## Kiss those Math Headaches GOODBYE!

### Everyday Life Sparks Mathematical Puzzles

So here’s the situation: you’re at the breakfast table, enjoying a bowl of steaming-hot steel-cut oats and maple syrup, and you just poured yourself a mug of black coffee. But then you realize you want to pour some milk in the coffee (sorry, purists). But the milk is in the frig, six feet away. So of course you walk to the frig, grab the milk, bring it to the table, pour some in your coffee, return the milk to the frig and sit back down. Question: could you have done this more efficiently?

Yes, of course. You could have brought your cup of coffee with you as you walked to the frig, poured the milk right there at the frig, returned the milk, and then walked back to the table.

“Morning Joe”

When I realized this this morning, I thought … hmmm. Had I used a bit of forethought, I would save myself an entire round trip from the table to the frig. And while I have no problem making that extra trip (hey, just burned 1.3 calories, right?), the experience made me wonder if anyone has ever developed a mathematics of efficiency for running errands.

I could imagine someone taking initial steps for this. One would create symbols for the various aspects of errands. There would be a general symbol for an errand, and there would be a special ways of denoting: 1) an errand station (like the frig), 2)  an errand that requires transporting an item (like carrying the mug), 3) an errand that requires doing an activity (pouring milk) with two items (mug and milk) at an errand station, 4) an errand that involves picking something up (picking up the mug), and so on. Then one could schematize the process and use it to code various kinds of errands. Eventually, perhaps, one could use such a system to analyze the most efficient way to, say, carry out 15 errands of which 3 involve transporting items, 7 involve picking things up, and 5 involve doing tasks at errand stations. Don’t get me wrong! I have not even begun to try this, but I’ve studied enough math that I can imagine it being done, and that’s one thing I love about math; it allows us to create general systems for analyzing real-world situations and thereby to do those activities more intelligently.

Of course, one reason I’m bringing this up is to encourage people to think more deeply about things that occur in their everyday lives. Activities that appear mundane can become mathematically intriguing when investigated. A wonderful example is the famous problem of the “Bridges of Konigsberg,” explored by the prolific mathematician Leonhard Euler nearly 300 years ago.

Euler in 1736 was living in the town of Konigsberg, now part of Russia. The Pregel River, which flows through Konigsberg, weaves around two islands that are part of the town, and a set of seven lovely bridges connect the islands to each other and to the town’s two river banks. For centuries Konigsberg’s residents wondered if there was a way to take a walk, starting at Point A, crossing each bridge exactly once, and return to Point A. But no one had found a way to do this.

One of the famous Seven Bridges of Konigsberg

Enter Euler. The great mathematician sat down and simplified the problem, turning the bridges into abstract line segments and transforming the bridge entrance and exits into points. Eventually Euler rigorously proved that there is no way to take the walk that people had wondered about. This would be just an interesting little tale, but it has a remarkable offshoot. After Euler published his proof, mathematicians took his way of simplifying the situation and, by exploring it, developed two new branches of math:  topology and graph theory. The graph theory ideas that Euler first explored when thinking about the seven bridges sparked a branch of math that’s used today to determine the most efficient ways of connecting servers that form the backbone of the internet!

Of course, there’s also the classic example of Archimedes shouting “Eureka!” and running through the streets naked after seeing water rise in his bathtub. In that moment, Archimedes, who had been trying to help his king figure out if the crown that was just made for him had been created with pure gold, or with an alloy, saw that the water displacement would help him solve the problem. In the end, Archimedes determined that the crown was not pure gold, and the king rewarded the great thinker for his efforts.

As I write this, I find myself wondering if any of you readers can think of other situations in which everyday life experiences led mathematicians or scientists to major discoveries. It would be enlightening to hear more of these stories.

And, if no such stories spring to mind, check out this site, which lists several such stories.  http://www.sciencechannel.com/famous-scientists-discoveries/10-eureka-moments.htm

In any case, the way that such discoveries occur shows that you never know where a seemingly trivial idea might lead … so it’s good to keep your eyes and mind open.

### Quick Easy Way to Untangle Confusion re: “Greater” and “Less”

Whenever I can find a memory trick that helps students get something straight, I use it. Students needs to remember so many things in algebra, so whatever help we can give them is well appreciated.

So recently I stumbled upon a memory trick that helps students tell which of two numbers is greater and which is less.

Let's Reduce Mistakes in Algebra!

You might be thinking:  greater and less?! Why would any student have trouble with that? Well, before students hit negative numbers and absolute value, there is generally little trouble. The greater numbers are the larger numbers, the lesser numbers are the smaller numbers. And kids basically know what we mean by larger and smaller whole numbers, when they are dealing with positive numbers and zero.

But when students encounter negative numbers, some things change.
While 10 > 5,   – 10 IS not > – 5. Instead:  – 10 < – 5.

As if that were not enough, absolute vale comes along and makes things still more confusing, since it takes the value of any number and makes it positive. So now:

abs. value of – 10 > abs. value of – 5

### Math in movies: Who says math isn’t cool?

Some people say that math isn’t “cool,” whatever that means.

I say that is just wrong. Take a look at a wonderful link I found.

It shows a wide range of movies that have math content as part of the story, and it directs you right to the scenes that have the math woven in (check out the Flash links). Check them out and see if you agree that math can be as entertaining as … as … showbiz!

I enjoyed browsing through these. A few of my favorites are:

A Beautiful Mind

October Sky

Contact

### How to Decrease Mistakes in Algebra – Part 3

When we left off, we were talking about the double-slash, a form of notation I’ve developed that helps students attain greater focus when simplifying algebraic expressions.

With greater focus, students make fewer mistakes. With the double-slash at their disposal, students avoid the mistake of combining terms that should not be combined. In the following example, students use the double-slash twice to simplify an algebraic expression:

+ 8 – 2(3x – 7)

=            + 8   //  – 2(3x – 7)

=            + 8  //  – 6x + 14

=            – 6x  //  + 8 + 14

=            – 6x  + 22

Let's Reduce Mistakes in Algebra!

By cordoning off the section with the distributive property:  – 2(3x – 7), the double-slash allows students to see it distraction-free. With this heightened level of focus, students are more likely to work out the distributive property correctly, then continue on, simplifying the whole expression with no mistakes.

### Invisible Misunderstandings: Square roots of 2 and 3

Would you say that the square root of two is an important number in math? Hmmm … and would you agree that the square root of three, while perhaps not quite so important, is still a quantity whose value students should be able to estimate?

Why not, right? After all, these numbers play key roles in the 30-60-90 and 45-45-90 “special triangles.” And therefore they both appear a lot in geometry, and a great deal in trig. And on top of that, root two, widely believed to be the first irrational number discovered, shows up in a wide range of other math contexts as well.

square root of 2 w/ "parent" triangle

### Who Invented the Coordinate Plane?

A fly …

Who would think that a mere fly could play a major role in the history of human thought?

But when it comes to the development of Algebra, that’s the story. I’ll explain how this works just a bit later in this blog. But it is all related to what is happening now in algebra classes all around the world.

For it’s spring, that time of year again when we get out the graph paper and the ruler. Kids are working on the Cartesian coordinate plane.

One about I like about the coordinate plane is that there’s an interesting story about how it was discovered, or should I say, invented. [Hard to know the right word for an intellectual Invention like the coordinate plane.]
(more…)

### New Approaches to Help Students Overcome Math Struggles

No one would attempt to climb Mount Everest in a day.

But when we teach math, we often expect something similar from students. We expect them to learn a complex, multi-step process in one lesson, in one hour. We expect them to go from no awareness of the process, to awareness to competence to mastery. And we don’t take account of the fact that many math process requires a long ladder of thought steps. In edu-jargon, this process of taking all of the little steps into account — and teaching each step individually — is called “scaffolding.”

Like climbing Everest, doing Math requires many STEPS

I have long found “scaffolding” important in working with students who struggle with math in general and algebra in particular. (more…)

### “Algebra-for-All” Strategy, Good or Bad? Get the News

Over the past 10 -15 years, many states have mandated tough new requirements that ALL students (special education students as well as mainstreamed students) take and pass Algebra 1 (sometimes higher math courses, too) in order to graduate from high school.

While that may not sound very challenging for students who do well in math, these mandates have placed major hurdles before students who struggle with math in general — and algebra in particular.

New studies have been coming out on the impact of this so-called “Algebra-for-All” teaching push. I just found an interesting article on this topic at this site.

I’m now including a general link to this math news portal — in my blogroll — as it contains a wide range of articles for math educators. Its name on the blogroll is Math Education News. Feel free to check it out any time you drop by the blog — or any time at all.

And do feel free to share your comments on the current “Algebra-for-All” push. Do you find that it is working where you live and work? Or not working? Any suggestions on how to tinker with mandates to make them work? This is an important topic since algebra is the critical “gatekeeper” course to all higher math. And what’s more, major studies have found that success in algebra is one of the key predictors of matriculation into college.

So a lot is at stake when it comes to algebra. And a lot rides on how well we as a nation help children succeed in this course.

Share your thoughts; we’re all curious to hear what you think.

The “fit” for each situation is the following ratio:
(Area of Inner Figure) ÷ (Area of Outer Figure)

For the square peg in a round hole —
Call the radius of the circle r.
Then the diagonal of square “peg” = 2r
Notice that by slicing the square along its diagonal,
we get a 45-45-90 triangle, with the diagonal being
the hypotenuse and the sides being the two equal legs.
Using the proportions in a 45-45-90 triangle,
side of square peg = r times the square root of 2
Multiplying this side of the square by itself gives
us the area of the square, which comes out as:

This being the case,
Area of square is: 2 times radius squared, and
Area of circle is: Pi times radius squared, and so …

Cancelling the value of the radius squared, we get:
Ratio of (Area of square) to (Area of circle) is:
2÷Pi = 0.6366

For the round peg in a square hole —
Call radius of the circle r.
And since the diameter of the circle is the same length as
the side of the square, the side of the square = 2r
Multiplying the side of the square by itself to get the
area of the square, we find that the area of the square
is given by: 4 times radius squared.

This being the case,
Area of circle is: Pi times radius squared
Area of square is: 4 times radius squared, and so …

Ratio of (Area of circle) to (Area of square) is therefore:
Pi ÷ 4 = 0.7854

Of the two ratios, the ratio of the circular peg in a square hole
is greater than that of the square peg in a circular hole.

Therefore we can say that the circular peg in a square hole
provides a better fit than a square peg in a circular hole.

### FUN MATH PROBLEM — Circling the Square & Vice-Versa

From time to time I will post interesting math problems.

Feel free to try these problems. Share them with friends and colleagues. Use them however you see fit!

I will post the answer to the problems two days later, after people have had time to respond.

To post your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.

The problem: Which provides the fuller fit? Putting a circular peg in a square hole, or putting a square peg in a circular hole? To get credit, show all work, and justify your answer by expressing each “fit” as a percent.