Create Fraction Shapes and Filling Rhind Papyrus Unit Fraction Holes

“Five out of four children have trouble with fractions.” I love this quote which is a parody on “4 out of 5 dentists recommend Crest toothpaste …..” When I survey adults, high schoolers, and middle schoolers about what were or are their biggest challenges in math, always their answer is ‘fractions’: “I hate fractions.”

In many ways, I believe that the love and respect of fractions is critical for our future as academicians as well as life participants. When I ask the children for their favorite number, it always is a whole number, never a fraction. I always antagonize them by telling them that my favorite number of the day is 2/3 or 1/7, etc. I also strongly encourage children to “play with their food.” The context here, of course, is fractions. I tell them the story of my first steak dinner at a restaurant when I cut up the meat into 20 relatively equal slices and exclaimed how each piece was $.50; a $10 meal. The most fun I have is trying to cut a piece of food into equal portions of odd quantities such as sevenths or ninths. Successful apportionment can be very satisfying. This is stretch, but from a health perspective, this process slows down their digestion and better focuses children on healthy eating. One parent said that when trying to get their child to eat carrots, asked them to eat 3/5th. After they did, the remaining 2/5ths did not go to waste.

I gave them another circle shading packet where they will continue to shade from fractions, identify fractions from shading and explore equivalent fractions. They were most interesting in exploring how they can create their own fractions from any shape. I gave the a packet with square, triangular, and hexagonal grid paper with examples of how I created different shapes with shading and identified the fraction and equivalents, if any. They especially loved the spiral shaped fraction I created on hexagonal paper. I think what piqued their interest most was the tiny hexagonal paper on which they wanted to explore detailed shapes. Many of them were intrigued to explore the total number of pieces that make up a whole page. For example, the square grid was 25 squares vertically and 16 squares horizontally. They were able to see that four columns totaled 100 squares so 16 totaled 400. 

3rd-6th graders continued to explore the amazing properties of the Rhind Papyrus. I rolled out a good copy of the actual Egyptian parchment dated back to 1650BC. It was tanish color and 5 meters long; so was mine. They were able to see how 2,4,8,16,32,64, and 128 unit fractions could have a sum of exactly 1/2 using the Egyptian method of using the sum of non-equivalent unit fractions to equal a unit fraction. I introduced them to the wonders of the graphing calculator as they each were able to add many different sets of unit fractions to verify that they in fact equaled the desired unit fraction. For example, 1/3 + 1/6 = 1/2; 1/4 + 1/12 = 1/3; 1/7 + 1/42 = 1/6, etc. The Egyptian algorithm of 1/n = 1/(n+1) + 1/(n(n+1)) was understood by a large “fraction” of the 3rd-6th grade Mathletes. They naturally wanted to explore what would happen if they added triangles of these fractions and sets of four or eight horizontally placed fractions. See the attached pdf called Rhind Papyrus Triangle Sums.

Some of your children were so intrigued by the graphing calculator that they said they wanted one for the holidays. They are pricy ($125+) so I buy them on eBay. Also, they can explore these fractions with a scientific calculator comparing the decimal answers of fraction sums. The graphing calculator allows them to convert decimal answers to simplified fractions immediately by hitting “Math” and “Enter x2.” 

Remember, all Mathletes Challenge work is optional but encouraged. All Mathletes can use the Fraction Shading Circles.pdf and the Create Your Own Fraction Shapes (Square, Triangular, Hexagonal).pdf. The 3rd-6th graders can also continue to explore the Rhind Papyrus algorithms and show me any work they create on their own. With the upcoming Thanksgiving holiday, I expect the Mathletes may want to share this with family and friends.

Finally, I also introduced the 5th-6th graders to a very challenging process of finding the holes on my Rhind Papyrus document. First, we had to begin to explore the standard algorithm for multiplication. For many of the Mathletes, this was their first introduction to these concepts. “Egypt was not built in a Day” so be prepared for many failures before mastery. I created a packet of these problems that they could explore. I would recommend only doing one or two of these problems over the next few weeks. The most important aspect of this method is the understanding of place value. Also, there are many ways to use creativity. See the pdf for 5th-6th graders only called Multiplication Rhind Papyrus Unit Fraction Holes.pdf. For example, on page 1, when multiplying 615,362,442 by 615,362,443, the children saw that after they multiplied the first 3 by 615,362,442, they could copy their answer on the sixth row down because the multiplier was also a 3. They could simply copy 615,362,442 on the 8th row down because the multiplier was a 1. After they multiplied the second multiplier “4” they could take that answer and divide it by 2 to get the answer to the multiplier of 2. There are literally dozens of different ways to generate these rows of products using creative processes. Believe it or not, the most mistakes children make is when adding the rows. This would seem to be the easy part but for 47 years I have witnessed this phenomenon first hand. This is where the Mathlete Method of Circling Carries is critical. I also leave a blank row under the horizontal line for the additive carries. NOTE: if your child is frustrated with this algorithm, take a break and move on to creating more fraction shapes. I will revisit this algorithm throughout the year until they achieve mastery.