Kiss those Math Headaches GOODBYE!

Archive for April, 2011

Answer to Fun Math Problem #2


ANSWER TO FUN MATH PROBLEM #2

The problem, once again, reads as follows. Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange: the minute and the hour hand have exchanged places from the positions they had just before you went to lunch.

The question is:  how long were you away?

I received several answers to this problem, but the first person who got it right was Adrian W. Langman, of Port Angeles, TX.

Here is Adrian’s answer, in his own words:

It’s probably true that the hour hand is near the 12 at the beginning and near the 1 at the end. So it’s about 5 minutes after noon at the beginning, and just a bit after 1 at the end.

(Or, it could be a tad before 11 in the morning at the beginning, and about 5 minutes before noon at the end, if the worker is an early riser.  But this problem is just the geometric mirror image of the one hypothesized above, so the duration of the lunch break will be exactly the same.)

Obviously the lunch break is about 55 minutes. But to find the exact length, let M be the number of minutes past noon at the beginning. I’ll use the obvious coordinate system – the origin at the center of the clock, the clock hands radial lines, the 12 at 0 degrees, and the 3 at 90 degrees.

At M minutes past noon, the minute hand is at M/60 x 360 degrees, i.e. 6M degrees, and the hour hand is at M/60 x 30 degrees (since it’s M/60 of the way from the 12 to the 1, which is at 30 degrees), i.e. M/2 degrees.

So at the end of lunch, since they’ve switched places, the hour hand is at 6M degrees and the minute hand is at M/2 degrees.

Since the minute hand is at M/2 degrees, it is 1/6(M/2) minutes past 1 o’clock, i.e. M/12 minutes past 1 o’clock.

Since the hour hand is at 6M degrees, it’s at 6M-30 degrees past the numeral 1, so it’s 2(6M-30) minutes after 1o’clcock.

Setting M/12 = 2(6M-30) and solving for M yields 720/143 (which is approximately 5.035).

So you left at 720/143 minutes after 12, and returned at 60/143 minutes after 1 o’clock.

So you were gone for 60 + 60/143 – 720/143 minutes,

i.e. 7920/143 minutes, i.e. 55 and 55/143 minutes, which reduces to 55 and 5/13 minutes.

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“Math Cafe” Gives Students Options for Math Success


Sometimes when I tutor I tell students that they are “hanging the in Math Cafe.”

I explain that when I tutor, I like to offer students a “menu of math options.”

Math Cafe

Math Cafe, Open 24/7 = 3.42857 ...

Instead of showing students just one way to work a math problem, I try to present a “menu” of approaches, and I tell students that they need to listen so they can decide which approach works best for them.

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Fun Math Problem #2


Here is the second in my series of “Fun 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 provide your response, simply send an email to me @ info@SingingTurtle.com
and make your Subject: Fun Problem.
Please show how you worked the problem. Thanks. I will post the names of the first three people who get this right.

The Problem:  Before you go out to lunch, you glance at the clock above your desk. When you come back from lunch, you glance at the clock again, and you notice something strange. The minute and the hour hand have exchanged places from the positions they had just before you went to lunch.

The question is:  how long were you away?

Rusting clock face

Image by The Hidaway (Simon) via Flickr

eVersion of Algebra Survival Guide ARRIVES!


The eVersion of the Algebra Survival Guide is HERE!

Amazon.com just listed the eBook TODAY, and so you can now get this book for just $9.95, and read it on ANY of these devices:

Kindle / iPhone / iPad / Android / Blackberry / PC / MAC

The standard price of the paperback book is $19.95, but now you can get the eVersion of this book, and have it with you electronically, for HALF the price  $9.95!

Algebra Survival Guide, now in convenient eVersion!

The Algebra Survival Guide, which debuted in 2000, has sold more than a quarter of a million copies. It has also garnered a Parents Choice award, and it has been used in school districts all across the country as the cornerstone curriculum for Algebra Professional Development Workshops.

The book is read and used by struggling students, teachers, tutors, homeschoolers, and parents. It is an easy book to read, as it is written in a friendly Q&A conversational style. The companion workbook, soon to be available in an eVersion as well, provides thousands of additional practice problems.

Check this book out on Amazon.com at this site!

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.

First letter of a text about the square root o...

square root of 2 w/ "parent" triangle

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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.]
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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.”

Mount Everest from Kalapatthar.

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…)