## Struggles

My colleagues and I decided to enrich our CMP 3 curriculum with one of Cathy Fosnot’s Math in Context Units, Best Buys, Ratios and Rates. It has been a rewarding experience and changed the way that I think about teaching.

One thing I have regained is the benefit of “struggle.” Students that are willing to struggle are able to gain more from a problem than students that insist on having an answer first. It is sometimes tempting to teach what students need to know before they have had this experience.

I have always enjoyed teaching this way, but time constraints and the need to finish the standards in a timely way often scare me into a different teaching style. I gave my students some extra time today. I wanted to finish the problem and share out by the end of the class period today, but students were engaged and learning. I let them go on with the problem for the whole period. One student looked at me so excited and said, “I’m so excited, it feels so good to struggle and then get it.” She had just discovered what I was trying to teach. I didn’t do any of the talking, she did all of it and she learned!

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## 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.

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