## Egyptian Fractions

I discovered Egyptian Fractions while studying number theory at PROMYS. It was one of the long term projects that teachers could research, but didn’t really interest me at the time. When I saw a group of teachers present their research at the end of the summer I saw the implications for teaching fraction operations.

Egyptians used only unit fractions. To represent $\frac{3}{8}$ the Egyptians would have used the fraction $\frac{1}{4}$ and $\frac{1}{8}$. Together these fractions are equivalent to the original total, but use only unit fractions to express the sum. Once students have some knowledge of fraction addition they can begin to attack these problems. My students had also covered multiplication of fractions which made the process easier. Here is the activity they did:

Before introducing this activity in class students watched a five minute video about Egyptian Fractions. If you have access to United Streaming from Discovery Education, this video was a great way to start students thinking about how the Egyptians work. Our students study Ancient Egypt this time of year in history as well so this is a great activity.

One process introduced in the video is the “loaf of bread method.” Many of my students understood this method and stuck with it through the activity. If an Egyptian wants to share 5 loaves of bread with eight people he will start by splitting 4 loaves into halves. Then the leftover loaf will get split into eighths. Each person will then get half a loaf and an eighth of a loaf.  $\frac{5}{8}=\frac{1}{2}+\frac{1}{8}$

In this activity students were forced to work together to discuss methods. One single method will not work for all of the fractions.  This is one of those problems that students cannot divide and conquer.

## Ordering Fractions

I know. How dull! If I find it boring students must find it even more boring. As I mentioned before I’m kind of a number theory nut. After learning about the Farey sequence during my summer in PROMYS I knew it would be a great addition to my fraction bag of tricks. I tried it and students did it, but they weren’t getting where I wanted them. I don’t think I was asking the right questions.

I recently found this activity and tried the Farey sequence again. Students loved it. They wanted to try to get the next sequence. I shared it with a colleague and she couldn’t believe one of her students asked if it would be all right to do F9.

Here is F4:

from nrich.maths.org

I love lessons like this one. There is so much mathematics packed into it. We have mixed ability classes so students that are struggling with multiplication are sitting next to students that can solve equations in there heads. This problem allows students to move through at their own pace. I want students to practice comparing fractions. The fact that they are discovering patterns and symmetry along the way is a bonus.

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