Metal brainteasers

Monkey Pod games produces high-quality, yet affordable wooden and metal brain teasers and puzzles. I got to review their Metal Brain Teasers Gift Set II. It has four things in it.

These were lots of fun to try to figure out. There's a double horseshoe with a ring that I was able to figure out eventually. This set of two pins was the easiest to solve:

The other two were harder (free the heart, and separate the two M-shapes). My hubby spent quite a while with the M-shapes but eventually got the solution for that one from their website.

Like I've said before, they make excellent gifts. This set comes in a nice wooden box but you could, if you wanted to, split the gifts and give just one brainteaser to three different children—or adults.

They also sell two other metal brainteaser sets:

Monkey Pod Games also has tons of wooden puzzles and brainteasers, some of them less than $10. Check them out! Hey, I got to playing around with Blogger's Amazon tool and found some images for you real quick... click for more info.

Math Teachers at Play #36

I'm terribly sorry I forgot to mention Math Teachers at Play carnival this last time it was published (which is over a week ago). I noticed it in the morning, was gone for part of the day, and then forgot about it. The carnival looks great and is filled with lots of nice posts about all sorts of things, related to mathematics... please go check it out, and enjoy!


Math Teachers at Play carnival #36

Math genius at age 12

Some of you might enjoy reading this article... a 12-year old boy (Jacob Barnett) is about done studying college-level studies in astrophysics and they want him to start doing research. He's also mildly autistic.

Genius at work: 12-year-old is studying at IUPUI

IUPUI is interested in him moving from the classroom into a funded researcher's position.
"We have told him that after this semester . . . enough of the book work. You are here to do some science," said IUPUI physics Professor John Ross, who vows to help find some grant funding to support Jake and his work.

Conceptua Fractions free tools

I've heard a lot of good things about the free fraction tools at Conceptua Fractions. They are interactive online tools that use visual models to teach all fraction operations. They have a tool for identifying fractions, another for equivalent fractions, another for adding fractions with like denominators, another for adding with unlike denominators, and so on.

These tools exceed anything I've seen in the past... truly comprehensive and excellent. They always include several different visual models, and you can switch between them in an instant... the pie, a bar, a rectangle, a number line, or a dot model. You hover over the area to see the slider, and then you can drag it to partition your model into however many parts you want (well, up to 90 or 100 or so). Then you color or uncolor the parts with a single click inside the parts.


I want to mention that these tools are meant for the teacher, to illustrate the concepts. You would best use these tools when teaching the concepts, whether  in whole-class instruction, in a homeschool,  or in tutoring. They do have a few example problems, just a few of various kinds, to show what you can do with the student(s), but they don't provide quizzes or a slew of ready-made problems for the student to complete.

Ok, enough talking. Go try them out -- they're free. For each tool, it's maybe easiest to first look at the sample problems or watch the video that shows how to use it. Then you can go experiment with the tool itself. Lessons plans are also provided.

Free money worksheets

I'm glad to announce these additions to the worksheet generators at HomeschoolMath.net. Use them to generate customizable free worksheets for counting coins and bills.

US money worksheets (cents & dollars)
Canadian money worksheets (cents & dollars)
Australian money worksheets (cents & dollars)
South African money worksheets (cents & rand)

I have two more coming up... British and European.

A homeschooler wins Intel's competition

Just an interesting tidbit... 17-year-old Evan O’Dorney has won the first prize of $100K in The Intel Science Talent Search 2011 competition. He is and has been homeschooled. His entry to the competition had to do with math.

...he compared two ways to estimate the square root of an integer. Evan discovered precisely when the faster way would work. As a byproduct of Evan's research he solved other equations useful for encrypting data.

Read the details

One-minute cool math video

This is a cool 'n' short math video (HT Denise) from a math blog Love 4 Math. My kids laughed... good fun! Good job!

Math Mammoth placement advice

As you may know, I provide placement tests for Math Mammoth. These tests are end-of-year or exit tests for a given grade. They will work, whether you're planning to place the child in the grade-level series (Light Blue), or to use the topical books (Blue).

Many people don't know what to do if their child doesn't totally "flunk" the test but almost passes, or if their child does well in other areas, but misses the problems in some particular topic, such as fractions or multiplication. On that page, I provide advice for these situations.

Recently I also added suggestions as to what to do if the student does well in other areas, but can't do the word problems in the test.

I also provide personal guidance to anyone who sends in the test results. Over the years, I've done quite a bit of that type of "counseling". So your child's case, however "disastrous" it might be, is probably not unique.

Plenty of people write to me how their child "bombed" the test, or is many grade levels behind, and they need to know what to do. Nearly universally I can find a solution using my Blue Series books as supplements.

So, don't hesitate to contact me about the placement, if you need help.

Measuring obesity with BAI

I'm not real super interested in obesity or calculations of people's weights, fat levels etc. but I ran across something half interesting that might be of interest to you.

It is a new way to measure obesity or if a person is underweight, normal weight, or overweight -- and it doesn't even use your weight in the calculation!

Murray has blogged about it at New measure of obesity – body adiposity index (BAI)


He also made a calculator for it that compares your "results" as far as BMI and this new index, BAI.

This new index uses your height and a measurement around the hips, and it's still somewhat experimental in the fact that it hasn't been thorougly tested in all races.


The formula is

  hip (cm)
-------------   −  18
[height(m)]1.5

The resulting number should be a close estimate to your body fat percentage.


Just for the record, or if you are curious, I am of normal weight, and supposedly have about 29% body fat according to that calculation. Which, 29% body fat sounds high but it seems to be perfectly normal for females. It makes me think of this age-old saying "Human body is 70% water." That just can't be, not for females anyway. I would be 70% water, 29% fat, and 1% all the other stuff...???
 
Ok, upon checking I find that indeed, the human body is NOT 70% water. Females average 55% water and males 60% water, but obese people may only be 45% water.

A problem with a chord: find the radius

Today I had the opportunity to solve a real math problem involving a circle and a chord of known length in it. I had to find the radius. It wasn't a textbook problem or a puzzle on some website, but a math problem I needed to solve for my own needs.

For a tiny while I thought I could find the answer online, but I didn't, so I'm writing it out in case someone else needs it -- they should be able to find this solution by searching the Internet.


I wanted to make a kind of "moon-sliver shapes" in CorelDraw, to use as watermarks in my new books. I have the height and the width of the "sliver".

Here's the problem mathematically:
I have a chord of a circle, 17 mm in length in my example, and the other distance marked in the image is 5 mm. I need to find the radius of the circle, AND the angle measure of the arc of the circle that makes the sliver's rounded part.





At first, like I said, I searched around if there was some theorem or formula that would tell me what I needed directly. I didn't find any, but I did realize that I can use this theorem to solve my problem:

If two chords intersect, the product of the segments of one chord equals the product of the segments of the other chord (see proof).


My problem looks like this:




My chord intersects the diameter of the circle, which is a chord too. The two parts of the first cord are 8.5 and 8.5, and the two parts of the other are 5 and d − 5. Thus I get the equation

5(d − 5) = 8.5²

From this it is quick to solve that d = 19.45. Then, the radius is of course half that, or 9.725.

On to the second part of my problem: to find the angle measure.

In this picture, I have a right triangle. I can therefore find the unknown angle α by using simple trigonometry, in this case the tangent.



The length 4.725 comes from the fact that the radius is 9.725, and then I subtract 5 from that.

The equation is tan α = 8.5/4.725, from which α ≈ 60.931°. The actual angle I want is double that, or about 121.862°.

I had to repeat this calculation several times for slivers of different "heights." (Or, actually I let Excel calculate the rest. Oh, how I love Excel!)

Here's one of the final pictures I made for my book:

Lulu sale again

Lulu has a 20% off sale again... so you can use it to purchase Math Mammoth printed books.

20% off any book order
Enter code: GIANT305

Enter coupon code GIANT305 at checkout and receive 20% off your book order. The maximum savings for this offer is $100. Sorry, but this offer is only valid in US dollars and cannot be applied to previous orders. You can only use this code once per account, and unfortunately you can't use this coupon in combination with other coupon codes. This great offer expires on March 7, 2011 at 11:59 PM, so don't miss out! While very unlikely, we do reserve the right to change or revoke this offer at anytime, and of course we cannot offer this coupon where it is against the law to do so.

See the offer details online

The value of manipulatives

(Note: I'm "resurrecting" an old post, with added information and a video. The topic is still very much valid.)

Manipulatives are IN, in math education. But do they TRULY facilitate learning to such an extent as people promoting them claim?

The entry at Text Savvy, Hands-on, Brains-Off, explains some of the pitfalls in manipulative use. Very enlightening!

Quoting from an article at Education Week (Studies Find That Use of Learning Toys Can Backfire):

In a similar series of experiments at the elementary-school level, the researchers found that children taught to do two-digit subtraction by the traditional written method performed just as well as children who used a commercially available set of manipulatives made up of individual blocks that could be interlocked to form units of 10.

Later on, though, the children who used the toys had trouble transferring their knowledge to paper-and-pencil representations. Mr. Uttal and his colleagues also found that the hands-on lessons took three times as long as the traditional teaching methods did.

The video below also illustrates how manipulative use can lead to problems.

The girl solves an addition problem involving thousands by drawing thousand-blocks, hundred-sheets, etc. on the board, taking 8 minutes. Then she says that at home she has been taught to stack the numbers and add. She solves it that way, too, taking 1 minute. But she gets two different answers.

(Hat tip to Denise)



Clearly, manipulatives aren't the best way to solve such problems, and children shouldn't be led to believe so. Not that this girl believed that way... she seems to understand which way is easier and quicker.

My take is that manipulatives aren't the ultimate answer to math teaching.

They're a reasonable starting point. But children should NOT be taught to rely on them. Children need to be taught and shown how to transfer all of that concrete play into the abstract.

In my books, I often instruct concepts with pictures, which essentially take the place of manipulatives. Then the student does a bunch of problems that have the same pictorial representation in them, including having to draw  those same pictorial models to illustrate the math problems. THEN after that, the student goes on to totally abstract representation.

Here's one example from my book Add & Subtract 2-B: Adding with Whole Tens. The student first adds using the pictorial model, and then with numbers only.

I don't know if that approach suffers from the drawbacks mentioned in the article above. I hope not; from the feedback I get, it seems to work well.