Saturday Pair: 4 and 38

This group is from 2002.

Problem 4

 

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

answer 04 and answer 38.

Algebra in the Third Grade?

I got the NASCO 2011 catalog in my mailbox this morning. If you haven't dealt with them, they are a pretty typical school supply catalog - prices are double what you can find if you search diligently or ten times the scrounger's price. What it is really good for is as a wish list, a source for ideas. "How does that toy work? I could do that in the garage for $2 in materials."

My problem with them is that sometimes they really miss the mark. Here's the cover:

Really? Third grade and up? I rail all the time about my students arriving in 9th grade without a clear grasp of fractions and clueless about operations with fractions (sans calculator), limited fluency with percents and decimals, and an unsteady grasp of their multiplication tables and other basic stuff. I had usually blamed the 3rd through 6th grade teachers for not really understanding arithmetic and passing on any math phobias, but this seems like a major problem right here.

If the expectation is that this kind of thing is possible for 3rd grade and up, maybe it shouldn't be a surprise that fractions and arithmetic aren't getting as much attention.

My questions:

• Is this typical? Do third grade teachers really do this?
• Is it just a stupid cover by a catalog desperate to sell overpriced AlgeBlocks to a gullible school system?
• Shouldn't we be solidifying knowledge to the point of automaticity instead of spreading algebraic materials ever lower?
• I'm pretty sure that a few third graders could get this but is it appropriate for that level? 
• Is it possible without manipulatives at this age?

 Then, there's the other debate:

• Are manipulatives appropriate? 
• Does the use of something tangible and obviously fixed in size get in the way of learning an abstract idea about a variable?

The Trader's Puzzle - Balance Weights

What are the weights of the four rings to as to measure any desired weight from a quarter-pound up to ten, in quarter-pound increments?

Archery Puzzle

How close can the young archer come to scoring a total of 100 - using as many arrows as she pleases.
The rings are numbered 16, 17, 23, 24, 39, 40

The Milkman's Puzzle

Honest John says that what he "don't know about milk is scarcely worth mentioning," but he was flabbergasted the other day when he had nothing but two ten-gallon cans of milk, and two customers with a five and a four quart measure wanted two quarts put into each measure.

It is a juggling trick, pure and simple, devoid of trick or device, but it calls for much cleverness to get two exact quarts of milk into those measures employing no other receptacles of any kind except the two measures and the two full cans. You can try the problem with the fullest assurance that it is a legitimate and not a silly catch.

Sam Loyd, Cyclopedia of Puzzles, 1914

see the answer.

The Weight of a Brick

If a brick balances with three-quarters of a brick and three-quarters of a pound, how much does a brick weigh?
~Sam Loyd, Cyclopedia of Puzzles, 1914

I won't bother with a separate post: answer

Cats and Kittens Puzzle

Seeing that four cats and three kittens weigh thirty-seven pounds, while three cats and four kittens weigh but thirty three pounds, we are asked to tell the respective weights of cats and kittens.

- Sam Loyd, Cyclopedia of Puzzles, 1914

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.