The cool thing about knowing how to label the fractions inside of an inch is that you can use it as a calculator to reduce fractions! Follow along and I'll show you how. This instructable may be a little hard to follow if you don't read it through all the way to the end before trying it out.
Step 1: Start Your Inches!
Pencil or Pen
A few definitions that might help you understand this instructable a little better:
Numerator: The number above the line in a common fraction
Denominator: The number below the line in a common fraction
The easiest way I have found to explain an inch is to draw it out, starting with a mark representing zero and a mark representing 1. These DO NOT HAVE TO BE ACCURATE!!! I find it is a lot easier if you make your "inch" BIG to give you more room to write in the fractions. SO:
On the paper, Draw a rectangle as shown in Picture 2 and 3. This is going to be our "ruler". Remember to make it big! On the ruler, make a mark, also as shown in Pictures 2, 3, and 4. This will be out 1 inch mark.
Step 2: Cut & Double- Halves
To explain this further, lets talk about the "Cut and Double" for a minute. When we marked the 1/2 inch on our ruler, we CUT the inch in half, and DOUBLED the denominator of the previous fraction. See Picture 2. For example, we cut the inch in half and made a mark. Then we took the previous fraction- ONE INCH which converted to a fraction with a numerator and denominator is 1/1- and DOUBLED the denominator. 2 x 1 = 2, so our new fraction will be 1/2. See Pictures 3 and 4. Confused? Maybe a little... Lets do it one more time in the next step and see if we start to catch on.
Step 3: Cut & Double- Quarters
CUT again! For each section of your inch, cut it in half as shown in Picture 1. Notice there are TWO marks now instead of one, because we have TWO sections- one on each side of the 1/2 inch mark.
DOUBLE again! The denominator of the last fraction is 2, so 2 x 2 = 4 as shown in Picture 2. That means our new fraction is going to be 1/4 as shown in Picture 3.
Here's where it gets a little different. There are TWO unlabeled marks on the ruler. We are going to count them off in quarters- the first mark being 1/4, the second mark would be 2/4, but it's already labeled as 1/2, and the 3rd mark will be labeled as 3/4. Count by 1/4'ths- 1/4, 2/4, 3/4, 4/4. See Picture 4.
Step 4: Cut & Double- Eighths
DOUBLE again! The denominator of the last fraction is 4, so 2 x 4 = 8 as shown in Picture 2. That means our new fraction is going to be 1/8 as shown in Picture 3.
Fill in the missing fractions by counting each mark as an eighth- 1/8, 2/8, 3/8, 4/8, 5/8, 6/8, 7/8, and 8/8. You will ONLY LABEL THE FRACTIONS WITH AN ODD NUMERATOR! See Picture 4.
Step 5: Cut & Double- Sixteenths
DOUBLE again! The denominator of the last fraction is 8, so 2 x 8 = 16 as shown in Picture 2. That means our new fraction is going to be 1/16 as shown in Picture 3.
Fill in the missing fractions by counting each mark as an eighth- 1/16, 2/16, 3/16, 4/16, 5/16, 6/16, 7/16, 8/16, 9/16, 10/16, 11/16, 12/16, 13/16, 14/16, 15/16, and 16/16. You will ONLY LABEL THE FRACTIONS WITH AN ODD NUMERATOR! See Picture 4.
Step 6: Tips, Tricks, and Continuing On.
Take a minute and look at the fractions. Do you see any patterns? There are two that stand out that can come in handy to check your work to make sure you drew your inch correctly...
1. Look at picture one again. What do you notice about ALL of the numerators?! THEY ARE ALL ODD. If you have an even number as a numerator, it needs to be reduced or you haven't got it in the right spot!
2. Look at the last fraction in each set as shown in Picture 2. Notice that in each fraction, the numerator is ONE LESS than the denominator!
You can use your completed inch as a calculator for reducing fractions. If you were to write ALL of the fractions down every time you did a set, your Inch would look like Picture 3. Each mark on the ruler that ends up with multiple fractions can be reduced to the top most fraction in the set!
Continuing on! You can continue the Cut and Double forever! Each time just split the last section in half and double the denominator of the last fraction. After 16ths, you would have 32nds, 64ths, 128ths, 256ths, 512ths, 1024ths, 2048ths, and on and on and on... But it does get a little hard to draw. :)
I'm sure there are a lot more cool things you can do with a ruler, and I'd love to hear about them! In the meantime, the best way to get good at this is to practice practice practice! After doing this about 20 times, my students can just about do it in their sleep- and their ability to read a ruler shows in their metals projects.
Once you have your Inch down, you can try some of my other projects. Making a Perfect Paper Cube is great practice at reading and using a ruler! See it here: https://www.instructables.com/id/Perfect-Paper-Cube-Laying-out-a-project-using-pa/