Difficulty: Tough to determine the pattern quickly, but as usual, all kinds of stuff cancels.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: ACB is 50°. If that isn't showing, refresh the page. Sorry about the mistake when first posted. Arc lengths, anyone?
Problem 37
Difficulty: Fairly difficult.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: Easy. Of course, the only thing worse than fractions for the average kid is compound percentages. That little difference.
Problem 38
Difficulty: Fairly difficult for your students.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Starting a new set! This group is from 2002. As with the previous ones, I'll post two at a time so you won't be tempted to work through all 41 at once. You have to eat, you know. Just looking out for your basic health.
Problem 1
Difficulty: Easy and fun. I really don't get why students freak out over these simple fraction problems. Okay, I do understand ... not enough practice. So here's another practice problem.
Problem 41
Difficulty: Until you see it, "What?" When you see it, "D'oh!" Simple geometry and two nifty trigonometric doodads.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: easy. Notice we didn't specify which color.
Problem 23
Difficulty: Very clever geometry question.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: pretty easy. Kids will get wrapped up in the P(A) and if-then, but it's an example of a problem that is much easier if you imagine the grid of all possibilities. Definitely a candidate for the SAT.
Problem 27
Difficulty: Medium. It's funny how easily the kids get thrown by an infinite series.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: easy-ish. It's similar to a lot of SAT questions in that you don't actually solve for a, b or c but use them all in concert to find some other value.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: easy-ish. Mostly, it's an interpretation question.
I love this next problem. So very cool. Problem 29
i.e., find the shaded area. Difficulty: Medium.
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Difficulty: so easy the kids kept second-guessing themselves. ;-)
Problem 35
Difficulty: Hard (last page)
Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest
Since there are so many truly able mathematicians among my readers, I've decided to give them something to keep busy with! This set will start at the hardest and work backwards.
Difficulty: hard. Standard instructions for this series: No calculator allowed. Express answers in reduced form. Rationalize denominators. Radicals must be reduced. All numbers are base ten unless otherwise specified. Do not approximate radicals or π. Leave such answers as 1025π or √39, for example. Source: UVM Math Contest Answer is here.
This geometric city park is perfectly square with a square koi pond in the middle. There's a circular walking path that is tangent to both squares at the indicated points. The pond is 3 feet deep. The area of the park is 4 acres.
What is the volume of the koi pond in gallons? (You’ll need to Google a few conversions.)
In the diagram, the 4 in. square overlaps the 3 in. square in such as way that the corner of the larger square is at the center of the smaller square. The 4 in. square has been rotated so that its side trisects the side of the 3 in. square. What is the area of the shaded portion?
Proposition – If the water lily is ten inches above the water, and disappears under the surface at a point distant twenty-one inches, what is the depth of the lake?
Click title to Read further The poet Longfellow was a fine mathematician who often spoke about the advantage of clothing our mathematical problems in such attractive or congenial garb as would appeal to the fancy of the student in place of following the dry, technical language of the textbooks. He would connect the proposition with some familiar subject which best explains the problems to be solved.
A clever kindergarten illustration of a mathematical theorem leaves a clearer and more lasting impression upon the mind of a student than a whole term of uncongenial study.
He always held mathematics to be the most important branch of knowledge taught in our colleges and high schools, for the reason that it enters so largely into all of the arts and sciences, and yet the average student graduates with such an undying aversion to figures that he speedily dismisses all recollections of them from his mind.
The water lily problem is one of several introduced in Longfellow's "Kavanah", written while occupying the Chair of Modern Languages in Harvard University, 1849. It is so simple that anyone, even without a knowledge of mathematics or geometry, could solve it with a pair of compasses or rule, and yet it illustrates an important geometrical truth in a never-to-be-forgotten way, which many graduates have never grasped at all.
I forget the exact language of the problem, as he described it to me personally during a discussion of the subject, but he told of a water lily growing in a lake; the flower was one span above the surface of the water, and when swayed by the breeze, would touch the surface at a distance of two cubits, from which data it was desired to compute the depth of the lake.
Now, let us suppose, as shown in the sketch, that the water lily is ten inches above the surface of the water, and that if it were pulled over to one side it would disappear under the surface at a point distant twenty-one inches from where it now stands, say just where the young lady is supposed to have drawn it, which shows that the two flowers are anchored to the same root at the bottom of the lake, what is the depth of the water?
Sam Loyd "Cyclopedia of Puzzles", Lamb Publishing, New York, 1914
Last year, one of the fish developed a fungal infection:
We don't care much about this particular one as it was a 12 cent feeder fish but the same pond has other fish. The koi, for example, are quite valuable. If we did nothing, this one might infect the rest. So we needed PIMAFIX, but how much? More below.
That's not too detailed a picture. You can give them this one following or let them search it out on the Interwebs. They need to find the text at the bottom of the bottle that describes the dosage proportions.
Next comes the question, "How big is the pond?" "Well, I don't really know. Certainly not the number of gallons or liters. But it is a pretty decent circle.
The sides are pretty steep. I guess we can assume the general shape of a cylinder. Maybe drop a percentage off the total?
So, how to measure? I didn't bring my math teacher's 100 meter tape so I walked around it and counted steps. One lap around the pond took me 65 steps, and my steps are pretty close to one yard long. I walked very close to the edges and the side walls are steep. When I drained the pond years ago to get out the boat motor someone had thrown in (don't ask) the bottom was roughly a flat bowl shape. The depth is 6 – 8 feet, depending on the weather.
I've got the blue bottle at school: Bottle of PIMAFIX Net 16 fl. Oz (473ml) Treats up to 2,400 US Gallons (9.085L) Add 2 teaspoonfuls (10ml) for every 50 US Gallons (190L) or 1/4 cup (60ml)for every 300 US Gallons (1,135L) of water
Take it away! Give as much or as little as your kids can handle or need. Good Luck.
p.s. The fish didn't make it. But the other one that developed it did, and the one that developed the same infection this year also seems to be doing well. It's a pretty common situation whenever the slimecoat is scraped off or damaged. Breeding frenzy and the rubbing that results from it can be a cause. This year's fish has a large cut, probably an owl or other bird, but the cut and the fungal infection are both improving.
Those of you who know me know that I love data visualization done well. I am always looking for, and hopefully finding, graphics that clarify or that bring a new perspective to data. Tables of numbers are rarely helpful. An appropriate graph or visual, on the other hand, leads everyone to the classic indicator of epiphany ... "Hummm, now that's interesting ..."
These are some that I consider the game changers ...
Below the fold so the graphics don't kill one-time visitors:
Snow's map of Soho, locating the source of one of the cholera outbreaks in London (the Broad Street Pump) and, for the first time, visually showing cause-effect for disease. This led him to use a microscope on the water and see the little white blobs.
Not that they believed him, though. It would be years before everyone accepted that the nearby septic pit had contaminated the well and was the source of the cholera.
Then we have earthquake data, geographically presented, showing clearly the Ring of Fire and giving strong evidence for tectonic plates. From USGS ...
I love how the earthquakes outline the plates. Students can even guess direction and speed of the plates by looking at the concentration of spots. India is moving north, for instance
Going historical again: Minard's graph of Napoleon's March to Moscow and Back showing troop numbers by width of the line (400,000 down to 10,000) and then correlating the temperatures to the trip back. "The Graph that made a Nation cry."
On the trip back, the vertical lines trace down to the temperature graph at the bottom. December 6th was -38oC, -34oF. Remember, these are men walking a thousand miles in the snow and that cold.
Of course, there's the infamous Time Magazine 'Sexual Relationships at Jefferson High' graph that freaked out a nation.:
and the kids wonder why we worry about disease and contagion ...
David Chandless of InformationIsBeautiful.com is a master at making a point visually:
Here's an interesting new one from Slashdot. This is firewall data arranged by date and attack source. The video contains a decent explanation of the characteristics.
Here's the YouTube link:
This is the kind of thing I point to when kids ask "What will we ever do with this?"
Okay, now it's on to circles and sectors. I certainly hope that our fearless graphic artists have got their Ps dotted and their Qs crossed, and their Is and Ts minded.
Or whatever.
Drumroll ...
Here it is -- the circle of sevenths. Those seven pieces are supplied separately. I guess no one ever thought anyone would ever think of reconstructing the circle. I know you can't read those numbers on the compass, so I'll do that.
In order from 0o North: 52o, 102o, 150o, 210o, 258o, 308o, 360o Low 48o, High 60o
So my questions:
At what point does being accurate matter in this 21st century?
Is right vs wrong being supplanted by "Don't I get partial credit for making seven pieces that are sorta the right size?" I know that my copy of Fireworks can make a line, copy it and then rotate each one 51.40 - why can't these designers do the same?
What happened in this country? We used to be ultra-precise about things. Is all software doomed to be forever "buggy" and "barely out of beta?"
Why is the lack of true understanding on my students' part no longer a surprise to me? And why have I started to be resigned to it?
