One SAT problem


SAT has a surprisingly good math section, for a standardized test. However, if you can solve these problems fast enough to meet the time limit, they become routine and boring at your level - merely exercises, not problems. My approach is to use these problems earlier in children's math careers, and spend good time playing with each one. It is a fun thing to do in pairs, with friends.

Yesterday we worked on this activity with two girls, S and K. I wanted to write up what we did for one problem, as an example. Please let me know if you can't see pictures and tables in this email.

The problem comes from Kaplan's "Purple Book of Practice SATs" p.13, and sounds quite dry - but read on!

If p is an integer greater than 1, such that p divided by 4 yields a remainder of 0, which of the following could be a prime number?
(A) p/4
(B) 2*SQRT(p) (that's square root)
(C) p/3
(D) p
(E) 2p


The first thing is to read the problem and answers. As you read, please note how many deep terms are involved in this relatively short problem statement. Reading math is a separate skill! The girls knew what integers greater than 1 were, but I asked them to make a list of examples for p: numbers which give 0 remainder when divided by 4. They produced this list:
4
8
12
16
20
24
28
32

Next, what are primes? K remembered the definition, "Numbers that can only be divided by 1 and themselves" with some prodding, but I wasn't sure the definition was strong enough to support problems solving. We also wanted to look up the definition online, and I gave girls the choice between Wikipedia and Wolfram's Mathworld. They noted Wikipedia is hard to read, and we headed to Mathworld, while I told them a joke about mathematicians and their "useful" definitions:

A man flying a hot air balloon got lost. So he descended and asked a woman walking in a field “Where am I?” She thought for some time and then replied. “In a hot-air balloon”. Immediately the balloonist realized she was a mathematician, for three reasons (1) She thought before replying (2) What she said was absolutely correct and (3) Her reply was totally useless.

It took us some time to find the right page in Mathworld, as it kept sending us to the pages for symbols of derivatives, and Marsienne primes. This page http://mathworld.wolfram.com/ Primes.html said:

"The set of prime numbers, sometimes denoted
P
P
(Derbyshire 2004, p. 163), and implemented in //Mathematica// as Primes." - yep, mathematicians made this definition, and we laughed about the balloon joke again. Following the link, we got confirmation of the definition K remembered, at http://mathworld.wolfram.com/ PrimeNumber.html

We did the Sieve or Eratosthenes activity on paper, with colored pencils, and then headed to Wikipedia for their nice animation of the same: http://en.wikipedia.org/wiki/ Sieve_of_Eratosthenes

Sieve of Eratosthenes: algorithm steps forprimes below 120(including optimisation of starting at squares)
Sieve of Eratosthenes: algorithm steps forprimes below 120(including optimisation of starting at squares)


After this, returning to our SAT problem, girls immediately noticed that p can't be prime ever, so (D) can't be a correct answer. I asked them to make a table for the values of 2p next to their values of p, which they started on paper like this:

p 2p
4 8
8 16
12 24
16
20
24
28
32


However, I sensed the reluctance, because it was a rather mindless grind, and switched to programming a spreadsheet. This was new and interesting for S, while K liked spreadsheets from before. I have to confess I absolutely LOVE to witness the incredible joy and relief of students first seeing how a spreadsheet simply propagates a formula through a column! Especially when they just spent long minutes doing it by hand. Instead of doing it slowly themselves and getting no new insights until the end of the tedious, error-prone labor, students can now focus on patterns and other DEEP mathematics. I

Math ed people hotly fight about the spreadsheet practice. Some believe in going for formal number patterns immediately, and others want kids to work with images and manipulatives first, and then formalize patterns from spreadsheets. As you can see from these activities, I work both ways, to make sure kids learn both methods and have a choice for their problem solving. Namely, for 2p and 3p and later p/3 we worked with number patterns only, but we did some picture work for square roots, and some storytelling work for the definition of primes, before looking at number patterns.

Once we programmed the p and the 2p patterns together, the girls noticed that the 2p pattern had all the numbers from the p column. I colored the two columns as they were pointing the numbers out, which you can see in the first sheet here: http://spreadsheets.google. com/pub?key= tMlrKgJIMsaq1zHefxLdu1A& output=html

I asked the girls what will happen to the pattern if we did 3p instead, they predicted it correctly, and we colored that one as well, in gray. These patterns have to do with proportions:


p
2p

p
3p
4
8

4
12
8
16

8
24
12
24

12
36
16
32

16
48
20
40

20
60
24
48

24
72
28
56

28
84
32
64

32
96
36
72

36
108
40
80

40
120
44
88

44
132
48
96

48
144
52
104

52
156
56
112

56
168

Somewhere during this part, the girls realized 2p can't be prime, so the answer (E) does not work. They also urged me to program the rest of the answers into the spreadsheet. However, they weren't quite sure what square roots are, or at least how they work in this context. We did a quick graph paper activity, drawing squares with sides 2, 3, 4, 5, finding their areas 4, 9, 16, 25, and then writing down square roots as the inverse:
SQRT(4)=2
SQRT(9)=3
SQRT(16)=4
SQRT(25)=5
I wanted to draw squares to bring up the name of the operation, "square root."

You can see the rest of the spreadsheet in the "All formulas" tab here http://spreadsheets.google. com/pub?key= tMlrKgJIMsaq1zHefxLdu1A& output=html

p
2p
p/4
2*sqrt(p)
p/3
4
8
1
4
1.33333333333333
8
16
2
5.65685424949238
2.66666666666667
12
24
3
6.92820323027551
4
16
32
4
8
5.33333333333333
20
40
5
8.94427190999916
6.66666666666667
24
48
6
9.79795897113271
8
28
56
7
10.5830052442584
9.33333333333333
32
64
8
11.3137084989848
10.6666666666667
36
72
9
12
12
40
80
10
12.6491106406735
13.3333333333333
44
88
11
13.2664991614216
14.6666666666667
48
96
12
13.856406460551
16
52
104
13
14.422205101856
17.3333333333333
56
112
14
14.9666295470958
18.6666666666667

The girls weren't sure what to do with infinite decimals. They obviously did not pay attention to the "integer" part of the definition of prime numbers. I asked them to make some examples of types of numbers they knew, and here are some examples they named:

- irrational numbers
- odd numbers
- even numbers
- fractions
- numbers equal to two (like 8/4)
- integers


S also tried to work out how the prime definition would work with fractions - an interesting investigation I'd like to pursue some time soon.

Their example list so much reminded me of Borges's Chinese Encyclopedia - which I immediately shared, at this link: http://www.multicians.org/ thvv/borges-animals.html With many giggles, we read:

~*~*~*~*~*
In "The Analytical Language of external image tinyglob.gif John Wilkins," Borges describes 'a certain Chinese Encyclopedia,' the Celestial Emporium of Benevolent Knowledge, in which it is written that animals are divided into:
  1. those that belong to the Emperor,
  2. embalmed ones,
  3. those that are trained,
  4. suckling pigs,
  5. mermaids,
  6. fabulous ones,
  7. stray dogs,
  8. those included in the present classification,
  9. those that tremble as if they were mad,
  10. innumerable ones,
  11. those drawn with a very fine camelhair brush,
  12. others,
  13. those that have just broken a flower vase,
  14. those that from a long way off look like flies.
This classification has been used by many writers. It "shattered all the familiar landmarks of his thought" for Michel Foucault. Anthropologists and ethnographers, German teachers, postmodern feminists, Australian museum curators, and artists quote it. The list of people influenced by the list has the same heterogeneous character as the list itself.
~*~*~*~*~*
Girls also discussed which types of numbers they liked and did not like. Both did not care much for fractions, for example. I joked that the person who defined primes probably did not, either - as primes can only be positive integers. Actually, a lot of number theory deals with integers, so there is some truth in that joke. They ended up quickly concluding that 2*SQRT(p) is either even or irrational, so it can't be prime, and that p/3 is either even or a fraction - ditto!

Upon checking their answer, S and K went to jump on the trampoline while I got them a snack before working on the next problem. Aerobic exercise and food make a huge difference in math success.