## Tuesday, November 23, 2010

### Another Cone in a Sphere Problem? - A Guide for the rest of us...

Students who have been out of geometry for a year or so and are preparing for standardized test like Math I Subject Test or SATs/ACTS need occasional review. The following is similar to several other cone problems I've posed before but even our strongest Algebra 2 through Calculus students lose their "edge" when it comes to "solid" geometry questions (yes, believe it or not, my terminal course in high school was called Sold Geometry and we covered topics like spherical trigonometry!).

A right circular cone of height 16 is inscribed in a sphere of diameter 20. What is the diameter of the base of the cone?

Reflections....

1)  Are these kinds of problems somewhat hard merely because students forget? I can think of several more reasons:

• The problem itself is somewhat challenging, however it's far from over their heads!
• The student never experienced a question like this in Geometry; perhaps questions like these were in the B or C or D exercises in the text and were never assigned or only for the "honors" students? Do you recall seeing a problem similar to this in the textbook from which you taught?
• The student did not take a formal course in geometry
• The topic was covered in a cursory manner or perhaps not at all because of time crunch. That's the whole point of a standardized curriculum, isn't it? To know what is needed to be covered and plan accordingly. Of course, I'm  a realist enough to know the myriad of reasons why the best laid plans oft go .........
• Students don't remember how to start because key geometry strategies were not explicitly stated and reiterated ad nauseam. Were your students asked daily to begin by reciting the key strategies such as those for circle and sphere problems? Were they placed on index cards or blocked out in a particular section of their notebook?:
• DRAW THE BEST DIAGRAM YOU CAN (and believe me, I'm no artist!)
• Always locate the CENTER of circles, spheres and label the point
• Label the measurements of all segments (angles) - I know, everyone does that!
• Successful problem-solving in mathematics is based on finding relationships! Were guiding/leading questions asked
• What do the cone and sphere have in common?
• TRUE  FALSE  The height of the cone is the same as the diameter of the sphere.  EXPLAIN!
• Was the student exposed to the strategy of comparing the 2-dimensional analogue of the 3-D problem? Would it be a right triangle in a circle? Equilateral triangle inscribed in a circl or???
• Oh and yes...
• Draw the radius of the sphere (or circle) so that it is the hypotenuse of some right triangle!

"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

## Monday, November 22, 2010

### 11-22- A Remembrance - Soon It Will Be Half A Century

And the night comes again to the circle studded sky
The stars settle slowly, in lonliness they lie
'Till the universe expodes as a falling star is raised
Planets are paralyzed, mountains are amazed
But they all glow brighter from the briliance of the blaze
With the speed of insanity, then he died.

From Crucifixion, Phil Ochs

"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

## Tuesday, November 16, 2010

### CONTEST! Just Another "Rate-Time-Distance" Problem?

CONTEST IS OFFICIALLY OVER AND THE WINNER IS ----- NO ONE! Guess I should have offered a 64GB 3G IPad! to be awarded on Black Friday...

The floor is now open for David, Curmudgeon, and my other faithful readers to offer their own solutions.

And the next contest is...

This is a contest so students must work alone and this needs to be verified by a teacher or parent. No answer will be posted at this time. Deadline is Wed 11-17-10 at 4 PM EST.

Here's a variation on the classic motion-type problems we don't see as often in Algebra I/II but still appear on the SATs. I found this in some long-forgotten source of excellent word problems to challenge NINTH graders!

Barry walks barefoot in the snow to school in the AM and back over the same route in the PM.  The trip to school first goes uphill for a distance, then on level ground for a distance and finally a distance downhill.  Barry's rate on any uphill slope is 2 mi/hr, any downhill slope is 6 mi/hr and 3 mi/hr on level ground.  If the round trip took 6 hours (hey, these are the old days in the 'outback'), what was the total number of miles walked?

First five correct answers  with complete detailed solutions emailed to me at dmarain@gmail.com will receive a downloaded copy of my new book of Challenge Problems for the SATs and Beyond when it becomes available. Both the student and teacher(s) will receive this.  (Illegal to reproduce or send electronically!). Read further...

Submission by email must include (Number these in your email and copy the validation as well).

1.  Answer and complete detailed solution. If answer is correct but method is sketchy or flawed,      the submission will be rejected.
2.  Full name of student
4.  Math course(s) currently taking
5.  Math teacher's name(s) and parent's name(s)
6.  Name, Complete Address of School; Principal's Name & Email address (if known)
7.  Email addresses of teacher(s),  parents, student
8.  Phone number (in case I need to call you) - Optional
9.  How your or your teacher or parent became aware of MathNotations.

VALIDATION

I certify that my student (child) did the work independently.

--------------------------------------------------------------------------------

Name of Teacher or Parent (if work done at home)

"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

## Wednesday, November 10, 2010

### Algebra 2/Precalculus "Extended" Activity Based on an SAT-Type Question

Consider the following problem:

If -5 ≤ x ≤ 4, and f(x) = 2x2 - 3, how many integer values are possible for f(x)?

One can simply view this as a more challenging question to pose to your honors/accelerated students, but, for me, it's an opportunity for all your students to think more deeply about important concepts. I feel strongly that our role here is to ask the key questions which will guide them toward understanding the "big ideas" underlying this problem. In fact, we can turn this question into an extended activity: 15-20 minutes).

Here is one idea for creating the environment currently being recommended. Please keep an open mind before concluding that there is simply not enough time for these explorations...

1.  Sketch the graph of the function on the given domain from recognition of quadratic functions and by making an x-y table with 4-5 points. WRITE YOUR INFERENCES FROM THIS. For example, from the sketch we believe that the greatest y-value on this domain is ___.

2.  Using the TABLE feature of your graphing calculator, with TblStart = -5 and ΔTbl = 1, display the Table.  Now turn TRACE on and analyze the graph on this domain. Does this alter or confirm your conjecture from Step 1?  YES   NO

3.  The following statement is plausible but FALSE.

The domain consists of 10 integer values. Therefore there are also 10 integer values for f(x), so the answer is 10.

Explain why this is wrong. There is more than one error!

4.  The correct answer is 51. Depending on the class, a few, if not several,  students should be able to come up with the correct answer and provide a thorough explanation.

5.  Group Discussion:

• Ask students how they might have approached this question if it appeared on a standardized test? Plug in x-values? Use the graphing calculator? Guess? Skip it?
• Ask the group what made this questionable formidable for some students? How important was understanding what was asked for?
• Review one successful approach to solving the problem by calling on individual students to give the "next" step.

NOTE: This  problem also presents a highly teachable moment for students to see an application of the Intermediate Value Theorem in Precalculus (or more intuitively in Algebra 2).  Help them make the connection! Is this easy for us to do?