Remainders and number theory for grades 6-12

Mr. Canastar had 3 identical decks of cards, and told his class that each contained more than 100 but less than 200 cards. He told Yohan to count the 1st deck by 3's and there were 2 left over. He told Matt to count the 2nd deck by 5's and there were 3 left over. CC counted the 3rd deck by 7's and there were 2 left.

He challenged the rest of the class to figure out how many cards were in each deck. Can you? And explain your method too!








If interested in purchasing my NEW 2012 Math Challenge Problem/Quiz book, Revised Version 1.1, click on BUY NOW at top of right sidebar. 175 problems divided into 35 quizzes with answers at back. There are now detailed solutions, hints and strategies for the first 8 quizzes. Suitable for SAT I, Math I/II Subject Tests, Math Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Special Limited Price $7.99. 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!

PerCent Challenges From Middle School to SATs

Never too late in the school year to review percents, right? Well, even if you don't agree, here goes... 

First a problem similar to the one I posted on Twitter the other day.

Middle School Level?

The cost of a meal including a 10% tip was $13.75. What was the tip, in dollars?
Ans: $1.25

SAT-type (Higher level of difficulty)

The cost of a meal is $M. With an x% tip included,  the bill came to $T. Which of the following is an expression for x in terms of M and T? (A) T/M (B) (T-M)/M (C) 100T/M (D) 100(T-M)/M (E) (T-M)/(100M)
Ans: D


Thoughts and Questions...

What % of your middle school students could handle the first question? For that matter, what % of your secondary students would solve it?

Can you predict which of your students would be able to solve the first question mentally or with some quick trial-and-error (ok, G-T-R), using their calculators. I chose 10% to make this possible. Do you get upset when students do this? Should you?

What do you predict would be the difficulties your algebra students might confront in the 2nd problem?

Is it easy to eliminate some of the answer choices and to make an educated guess from the rest?
(NOTE: I composed the question and the answer choices and I know some of you could improve upon my efforts!)

NOTE: The 2nd question is representative of the harder problems on the SATs and there are many of these in my new Challenge Math Problem/Quiz Book mentioned below.









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 Contest practice and Daily/Weekly Problems of the Day. Includes multiple choice, case I/II/III type and constructed response items. Price is $9.95. 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!

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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