the United States IS divisible by 11 as the odd digits (digits 1, 3, 5) are 1, 0, 5 and add up to 6 while the even digits (digits 2 and 4) add up to 6. Therefore 10065 is divisible by 11.
Therefore:
In the number 6X92114. the odd digits are: 6 9 1 4 and add up to 20
The even digits are x 2 1 and add up to x + 3
If we set x + 3 = 20 then x would be 17 which cannot fit in one digit
So we have to think about what happens when the numbers add up to more than 11
For example 990 is divisible by 11 as is 979 and 968 -- notice that in 979 the sum of the odd digits is 11 more than the even digit (9 + 9 is 11 more than 7). This is also true in 968 as the sum of the odd digits is 17 which is 11 more than the even digit of 6.
For 6X92114, if we set x + 3 = 20 then x would be 17 which cannot fit in one digit but just like 968, we can use a 6 instead of a 17 (notice that 6 is 11 away from 17!)
Saturday, September 29, 2012
Promoting Academic Integrity: Prof Somerset Syracuse U
In Statics and Dynamics, the one Mechanical Engineering class we had to take was MEE 225 first semester sophomore year in the Fall of 1982. Our professor, James Somerset, programmed his computer to generate unique exams for each student. There were 25 problems and you had to match them with answers in a 36 answer grid -- there were 11 extra answers. We all got our own data. For example, a _A__kg car is traveling at __B_m/s. We also got different As and Bs.
This promoted academic integrity as there was absolutely no way to cheat as we were all doing different problems!!
This promoted academic integrity as there was absolutely no way to cheat as we were all doing different problems!!
Saturday, September 22, 2012
September 2012 Brain Teaser Solution
Q: When a four digit number wxyz is multiplied by 4, the digits are reversed and it becomes zyxw. All the digits are different. What is the number?
A: 2178
If the number is still 4 digits when multiplied by 4, then the number has to be less than 2500 (as 2500 x 4 = 10000 and you would then have 5 digits).
That makes w = 1 or 2
But when multiplying by 4, the last digit will never be 1 so when we reverse to zyxw, the w must be 2 so that the last digit will then be 2. The multiples of 4 that have a last digit of 2 are 3 x 4 and 8 x 4.
Since we are multiplying a number in the 2000s by 4, the number is becomes will start with an 8.
w = 2 and z = 8
The second digit x must be 4 or less as it says above, the number must be less than 2500.
Because 2250 x 4 = 9000, x must be 0 1 or 2
Through some trial and error, you can find that the number is 2178 and its reverse is 8712.
If the number is still 4 digits when multiplied by 4, then the number has to be less than 2500 (as 2500 x 4 = 10000 and you would then have 5 digits).
That makes w = 1 or 2
But when multiplying by 4, the last digit will never be 1 so when we reverse to zyxw, the w must be 2 so that the last digit will then be 2. The multiples of 4 that have a last digit of 2 are 3 x 4 and 8 x 4.
Since we are multiplying a number in the 2000s by 4, the number is becomes will start with an 8.
w = 2 and z = 8
The second digit x must be 4 or less as it says above, the number must be less than 2500.
Because 2250 x 4 = 9000, x must be 0 1 or 2
Through some trial and error, you can find that the number is 2178 and its reverse is 8712.
Friday, September 07, 2012
Regents Scoring and new 3 - 8 Math sample items
http://cityroom.blogs.nytimes.com/2012/09/06/at-a-new-school-a-math-lesson-from-the-mayor/#postComment
Jeff S is correct: if students get 30 points out of 87 (34%), it gets scaled up to a 65.
Here is the conversion chart for the most recent Integrated Algebra Regents -- it is still thankfully up on line as the actual exam ;)
http://www.nysedregents.org/IntegratedAlgebra/612/ialg62012-cc.pdf
Unfortunately, the latest 3 - 8 exams have not been posted due to the changeover to the Common Core. The latest exams for grades 3 - 8 are 2010: Here is an example: http://www.nysedregents.org/Grade5/Mathematics/home.html
However, there are new sample items for Grades 3 - 8 and they look more challenging than the old exams -- check them out:
http://www.p12.nysed.gov/assessment/common-core-sample-questions/ (scroll to the bottom)
Sunday, August 19, 2012
August 2012 Brain Teaser Solution
Q: A group of four people has to cross a bridge at night. A flashlight is needed to light the path and they only have one flashlight. No more than two people can cross the bridge simultaneously, and it takes different time for the people in the group to cross the bridge: Abby crosses the bridge in 1 minute, Bob crosses the bridge in 2 minutes, Cathy crosses the bridge in 5 minutes, David crosses the bridge in 10 minutes. How can the group cross the bridge in 17 minutes?
With many thanks to:
http://www.math.utah.edu/~cherk/puzzles.html
A: Abby and Bob go across: 2 minutes
Bob returns with flash: 2 minutes
Cathy and David go across: 10 minutes
----------
14 minutes
Now Abby, Cathy and David have crossed, so one of these needs to return
with the flashlight and afterwards go across with Bob in 3 minutes:
With many thanks to:
http://www.math.utah.edu/~cherk/puzzles.html
Tuesday, July 24, 2012
July 2012 Brain Teaser Solution
Q:What are the largest and smallest 5-digit numbers that satisfy the following conditions?
1. Each digit of the number is a prime digit.
2. Each successive pair of digits forms a 2-digit number that is NOT a prime number.
3. Each of the prime digits must appear at least once in the 5-digit number.
A: Largest: 35772
Smallest: 32257
Definition of Prime Numbers from Coolmath.com
Prime digits are 2, 3, 5 and 7. 1 is not considered a prime number or digit.
Taken as pairs, the only combinations that satisfy condition 2 are:
22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75 and 77.
Of that list, 33 is the only number that contains a 3 as the second digit and so for condition 3 to be satisfied, 3 can only appear at the start of the 5-digit numbers.
Many thanks to Braingle
1. Each digit of the number is a prime digit.
2. Each successive pair of digits forms a 2-digit number that is NOT a prime number.
3. Each of the prime digits must appear at least once in the 5-digit number.
A: Largest: 35772
Smallest: 32257
Definition of Prime Numbers from Coolmath.com
Prime digits are 2, 3, 5 and 7. 1 is not considered a prime number or digit.
Taken as pairs, the only combinations that satisfy condition 2 are:
22, 25, 27, 32, 33, 35, 52, 55, 57, 72, 75 and 77.
Of that list, 33 is the only number that contains a 3 as the second digit and so for condition 3 to be satisfied, 3 can only appear at the start of the 5-digit numbers.
Many thanks to Braingle
Wednesday, June 27, 2012
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.
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.
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