Make One Whole
What unique unit fractions combine to form 1 whole?
The Egyptians used a sum of unique unit fractions to represent other fractional values. For example, they could use ½ + ¼ to represent the value ¾. The Egyptians would not have used this representation for whole numbers, but it’s interesting to explore the different ways to make 1 whole with unique unit fractions.
How can unique unit fractions be combined to form 1 whole?
- Can 1 be represented as the sum of two unique unit fractions? How do you know?
- How could you represent 1 as the sum of three unique unit fractions? How about four unique unit fractions?
A unit fraction is a fraction with 1 as the numerator. For example, ½, ⅓, and ⅛ are unit fractions.
A unique unit fraction is a fraction that is different from the others. For example, ½ and ¼ are unique unit fractions, but ⅓ and ⅓ are the same, so they are not unique unit fractions.
- How can you model a unit fraction? How much of the whole will be left?
- How could you use equivalent fractions to choose unique unit fractions?
- Try to find a combination of unique unit fractions that have a sum close to 1 ½. How close can you get?
- What fractions can be made by combining two unique unit fractions? What patterns do you notice among the addends and sums?
In this task, students find unique unit fraction pieces that fit together to make a whole. Students might start with a set of unique unit fractions and experiment with those that combine to make a whole, or they might start with 1 whole and take off unique unit fractions until they are left with a unique unit fraction. Exploring this puzzling problem will engage students with equivalent fractions to find the missing unit fraction piece to complete their whole.
There are many different combinations of unique unit fractions with a sum of 1. Students may find they have to exclude several options because the fraction that remains is not a unit fraction, or may not be unique. Students may choose to:
- Model several unique unit fractions of the same-size whole and use them to build a whole. To help find the final piece, students might build a whole using a denominator that will allow them to model equivalent fractions and determine if they can make a unit fraction for the amount that remains.
- Start with a pair of unique unit fractions. Using equivalent models, students might find the fraction that represents the remaining amount needed to make 1 whole and build another unit fraction close to that difference. They could iterate this process until they find the last unit fraction needed to make 1.
Multiple apps can be used to explore this problem.
- In the Fraction app, students might model two unit fractions, then create a model for a whole using a common denominator. Overlaying a model of 1 with the new denominator over their first two models, students can see how much more they need to make a sum of 1. Here’s the beginning of a solution.
- The Geoboard app can also be used to model fractions. Students may begin modeling unit fractions by first dividing a model in half, then modeling a unit fraction in the remaining area of the whole. Here’s the beginning of a solution that uses this visual representation.
- Students can also model fractions with up to 60 equal parts in the Math Clock app. Students might use their partitioning to find equivalent unit fractions and shade unit fraction sections to build 1 whole. Here’s the beginning of a solution with a student using 12ths as the basis for the unit fractions.
To extend students’ thinking about their own and other possible combinations of unique unit fractions, ask: What do you notice about the denominators of your unit fractions? Could you use this to help you find other combinations of unique unit fractions that have a sum of 1? They might notice that all (or most) of the denominators share at least one common factor. Or, the denominator of their least unit fraction might be a multiple of all (or most) of the denominators of the other unit fractions.