SAT Question Of The Day – estimating for hard geometry

From the official SAT site, here’s the Question of the Day for 9/25/14:

http://sat.collegeboard.org/practice/sat-question-of-the-day?questionId=20140925&oq=1

sat925

The answer is C.  Choose it (on the site, using the link above), and you will see the explanation involving a 30-60-90 triangle.  This is definitely the way to go, if you ‘remember how’.

The problem is, so many students will look at this question and say ‘I don’t remember how to do this’ and then skip it.  DON’T SKIP IT!!!!

Just ‘measure’ it!  OK, it’s not really measuring.  But consider this:  The diagram does NOT say ‘not drawn to scale’.  Therefore, we can assume it IS reasonably drawn to scale.   Look at the diagram.  Triangle ABO does look equilateral, O does look like the center of the circle, and the circle does look like a circle.  So, take the given measurement – that the sides of triangle ABO are 6 – and run with it!  Look at segment AB below.  It is 6 units long.  Measure it with your fingers and then use the same distance to guess at the length of segment BC.  Below, I’ve shown that 6 units, placed twice end-to-end along segment BC, is just a little longer than segment BC.  So, segment BC is shorter than 12.  If you do it carefully, you’d probably guess segment BC is about 10-11 units long.

2014-09-27_104226ans

Look at the choices.  Calculate decimal approximations for A, B, and C by typing the answers into your calculator.  Which one is closest to our estimate? C! And it’s the right answer! 

 

 

2014-09-27_104256

You’d be surprised how often you can really do this and get the right answer.  Follow me, and you’ll see lots of these!

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Precalculus – Domain of a function

Q: What is the domain of this function?

 

Answer:

   ‘Like’ if you got it!

Explanation:

The domain of a function is a description of the x-values that may be plugged into the function. For a square root function, the restriction is that the number under the radical must not be negative. Therefore,

 

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So, how much change should I get?

I went to Costco today.  I almost never get out of Costco for less than $100, but I wasn’t keeping a running total in my head.  It’s been one of those days, the kind where you just don’t even look at the prices (gasp, did I just admit that?!).   The cashier said the total was $118.87, and that was fine.  I handed her $120.00 and got to work packing my items in the cart.  The cashier was soon in total dismay when she discovered she had punched in ‘200.00’ instead of ‘120.00’.  Now, she’d have to get the manager!  The change amount was wrong and there was no telling how much it really should be!  And the line was so long already!  This was going to really slow her progress…  I piped up, ‘Just give me one dollar and 13 cents.’  She didn’t know whether to believe me.  She stared at the register display, which said ‘$81.13’, and just couldn’t figure out what to do.  I continued, ‘You put in 80 dollars too much, so take 80 dollars back from the change amount.’  Her face said she believed I was telling the truth, but she was still puzzled.  I got the feeling she didn’t know how to take $80 away from $81.13, at least not on the spot, in her head.  She asked, ‘So how much is it?’  I repeated, ‘one dollar and 13 cents.’  She must have decided I was being honest, since I was asking for such a small amount.  She thanked me, presumably for saving her the call to the manager, and I was on my way with the correct change.

True story.  I wish it weren’t true.  I don’t blame her, though.  She’s a product of a broken system.  I wish our country put a much, much greater value on how to do basic math.  You don’t have to be one of the ‘smart ones’ to know how to do basic math.  Most everyone can master it, if it is made important.  My grandmother did computation by hand flawlessly, even at age 90.  Even though she once asked me where Mars was, and upon being told it was a planet out in space, asked whether we could drive there.  Growing up, my grandmother never had the educational opportunities children in our country have today.  She went to work after 7th grade.  She knew how to read, write, and do her arithmetic.  She did it well, and was proud of it.  Because it was made important.  It’s not that hard.

Well, it shouldn’t be that hard.  Actually, in today’s society it is hard.  It’s hard because we (well, not me, and hopefully not you either, but too many people) tell our little girls they are bad at math.  WHAT??  We (same ‘we’ as before) watch talk shows where our idolized celebrities tell us it’s cool to be ‘clueless when it comes to numbers’.  We hear other celebrities admit with embarrassment that they are ‘math nerds’.  We give tons and tons of money to sports ‘heroes’ while our teachers and researchers can barely scrape together enough food to eat. 

And so, one day at a time, one student at a time, I try to change this collective mentality.  I tell my students and their parents that I’m in the business of putting myself out of business.  Please, go ahead and get your math sorted out perfectly.  Understand it so well that you won’t need me anymore.  I’ll be glad to get a different job!

I’m only one person, though I’m thankful to know so very many others who change the world daily, much more than I.  So, how about you hop on this train, too?  You don’t have to be a teacher or an accountant or a scientist to be on board.

Here’s something easy you can do in a few minutes, that just might make a difference.  The next time you see your 9-year-old niece, ask her to count by threes for a while.  Then write down the numbers in a list.  For the two digit numbers, ask her to add the digits together, and add the digits of the resulting total.  Get her to notice that the digits always eventually add up to 3, 6, or 9!  Try it with a three digit number, too:

12:   1 + 2 = 3

15:   1 + 5 = 6

18:   1 + 8 = 9

66:   6 + 6 = 12;   1 + 2 =3

819:   8 + 1 + 9 = 18;   1+8 = 9

Then point out that when the digits of a number add up to a number that is divisible by 3, then the original number is also divisible by 3.  (Use the word ‘multiple’ instead if it works better.)  Wait for it… ‘COOL!’  Now, ask her if the number 56,418 is divisible by 3.  She’ll be so proud to know the answer.  That’s how it works.  And welcome aboard!

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Fun stuff!

Take a look at this set of beautiful animations that illustrate / explain common mathematical concepts much better than words ever could: 

http://www.iflscience.com/brain/math-gifs-will-help-you-understand-these-concepts-better-your-teacher-ever-did

I especially love the water-pouring one with the squares on the sides of the right triangle.  This illustrates the Pythagorean Theorem (a² + b² = c²) in such a beautiful way.  You may have seen this diagram before – the three sides of a right triangle drawn into squares.  BUT the animation of pouring water from the two smaller squares into the larger square, filling it perfectly, is among the purest illustrations that math is truly an art.

What else?  The one about the exterior angles of a convex polygon always adding up to 360° is so simple and so perfect.  The algebraic proof of this is not too hard to understand, but it’s a lot of writing.  In a few seconds, you can see a great illustration of what happens when the polygon’s sides shrink down to nothing.  What is left is a bunch of angles that clearly total 360°.

I’ve chosen the two above examples to talk about mainly because they would be of the most illustrative use to my own students.  Still, I love each and every one of the lovely little animations there!

Here’s the link: http://www.iflscience.com/brain/math-gifs-will-help-you-understand-these-concepts-better-your-teacher-ever-did

Let me know what you think!  Did something suddenly make sense to you?

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ACT, Geometry – diagonal of a rectangular prism

Q: What is the (straight-line) distance between points A and B on the figure below?

rectangularprism2

Answer:  7

Explanation:  We are finding the length of the green segment below:

 

rectangularprism2wline

This requires the three-dimensional version of the distance formula:

  2014-09-12_225150

There is another way to do this, using a progression of two right triangles, with the green segment being the hypotenuse of the second triangle, but I find the above formula much simpler.  It is just an extension of the ‘usual’ (2-dimensional) distance formula we are so familiar with from Algebra and Geometry.

BUT if you want an even easier way, and I know you do, use my TI-84 Plus program ‘DIST3D’.

 

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ACT , Algebra 1 – system of equations

MathPro Q&A Forum (ACT)

 Q: For what value of A will the following system of equations have infinitely many solutions?

3x – 2y = 4

Ax + 4y = -8

 

Answer: -6

Explanation: In order for the system to have infinitely many solutions, the two equations must describe the same line.  (Recall that the solution(s) of a system are the points where the graphs of the two lines meet.  For the two lines to meet in infinitely many points, they must be the same line.)

 To make the second equation the same as the first, multiply both sides of the first equation by -2:

 -2(3x – 2y) = -2(4)

-6x + 4y = -8

So, A = -6.

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SAT – number properties

Q: Find the largest number less than 4000 that is the square of an integer.

Answer: 3,969

Explanation:  The simplest, fastest way to do this is:

On a calculator, find the square root of 4000.  It is 63.245….  Round this down to 63.  Now, square 63.  It is 3,969.

This is the way you should be thinking for SAT problems.  If you decide instead to count down from 3,999, finding the square root of each number until you find a whole number, you will have to do this 31 times until you find the right one!  (And by the way, why 31?  Doesn’t 3999 – 3969 equal 30?  Was I wrong?  No!  This is another little SAT trick.  Always add 1 to include the first or last number, whichever one wasn’t counted in the original 30.)

 

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Calculus – Limits

Q: Find the limit:

Answer: -5

Explanation: Direct substitution (plugging in -1) yields 0/0, an indeterminate form. Factor and cancel:

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SAT – number properties

Q: The variables a, x, and b are positive integers. If ax = 7, xb = 28, and ab = 4, what is the value of a?

Answer: 1

Explanation:

a, x, and b are all positive integers, so the only possibilities for the variables are factors of 7, 28, and 4.

Since 7 is prime, a and x must be 7 and 1, but in which order? If x = 1, then b = 28, and then we can’t find an integer value for a from the last equation. So, x must instead be 7.

It follows that a = 1 and b = 4.

The answer is 1.

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Geometry – polygons

Q: A hexagon has interior angles measuring (in degrees) 137, 100, 150, 114, 90, and x. What is the value of x?
 
 Answer: 129
Explanation:
The total measure of all the interior angles in a polygon is (n-2) * 180, where n is the number of sides of the polygon.
A hexagon has 6 sides, so n = 6. The total is (6-2) * 180 = 720.
Add up the five given angle measures: 137 + 100 + 150 + 114 + 90 = 591.
The measure of the remaining angle is 720 – 591 = 129.
My TI-84 Plus program POLYGON will find this total for you.
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