*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.**

**--------------------------------------------------------------------------------------------**

**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}= 1/10

^{2}= 1/100 = .01 Note that there are

*two*decimal places, etc.

10

^{-1000}- 10

^{-997}= 1/10

^{1000}- 1/10

^{997}

Using 10

^{1000}as the common denominator, we obtain

1/10

^{1000}- 10

^{3}/10

^{1000}=

-999/10

^{1000}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:

Was #1 supposed to be a right triangle?

Jonathan

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

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