Algebra 2/PreCalculus Investigation CCSSM

Hi All! Long time, no blog...

I've been spending most of my 'spare' time on Twitter and my Facebook Fan Page but I decided to share some of the activities I've developed with the few faithful readers who might still remember me! For those of you looking for richer problem-solving which conforms to the Mathematical Practices of the State Standards, here's one you might find useful. Please let me know how you adapt it for your students and how they responded. I hope to be writing more of these.

Note: If the image below is cut off, click on it and zoom in if needed.






















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 and includes detailed solutions, strategies, tips, hints and key facts for the first 8 quizzes as well as answers for all quizzes 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 WILL BE ABLE TO SEND THE ATTACHMENT!

Almost-National Math Curriculum... Should I Feel Vindicated?

Excerpt from my local paper, The Record, 9-3-12

For the first time there is broad national consensus on the most necessary skills, so a third grader in Paramus and his camp buddy in Peoria will face roughly the same expectations when they walk back through the school doors this week.

"It's almost the entire country coming to an agreement about what kids should learn." Many parents in affluent suburbs might assume that widespread worries about weaknesses in the American education system relate to poverty and think their own kids' schools are doing just fine but this shift aims to raise the bar for everybody. Some studies for example conclude that even advantaged U.S. children with college educated parents can barely compete internationally in math.

How long have I waited to read these words?
How long have I been advocating this and supporting Prof. Wm. Schmidt's recommendations?
How many of my blog posts have been dedicated to this topic in the past 6 years?
How many of you supported me? Opposed me? Argued that this will never happen?

So do I feel vindicated?  I'll let my readers guess!




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!

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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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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So why am I publishing so much recently...

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...

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

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...

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...

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

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

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==?=

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

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

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

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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SAT CHALLENGE : Counting Non -Multiples of 7

Twitter Problem posted 4-21-12

How many pos integers less than 1000 are not multiples of 7?

Middle school problem?
Strategies you teach your students?
Calculator appropriate?
"Big Ideas" here?

Ans: 857

Sketch of one possible method:
1000/7=142.857... ---> 142 multiples of 7 less than 1000 ---> 999-142 = 857 non-mult

The devil is in the details of course which I intentionally omitted! Why didn't I mention that the largest mult of 7 less than 1000 is 994?  Would most solutions involve finding 994 first?

Someone out there is thinking about the repeating decimal expansion of 1/7 = 0.142857142857… and why the ans to our problem is 857. A coincidence?

Too bad we have no time in our classrooms to explore and go in depth. If we spend time doing that we'll never cover all the required topics in the Core Curriculum. Yes?

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A Square Gets Between 2 Kissing Circles

I saw a problem similar to the following on some website recently. The problem stayed in my head but not the site. If you recognize it, please comment so that I can provide proper attribution.

Circle I and circle II each of radius 10 are tangent to each other and to a common external tangent line T.
A square ABCD is drawn between the circles such that A,B are on circles I and II respectively and C,D are on line T.

(a)  Draw the diagram from the above description.
(b) Show that a side of the square is 4.

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(E) Cannot be determined...

I've posted the following geometry classic before but it seems relevant now with SATs and other standardized tests looming.

Given 2 concentric circles, segment AB is a chord of one and a tangent segm of the other. If AB=10, show that the pos difference of the areas of the circles CAN BE DETERMINED!
Explain.

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SAT Logic and Semantics Twitter Problem

Posted on Twitter 4-14: I have 3 cards with a blue dot and 3 cards with a red dot. If I have no other cards, how many cards "do" I have?

Too easy for most secondary students?
Too ambiguous?
How would it be modified for SATs?
How would 3rd or 4th graders respond?
What do think my underlying purpose is?
What are the "Big Ideas" here?
Hiw would you present this in a 4th grade vs a 10th grade classroom?
After discussion how would you assess understanding?

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Choosing Pairs of Cards on SAT

A set of 5 playing cards consists of a 10, Jack, Queen, King and Ace. If 2 of the cards are chosen at random, what is the probability that neither card is a king nor an ace?

Ans: 3/10

Explain using at least 3 different methods!


Interested in seeing 175 more of these kinds of problems with answers? Look at the top right sidebar for info on my new Math Challenge Quiz/Problem Book...

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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!

SAT TWITTER PROBLEM

Students taking May SAT MAY want to try today's Twitter problem I just posted at twitter.com/dmarain

If n is a positive integer, then the expression n(n+3) + (n+3)(n+8) must be divisible by
I.   2
II.  4
III. 8
EXPLAIN!

This is a typical "cases" type but I omitted the usual choices like
(A) I only
etc...

Might be worth some discussion to consider more than the typical student's "plug-in"  approach. That's why I added "EXPLAIN! "
There is some rich mathematics to be unearthed here IMO...

Interested in 175 more of these types with answers? Try my new Math Challenge Problem/Quiz Book. Look at top of right sidebar.

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The Third Wheel...

Two wheels with  diameters 18 and 8 are touching and are on level ground. Show that the diameter of a 3rd wheel on the ground which touches the other 2 is 2.88.


For info on my  NEW MATH CHALLENGE QUIZ/PROBLEM BOOK check top of right sidebar!

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Any child struggling with mixed numerals=?UTF-8?B?Pw==?=

The answer may be "no" in some parallel universe but here on earth the title of this post is rhetorical.

So we show children a diagram of 4 identical pizzas each divided into 8 equal slices or for the younger set we have manipulatives. We would probably not use so many pieces when introducing this but I needed an example which could also appear on the next state test.

We cross out or shade all the slices in 3 of the pizzas and 5 of the slices in the 4th pizza, representing what a group of kids ate.

What are the questions we ask or might appear in the text or on the worksheets or on the state mandated tests?

What do you believe are the major stumbling blocks for most children and what can we as educators or parents or tutors do to help?

Here are some thoughts...
Is the issue more conceptual or procedural?

How would you rank the importance of how each question is worded?

You want the answer to be both the improper fraction 29/8 and the mixed numeral 3 5/8. How should the questions be worded?  Hey, there's no universal remedy here! Some children will misunderstand the questions no matter how they're expressed or simply have not yet made sense of the ideas. BUT on an assessment the wording must be mathematically correct and age-appropriate, right?

How would you react to the child who responds 29/32? Is (s)he wrong? How could the question be asked for which this correct? Is the child confused or was it the question itself?

Whether you're a 3rd grade teacher, a professor of math/math ed, a math staff developer or coordinator/administrator I hope you'll weigh in on this with your reflections and/or anecdotal experiences.

I consider this issue to be of vital importance in the development of the concepts and skills of fractions and part vs whole.

What do you think?
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If a hen and a half can lay...

Share your teaching methods for the classic problem in the title:

If a hen and a half can lay an egg and a half in a day and a half, how many eggs can 3 hens lay in 3 days?

Would you ask students for an immediate intuitive guess and expect many to say 3?
The answer is 6 so the purpose of this post is reflection on sharing instructional strategies.

What are the BIG IDEAS here? Do you use one basic strategy in teaching all ratio problems?  Does dimensional analysis work for middle schoolers?  Should students always reduce everything to a single unit ratio like 1 hen per day? How many ways could your students devise if we tell them the answer is not 3?

Not much sharing going on like in the old days of this blog but maybe it's time..


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Shameless self-promotion!
Consider buying my Challenge Math Problem book designed for standardized tests, Problems of the Day, etc. Go to top of right sidebar for more info.

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State Testing Review Time In Your District=?UTF-8?B?Pw==?=

This is that special time of the year when some districts, particularly in elementary grades, hand out practice materials for the test and teachers are expected to devote the majority of class time to it.

Here's my question...
Does anyone out there feel that these materials seem to be somewhat different and of a higher level of difficulty than regular classroom materials/tests?  If this is the case then what are the implications for the child, the teacher and the district?  I do have strong feelings about this but I'll wait for your comments first. WHAT ARE YOUR THOUGHTS?

Also I have to repeat a tweet I just saw from the brilliant Timandra Harkness from the UK. She made my day...

"I'm decorating my bathroom with those new fractal tiles. I think it's going to take forever."

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PerCent "More" - Applying RATIOS DEEPLY

PROBLEM
Two thin cylindrical steel  disks have diameters of 35 in and 25 in. The area of the base of the larger is what % more than the smaller?

We would hope juniors in Alg 2 or Precalc would know the basic setup for % more, % increase/decrease or % change types, particularly since this is a middle school concept. Of course we know this is often not the case!

After having students work in small groups for a few minutes and watching them pushing calculator buttons you have someone come up and explain, asking questions and reviewing basics. As is typical, some students will use the diameters instead of the radii and get the right answer anyway. What are the "BIG IDEAS" here?

Write on the board (35/25)^2 = 49/25, then 24/25 = 96%.  No explanation. You give students in small groups 2 minutes to make sense of this and have 2 groups take turns explaining it to the class.A mental calculation?

Before you kneejerk reflexively react to this with " Even some of my honors students would struggle with that", I would like my readers to reflect on our obligation to stretch their minds and promote conceptual understanding.

IN NO WAY AM I SUGGESTING THAT IS THE METHOD MOST STUDENTS SHOULD USE!

SO WHAT ARE THE KEY MATH CONCEPTS USED IN MY EXPLANATION?

Of cou

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Investigation for "Squares"

CHALLENGE YOUR GEOM STUDENTS OR YOUR MATH TEAM

In square ABCD of side 1, E is the point on diagonal AC such that AE=1.

(a) Explain without numerical calculation why
√2 < BE + DE < 2
(b) Show that BE+DE = 2(√(2-√2)) ≈ 1.531 without using Law of Cosines
(c)  Be a math researcher! How might you generalize this?

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NEW MATH CHALLENGE QUIZ/PROBLEM BOOK AVAILABLE

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, SAT Math I/II, Math Contest practice or daily/weekly Problems of the Day. Questions include multiple choice, cases I/II/III type and constructed response. Price is $9.95 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT! (dmarain "at" "geemail dot com")

My beloved wife of 42 yrs passed away on 2-28-12 after battling pancreatic cancer over 8 months. She gave tirelessly to those in need all of her life and never asked for anything for herself. She can finally have the rest she has earned. I, her 7 children, 4 grandchildren and all those whose lives she touched feel a gaping hole in our hearts...

FIT THE WORLD'S PRODUCTION OF GOLD IN YOUR CLASSROOM

Visit this link for rich applications of math:
http://money.howstuffworks.com/question213.htm

The particular article in the link addresses dimensional analysis and fundamental science in relation to math. MOST IMPORTANTLY IT MAY ENGAGE EVEN THE LESS MOTIVATED!

Try it with your prealgebra students as an application of unit conversions, exponents, scientific notation and basic geometry concepts.

Students will need to convert mass (kg) to volume (liters) using the specific gravity of gold, then to cm^3. This will allow them to see how the world's estimated annual production of gold will approximately fill up a cube 14 ft on an edge or, equivalently, a rectangular room 14 ft by 28 ft with 7 ft high ceilings.

By requiring students to research the data and conversion constants, you can integrate the web into the assignment.

A scientific calculator is an appropriate tool here, however, require students to WRITE all steps. It is very easy to lose track of the process when we do a series of 5 or more calculations. The result will appear to be incorrect but the student will be hard pressed to find the error.  It's human nature to press CLEAR and keep pressing more buttons until it works, then forget the exact sequence of keystrokes! 
RECORD EACH STEP ON PAPER WITH APPROPRIATE UNITS. BE A RESEARCHER!

Let me know if you try it and how it works out.

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GEOM CHALLENGE 2-7-12

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 $9.95. 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!]

If interested in purchasing my new 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/Math Contest practice or Problems of the Day/Week.

Price is $9.99 and secured pdf will be emailed when purchase is verified. DON'T FORGET TO SEND ME AN EMAIL FIRST SO THAT I CAN SEND THE ATTACHMENT!

"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

Explain the SUPER pattern: 2-9-12-15-21

First, congrats to the NY/NJ Giants on another extraordinary accomplishment!
Kudos to both teams and Eli-te and Tom B in particular.

My opinion, but it drives me nuts when I hear that the Miracle Catch 4 years ago was just luck. It wasn't luck that enabled David Tyree to jump higher than Rodney Harrison. It wasn't luck which enabled him to hold onto the ball despite every attempt by the defender to rip the ball from his hand or his helmet. Not to mention Eli's "refuse to lose" attitude. The throw by Eli and catch by Mario Manningham last night was the result of endless practice, skill, and a will that was stronger than the opposition's. Why was Manningham able to make the HIGH PRESSURE play but Welker was not? I say it was DRILL,SKILL,WILL -- necessary ingredients for success in life!

Alright, you haven't read a blog post from me in eons so I cannot disappoint...
I do hope y'all have been following me on twitter.com/dmarain. I've tweeted numerous challenges for your students. Hope you've enjoyed them...

MATH CHALLENGE GRADES 3-8

LIST THE WAYS A FOOTBALL TEAM CAN SCORE 21 PTS. DO THE SAME FOR 17 PTS.
Repondez s'il vous plait! (supply your own accents!)


By the way, one of my daughter's friends had boxes of 9-0 and 9-3 in either order. She won $75 for the 1st quarter and $150 for quarter 2! My buddy had 0-8 in either order for the final score. If the Giants kick the field goal making it 18-17 and the Patriots kick a field goal, making the final score 20-18, he would have won $2500! He's a big Giants fan but talk about mixed emotions!


"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