## Represent the whole as the sum of unit fractions

## Mod 5 les 1 decompose a whole as a sum of unit fractions

Please see the Number Talk Explanation for a more detailed overview of the method. For this Number Talk, I’m encouraging students to use a number line model to reflect their thought. Students today shared their plans for each mission with their peers (sometimes within their group, sometimes with someone across the room). It was wonderful to see students encouraging others to try new approaches, as well as students checking each other’s work for potential errors!

With their number lines, I invited students to join me on the front carpet. On the surface, I then drew a number line and marked 0, 1, and 2 on it. On their own number lines, I asked students to do the same.

After that, I gave each student one of the following numbers and asked them to figure out where each number would go on the line. I asked students to turn and talk about their thoughts after they had had time to position each number. I also requested that a student volunteer demonstrate to the class why and where the number will be put. I didn’t need to ask students for decimal equivalents at this stage. “50/100 is located here since 50/100 equals $0.50!” they immediately explained.

## Write fractions as sums – section 7.2

If the sum of its divisors (including 1, but excluding ) equals itself, it is called a perfect number. As an example, 6 (as shown above) and 28 (with divisors 14, 7, 4, 2, and 1) are both perfect numbers:

There are a slew of other perfect numbers available (496, 8128, 33550336, and 8589869056, for example). Although we may decompose these numbers in the same way as above, the requirement that each denominator be 99 or less restricts which ones we can use. For instance, 496 is not allowed because two of its divisors are three digits:

So let’s put perfect numbers aside and take a different approach. We can save space by omitting the unit numerators and writing fractions like,, as,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,,, By decomposing each individual unit fraction into a total of unit fractions, we can clearly make sums of unit fractions adding up to 1 longer. Let’s take a look at how a unit fraction can be broken down into two distinct fractions. Assume that the denominator of the unit fraction can be factored as, where and are both 1. As a result, decomposition into two unit fractions is possible:

### Write 1 as sum of egyptian unit fractions

A unit fraction is a rational number written as a fraction with one as the numerator and a positive integer as the denominator. As a consequence, a unit fraction is the inverse of a positive integer, 1/n. 1/1, 1/2, 1/3, 1/4, 1/5, and so on are examples.

In modular arithmetic, unit fractions are useful because they can be used to reduce modular division to the calculation of greatest common divisors. Consider the case where we want to divide by a value x modulo y. x and y must be relatively prime in order for division by x to be well defined modulo y. We can then find a and b using the extended Euclidean algorithm for greatest common divisors.

Triangle groups are defined as Euclidean, spherical, or hyperbolic in geometric group theory based on whether a related number of unit fractions is equal to one, greater than one, or less than one.

All probabilities in a uniform distribution on a discrete space are equal unit fractions. Probabilities of this kind often appear in statistical equations due to the theory of indifference. [number six] Furthermore, Zipf’s law states that the likelihood of selecting the nth item from an ordered sequence is proportional to the unit fraction 1/n for several observed phenomena involving the selection of items from an ordered sequence. [nine]

### Unit fractions of a whole – lesson 8.3

Students will rewrite a basic logical expression and analyze the arithmetic of these expressions in this assignment. Egyptian fractions offer an interesting framework for students to use logical expressions, both historically and mathematically. Students rewrite rational expressions (A-APR.D) in order to deduce knowledge about rational numbers, similar to the regular A-APR.4, in which students use polynomial identities to explain numerical relationships.

Students will work to rewrite fractions as a sum of distinct unit fractions, which will require the use of the identities of rational functions given. Students would need to rewrite a basic rational expression and research the arithmetic of these unit fractions with a variable in the denominator in order to solve the problem. The role has a moderate degree of cognitive demand and requires students to model several of the Criteria for Mathematical Practice, depending on how it is applied (e.g., levels of scaffolding). As a result, teachers should be mindful that a completely open-ended implementation of this assignment would necessitate a considerable amount of time and supervision. The rest of this commentary goes through some implementation information.