Monday, April 30, 2012
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
Sent from my Verizon Wireless 4GLTE Phone
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
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:04 AM 0 comments
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.
Sent from my Verizon Wireless 4GLTE Phone
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.
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 6:42 AM 0 comments
Labels: cone, geometry, ratios in geometry
Sunday, April 29, 2012
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.
Sent from my Verizon Wireless 4GLTE Phone
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.
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 6:52 AM 2 comments
Labels: geometry, SAT-type problems
Saturday, April 28, 2012
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) 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!
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:56 AM 0 comments
Labels: geometry, SAT-type problems
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!"
Sent from my Verizon Wireless 4GLTE Phone
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!"
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 5:57 AM 1 comments
Friday, April 27, 2012
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?
Sent from my Verizon Wireless 4GLTE Phone
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?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 2:57 PM 0 comments
Labels: exponents, SAT-type problems
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.
Sent from my Verizon Wireless 4GLTE Phone
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.
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:09 AM 2 comments
Labels: geometry, regular polygons, tiling patterns
Thursday, April 26, 2012
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?
Sent from my Verizon Wireless 4GLTE Phone
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?
Sent from my Verizon Wireless 4GLTE Phone
Posted by Dave Marain at 7:38 AM 2 comments
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