Tuesday, May 29, 2012
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?
Sent from my Verizon Wireless 4GLTE Phone
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?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 4:24 PM 0 comments
Labels: geometric sequence, investigations
Tuesday, May 22, 2012
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?
Sent from my Verizon Wireless 4GLTE Phone
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?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 6:38 AM 12 comments
Thursday, May 17, 2012
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.
Sent from my Verizon Wireless 4GLTE Phone
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.
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:04 AM 3 comments
Labels: growth rate
Tuesday, May 15, 2012
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!
Sent from my Verizon Wireless 4GLTE Phone
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!
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:06 AM 1 comments
Saturday, May 12, 2012
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
Sent from my Verizon Wireless 4GLTE Phone
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
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 6:51 AM 0 comments
Labels: anagram, SAT-type problems
Friday, May 11, 2012
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.
Sent from my Verizon Wireless 4GLTE Phone
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.
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 8:02 AM 0 comments
Labels: binomial probability
Wednesday, May 9, 2012
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
Sent from my Verizon Wireless 4GLTE Phone
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
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 9:35 AM 0 comments
Tuesday, May 8, 2012
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!
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:44 AM 0 comments
Labels: math challenge
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