Reciprocals, Square Roots and Iteration -- The gift that keeps on giving!

OVERVIEW

SEASONS GREETINGS!

While gifting and regifting this holiday season, here's my gift to all my faithful readers without whom I'd have no reason to put finger to touch screen...

The following series of problems does not on its surface involve anything more than basic algebra, but it is intended to provoke students to reflect on the interconnectedness of number and algebra.

The extension at the bottom goes beyond what might be expected from the beginning of this exploration.

Math educators can adapt this for Algebra 1 through AP Calculus students...

THE PROBLEMS

What are the number(s) described in the following?

1. A number equals its reciprocal.

2. A number equals 25% of its reciprocal.

3.  A number equals twice its reciprocal.

4.  A number equals the opposite of its reciprocal.

5.  A number equals k times it's reciprocal. Restrictions on k? Cases?

Answers:
1. 1,-1
2. 1/2,-1/2
3,  √2,-√2
4. i,-i
5. k>0: √k,-√k; k<0: i√k,-i√k; k=0:undefined

OVERVIEW and much more...

• So why don't we just solve the equation x^2=k? See extension below for one reason.
• Why not ask the students what the graphs of, say, y=x and y=2/x have to do with #3. They might find it interesting how the intersection of a line and a rectangular hyperbola can be used to find the square root of a number!
• Extension to Iteration
Ask students to explore the following iterative formula for square roots:

(*) New = (Old + k/Old)/2

Have them try a few iterations for k=2:
x1=1 (choose any pos # for initial or start value; I chose 1 as it's an approximation for √2 but any other value is OK!)
x2=(1+2/1)/2=3/2=1.5
x3=(1.5+2/1.5)/2=17/12≈1.417 Note how rapidly we are approaching √2)
x4= etc
[Note: Plug in √2 into the iteration formula (*) to give you a feel for how this works!]

Students may want to explore further and they might be curious about where this formula came from, how it's related to Euler, Newton, Calculus and Computer Science. For example, they could  implement this on their graphing calculator or program the algorithm themselves!

Two overlapping circles of radius r... - A Common Core Geometry Problem

OVERVIEW
Intersecting circle problems are always interesting and often challenging whether you find them in the text, on SATs or on math contests. The general case involves trig and formulas can be found online.

The objectives of the problem below include:

• Drawing a diagram from verbal description
• Dissecting or subdividing an unknown region into more common parts
• Applying circle theorems and area formulas
• Solving a multistep problem (developing organizational skills, attention to detail)

THE PROBLEM
Two circles of radius r intersect in two points in such a way that the overlap is bounded by two 90° arcs. If the area of the common region is kr^2, determine the value of k.

Answer: (Pi-2)/2
Note: Please verify!

REFLECTIONS FOR MATH TEACHERS
[Note: These are discussion points --- not short answer questions with simple answers!]

• Should the diagram have been given to eliminate confusion?
• Does this problem appear to have any practical application?
• Have you seen a similar problem in your geometry texts? On standardized tests like SATs?
• In similar problems, were the arcs 60° or 90°?
• How would you introduce this problem? Is it worth the time to have students cut out congruent paper or cardboard circular disks, keep one fixed and move the other until it approximates 90° arcs?
Better to use geometry software?
• Assign this for homework? As a group activity in or out of class? As a demo problem with a detailed explanation provided by you?
• How much time would be needed for classroom discussion of this problem?
• Would you plan on providing extensions/generalizations?
• Too ambitious for "regular" classes? Appropriate only for Honors?
• So what makes this a Common Core activity? Are you guided by the Mathematical Practice Standards?

The Myth of Developing Math Skills Without Effort and Practice - A Rant

Every research study I read reinforces my belief in the children's fable, "The Emperor's New Clothes".

Why is the truth about the need for practicing math skills so evident to everyone EXCEPT those who actually develop and implement education policies in this country, the so-called 'experts'? Is it arrogance, short-sightedness or simply a reflection of a society which has lost its way? Perish the thought that there could be a profit motive in promoting new approaches to learning math...

Think of your most "talented" students/children for a moment. They may think more quickly, display more insight, have greater abstract or spatial ability. But do they ever make mechanical arithmetic errors? No? Then they are truly the exception. Because that's not what I observe. I see a generation of youngsters who are now better at problem-solving yet lack proficiency with the, should I utter the word, BASICS. Why? You all know why!

THIS IS NOT A CALL TO BASICS. DON'T ANYONE MISINTERPRET THESE WORDS. IT IS THE SAME CALL I'VE ECHOED FOR DECADES. BALANCE CONCEPTUAL DEVELOPMENT/APPLICATION WITH SKILL/PROCEDURAL PRACTICE TO BUILD THE FOUNDATION REQUIRED FOR SUCCESS IN MATHEMATICS. BALANCE...BALANCE... BALANCE.

Of course we want students to inquire. Of course we want out students to use tools to analyze the vast amounts of data they now have at their disposal. Of course we want students to understand the WHY as well as the HOW. Of course we want to reach a variety of learning styles. Of course we want to use multiple representations in class. Of course we want mathematics to be interesting and useful.

BUT THERE IS NO SUBSTITUTE FOR PRACTICE. FOR EFFORT. FOR TIME. FOR APPLYING ONESELF. FOR SELF-DISCIPLINE. IN FACT, MATHEMATICS HELPS TO DEVELOP THESE QUALITIES!

So why isn't the obvious visible to the  researchers and policy makers? I'm sure they'll tell you...

Continuation of A Very Inconvenient Math Truth

Pay 2 of my response to Prof Willingham...
Here is the link to his original post:

http://www.danielwillingham.com/1/post/2013/12/what-the-ny-times-doesnt-know-about-math-instruction.html

My reply (unfortunately my reply was duplicated several times. My bad. ..)

(a) 9+9+9+9+9+9+9+9+9+9 vs
(b) 10+10+10+10+10+10+10+10+10
Is their equality a coincidence?

On your grid paper, make 10 rows of
* * * * * * * * *
Which addition problem does this represent?

Should how you could represent the other addition problem without drawing any more stars. [Rotate paper 90°]

Now write both as multiplication sentences...

We can pontificate about all of this ad nauseam but in the end teachers have to be trained to provide an environment which BLENDS explicit and implicit instruction. I learned much about arithmetic and number sense from playing Monopoly but I didn't learn everything that way! Some concepts/skills/procedures had to be clearly demonstrated to me. I was observing my precocious 6-yr old grandson learning to play Monopoly. From playing a couple of times he decided to buy every property he landed on. When he ran out of money I told him he'd have to wait until her could collect $200. "No problem PopPop. Just let me be the banker!" Will he improve his understanding of the game without formal instruction? Of course. Will he also develop some misconceptions if not corrected and given a clear explicit explanation? Of course. INFORMAL LEARNING CAN GO ONLY SO FAR IN MATHEMATICS. This must be balanced with the child developing proficiency with skills/algorithms, attention to detail and recognizing the appropriateness of approximate vs exact results. I'm only scratching the surface here. But I do know that none of this happens by accident. CCSS are necessary to raise the bar but without the"heavy lifting" required to train/prepare teachers, it will be futile. But nothing substantive will occur until the education of our children is genuinely seen as an investment instead of an expense. When we truly put our money where our mouths are...

Another Very Inconvenient Truth About Math Education

I just responded to Daniel below. He was commenting to a NYT editorial on math education. I've included some of his response.

Daniel Willingham
What the NY Times Doesn't Know About Math Instruction
12/09/20131 Comment

"A New York Times editorial on December 6 called for improved math instruction, calling the current system "broken." Although I agree we could be doing a better job of teaching math, the suggestions in the editorial showed a striking naiveté about what it will take to improve."

Now for my reply...

COMMENTS

Dave Marainlink12/09/2013 8:38am

Some excellent points made here but sadly one could read editorials and letters on this topic from every decade for the past 40 years and I have and little has changed. Experienced math educators have forever recognized the problems identified here but what substantive change has occurred other than more testing and expecting more accountability from teachers who are expected to change water into wine.

Helping young children develop conceptual understanding of numbers, operations, relationships (spatial as well) requires specialized training that is not currently the norm in teacher preparation. We are very good at appointing commissions to draft world class standards and creating more ambitious testing but not very good at providing prospective and current teachers with the training necessary to implement these ambitious changes. THERE ARE NO SHORTCUTS HERE. It requires a sea change in teacher preparation and an investment of time and money that no one up to now has been willing to make. The money is out there to make this happen but saying we want to be the best is very different from preparing to be the best. There will always be fads and theories about how to improve the education of our children. But it's not rocket science to figure out that changing standards and assessments is putting the cart before the horse. Teaching children HOW to think isn't easy but it is doable. A young NFL player who happens to have majored in math was asked how math could help him with football. He replied, "It's all about problem-solving." Perhaps we should be listening more to young people like this...

The square root of x+1 equals x+1... A Common Core Investigation

OVERVIEW
Fairly straightforward radical equation in the title but there is so much hidden potential here for students in Alg 2/Precalculus.

REFLECTIONS
• The solutions to the equation above are -1 and 0. No big deal, right? The usual algorithm --- just square both sides and solve the resulting quadratic by any one of several methods. Done. Cheerio. But wait...

• We can encourage students to "make it simpler" by substituting 'a' for x+1 obtaining a^(1/2)=a, square both sides yielding a=a^2 which gives 2 easy solutions 0,1 and then x+1=0,1 producing the final result. Not that big a deal though except...

• A graphical interpretation of these equations is illuminating and illustrates multiple representations/The Rule of 4. You could demo this with the graphing calculator displayed on your smart board or have the students graph by hand or on their device. The graphs of y=x^(1/2) and y=x intersect at x=0 and x=1 then, by translation, the graphs of y=(x+1)^(1/2) and y=x+1 will intersect at x=-1 and x=0. Students should be asked for this conclusion BEFORE checking the graphs to verify!

• Is that all there is? Hardly! The current trend on assessments and hopefully in texts is to have students analyze a family of equations using a parameter. But first we can generalize numerically:
Solve
(i) (x+4)^(1/2)=x+2
(ii) (x+9)^(1/2)=x+3

Are there still 2 solutions for each of these? Solving just a couple of these and recognizing extraneous or apparent solutions would traditionally have been the WHOLE lesson! Not any more...

By the way -  why the "4" and "9"? Did I change the pattern from the original equation?

• Now for the parametric form:

(x+k^2)^0.5=x+k

What questions should STUDENTS be asking themselves BEFORE WE ASK THEM?

• Students can certainly be asked to solve the latter equation for x in terms of k. Some will struggle with the procedure/algorithm. Hopefully someone in each group (or the whole class!) will obtain x=0 and x=1-2k. BUT WILL THEY CHECK THE 2nd SOLUTION! The use of a  parameter goes beyond making a better standardized test question. Now the student has to recognize that, in order for there to be 2 solutions, k must be less than or equal to 1 which was suggested by the numerical examples above.

• Of course I'm anticipating most teachers' reactions to an exploration like this. I've provided much more than can reasonably fit in a 40 min lesson. Use it as you see fit or just ignore it. It will go away or will it?