1,-3,9,-27,... Investigation

No matter where you introduce a lesson on geometric sequences, we can always begin with definitions, rules and formulas. 

The alternative is to build on student intuition and natural curiosity by asking them to write their own observations and questions they would like to have answered.

Imaginary Scenario (or is it?)

Jack: Mom, all the terms are just powers of 3 or their opposites, right?

Mom (Jane): Write your hypothesis, test it and let me know.

If your students or your son is not 15 year old Jack Andraka, here are some suggestions...


1.  What are the next 3 terms?
2.  If the 99th term is x, write an expression for the 100th term? (Recursive thinking)
3.  Which terms are positive? Negative?
4.  Write an expression for the nth term.
5.  How would we graph the sequence?
6.  Are the terms of the sequence increasing? Decreasing? Both? Neither?
7.  Which terms of the sequence are greater than a million? A trillion? Less than -1000000?

Another Imaginary Scenario (or is it?)

Uh, show me where this topic is in the CCSSM.
Uh, where does it say I have to ask all these questions?
Jack who?

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THE FULL MONTY HALL REVEALED

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

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...

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

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

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

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...

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

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...

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

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

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

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...

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

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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