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!

Sent from my Verizon Wireless 4GLTE Phone

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?
Sent from my Verizon Wireless 4GLTE Phone

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?

Sent from my Verizon Wireless 4GLTE Phone

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.

Sent from my Verizon Wireless 4GLTE Phone

(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.

Sent from my Verizon Wireless 4GLTE Phone

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?

Sent from my Verizon Wireless 4GLTE Phone

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...

Sent from my Verizon Wireless 4GLTE Phone

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!