## Archive for June, 2017

### Friday Word Puzzle

Sometimes, a small word is contained in a longer word. For example, you can see the three-letter word *rid* tucked nicely inside *F riday* in the title for this post, and

*zip*can be found in the middle of

*mar*.

**zip**anSome folks have told me that the following word-in-a-word is particularly appropriate for this blog…

…since my puns put the UGH in LAUGHTER.

Words within words are the basis of today’s puzzle.

Complete each of the nine words below by placing a three-letter word in the blank. The three-letter words that you use all belong to the same category. But there is a tenth three-letter word from the same category that is not used below. What is the category, and what is the missing word?

- OB _ _ _ D
- C _ _ _ PY
- M _ _ _ OT
- PH _ _ _ M
- H _ _ _ SE
- VE _ _ _ D
- CA _ _ _ OU
- EL _ _ _ SE
- LE _ _ _ E

When I started to create this puzzle, I was hoping to give you a similar list in which a short math word was found in a longer word. I found several, but they seem pretty darn hard, and the missing words aren’t always obviously mathy. But for fun, you can try your hand at these, too…

- D _ _ _ Y
- AS _ _ _ E
- SE _ _ _ H
- BU _ _ _ _ SS
- EL _ _ _ _ TH
- C _ _ _ _ RA
- RU _ _ _ _ NT
- SH _ _ _ _ BLE
- BRA _ _ _ _ ILD
- HU _ _ _ _ D
- PR _ _ _ _ _ IFY
- RE _ _ _ _ NT
- WA _ _ _ _ ELON
- DE _ _ _ _ OR
- HO _ _ _ _ SS
- H _ _ _ _ HOG
- IM _ _ _ _ ST

ANSWERS

- OB eye D
- C hip PY
- M arm OT
- PH leg M
- H ear SE
- VE toe D
- CA rib OU
- EL lip SE
- LE gum E

The three-letter words are all parts of the body. The tenth word in that category is *jaw*, which never appears in the interior of a longer word (only at the beginning or end, such as * jawbone* or

*lock*).

**jaw**- D add Y
- AS sum E
- SE arc H
- BU sine SS
- EL even TH
- C hole RA
- RU dime NT
- SH area BLE
- BRA inch ILD
- HU more D
- PR equal IFY
- RE side NT
- WA term ELON
- DE mean OR
- HO line SS
- H edge HOG
- IM mode ST

### The Amazing National Flag of Nepal

The National Flag of Nepal is unique.

Here are several trivia questions about the Nepalese flag:

- It is the only non-quadrilateral national flag in the world. What is its shape?
- It is one of only three national flags where the height is not less than the length. What are the other two?
- What is the sum of the three acute interior angles within the flag?

Question 3 may be difficult to answer without knowing more about the exact dimensions of the flag. For help with that, we turn to the Constitution of Nepal, promulgated 20 September 2015, which contains the following geometric description for the construction of the flag:

**SCHEDULE – 1
The method of making the National Flag of Nepal**

- Method of making the shape inside the border
- On the lower portion of a crimson cloth draw a line AB of the required length from left to right.
- From A draw a line AC perpendicular to AB making AC equal to AB plus one third AB. From AC mark off D making the line AD equal to line AB. Join B and D.
- From BD mark off E making BE equal to AB.
- Touching E draw a line FG, starting from the point F on line AC, parallel to AB to the right hand-side. Mark off FG equal to AB.
- Join C and G.

The traditionalist in me wishes that “line segment” were used instead of “line,” or that the overline were used to indicate those segments, and that a few more commas were inserted to make it more readable. Consequently, the math editor in me feels compelled to rewrite the directions as follows:

But the American in me — given how many times someone in the United States has tried to legislate the value of π — well, I’m just excited to see accurate mathematics within a government document.

The description continues for another 19 exhilarating steps, explaining how to construct a crescent moon in the top triangle, a twelve-pointed sun in the bottom triangle, and a border around the shape described above. Those steps are omitted here — because you surely get the gist from what’s above — but the following “explanation” that appears below the method is worthy of examination:

The lines HI, RS, FE, ED, JG, OQ, JK and UV are

imaginary. Similarly, the external and internal circles of the sun and the other arcs except the crescent moon are alsoimaginary. These are not shown on the flag.

The entirety of this construction, as any classical geometrician would hope, can be completed with compass and straightedge. I cheated a bit and used Geometer’s SketchPad, with this being the resultant mess:

The rough part was placing *C* so that *AC* = *AB* + 1/3 *AB*. Geometer’s SketchPad could have easily measured *AB*, calculated 4/3 of its length, and then constructed a “circle by center and radius,” but that felt like cheating. Instead, I…

- located
*Q*, which is halfway between*A*and*D*; - constructed circle
*A*with radius*AQ = AP*; - constructed circle
*P*with radius*PD*; - constructed circle
*D*_{1}with radius*DA*; - located the intersections of circle
*P*and circle*D*_{1}at points*X*and*Y*; - constructed a line through
*X*and*Y*; - located
*R*, which is 1/3 of the way from*D*to*A*; - constructed circle
*D*_{2}with radius*DR*; and, - located
*C*, so that*CD*= 1/3*AB*.

Now, you could use that information to determine *CF* and *FG*, and then use the arctan function to calculate the measures of the two acute angles in the upper pennon. If you were so inclined, you’d find that their measures are 32.06° and 57.94°, respectively.

But the question above asked for the **sum** of the three angle measures. Without any work at all, it’s clear that the sum of those two angles must be 90°, since the construction described above implies that Δ*CFG* is a right triangle.

And because *AB* = *AD* by construction, then Δ*DAB* is an isosceles right triangle, and the measure of the third acute angle must be 45°.

And that brings us to a good point for revealing the answers to the three questions from above.

- Pentagon
- The flags of Switzerland and Vatican City are square, so the height and width are equal.
- 135°

If you’re looking for more flag-related fun, check out the MJ4MF post from Flag Day 2016 about converting each flag to a pie graph.

### A Ton of Money (or Maybe More)

One of my favorite resources from Illuminations is Too Big or Too Small, a collection of three classroom activities that develop number sense, one of which is the following problem:

Just as you decide to go to bed one night, the phone rings and a friend offers you a chance to be a millionaire. He tells you he won $2 million in a contest. The money was sent to him in two suitcases, each containing $1 million in one-dollar bills. He will give you one suitcase of money if your mom or dad will drive him to the airport to pick it up. Could your friend be telling you the truth? Can he make you a millionaire?

This problem is from the book *Developing Number Sense in the Middle Grades* by Barbara Reys and Rita Barger, published by NCTM in 1991. So it’s not new, but it’s still good.

My first attempt to use this problem with students was dreadful (details below), but I’ve used this problem successfully many times since. Yet something about it always bothered me. I’m not opposed to fictitious scenarios if they get students interested. But this scenario, in which a friend claims to win $2,000,000 and needs a ride to the airport, seems too contrived and not adventurous enough. Luckily, I recently had food poisoning and spent an entire Saturday on the couch watching bad movies. While watching *Rush Hour* (1998), I found a scenario that I liked a whole lot better…

In the clip, the kidnapper asks for the following:

- $20 million in $50’s
- $20 million in $20’s
- $10 million in $10’s

Now the questions of “How much would that weigh? How big a case would you need to carry all of it?” seem a little more meaningful.

I’ll channel my inner Andrew Stadel here. For both the weight and volume:

- Give an estimate that you know is too low.
- Give an estimate that you know is too high.

Now, do the calculations, and see how close your intuition was.

When I first used this task with students, I was anticipating a great discussion about how to estimate the weight and volume of the money. I suspected that some students might estimate that you could fit 5 or 6 bills on a sheet of paper, there are 500 sheets of paper in a ream, a ream weighs about 5 pounds, yada, yada, yada. Instead, one student raised his hand and said:

A dollar bill weighs exactly 1 gram.

I asked how he knew that. “Do you collect money? Are you a numismatist?”

No. That’s how drug dealers measure cocaine. They put a dollar bill on one side of a scale, and they put the cocaine on the other.

“Oh,” I said.

Some days, your students learn something from you. And some days, you learn something from them.

After you estimate the weight and volume, check your answer by clicking over to reference.com.

If you use this video clip and activity in a classroom with students, I’d love to hear how it goes. Please post about your experience in the comments.