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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