For this meeting, Sarah invited us to explore the weird and wonderful world of numbers between 0 and 1. We started with a notice/wonder on this set of equations (suggested by Eric):
What do we notice?
- There is a 1 in all the tops.
- If there’s an even number on the left, there’s an odd number on the right and vice versa.
- The bottom number on the left of the equals sign multiplied by the number on the right of the equals sign equals 1.
- In each row of equations, the numbers seem to flip or swap the denominator and the decimal digits.
- When the bottom increases by 2 (like 2 to 4), the quotient decreases by 2 (like 0.5 to 0.25).
- There are 2’s and 5’s in the decimals and in the denominators of the fractions.
Before we got too far into wondering, Sarah invited us to play a game called “What Would I Think If I Didn’t Know? Many of us know a lot about numbers between 0 and 1, but many of the things we know are things we’ve been told or been taught and did not get a chance to figure out ourselves. However, just because we didn’t get a chance to figure these things out for ourselves the first time, that doesn’t mean we can’t still have the experience of discovery if we’re willing to pretend we don’t know the things we’ve been told. When we play, we can decide what we want to know and what we want to pretend we don’t know and see what kind of adventure that takes us on. Here’s an example of how this game can work:
After considering this game and how it might guide us in exploring the equations we started with and where it might lead us from there, we came back to generating some wonderings.
What do we wonder?
- The 4’s feel like they are coming out of nowhere. Where are they coming from?
- Would the pattern continue if the denominators continued to be squared?
- Why is it 0.04 and not 0.4 in the bottom right equation?
- What if you cubed the 2 in the denominator of the first equation? Would 1/8 = 0.35? (In other words, if the pattern in the denominators is 2, 4, 8, would the pattern in the decimals be 0.5, 0.25, 0.35?
- If I know the decimal for 1/3, how could that help me find the decimal for 1/9? Is it cutting the decimal in half? How could I cut 0.33333…. in half? Could I put more decimal points in a decimal?
- What numbers do these equations work with or not work with?
- What if we compared 1/3 and 1/6? What about ¼ and 1/16?
- What about even and odd numbers?
In separating into groups, one group decided to play with the pattern, asking questions like what would happen if we keep doubling or squaring the denominators. The other group decided to play with the numbers, asking questions like what do these equations look like when we put in other kinds of numbers?
Some of the work and discoveries from the groups:
Playing with the Pattern:
One pattern this this group explored was multiplying the denominator on the left by 2 and multiplying the decimal on the right by ½. They also noticed that they could describe this as the left side having denominators that are powers of 2 and the right side being divided by 2. One thing they noticed was that each time they multiplied the denominator by 2 on the left, the decimal on the right got longer by one digit:
At first glance, it appeared that this pattern did not hold when the starting equation involved a repeating decimal, but when we looked at the number of numbers before the decimals started repeating, the pattern was still there!
Playing with the Numbers:
This group started out with noticing how, in pairs of equations, the digits in the denominator of the fraction in one equation were the same as the digits in the decimal in the corresponding equation (with sometimes some leading zeroes). They experimented with seeing if they could predict the digits that would show up in the decimals or find a fraction that would give a decimal with specific digits in it:
Through experimentation, this group was able to use some fraction/decimal equivalents to find decimals for other fractions without using a standard procedure. However, a wondering that remained at the end of their explorations was whether there was a way to get to a first equation involving a decimal and a fraction without using a procedure they had been taught. This group discovered that sometimes, when you pretend you don’t know what you know, you can get to a point where you don’t have to pretend anymore because you arrive at questions you really don’t know the answer to! One such question – what kinds of fractions have decimal representations that repeat (like 1/3 = 0.33333…..) and what kinds of fractions have decimal representations that don’t repeat (like ½ = 0.5)?
A bonus: All this exploration of fractions and decimals led to discovering the exciting decimal equivalent of 1/81. Try it on a calculator! (Depending on how many digits your calculator shows, you may get different results. On an iPhone, try turning your phone sideways to see more digits. Or try looking it up at Wolfram Alpha.)
