Once upon a time and far away there was a wonderful place called Number Line Land. It was part of a larger world known as *Mathematica*. The creatures who lived in this place were like nothing you've ever seen before. There were two main types: **heptoads** and **tetrogs**. They lived in a strange habitat, strange that is, for you and me. For you see, they lived on number lines. (That's why it's called Number Line Land.) Each creature had his/her own number line to call home. They lived in nests positioned at the *origin*, the "0" point. The terrain to the right of their nests was *positive*, sunny and safe, while to the left all was *negative*, dark and dangerous.

Now the heptoads and tetrogs had a very unusual way to get around on their lines.
Heptoads always jumped in 7's, that is, each jump was 7 *nuluns* in length. [The nulun
was their basic unit of length, just like the meter and foot are units of length for us.]
You may be wondering: why 7? That's because some Greeks once taught math in
their schools, and *hepta*- is the Greek prefix for 7. So I'll just bet you know how far
the tetrogs would jump. Right, 4 nuluns per jump, because *tetra*- means 4 in Greek
numbers.

When a heptoad and a tetrog got married and had offspring, there were two
more types of individuals that could form a family: **hepta-tets** and **tetra-heps**. (Be
patient, you'll get used to it soon.) These individuals would combine the jumping
skills of their parents. When going to the right, a young hepta-tet would always
jump 7 nuluns at a time; yet when going left, it would always jump 4 nuluns. It's
only natural to assume that tetra-heps did it in the reverse way: 4 to the right and
7 to the left. Don't ask me why they did it this way, it was just instinct with them.

The pace of life was pretty relaxed in Number Line Land. The creatures spent most of their time just eating during the day and sleeping at night. Each morning the Great Spirit of Material Sustenance would place baskets of food on the number line numbers and the owner of that line just had to go to the basket and eat all he or she wanted. But with that, life was still not so easy as you might think for them. They could only eat food that was found at the end of a jump or a sequence of jumps. This means that heptoads could only eat on a day in which food baskets were placed on a multiple of 7, while tetrogs feasted on baskets placed on multiples of 4.

Let's do a little math here. For lines no longer than 50 nuluns to the right, a heptoad has only 7 points at which he could eat: 7, 14, 21, 28, 35, 42, and 49. On the same sized line a tetrog would have 12 points on which she could eat. What are they? And at which number could either a heptoad or a tetrog obtain nourishment? (Hard though it may have been, things were a little easier for tetrogs, I'm sure you can see why.)

But things were less difficult for hepta-tets and tetra-heps. While the others could not reach food placed on the number 10, hepta-tets could reach it by moving two 7-nulun jumps to the right, followed by one 4-nulun jump to the left. Here's why:

The Greeks had taught the Commutative Property of Addition very well.

What about the tetra-heps? Well, these creatures could also reach food on number 10, it just required more jumping around. Six 4-nulun jumps to the right had to be combined with two 7-nulun jumps to the left.

Of course, a tetra-hep could mix up the jumps in different ways so as not to stray too far away from the food. Can you find how many different jump sequences there are? And which one(s) of those sequences do you feel best help the tetra-hep stay close to the food to protect it from poachers?

It's time for some more math now. Help Hestor, the hepta-tet reach his daily meals if Monday it was found on 15, Tuesday on 20, Wednesday on 25, Thursday on 30, and Friday on 35. Then help Tessie, the tetra-hep, to reach her food if it were placed on the same numbers on those days? Finally, decide which one, Hestor or Tessie, had a harder time on a given day to get their food?

__One final note:__ a special law must always govern the jumping behavior of the young. Until
an individual reaches his/her 18th year, it is forbidden to jump into that scary territory of the
**NEGATIVE** numbers!

Recent history in Number Line Land has turned a bit tragic, I'm sorry to relate. Just as things are going recently on Planet Earth regarding the ozone layer in the upper atmosphere, the same thing began to happen there as well. The disappearance of this protective shield allowed greater and greater amounts of solar radiation to attack the flora and fauna (i.e. the plants and animals) in N.L.L. This caused some species to die out altogether while others just mutated (i.e. changed form).

Fortunately for our story, the heptoads, the tetrogs, and their mixed offspring were
among the lucky ones: they just changed. And how they changed! Heptoads turned
into **octoads** and tetrogs became **pentrogs**. This is because after 7 comes 8, and
*oct-* is the Greek prefix for 8. Likewise, as 5 comes after 4, *pent*- is the Greek prefix
for 5.

I'm sure you can figure out now how their jumping abilities were altered. Sure, octoads made leaps 8 nuluns long and pentrogs could go 5 nuluns; both could go either way, of course.

As before, children came out mixed in their jumping patterns if they had mixed
parentage. We now have **octa-pents** (8 nuluns to the right and 5 to the left) and
**penta-octs** (who have a 5-right/8-left jumping style).

In spite of longer jumps, reaching the food baskets was the same kind of problem that is was in times past. For example, if food were on Point #10, an octa-pent required at least eleven leaps to land on the basket; yet a penta-oct would need at least fifteen. (Can you figure out how many of each kind of jump each creature performed in order to dine on the delicious food?)

Pity the poor octoad! While the pentrog can eat from a 10's basket in 2 simple steps, the octoad can never reach the goodies.

If you have already read the story about Apple Bobby and the
Teacher's Guide that is given there, you will recognize that this story
is very much the same as the other one, yet at the same time, it's an
extension of it. Now it is more appropriate for the middle school
student. This one admits the use of negative numbers, whereas the
other did not. Yet the basic mathematical foundation still rests on
the same number theory concept that was used earlier: equations of the
form **ax - by = c**.

I have not given answers here, and that was a deliberate decision on my part. This encourages you and your students to work together to find them, and discuss among yourselves as to their correctness. However, if you wish to consult with me, please send me an e-mail.

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