Quarantine math help

[Charlie Velasquez]

Need some help.
My grandkids- middle school age, strong math skills- are doing the home schooling thing. I thought I would send some practical math problems, then video chat to discuss answers. Some I thought of are quarantine related. The virus lends itself to a multitude of number problems, but I don’t want to do anything involving number of deaths or respirators or similar.
But the anticipation that the Texas governor will soon relax restrictions and start up the state gave rise to a couple of social distancing questions that aren’t morbid.
The questions:
1. If the people in Texas spread out evenly, about how much space will each person have?
2. If we split the state into an 8’x8’ grid and put one person in each square, about how many could we fit in Texas?
But the grid method means we have more distance at the diagonals, so...
3. If instead of distributing the people one per square of the grid, we put them in center of an 8’ diameter hula hoop, could we fit more, less, or about the same number of people? If more or less, what would be the difference?


#1 & #2 are simple plug and play multi step problems. Look up area and population and do the arithmetic.
I would expect them to discuss and find the area and population on their own.


#3 ....... is a thought problem for them. The area of a circle will fit some into the interstices of the adjacent circles. How many more circles could we fit than squares?

To examine #3 I really didn’t use the area or population data, just did some concrete samples.
My numbers suggest about 15% more. Does this sound right?

[Mike Henderson]

The problem with the answer to #3 is a packing problem. The 8' diameter circle fits inside the 8x8 square so you can only pack one circle in each square. So the answer is the same in either case - same number of people. My assumption is that you don't get to put people into the interstices between the circles because you wouldn't have the same social distancing.

Mike

[Jim Koepke]

Question #4 could be how many will be standing in rivers, creeks or lakes?

jtk

[Dave Mills]

I like question #3 also because you can discuss how one would calculate the exact difference. It could be by doing some high level math to compute it, or by pouring marbles into a tray, and whether the edges have an impact on your calculation (and yes, the lakes and rivers!), for something the size of Texas versus the size of one room in a house.

Pic from Wikipedia (https://en.wikipedia.org/wiki/Circle_packing)

[Gary Ragatz]

The problem with the answer to #3 is a packing problem. The 8' diameter circle fits inside the 8x8 square so you can only pack one circle in each square. So the answer is the same in either case - same number of people. My assumption is that you don't get to put people into the interstices between the circles because you wouldn't have the same social distancing.

Mike

But you can offset the circles to use some of the "wasted" space. Not worthwhile in a case of beer, but if your packing problem is thousands of units in both dimensions, it will help.

Edited to add - see Dave's picture.

[Mike Henderson]

But you can offset the circles to use some of the "wasted" space. Not worthwhile in a case of beer, but if your packing problem is thousands of units in both dimensions, it will help.

Edited to add - see Dave's picture.

Yep, I agree. My mistake.

Mike

[Roger Feeley]

Here's an old puzzler from Click and Clack.

Imagine that you could count all the hairs on the heads of everyone in the world. Take the number of hairs on the first person and multiply them by the number of hairs on the second person and so on for every person in the world.
How many digits are in the answer?

Answer: 1 digit. Answer is zero because someone is bald.

[Bill Dufour]

https://www.purplemath.com/

Good site for algebra and above.
Bill D.

[Bill Dufour]

For you home schoolers can your 8th grade and above kids pass this test? Why not?
Bill D.

http://www.indiana.edu/~p1013447/dictionary/8thgradeexam.htm

[Lee Schierer]

[QUOTE=Gary Ragatz;3010602]But you can offset the circles to use some of the "wasted" space. Not worthwhile in a case of beer, but if your packing problem is thousands of units in both dimensions, it will help.

Edited to add - see Dave's picture.[/QUOTE

You need to clarify that the area is a number and not related to actual boundaries of Texas, in which case you would get 148,288,570,982 whole circles. This is based on an area of Texas = 267,339 square miles

[Gary Ragatz]

You need to clarify that the area is a number and not related to actual boundaries of Texas, in which case you would get 148,288,570,982 whole circles. This is based on an area of Texas = 267,339 square miles

Lee,

You're right that the area of 148,288,570,982 circles with a diameter of 8' is equal to 267,339 square miles. But Charlie's original question was how many of these "social distancing circles" can you fit into the available area - so, you still have to consider the wasted space between circles. Regardless of the specific boundaries of the space, you can't pack circles together without "wasting" some of the area available. Squares or hexagons, yes - but not circles.

[Bryan Lisowski]

If I asked my 8th grade son to do a math question like this his head would explode, literally blow right off his shoulders. Obviously they must be teaching kids in TX better than NJ.

[Charlie Velasquez]

If I asked my 8th grade son to do a math question like this his head would explode, literally blow right off his shoulders. Obviously they must be teaching kids in TX better than NJ.

The answer is not so important, rather the talking about how we might go about finding the answer.
But, once the discussion yields a plan, the rest is just arithmetic, and a calculator does all the tedious stuff.

In my classroom I had some basic rules of math, the rules were so simple the kids thought they were idiotic. But after awhile they learned to embrace them and their ability to solve problems improved.
The very first rule I told them was, “We don’t do hard problems. If I ever give you a hard problem, you don’t have to do it.”
That did not mean they didn’t have to do a problem, they could make up a similar problem with easy numbers (but they couldn’t use the numbers 0, 1, or 2)
Another rule was, we always draw a picture. Numbers are figments of our imagination, they only have meaning in the context of something else.

Using these they can come up with a plan.... maybe.
This is what I did. Drew a simple picture of Texas.... too hard, so I made Texas a rectangle.
Made the size of Texas an easier number, 400 feet tall (easily divided into 8x8 grid), the width didn’t matter.
Stacked circles, one in each grid section as Mike described. That is equal to the number of squares and gave me 50 rows of circles.
Then stacked them as in Dave’s picture. One can see the line of the tangents of circles forms a hypotenuse of a right triangle. So the length of the hypotenuse divided by 8’ gives the number of rows.
So now it is just a problem of finding the hypotenuse of a right triangle.
Got 57 rows. About 15% more
Did it again with Texas 800 feet tall, again ~15%. Tried weird numbers, always ~15% more rows.
The number of circles in each row would basically be the same number as the number of squares, minus one on every other row; negligible over the breadth of Texas.
So ~15% was what I came up with.

When we go over problems it is important I have at least one graphical solution that I can steer them towards in case we slow down. But anything that makes sense works. Lots of times the kids come up with totally creative ways to solve stuff.

Lee, I was getting about 133 920 798 660 (tel:133 920 798 660), give or take.....

[Lee Schierer]

Here's how to determine the area of the little triangular spaces between circles. Not exactly middle school math.

Connect the centers of three touching circles. Makes an equialteral triangle with side s = 8 feet. Area of this triangle is sqrt(3)/4 * s^2. Now the three circular wedges, when put back to back, make a semi circle with radius 4 ft. Area of this semi circle is Pi/2 * 4^2. Just subtract them to get the bit inside.

[Lee DeRaud]

Whether it's 133 billion or 148 billion, it's still both (1) an order of magnitude more than the world's population and (2) way too many friggin' Texans.

(Bonus points for mentioning John Brunner's novel, 'Stand On Zanzibar'.)

[Malcolm McLeod]

... way too many friggin' Texans. ...

Yeah, the lift lines would be a bit long at the CO ski resorts. ...How's the CA skiing??

[Lee DeRaud]

How's the CA skiing??Never touch the stuff, myself. When I moved here from Denver in '74, my first bit of culture shock was hearing the weatherman report local ski area snow depths in inches instead of feet.