It's not Psuedocontext, it's just wrong.

Credit for the Bad parts, too.

There are many errors in the packaged, bought and paid for, courses from Florida Online. Unfortunately, the "developers" seem to have taken that "Beta" software approach ... if it isn't "Threatening-the-President-bad", then we won't bother fixing it.

Take this problem, from the Semester Exam:

Triangle ABC is congruent to triangle DEF. In triangle ABC, side AB measures 9, side BC measures 3x+18, and side CA measures 7. In triangle DEF, side DE measures 9, side EF measures 2x+26, and side FD measures 7. What equation would help you to solve for the side length of BC and EF? Explain your reasoning using complete sentences. (10 points)

At first blush, you say "They're congruent. BC is congruent to EF, so 2x+26=3x+18. Done."

What's the big deal?

Solve it.

Pretty easy ... x = 8.  So the sides of a triangle are 9, 7, and 42. Shit.

Any student who takes that obvious next step, and thinks for more than a second about a different topic just studied, i.e. comparing sides of a triangle and scissors theorem, is now convinced he's done something wrong.

It's not Psuedocontext, it's just wrong.

Pallet Load configurations

Found this interesting ... a pallet manufacturer's images of various loading schema:

Lake Puzzle

see the answer here.

Koi Pond Puzzle

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.)

Answer in this backdated post.

Overlapping Squares Puzzle

Overlapping Squares

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?

answer, here, in a backdated post.

The Lily-Pad Problem

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

What can you do with this? Fish Fungus

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.

Game-Changing Graphics

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:

More SmartBored Fun

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 0^o North: 52^o, 102^o, 150^o, 210^o, 258^o, 308^o, 360^o
Low 48^o, High 60^o

So my questions:

1. At what point does being accurate matter in this 21st century?
2. 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?
3. 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?"
4. 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?

Find the Area of a Triangle

In discussing the following problem over on Math Notations, there's been an interesting mix of methodology.

In the coordinate plane, what is the area of ΔPQR given the coordinates P(4.5,4.5), Q(8.5,8.5), and R(6,0)?

The first level of difficulty lies, of course, in getting the image correct. Leaving this up to the kids makes for a tougher problem. Here's the diagram.

So the question fairly begs for a solution and it was interesting to me how many different ideas came up. Before you scroll down and see what the others did, try this problem yourself.

Dee-dee-dee-dee, dee-dee-dee, dee-dee-dee-dee-DUM-da-dee-dee
(Badly transcribed Jeopardy theme tune)

Here we go!

All of them are valid but operate under different frames of reference - your starting point, I think, depends a lot on what you're working on in math or the most recent similar problem you've solved.

Here's probably the simplest version. Big triangle minus the yellow one:

(1/2)(6)(8.5) - (1/2)(6)(4.5) = 12

I did the problem mentally and didn't twig on the two triangles. Instead I found a base and an altitude. 1/2* 4(root2) * 3(root2) like so:

The most interesting one was done by a student:
Translate the points P and Q down the line y=x. The orange and red triangles are still equal areas because they have the same height and equal bases.

Now find the area of the orange triange with a height of 4 and a base of 6.

Ain't that cool?

To repeat my comment on MathNotations:
We all approach problems differently depending on what frame of mind we're in at the time.

A basic math student would probably use the 2triangles method. An algebra student who just got through pythagorean theorem, distance formula and simplifying radicals might gravitate to the other. The algebra student who's been translating stuff might think of that - though the idea that sliding the two points down y=x doesn't change the area is NOT something that many students can do in their heads! Someone else might go to the trouble of flipping out Heron's formula or 1/2ab*sinC if the points were set up differently. Still others use Pick's formula since they can see the points in the diagram.

If your student weren't able to immediately follow your using PQ as a base and finding the altitude to it using perpendicular slopes (the problem, it seemed to me, was set up to push the solver in that direction) then he isn't really comfortable with algebra.

Don't knock your initial instincts until they get in the way of solutions. Students do like to see patterns that flow through many courses and like to see that old ideas are springboards to new solutions. We tell them all the time to "use critical thinking" and "Choose the best method," so we should show them different methods when the opportunity arises.

Hindsight is 20-20 in mathematics, too. For the record, my first instinct was 1/2 PQ * altitude. Had P and Q not been on y=x, I might have changed course but the solution was easy enough.

Here's some variations on this theme which may push the solver in different directions depending on what he's just been working on:

P(4.5,4.5), Q(8.5,8.5), R(5,1)
or
P(4.5,5.5), Q(8.5,9.5), R(6,1)
or
P(1.5,6), Q(4.5,4.5), R(6.5,8.5)

Just a few thoughts on a Saturday morning early Sunday afternoon.

Overlapping Squares Puzzle Answer

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?

Rotate the 4” square until the sides of the squares meet at a right angle (dashed square above). We create a shaded triangle and a white triangle.

The triangles are congruent (properties of squares, the angles are congruent and the heights are both 1.5 and the bases are 0.5)

Take that small shaded triangle and move it to cover the small white triangle.

The area of the shaded region is congruent to one-fourth of the 3”square.
Area = 2.25

Koi Pond Puzzle Answer

Start by assuming the area of the big square is 4 square units instead of acres – it will make the calculations easier. Therefore the sides are 2 units x 2 units.

With a rotation, you can see that the diagonal of the pond and the diameter of the circle are both 2 units.

By Pythagoras, this isosceles right triangle with hypotenuse = 2 has legs √2/2. Thus the area of the smaller square is √2/2 * √2/2 = 2 units² = 2 acres.

SO the pond is 2 acres, or 87120 ft².
(1 acre = 43560 ft²)

Volume = 87120 ft² * 3ft = 261360 ft³
* 7.48051948 gals/ft³ = 1,955,109 gallons