Showing posts with label exponents. Show all posts
Showing posts with label exponents. Show all posts

Wednesday, November 27, 2013

How (m^2)/(n^2)=(m/n)^2 is Fundamental to Geometry!

OVERVIEW
The Common Core stresses the importance of students developing a deeper understanding of fundamental concepts and to discover/uncover the interrelatedness of mathematics. The discussion below can be used to demonstrate how a basic law of exponents is tied to the geometry of similar figures.
THE PROBLEM/INVESTIGATION
1) If the sides of 2 squares are in the ratio 2:1, show that their areas are in the ratio 4:1
(a) visually
(b) numerically by examining particular cases
(b) algebraically
2) If the sides are in the ratio 3:1, do you think the areas will be in the ratio 6:1 or 9:1? Now do parts a-c as in 1).
3) If the ratio of the sides is 3:2 show algebraically that the ratio of the areas is 9:4.
4) Show algebraically that if the ratio of the sides of 2 squares is m:n then the ratio of their areas is (m/n)^2.
Note: How does this result connect to the idea that the area of a square varies directly as the square of its side length?
4) If squares are replaced by circles using radii or diameters in place of "sides" show that the results of questions 1-4 are the same.
How does this result connect to the idea that the area of a circle varies directly as the square of its radius or diameter (or circumference)?
REFLECTIONS
• Squares and circles are of course special cases of similar figures. Beyond this investigation lies the BIG IDEA:
The areas of 2-dim similar figures are proportional to the squares of their linear dimensions.
Note: In 3 dim, we can replace 'area' by what?
• Do you see this as one of the fundamental theorems of Euclidean geometry? Is it sufficiently stressed in textbooks and in the standards? Of course you may not feel as I do about all this!
• So what is the geometry connection to
(m/n)^3 = (m^3)/(n^3)...
'.

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

Posted by Dave Marain at 2:57 PM 0 comments

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Thursday, February 9, 2012

FIT THE WORLD'S PRODUCTION OF GOLD IN YOUR CLASSROOM

Visit this link for rich applications of math:
http://money.howstuffworks.com/question213.htm

The particular article in the link addresses dimensional analysis and fundamental science in relation to math. MOST IMPORTANTLY IT MAY ENGAGE EVEN THE LESS MOTIVATED!

Try it with your prealgebra students as an application of unit conversions, exponents, scientific notation and basic geometry concepts.

Students will need to convert mass (kg) to volume (liters) using the specific gravity of gold, then to cm^3. This will allow them to see how the world's estimated annual production of gold will approximately fill up a cube 14 ft on an edge or, equivalently, a rectangular room 14 ft by 28 ft with 7 ft high ceilings.

By requiring students to research the data and conversion constants, you can integrate the web into the assignment.

A scientific calculator is an appropriate tool here, however, require students to WRITE all steps. It is very easy to lose track of the process when we do a series of 5 or more calculations. The result will appear to be incorrect but the student will be hard pressed to find the error. It's human nature to press CLEAR and keep pressing more buttons until it works, then forget the exact sequence of keystrokes!
RECORD EACH STEP ON PAPER WITH APPROPRIATE UNITS. BE A RESEARCHER!

Let me know if you try it and how it works out.

Sent from my Verizon Wireless 4GLTE Phone

Tuesday, December 1, 2009

Using WarmUps in Middle School/HS to Develop Thinking and Review/Apply Skills

My 500 or so subscribers may not have seen the following anagrams which have been in the right sidebar of my home page for the past month or so. No one has yet taken the time to solve them. They're not that hard! Pls email me at dmarain at gmail dot com with your answers.


VORTEC SCAPE


(1) Hidden Steps OR


(2) General Arrows



The following problems are similar to ones I posted recently...



Mental Math and No Calculator!


1) The following sum has a trillion terms:


0.01 + 0.01 + 0.01 + ... + 0.01 = 1000...0
How many zeros will there be in the sum?




2) The following product has a trillion factors:


(0.01) (0.01) (0.01) ... (0.01) = 0.000...1
How many zeros after the decimal point will there be in the product?



A Few Comments...
(a) You may want to adjust the "trillion" for your own groups but I'm intentionally using this number for a few reasons, not the least of which is to review large powers of 10 (Will most think: "A million has 6 zeros, a billion has 9 zeros, so a trillion has..."?).


(b) The second one is more challenging and intended for Prealgebra students and above but, using the "Make it simpler" and "Look for a pattern" strategies, make it possible for younger students.


(c) How many of you are reacting something like: "Is Dave out of his mind? My students don't know their basic facts up to 10 and he wants mental math with a trillion!" I have found that large numbers engage students since they know there is a way of doing these without a lot of work if you know the "secrets"! Besides, we either push our students or we don't. You decide...


(d) These questions review several important concepts and skills. You may want to use these to introduce or review the importance of exponents and their properties.

Saturday, June 13, 2009

An Equation Which May Be More 'Complex' Than It Appears!

Maybe I should rename this blog to Saturday 'Morning' Post. After all, no one reads that either anymore!

As the school year comes to a close (and I'm assuming it's already over for some), here's an innocent-looking equation which might be worth discussing with your advanced algebra/precalculus students now or next year. I might have considered saving this for our next online math contest but it's complex nature makes it more suitable for discussion in the classroom than on a test. Have you seen exercises like this in your Algebra or Precalculus texts? Do students often delve beneath the surface of these? It's kind of like a black box. We often feel we simply cannot reveal too much of the mystery here or we will not finish required content. Well, you know my philosophy of 'less is more' and I don't even live in Westport, CT. (Ok, that's a post for another day!).

SOLVE (by at least two different methods):

2a-3/2 - a-1/2 - a1/2 = 0

Preliminary Comments/Questions/Issues

  • Is the term solve ambiguous here, i.e., should we always specify the domain to be over the reals or over the complex numbers or is that understood in the context of the problems? I'm guessing that most advanced algebra students learn that the domain of the variable or solve instructions may impact on the result, but, that is precisely one of the objectives of this problem.
  • Should students immediately change all fractional exponents to radical form? OR use the gcf approach (which requires strong skill)?
  • It's not hard to guess that 1 is a solution but is it the only solution? Can we make a case for -2 being the other solution? The graph doesn't reveal this and surely, -2 doesn't make sense or does it....
  • Is there ambiguity in raising a negative real number to a fractional exponent (never mind raising i to the i)? Why? Isn't there a principal value for such an expression? How is it defined? This problem raises fundamental and sophisticated issues about numbers which can be taken as far as one chooses to go Just how complex can complex numbers get?
  • What is the role of the graphing calculator here? Mathematica? Wolfram Alpha? In addition to verifying solutions or determining answers, can these tools also be useful in clarifying ideas or raising new questions?
  • Students (and the rest of us) are now capable of quickly filling in the gaps in their knowledge base by visiting Wolfram's MathWorld or Wikipedia for more background. Should this impact on how we present material? Typically, in the pre-web days teachers would avoid opening up a can of worms like complex solutions here, but, with your more capable groups, the sky's the limit now IMO...

Friday, April 17, 2009

Classic Exponent Challenge for SATs, Algebra 2, Math Contests...

Don't forget to register for the upcoming 2nd MathNotations Free Online Contest for secondary students. Click here for more info.



The first 4 terms of a sequence are 2, 6, 18, and 54.
Each term after that is three times the preceding term.
If the sum of the 49th, 50th and 51st terms of this sequence is expressed as k⋅349, then k = ?

Click Read more to see the answer, solution, discussion...


Answer: 26/3

Suggested Solution
The first three terms can be written as
2(30), 2(31) and 2(32). (***)
In general, the nth term is 2(3n-1).
The sum of the 3 desired terms would then be 2(348) + 2(349) +2(350). Factoring out 349, we obtain 349(2/3 + 2 + 6) = (26/3)(349), so k = 26/3.

Comments
(1) Too hard for SATs? Similar (but slightly easier) problems have appeared on the test.
(2) Could students use the "Make it simpler strategy" here to reduce the problem to the sum of just the first three terms? But this is the essence of geometric sequences (or exponential functions):
From (***) above, this sum would be
2(30) + 2(31) + 2(32) = 2 + 6 + 18 = 3(2/3 + 2 +6) = 3(26/3). The coefficient 26/3 would be the same for any three consecutive terms! Is this concept/technique worth developing?

...Read more

Tuesday, November 25, 2008

A "VERY BIG" Pre-Turkey Day Math Challenge for Middle or HS

Just a 'little' last-minute challenge before Turkey Day -- similar to many you've seen before on this blog and elsewhere...

Determine the exact digits of 1002008 - 1001004.


Comments:
Students in middle school or higher will often (or should) employ the "make it simpler and look for a pattern" strategy. Some students will be able to apply algebraic reasoning (factoring, laws of exponents, etc.) to evaluate. It's worth letting students, working in pairs, 'play' with this for awhile, followed by a discussion of various methods. Then challenge them to write their own BIG exponent problem!

HAPPY THANKSGIVING!

Monday, October 27, 2008

An Algebra 2 Math Contest Problem and a Warmup Too!

Not the most challenging contest problem but something to give to your students to develop logical careful thinking and some "basic skills." There's a slight 'twist' but nothing that will faze our math experts out there.

SOLVE

(x2 - 6x + 9)(x2 - 4x + 3) = 1

Monday, April 2, 2007

Motivating Zero, Negative and Fractional Exponents in Middle School and 1st year Algebra

Algebra teachers, like myself, are always looking for ways to help students make sense of exponents. We look through copies of the Mathematics Teacher, we go to the Math Forum and now we Google, Google ad infinitum (or some other search engine to be fair!). Here's an approach that I have found helpful. I assume the student has had some basic introduction to exponents and their properties. I call it the exponential function approach which sounds too challenging for middle schoolers but you decide if they can handle this. Students will use pattern-based thinking and graphs to make conjectures about extending powers of 2 to include zero, negative and even fractional exponents. Properties of exponents will then be used to 'justify' the conjectures. The juxtaposition of the numerical, symbolic, graphical and verbal descriptions are consistent with the Rule of Four that is now regarded as the most powerful heuristic in teaching mathematics.

Begin by making an x-y table - this is the critical piece.

Exponent (x)...........................Power (y = 2x)
3 ..................................................23 = 8
2 ..................................................22 = 4
1 ..................................................21 = 2
0 .................................................20 = ??

The instructor of course is prompting the students for the powers while they are taking careful notes. At the same time the instructor is plotting these results as ordered pairs and the students do likewise. It might be helpful to let 2 or 4 boxes represent one unit on the y-axis since, at some point, the y-values will be fractional. Similarly for the x-axis (play with it first).
At this point, the instructor asks a key verbal question (you may phrase it much differently depending on the level of the group and your preference):
[While pointing to the left and right columns]
"When the exponent decreases from 3 to 2, the corresponding power of 2 is divided by ___.
Repeat this phrase a couple of more times until you reach an exponent of 0, then -1 and voila! Keep going until x = -3, plot the points and the students are seeing an exponential curve in grade 7? 8? 9?
Motivating zero and negative exponents using a function model (tables!) seems to make sense to me because it begins to create a 'function' mind-set that can be carried through all subsequent math courses. It may also help students to 'see' that the range of the function consists of positive real numbers. If you're wondering why I didn't mention turning on the graphing calculators to make the TABLE and GRAPH, I hope you can guess why. It was important for me to have students do this by hand first, then I will turn on the overhead viewscreen and we can explore with technology. Just my opinion of course but students in my classes seem to make sense of this. Of course, I don't kid myself that this approach will lead to better grades on tests of this unit! Facility with the properties of exponents only comes from considerable skill practice with paper and pencil.
For fractional exponents, I'll begin the discussion but I will have to explore further on another post or leave it to your imagination. "Ok, boys and girls, if mathematicians believed exponents could be zero or negative integers, would you be surprised if they wondered about 21/2? From the table and the graph, 21/2 should fall between ___ and ___? Do you think it will be exactly 1.5? Why or why not?
I know many of you use the exponent properties to develop this topic, but I wanted to suggest an alternative. I usually follow this discussion with arguments like: " Hmmm, I wonder what
21/2 times 21/2 would be?" etc...

Sunday, February 25, 2007

SAT-Type Challenge: Exponents and Combinatorial Thinking

These SAT-type questions provide review for the multiplication rule of exponents as well as recognizing the need for using the counting principle vs. careful enumeration in an organized list. Both questions need to be given for the effect. Target: PreAlgebra and beyond...

(a) Consider the list 1,2,3,4,5
If a, b and c are assigned different values from the list above, how many different values will result from the expression abc ?

(b) Consider the list 1,2,4,8,16
If a, b and c are assigned different values from the list above, how many different values will result from the expression abc ?



Notes:
(i) To encourage use of exponent rules, do not allow calculator. What variations would make this even more powerful?
(ii) Possible extension: For (a), ask students to make a conjecture regarding the largest possible power, i.e., is it obvious which is the greatest among 512, 415, and 320 ?

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