Sunday, November 15, 2009

The Return of the WarmUp Challenges!

Just when you thought that MathNotations is on permanent hiatus or in hibernation, here are a couple of WarmUps/Problems of the Day/Test Prep/Challenges/// to consider for your students. 

Actually, I'm embarking on a new venture - an online tutoring website with live audio and video for OneOnOne math tutoring for Grades 6-14 (through Calculus II). In addition, I'm also working on setting up a small group (5-10 students) online SAT or ACT Course grouped by ability (a 600-800 SAT group, a 450-600 group, etc.).  If you're interested in getting more information about these before the official launch just contact me at dmarain at gmail dot com.


Update: Answers/comments are at the bottom...

1.   NOTE: ANGLE B IS A RIGHT ANGLE IN DIAGRAM BELOW - THANKS TO JONATHAN FOR CATCHING THAT OVERSIGHT!


















2.   If 10-1000 - 10-997 is written as a decimal, answer the following:


(a) How many decimal places are there, i.e., how many digits to the right of the decimal point?
(b) One can show that the decimal digits end in a string of 9's. How many 9's?
(c) How many zeros are to the right of the decimal point and to the left of the string of 9's?

Notes:
(1) If we write the negative exponent expressions as rational numbers, this is perfectly appropriate for middle schoolers and, in fact, I think they need more of these experiences!
(2) The "Make It Simpler - Look for a Pattern" Strategy should be second nature to our youngsters, but when they see questions like these on the SATs, how many of our students really think of it!
(3) The fact that some calculators return a value of zero for the expression in the problem is a teachable moment - seize it!!
(4) See below for an algebraic approach.



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ANSWERS


1. 9√3


2. (a) 1000   (b) 3   (c) 997


An Algebraic Approach to #2:
First, students need to be familiar with the basic pattern:
10-1 = 1/10 = .1 Note that there is one decimal digit.

10-2 = 1/102 = 1/100 = .01  Note that there are two decimal places, etc.


10-1000 - 10-997 = 1/101000 - 1/10997
Using 101000 as the common denominator, we obtain
1/101000 - 103/101000 =
-999/101000 from which the results follow (with some additional reasoning)...

Note: I could have worked directly with the exponent form by factoring out 10-1000 but I chose rational form for the younger student.

2 comments:

jd2718 said...

Was #1 supposed to be a right triangle?

Jonathan

Dave Marain said...

Absolutely, Jonathan! Thanks! I had it in my original diagram then lost it. Oh, well, I guess I'm out of shape!