MathNotations

THE FULL MONTY HALL REVEALED

2012-05-22T06:38:00.000-04:00

READ COMMENTS TO GET FULLER PICTURE!

I'm the host, you're the player.
I shuffle 3 cards, 2 of which have the word "LOSE" on them, one has "WIN".

You randomly select a card but you're not allowed to turn it over and I do not turn over my 2 cards.
AT THIS POINT, WHO IS MORE LIKELY TO HOLD THE WINNING CARD?

I look at my cards and reveal a losing card.
NOW, WHO IS MORE LIKELY TO HOLD THE WINNING CARD!

I ALLOW YOU TO SWITCH TO THE REMAINING FACE DOWN CARD. SHOULD YOU?

I WOULD!

Hey, I figured I'd try my "hand" at this classic too! An important point here is whether my model of the original puzzle is equivalent.

Your thoughts?

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Rates of Growth Imagined

2012-05-17T07:04:00.000-04:00

Which growth rate will make the Hulk taller?

A growth of 60% for the year OR
1% growth per week?
EXPLAIN!!

POLYANAGRAM

Fresh fruit is so expensive these days. I cannot find a _ _ _ _ _ _ _ _ _ _.

Fill in blanks, with two 5-letter words which are anagrams of each other.

First 3 correct answers to the math problem (with explanation) and the PolyAnagram will win my Challenge Math Book. Email me at dmarain at gmail dot com.

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SAT PLUS, DEF OF THE DAY, ETC...

2012-05-15T07:06:00.001-04:00

DYSFUNCTIONAL
DEF: ONE WHO STRUGGLES WITH FUNCTIONS

The x- and y-intercepts of a line are 2t^3 and 3t respectively. If the slope of a perpendicular line is 3/2, the positive value of t is ?

Ans: 3/2

RANDOM THOUGHTS
1. I've received several thoughts re my PolyAnagrams. I'm a word puzzle fanatic as you might have guessed by now and I enjoy writing these. Let me know if you'd like to see more or restrict a math blog to math!

2) I'm actually thinking of writing 50 of these and offering it on Amazon for a couple of bucks. My question for my readers is, would you buy it?

3) I'm still frustrated by reviews of this blog that no one comments that it is essentially intended for teachers. I use the problems as a vehicle for deeper reflection about our practice. That's why I usually ask a series of questions after the problem. Does anyone actually read these!

4) I noticed that my post about an explanation of one of my problems drew more readers than all others combined! Should I interpret that to mean that my readers want to see solutions more than answers? Pls comment!

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SAT List and Count and a PentAnagram

2012-05-12T06:51:00.000-04:00

For how many pairs (x,y) of positive integers is 2x+3y<24?

Ans:37

Ans to QuadAnagram:
FILER,RIFLE,FLIER,LIFER

Today's PentAnagram!
Complete the sentence with FIVE 4-letter words which are anagrams of each other.

Mr. Jones' students watched with ---- attention when he took a -----fall, onto the ----. But this was just ---- of a ---- he was setting

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An Explanation of the Probability Problem

2012-05-11T08:02:00.000-04:00

First, here's a restatement of yesterday's probability question :

Compare these 2 probabilities and explain method:

(a) Prob of rolling exactly 3 sixes in 5 rolls of a fair die.

(b) Prob of rolling exactly 3 sevens in 5 rolls of a pair of fair dice

Discussion :
Both are examples of binomial probability because they involve repeated independent trials each of which has 2 outcomes. The following explanation is intentionally detailed and 'repetitious'.

The prob of a 6 on each roll is 1/6. Each roll produces only 2 outcomes, either a 6 (prob=1/6) or not a 6 (prob = 5/6).

The prob of a 7 on each roll of a pair of dice is 6/36 or 1/6. Each roll of the pair has only 2 outcomes, either a 7 (prob=1/6) or not a 7 (prob=5/6).

Therefore, the probabilities of getting 3 successes in 5 trials is the same. Since the question asks for a comparison, we're done.

The actual prob is C(5,3)(1/6)^3•(5/6)^2 where C(5,3) is the 'MathNotation' for the number of ways of arranging 5 objects, one group of 3 identical objects and a separate group of 2 identical objects. This is not the usual way of defining combinations but I like this interpretation.

I guess the QuadAnagram was a bit challenging. Here's a hint for the ending:

...he's a bored L---R.

Email me at dmarain at gmail dot com with your answer.

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QuadAnagram Contest and maybe some math too

2012-05-09T09:35:00.000-04:00

Well if you tried yesterday's TriAnagram you know the rules. This time we're looking for FOUR 5-letter words to fill in the blanks. The words are all anagrams of each other.

John was so bored with being a ----- that he took his -----, went to the airport, saw his boss who was a regular ----- and now John is a bored -----.

Ok, some math..

Compare these 2 probabilities and explain method:

(a) Prob of rolling exactly 3 sixes in 5 rolls of a fair die.

(b) Prob of rolling exactly 3 sevens in 5 rolls of a pair of fair dice

We had 2 winners yesterday and each received my new New Math Challenge Book.

FIRST 3 TO SOLVE TODAY'S ANAGRAM AND MATH PUZZLE WILL RECEIVE MY BOOK AND THEIR NAME WILL BE PUBLISHED!
EMAIL ME AT dmarain at gmail dot com
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135 and 144 are very special but why...

2012-05-08T07:44:00.002-04:00

Update...
Mark James is our first winner today and he already has received his prize! Two to go...
Charles Drake Poole is our 2nd winner!
Joshua Zucker is our 3rd and final winner! Congratulations! First if you haven't seen my QuadAnagrams and Trianagrams on Twitter, I'll start you off with a fairly easy Triple- or TriAnagram.2

I opened my mouth ----- but my ----- braces still felt -----.

Object: Replace the dashes with 3 different 5-letter words which are anagrams of each other.

First 3 to email me at dmarain at gmail dot com with the solution to my TriAnagram and the unique property shared by 135 and 144 will receive a free copy of my new Math Challenge Problem Quiz Book.

Ok, back to asking your students the bigger question:

What makes 135 and 144 so special!

1) Have them work individually or in pairs?
2) Use calculator?
3) Get them started or ask someone for an idea?
4) What if they say 144 is a perfect square? Does the question imply that the properties must apply to both? Should I have made it clearer in the wording of the problem or is the word and sufficient to convey that?
5) The really unusual property I'm looking for is only shared by 0,1,135 and 144. Good luck finding it!

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All Tied Up - a Geometry Classic Challenge

2012-05-07T16:30:00.000-04:00

For exercise, a prisoner was chained to one corner (lower) of a 10 ft concrete cube located in the center of the yard. If the chain was 16 ft long and was not obstructed except for the cube, over how many sq ft of ground could he roam?

Ans: 210π sq ft

1. Give the students the diagram or have them draw it themselves?
2. Have them work individually or in groups?
3. How much time would you give them to work on this in class?
4. After discussion, how would you know if they 'got' it? Assessment?
5. Makes more sense to give them a variant of the problem for HW or ask them to design their own and solve it?

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Given the sum and product of 2 numbers...

2012-05-06T14:17:00.000-04:00

A fairly common standardized test question for Algebra 1,2 or SATs is something like

The sum of 2 numbers is 20 and their product is 64. What is the larger number?

This question requires the student to actually find the numbers as opposed to a question with the same given info but asking for the positive difference of the numbers.

Do you suggest to students that many of these types of questions can be handled by inspection with mental math? This is because the majority of standardized math questions involve simple integer values or adhere to the "Keep it Simple" philosophy!

From either of the given relationships students should be able to arrive at 16 and 4 as the values and proceed from there. For the 25% or so of questions which do not admit a simple solution there's always straight algebra or the "test each answer choice" strategy for Multiple Choice. By the way this is why item writers often shy away from direct "solve for x" types, preferring the "find the positive difference " type.

Please don't forget to make that critical connection to the graph of a linear-quadratic system. A quick sketch of the line x+y=20 and the rectangular hyperbola xy=64 suggests there are 2 pairs of solutions which involve the same numbers by symmetry, i.e., (4,16) and (16,4).

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SAT Mental Algebra

2012-05-06T06:43:00.000-04:00

Well, SATs are now over for this month but anytime we can exercise students' minds is not a waste of time IMO.

If x=2.76, what is the value of
(x-3)/(x-2) - (1-x)/(x-2)?
NO CALCULATORS - 30 sec...

(1) Would students think "there must be a trick here"?
(2) Do you see value in this quickie?
(3) It might be fun to have half the class use pencil, paper and calculator while other half does it mentally.
(4) Of course most students should be careful when doing standardized test questions so we're not advocating quick mental math methods for all questions!

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A Classic Algebra Challenge

2012-05-04T08:22:00.000-04:00

Once students learn the strategy for doing these kinds of questions, the SAT and other standardized tests seem rather easy!

x+y=10
x^2+y^2=10
Find x^3+y^3

Ans: -350

Notes:

(1) Before giving students this question you may wish to scaffold with finding xy first.
Ans: 45

(2) To promote connection-making and to deepen their thought processes, give them the answer -350 and ask:
(a) Without graphing. explain why the graphs of the 2 given eqns DO NOT INTERSECT!
(b) Then how can there be a solution!

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Questioning 0.9999...=1 or Heres to you Mr. Robinson

2012-05-03T07:07:00.000-04:00

Vi Hart must be having an effect on me! After proudly explaining for over 40 years why 0.9999... must equal 1 using the Density Property of the Reals (see my post Another Proof that 0.9999...=1), I just had an epiphany of sorts.

If 0.9999...=1, then (0.9999...)^2 must also equal 1 from the properties of the reals. But squaring a finite string of 9's (with or without a decimal point) produces a fascinating result:
(0.99)^2=0.9801
(0.999)^2=0.998001
(0.9999)^2=0.99980001 etc...
This sequence of decimals seems to suggest the existence of a non-real number which differs from 1 by an infinitesimal amount, so-called hyperreal numbers, leading to the non-standard analysis of Abraham Robinson. Who knows where the teaching of calculus might be today if Dr. Robinson had not died at the age of 55 from the disease that took my wife 2 months ago -- pancreatic cancer.

Well, maybe it's healthy to have one' roots shaken after many years. After all, my tag line for this blog for a couple of years involved how new ideas are often at first ridiculed, then vehemently opposed and finally accepted as obvious ...

NOTE: I omitted the hyperlinks in this article. I was getting too 'hyper'!

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841 is interesting because...

2012-05-02T07:26:00.000-04:00

Here's another quick exploration for middle schoolers and beyond. I believe it builds mental math and number sense skills and more.

With your partner write as many "interesting" observations about the number 841 as you can in the next 5 minutes. Yes, calculators are permitted.

If they've learned the Pyth Thm, you may want to suggest afterward that 841 could be the square of the hypotenuse of a right triangle - let them find the 3 sides (unless a team comes up with that! ).Talk about making connections!

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SAT CHALLENGE - ODD NUMBERS OF FACTORS

2012-05-01T19:30:00.000-04:00

How many positive integers less than 1000 have exactly

(a) 3 positive integer factors
Ans: 11

(b) 5 pos int factors
Ans: 3

(c) 7 factors
Ans: 2

Is this topic in the middle school core standards? Under divisibility? Factors?

Have you seen questions like these on state tests? SATs?

What strategy would you like your 6th-8th graders to use? Assuming they don't know a 'rule' for this problem, how can they best discover a pattern? Would it make sense for students to make a 2-column table of integers and number of factors?

Why am I addressing middle school curriculum when the title of this post refers to SATs?

Is this question not worth all the time it would consume?

Do you believe this question is only for the 'mathletes' who take math contests?

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So why am I publishing so much recently...

2012-04-30T07:04:00.000-04:00

1) To show myself that I still can
2) To let my faithful readers and fellow/sister bloggers know that I'm back
3) To have that feeling of accomplishment seeing my posts ranked #1 on Alltop again
4) To keep busy and distract my mind from other thoughts

Don't worry if you can't keep up with my manic publishing pace. I will soon be slowing down!

Dave Marain

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GEOMETRY: When is a cone half full...

2012-04-30T06:42:00.000-04:00

Ever wonder about practical applications of those 'some liquid is being drained from a conical tank' calculus problems?

Well, they do manufacture storage tanks with cylindrical tops and cone-shaped bottoms. Ask your students why, then share the following excerpt 'borrowed' from the website of a company which makes these:

"Cone bottoms provide for quick and complete drainage."

Alright already - enough motivation for a geometry problem! No calculus needed!

A conical storage tank with a maximum depth of 10 feet is completely filled with a chemical solution. Some of its contents are then drained from the bottom.

Ask your students:

(a) When depth of liquid falls to 5 ft, explain intuitively (no calculations) why much more than half the contents has drained out.

(b) Now for the geometry application...
What % of the total liquid has been drained when depth drops to 5 ft?

Ans: 87.5%

(c) (More challenging) What should depth be for tank to be half full? Give both one place approx and 'exact' answer.

Ans: approx 7.9 ft
I'll leave exact answer to my astute readers!

Note for instructor: You may want to explore different depths like 6', 7', 8' first to see how close we can come to half full.

QUESTIONS FOR THE INSTRUCTOR
WHAT ARE THE BIG IDEAS HERE?
DO YOU BELIEVE THIS CONCEPT IS ASSESSED ON SATs?
GIVE PRECISE WORDING OF THIS OBJECTIVE IN THE CORE CURRICULUM.
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13-14-15 triangle as special as 3-4-5

2012-04-29T06:52:00.000-04:00

Show that the area of a 13-14-15 triangle is 84. Compute mentally - 30 seconds tick tick tick...

I'm being silly with the ticking clock but it is possible to do this if you choose the "right" base! Unless of course you can mentally apply Heron's formula which is doable! Ok, so there's more than one way as always!

So what makes it special!? Somebody out there knows...

If you like these challenges consider purchasing my new Math Challenge Problem/Quiz Book - 175 questions - SAT format - with answers. Go to top of right sidebar to order.

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SAT GEOMETRY REVIEW Is it a Rectangle or a Triangle...

2012-04-28T07:56:00.000-04:00

A diagonal of length x of a rectangle makes a 30° angle with the base.

(a) Show that the area of the rectangle is
(x^2)√3/4.

(b) The formula in (a) is also the area of an equilateral triangle of side length x. What triangle is this the area of? Explain!

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A Passing Thought...

2012-04-28T05:57:00.000-04:00

I just tweeted this flight of fancy...

The next time a student says, "When are we ever going to use this?", try
"If you're referring to your brain, I was thinking the same thing!"

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SAT EXPONENT CHALLENGE 2012

2012-04-27T14:57:00.000-04:00

The mean of 3^(m+2) and 3^(m+4) can be expressed as b•3^(m+3). If m>0, then b=?

Ans: 5/3

On an actual College Board test, this would likely be multiple choice and perhaps a bit easier but s similar question appeared on the October 2008 exam.

Would you recommend to your students 'plugging in' say m=1?

Even if students avoid an algebraic approach, we as educators can still use this example to review exponent skills, yes?

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Geometry in the Tiling Patterns All Around Us

2012-04-27T07:09:00.000-04:00

I took this picture of a section the floor of the hospital where I volunteer and fortunately I wasn't dragged to the psych ward. Students see tiling patterns every day yet rarely think of applying their knowledge of geometry.

Assume each white square has side length 2 and that the shaded square is obtained by rotating one of the white squares 45 degrees.

Show that the overlap is a regular octagon of side length 2√2 - 2.

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When is 11 1/9% equal to 10%=?UTF-8?B?Pw==?=

2012-04-26T07:38:00.000-04:00

If # of left-handers are 11 1/9% of right-handers, what % of total pop are left-handed? (disregard ambidextrous)

Questions for Middle School Teachers
1) At what grade level would this kind of problem be introduced?
2) Would you allow use of calculator here or expect students to change 11 1/9% to 100/9% and 111 1/9% to 1000/9%? More importantly, am I out of my mind to think that students at any grade level including secondary would do this!
3) WHAT ARE THE BIG IDEAS HERE?
4) WHERE DOES THIS TYPE OF QUESTION FIT INTO CORE STANDARDS?

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ANOTHER SAT PRIME CHALLENGE

2012-04-25T07:57:00.001-04:00

If p is prime, which of the following could be prime?
I. p+7
II. 4p^2-4p+1
III. p^2-p

(A) I only (B) II only (C) III only
(D) I,II,III (E) none

What KNOWLEDGE must middle/secondary students have to solve this? In what grade is this taught?

Ask students: If "could" was replaced by "must" would the answer change? Explain.

For homework, ask students to write their own version of this problem. You may get some awesome questions you can use later on!

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SAT ALGEBRA MULTIPLE GUESS

2012-04-24T07:33:00.002-04:00

If a^2 = b^2 = c^2 = 4 and abc ≠ 0, how many different values are possible for a+b+c?

(A) 1 (B) 2 (C) 3 (D) 4 (E) 5

Ans: 4

For those of you who find this question trivial, remember that difficulty is very subjective.

Whst is(are) the BIG IDEAS here?

Can you predict which of your students will choose an algebraic approach vs "plugging in"?

Extension for students: Suppose we use 5 variables instead of 3.
Ans: 6

Generalize and explain!

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An Equal Number of Democrats and Republicans Are Locked In a Room

2012-04-23T08:03:00.002-04:00

An equal number of Democrats and Republicans are locked in a room (at least 2 of each). If 2 are released at random, what is the probability that there will be one from each party? Remember, your answer must be both mathematically and politically correct.

(a) What questions should your students ask before starting the problem? And if they don't..
(b) Is it worthwhile to give students 10 sec to make an intuitive guess?
(c) Do you think 1/2 will be intuitively guessed by a majority?
(d) What strategies do you want your students to use with an open-ended question like this?
(e) Would you have your students solve problem if there were originally 2 from each party, then, say, 3 from each?
(f) Show that if there are originally n from each party, the desired probability is n/(2n-1).
(g) As n increases beyond all bound...
(h) What do you see as the benefits of this inquiry?
(i) How would you extend this investigation? (j) How would you have done it differently depending with middle schoolers vs secondary?
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