On+Base+Five

= On Base Five =

A //**base**// is a grouping unit that is used recursively to build numbers. For example, in base ten, we consider this many tallies: |||||||||| to be grouped together into a single, higher level unit that we call "ten." Then when we get ten tens, we group that again into a single, higher level unit that we call "one hundred." Then we we get ten hundreds, we do it again. **Can you explain the same process in base five?** (In base five, we consider this many tallies;lllll to be grouped together into a single, higher level unit that we call”hand”. Then when we get five hands, we group that again into a single, higher level unite that we call “one clap” Then we get five claps, we do it again. Vanessa Niu)

In base five, the first number words are as follows: one, two, three, four, _ _ _. <-- At this point, we have counted up to base number of tallies, and so we need a name for that amount that is special. We could use "five," but that is a word very associated with base ten counting, so it can be better to choose a new name. Most of the time, classes choose "hand" for obvious reasons! **Can you write the next hand number words past hand?** one, two, three, four, hand, hand-one, hand-two, hand-three, hand-four. (Melissa Kilpatrick)

In base five, the Hindu-Arabic numerals for the number words above are: 1, 2, 3, 4, 10. **Why do we symbolize hand number of tallies with the Hindu-Arabic numeral 10? Can you write the Hindu-Arabic numerals for the next hand numbers past hand?** We do this becuase the "1" symbolizes that we have one complete base (one complete hand) and no left overs (as shown by the 0 after the one). If there was a 2 there it would mean we have two complete bases, and 3 would mean three complete bases, and so on. 1, 2, 3, 4, 10, 11, 12, 13, 14, 20 (Melissa Kilpatrick)

Determining numbers in base five
Let's say that we want to count seventeen times in base five, where seventeen is a base ten number. What will the number word and Hindu-Arabic numeral be in base five?

We can start by drawing out tallies and keeping track of the number of hands we draw as we go along: ||||| <-- that's one hand, ||||| <-- that's another hand (but we have not reached 17 tallies yet), ||||| <-- that's a third hand, and then we have ||, or 2 more tallies. So, we have counted 3 groups of hand tallies and 2 more tallies, which means seventeen in base ten is three-hand-two in base five. We write three-hand-two as 32 because this numeral represents 3 groups of hand tallies and 2 single tallies (just like in base ten, thirty-two is written 32 to represent three groups of ten and 2 single units).

Can you determine the base five number words and Hindu-Arabic numerals for these base ten numbers? Be sure to write tallies at least for the first few! (Melissa Kilpatrick)
 * eight- ||||| |||; hand-three; 13
 * twenty-two- ||||| ||||| ||||| ||||| ||; four-hand-two; 42
 * twenty-eight- [ ||||| ||||| ||||| ||||| ||||| ] |||; clap-three 103
 * thirty-seven- [ ||||| ||||| ||||| ||||| ||||| ] ||||| ||||| ||; clap-two hand-two; 122
 * forty-five- [ ||||| ||||| ||||| ||||| ||||| ] ||||| ||||| ||||| |||||; clap-four hand; 140
 * one hundred- [ ||||| ||||| ||||| ||||| ||||| ] [ ||||| ||||| ||||| ||||| ||||| ] [ ||||| ||||| ||||| ||||| ||||| ] [ ||||| ||||| ||||| ||||| ||||| ]; four-clap; 400

Something you might notice when you draw out twenty-eight (base ten number) tallies is the following: There are **hand number of groups**, each with hand tallies inside it, and 3 "loose" tallies. In other words, there are **hand number of hands**. If this is not clear to you, count each group of hand tallies in your picture (1, 2, 3, 4, ...hand...!). When we have hand number of hands, we have a new higher level unit in our base system. In class on 9/1/10, we said we would call this unit **clap**. So, that means that twenty-eight in base ten is the same as one-clap-three in base five. Notice that we have one clap and no "extra" full hands outside of this clap (just three extra tallies). So, we can symbolize this number in base five as 103. **What does each numeral in this symbol stand for?** The 1 stands for one clap, since in bases five it is in the "claps' spot" or in base ten in the hundreds' spot. The 0 means there are not any "extra" hands. The three then means that there are three individual little tallies left over that do not make a hand (Melissa Kilpatrick).

Counting in base five

 * What does counting by twos look like in base five? Counting by threes? By fours? By hands? How are the patterns for these kinds of "skip counting" in base five similar to or different from skip counting patterns in base ten?
 * Can you count by hands from 2? How high can you go? Do you see a pattern? Does this pattern change if you count by hands from three? From one? From four? Is it similar to or different from counting by bases in base ten? Why is it useful to be able to count by bases from any number?
 * 2, 12, 22, 32, 42, 102, 112, 122, etc. Everyone number ends in two because you're going up by whole bases each time and we started at two. Whatever number you started at that would be the number at the end each time. This is just like counting by tens in base ten. If we started at two it would go 2, 12, 22, 32, 42, 52, 62, 72, 82, 92, 102, etc. Counting by bases is useful because it saves you time from counting by ones. And once you know the pattern it goes even faster. -Melissa Kilpatrick

Building a Base System in Base Five...
On 9/27, we counted a lot of cubes in base five and named a new unit: hand number of claps is now called an "encore."

[posting opportunity: Answer one of the questions below. Be sure to stay in base five thinking in your responses.]
 * What does an encore look like?
 * How many hands are in an encore? There are clap number of hands in an Encore. An Encore is a cube that is hand by hand on all sides. If you think of the cube as hand "slices" of clap ones then the cube would be **clap** (number of ones in each slice) **X hand** (the number of slices) **/ hand** (you must divide by hand again becuase we said there were clap ONES in each slice and the question asks about hands). So since the hands cancel each other out we get Clap number of hands (Melissa Kilpatrick).
 * How much bigger is an encore than a clap?
 * How much bigger is an encore than a hand?
 * How much bigger is an encore than a "one"?