MathNotations

Posted by Dave Marain at 3:15 PM 0 comments

Labels: Common Core, geometry, higher-order thinking, math challenge, SAT strategies, SAT-type problems, standardized tests

Posted by Dave Marain at 7:44 AM 0 comments

Labels: math challenge

Posted by Dave Marain at 4:30 PM 2 comments

Labels: geometry, math challenge

Posted by Dave Marain at 7:17 AM 0 comments

Labels: geometry, math challenge

If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. Suitable for SAT I, Math I/II Subject Tests, Math Contests and Daily/Weekly Problems of the Day. Includes multiple choice, cases I/II/III type and constructed response items.
Price is 9ドル.95. Secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL (dmarain "at gmail dot com") FIRST SO THAT I CAN SEND THE ATTACHMENT!
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DON'T FORGET TO VISIT ME ON TWITTER AT twitter.com/dmarain

TODAY'S TWITTER PROBLEM - A CLASSIC GEOMETRY CHALLENGE
A regular octagon is formed by cutting congruent isosceles right triangles from the corners of
a square of side 1. What is the length of a side of the octagon?

[Ans: ≈ 0.414; also give "exact" answer!]

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur SchoDONTpenhauer (1778-1860) You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

Posted by Dave Marain at 8:14 AM 0 comments

Labels: geometry, math challenge

Here are the last two math challenges I just tweeted for middle schoolers and beyond. You may want to use this as a fifteen minute activity to improve reading, review basic terms and concepts, develop reasoning and writing in math. There was an error on the 2nd question as it originally appeared on Twitter. I then corrected it.

List the nine 2-digit primes which produce prime numbers when their digits are reversed.




List the SIX 3-digit primes which produce primes when their digits are written in ALL possible orders. 137 fails b/c 371 is not prime.

For both questions students should work in teams of 2-4.

For the first question, students should not be allowed to use a calculator!

For the second one, have them experiment with a calculator for a few minutes. If a student thinks they found one, their teammates must verify it! After 3 minutes ask: "Have you noticed that the numbers you're looking for cannot contain certain digits like 2. What digits and why? Discuss it and one member of the team must record the team's findings and provide a written explanation!

After 3-4 more minutes, have them refer to a table of primes online (or print it and hand out a copy to each team). If they don't find it within the 15 min time limit, have them finish it for extra credit for the next day.

Here is one of the numbers: 113. Good luck!


"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

"You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught." --from South Pacific

Posted by Dave Marain at 10:22 AM 2 comments

Labels: explorations, investigations, math challenge, middle school, primes

Students who have been out of geometry for a year or so and are preparing for standardized test like Math I Subject Test or SATs/ACTS need occasional review. The following is similar to several other cone problems I've posed before but even our strongest Algebra 2 through Calculus students lose their "edge" when it comes to "solid" geometry questions (yes, believe it or not, my terminal course in high school was called Sold Geometry and we covered topics like spherical trigonometry!).


A right circular cone of height 16 is inscribed in a sphere of diameter 20. What is the diameter of the base of the cone?


Reflections....

1) Are these kinds of problems somewhat hard merely because students forget? I can think of several more reasons:

• The problem itself is somewhat challenging, however it's far from over their heads!
• The student never experienced a question like this in Geometry; perhaps questions like these were in the B or C or D exercises in the text and were never assigned or only for the "honors" students? Do you recall seeing a problem similar to this in the textbook from which you taught?
• The student did not take a formal course in geometry
• The topic was covered in a cursory manner or perhaps not at all because of time crunch. That's the whole point of a standardized curriculum, isn't it? To know what is needed to be covered and plan accordingly. Of course, I'm a realist enough to know the myriad of reasons why the best laid plans oft go .........
• Students don't remember how to start because key geometry strategies were not explicitly stated and reiterated ad nauseam. Were your students asked daily to begin by reciting the key strategies such as those for circle and sphere problems? Were they placed on index cards or blocked out in a particular section of their notebook?:
• DRAW THE BEST DIAGRAM YOU CAN (and believe me, I'm no artist!)
• Always locate the CENTER of circles, spheres and label the point
• Label the measurements of all segments (angles) - I know, everyone does that!
• Successful problem-solving in mathematics is based on finding relationships! Were guiding/leading questions asked
• What do the cone and sphere have in common?
• TRUE FALSE The height of the cone is the same as the diameter of the sphere. EXPLAIN!
• Was the student exposed to the strategy of comparing the 2-dimensional analogue of the 3-D problem? Would it be a right triangle in a circle? Equilateral triangle inscribed in a circl or???
• Oh and yes...
• Draw the radius of the sphere (or circle) so that it is the hypotenuse of some right triangle!

"All Truth passes through Three Stages: First, it is Ridiculed... Second, it is Violently Opposed... Third, it is Accepted as being Self-Evident." - Arthur Schopenhauer (1778-1860)

You've got to be taught To hate and fear, You've got to be taught From year to year, It's got to be drummed In your dear little ear You've got to be carefully taught. --from South Pacific

Posted by Dave Marain at 8:37 AM 1 comments

Labels: 3-dimensional geometry, cone, cone in sphere, instructional strategies, math challenge

I've been working on a new website which I will share with you when ready but I haven't forgotten my faithful readers who may have forgotten me!

There are so many issues in mathematics education that it would take forever to update you on all of them, however, I know that you are already aware of most of these.


Some Significant Current Issues in Math Ed

• Moving Inexorably Towards Common Standards in Math
• Teachers Need a Clear Curriculum Map/Content Guide rather than Standards!
• Rapid Push Toward Including Several Open-Ended Questions on State or Common Assessments is Slowing Down. Can you think of the major reasons for this?
• Joel Klein's Education Equality Project whose goal is to close the Achievement Gap

Of course, most of you have already skipped down to the Challenge Problems!

The first can be tackled by middle schoolers, although many high schoolers may find it interesting and fall into a trap if not careful. The wording is challenging but your students may benefit from working in small groups.

Challenge Problem #1
a, b, c, d and e are positive integers with a ≤ b ≤ c ≤ d < e.
If a + b + c + d + e = 143, what is the least possible value of e?

Comments:
Is this merely a guess-test-revise question or is there a strategy/method your students can come up with? How would you extend this problem? change the "143" to a larger value? Change the set of integers to 4 values (a,b,c,d)? 6? k? This is an important issue. Otherwise students may see each problem as an isolated quickly solved puzzle!


The goal of the next question is to review geometry and algebra skills and concepts and to encourage a variety of approaches. I will give the answer -- the challenge for your students is to find AT LEAST THREE METHODS! The teacher may want to submit the best team's efforts to me for acknowledgment on this site.



Challenge Problem #2

P(5,1), Q(8,2) and R(a,b) determine an isosceles right triangle with point R above line PQ and ∠ PRQ the right angle. Determine the coordinates a and b. In your group, you must devise at least THREE methods!

Answer (6,7)
Methods???






"All Truth passes through Three Stages: First, it is Ridiculed...
Second, it is Violently Opposed...
Third, it is Accepted as being Self-Evident."
- Arthur Schopenhauer (1778-1860)

"You've got to be taught
To hate and fear,
You've got to be taught
From year to year,
It's got to be drummed
In your dear little ear
You've got to be carefully taught."
--from South Pacific
Note: These lyrics provoked considerable criticism back in 1949-50 but Rodgers and Hammerstein would not take them out. Do they still have relevance today?

Posted by Dave Marain at 8:10 AM 10 comments

Labels: geom, math challenge, middle school, update

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students.

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.). If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.


Update: Answers/comments are at the bottom...

1. NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!

2. If 10^-1000 - 10^-997 is written as a decimal, answer the following:


(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.



--------------------------------------------------------------------------------------------


ANSWERS


1. 9√3


2. (a) 1000 (b) 3 (c) 997


An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10^-1 = 1/10 = .1 Note that there is one decimal digit.

10^-2 = 1/10² = 1/100 = .01 Note that there are two decimal places, etc.


10^-1000 - 10^-997 = 1/10¹⁰⁰⁰ - 1/10⁹⁹⁷
Using 10¹⁰⁰⁰ as the common denominator, we obtain
1/10¹⁰⁰⁰ - 10³/10¹⁰⁰⁰ =
-999/10¹⁰⁰⁰ from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10^-1000 but I chose rational form for the younger student.

Posted by Dave Marain at 9:53 AM 2 comments

Labels: 30-60-90, geometry, math challenge, middle school, SAT strategies, SAT-type problems, warmup

Summer vacation is an appropriate time for fantasy. Enjoy the hiatus!



The following investigation is not intended to be a math contest challenge. It reviews fundamental principles of probability and you might want to bookmark it for the fall. We can also simulate the first problem using the programming capabilities of a graphing calculator. I may post a simple program for this later on.


The wizard will let Dorothy go home if she can pass three challenges.

He shows Dorothy 3 playing cards, 2 of which are black and one is red. He shuffles them and turns them face down. "Dorothy, here's your first challenge."

"You will pick a card. If it's red the game ends, you win the game. If it's black, I will remove the card and you will pick a card from the remaining two. If it's red you still win! Ah, but if it's black again you and Toto and your weird friends will remain here for at least one more month."

Well, Dorothy won the game and said, "Now, I want to go home!" But the crafty wizard said, "You weren't listening carefully, Dorothy. I never said you can go home if you won the game. You've only passed the first challenge. You must still pass two more." "That's not fair!" Dorothy protested but the wizard makes his own rules in Oz.

"Alright, Dorothy, you won the game but you knew the odds were in your favor since you had two chances to win. Here's your next challenge:

"What was the probability of your winning and you must give me two correct but different methods?"

Dorothy asked, "These are the remaining challenges, so if I get them right, I can go home, yes??"
"I will not lie to you, Dorothy. This is your 2nd challenge. There will still be one more."

Dorothy was upset but knew she had no choice but to trust him. She thought about the problem for a minute and replied, "The probability of my winning was 2/3. I know I'm right!"
"Very good, Dorothy, but you must explain that answer two different ways." Fortunately, Dorothy was a very responsible middle school student back in Kansas and had learned the methods of compound probabilities and the idea of complementary events (this is a fantasy after all!).

Dorothy was able to provide two correct methods. Can you?

"Very good, Dorothy! You only have one more challenge to conquer and you can go home.
This time there are N cards, one of which is red while the remaining cards are black. N is a positive integer greater than 1. Same rules as before. The cards are shuffled and laid out face down. You pick a card. If it's red the game is over and you win. If it's black, the card is removed and you try again. The game continues until you pick the red card. The only way to lose the game is if you pick all the black cards and the last card remaining is red."

"In terms of N, what is the probability that you will win? Oh, yes, you again have to show two different methods in detail on this magic board over here."

This time, Dorothy needs your help. She can guess the formula but she needs our help to show two ways to derive it. Please help Dorothy go home!

Posted by Dave Marain at 6:41 AM 4 comments

Labels: compound probability, investigations, math challenge, middle school, probability

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a^-3/2 - a^-1/2 - a^1/2 = 0

Preliminary Comments/Questions/Issues

• Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  
• Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  
• It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  
• Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
• What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  
• Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...
  

Posted by Dave Marain at 6:39 AM 7 comments

Labels: advanced algebra, complex numbers, exponents, instructional strategies, math challenge, math contest problems

Number puzzles always intrigued me and, perhaps, they are one way we can invite our students into the wonderful and exciting world of mathematics. Oh, alright, maybe that's a bit of a stretch, but, I suspect that if you give the following famous puzzle to your students in Grades 5 and up, they will try it even if you don't offer food or a 10 point bonus! Yes, calculators are allowed but after a few minutes of frustration they will be begging for a hint.

(Oh, and if you give them this problem at the beginning of class, you may as well forget the lesson!)

Find two 5-digit numbers whose product is 123456789.


If you solve it, don't post your answer immediately. I will probably publish a hint or the answer in a day or so. You can always email me with your solution at "dmarain at gmail dot com."

Click Read more for a hint and comments...


HINT: Rather than pressing random numbers into the calculator as some would do, encourage them to find the prime factors of 123456789. It's easy to show that this number is divisible by 3 and 9, but find finding the other factors will be challenging. I'll post another hint if you request it...

COMMENT: This beautiful puzzle was invented by Y. Yamamoto and has intrigued many puzzle enthusiasts for awhile now. Is there some profound meaning behind the solution or is it just a curiosity? Perhaps we'll have to wait for Dan Brown's next book to unlock the mystery! I will probably post the answer if I don't get a response within 24 hours. Probably...

If any of your students solve it, email me at "dmarain at gmail dot com" and let me know if I can post their names.

...Read more

Posted by Dave Marain at 6:31 AM 3 comments

Labels: math challenge, middle school, more, puzzle

Just a 'little' last-minute challenge before Turkey Day -- similar to many you've seen before on this blog and elsewhere...

Determine the exact digits of 100²⁰⁰⁸ - 100¹⁰⁰⁴.


Comments:
Students in middle school or higher will often (or should) employ the "make it simpler and look for a pattern" strategy. Some students will be able to apply algebraic reasoning (factoring, laws of exponents, etc.) to evaluate. It's worth letting students, working in pairs, 'play' with this for awhile, followed by a discussion of various methods. Then challenge them to write their own BIG exponent problem!

Posted by Dave Marain at 6:56 PM 4 comments

Labels: algebra, exponents, math challenge, middle school, patterns

The following problem is certainly appropriate for later in the year when geometry students reach this topic but it can also be used to review a considerable number of essential ideas in preparation for SATs, ACTs or just review in general. It's at the top end on the difficulty scale for these tests, but it's far from the AMC Contest!


Clarifications: Figures are not drawn to scale and the measure of ∠TAU is given in each diagram.

For each of the figures above, determine the following:
(a) the radius of each circle
(b) the length of minor arc TU in each circle

Have fun discovering a variety of approaches!

Variations? Generalizations? Choosing an angle other than special cases like 60 or 90 generally requires trig -- not that there's anything wrong with that!

Posted by Dave Marain at 7:18 PM 2 comments

Labels: circles, geometry, math challenge, tangents

Don't forget our MathAnagram for Aug-Sept. Thus far we have received a couple of correct responses. You are encouraged to make a conjecture!
Look here for directions. Here is the anagram again:

---------------------------------------------------------------------------------------------------------------
A former student sent me a wonderful reasoning problem involving mean, median, and mode, so it is accessible to middle schoolers. The question came from his teacher so I decided to revise it, put it in a different context but preserve the essence of the logic. The student will need to know some basics of scoring in golf but most of it should be clear. If not, this may help.

This kind of question will frustrate some but reasonable frustration can often lead to 'pearls of wisdom.' Clear thinking and careful attention to detail is necessary. Certainly basic knowledge of measures of central tendency is a prerequisite, but this question can also serve to review these ideas.

Have fun with it yourself and, if you can, try it as a 5-minute warmup in class, preferably with students working in pairs. Let us know if they make a 'hole in one'! Again, thanks to my student and his teacher for the original source of this challenge.

Alex played 18 holes of golf and we know the following information:
His maximum score on any hole was a '5' and he shot this on six holes.
His median score on the 18 holes was 4.
The mode was 3.
What was the lowest possible mean score he could have achieved on the 18 holes?
Express your answer to 2 places (rounded).

Posted by Dave Marain at 4:29 PM 16 comments

Labels: logic, math challenge, mean, statistics

Edit: #4 below has been corrected. I am indebted to one of mathmom's astute students for catching my error!

Posted by Dave Marain at 7:09 AM 13 comments

Labels: math challenge, middle school, number theory, remainders

Don't forget to submit your solution to this month's Mystery Mathematicianagram (ok, so I can't decide on a name yet!). We've received 3 correct solutions thus far and I will announce winners around the 20th.







As we wind down the school year, the problems below may come too late for students taking their final exams in geometry, but you may want to hold onto this classic puzzler for next year. I don't consider these overly challenging but I do feel they demonstrate some important mathematical ideas and problem-solving techniques. Further, encourage students to justify their reasoning since some may make assumptions from the diagram without verification. This will review some nice ideas from circles.

OVERVIEW OF PROBLEMS (see diagram)
For both questions, assume the circles are concentric, segment PQ is a chord in the larger circle and tangent to the smaller.

PART I
If PQ = 10, show the difference between the areas of the 2 circles is 25π.

PART II (the converse)
If the difference between the areas of the circles is 25π, show that the length of PQ must be 10.

Notes
(1) It is important for students to recognize that there are many possible pairs of concentric circles (varying radii) satisfying the hypotheses of these problems, yet the conclusions are unique! Some students will assume a 5-12-13 triangle is formed (not a bad problem-solving strategy), but stress that this is not the only possibility! Remember, we're not restricting the radii to integer values.

(2) There is a classic math contest strategy for these questions that mathematicians love to employ - the "limiting case." Can you guess what I mean by this phrase?

Posted by Dave Marain at 6:24 AM 4 comments

Labels: circles, geometry, math challenge, tangents

[Don't forget the Mystery Math Anagram for this month. Only two correct replies have been received thus far. I will announce the winners in a few days.]

Have you been wondering where the math challenges have gone on this blog? Here's one that I came across while reading David Baldacci's recent best seller, Simple Genius, just your usual tale of the dark world of mathematicians, codes and spies. Gee, math has become such an integral part of novels, TV shows and movies over the past few years, our students are going to think the life of a mathematician is really cool and exciting (which, as we all know, it is!).

Anyway, here is a paraphrasing of the problem (as long as I'm not copying the problem verbatim, the publisher granted me permission to discuss this):

Alex is as many months old as his grandpa is in years and about as many days old as his dad is in weeks. If the sum of their 3 ages is 140, how old is each?


Hint: This is a wonderful problem demonstrating the power of ratios. If you can solve it less than 20 seconds, then you're either an honorary member of Mensa or you could be the subject of Baldacci's next book!

Comments

(1) Like all riddles, the wording is somewhat convoluted and the mathematical assumptions are not explicitly stated. But that's part of the intrigue here. I will say that one needs to assume the ages are integers, but that's about it.

(2) In the novel, the problem is posed to a young mathematical prodigy named Viggie. While another mathematician in the room takes some time to solve it algebraically, Viggie comes up with the solution mentally in a few seconds. Can you!

(3) You may want to give this to middle school students, although the wording might frustrate them. You could demonstrate the idea with a concrete example or make it into a simpler problem:
Let's say that Alex is 96 months old, then his grandpa would be 96 years old. Now ask them to determine how old Alex's dad would be. This may be challenging enough...

(4) I'm naturally wondering what the source of this problem is. If anyone out there recognizes it, let us know its source!

Posted by Dave Marain at 6:03 AM 9 comments

Labels: age problem, math challenge, ratios, riddle

First a humorous aside from one of my friends on another message board. A friend emailed it to him so it's probably making the rounds of the web. In case you haven't seen it, here it is...

A recent study found that the average American walks about 900 miles a year.

Another study found that Americans drink, on average, 22 gallons of alcohol a year.

That means, on average, Americans get about 41 miles to the gallon.

Kind of makes you proud!



One of our new and devoted readers, Florian, contributed the following unusual digits by algorithmic construction problem. This is a wonderful example of a different type of solution, since a standard algebraic approach should prove fruitless. Florian is our resident computer scientist. That should help you understand how he devised this question.

Suppose a₁a₂a₃...a_n-16 represents an n-digit positive integer whose units' digit is 6. Find the least such positive integer satisfying the property that when the number is multiplied by 2, the result is 6a₁a₂a₃...a_n-1 , the n-digit number whose digits are the same as the original number except that each digit is shifted one position to the right and the rightmost digit '6' rotates to the leftmost position.

Have fun looking for this 18-digit number! Would a calculator be useful here?

Variations and Extensions:

Here is how one could modify this for middle schoolers:
(i) Give them the 18-digit number to start with (sorry, I'm not giving this away yet), have them multiply it by 2 using paper and pencil and see how long it takes for various students to see the surprising result. (Yes, Steve, they actually are expected to multiply with accuracy!)
I guarantee they will express surprise!
(ii) Now ask them to figure out how they could construct the digits of the mystery number, one digit at a time. Some will catch on quickly, others will need guidance.
(iii) What questions should occur to students as they are building this number? You may need to ask them if they believe this process eventually has to terminate.

Extension for the Very Highly Motivated (or for people like me who need to get a life!):

Construct the 42-digit number a₁a₂a₃...a₄₁5 (ending in the digit '5'), which when multiplied by 5 is of the form: 5a₁a₂a₃...a₄₁, in which the result has the same digits as the original number with each digit shifted one position to the right and the rightmost digit rotated to the leftmost position.

Note: Check my accuracy on this!

Posted by Dave Marain at 5:36 PM 5 comments

Labels: algorithms, math challenge, math humor

What is the sum of the digits of (googol + 1)(googol - 1) when expanded?

Comments:
(1) Google 'googol' if you need some background!
(2) Does the strategy of 'make it simpler' work well here?
(3) Can you invent a similar problem or, better, have your students devise their own!
(4) Oh, BTW, NO CALCULATORS!!
(5) I felt I needed a change of pace from the heavy math ed stuff from the past few days. You too?

Posted by Dave Marain at 3:31 PM 12 comments

Labels: math challenge, problem-solving

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