1/2 + 1/3 + 1/6 = 1 AND BEYOND
GR 5-8 LESSON OUTLINE DEEPENING FRACTION SENSE
Exploring Unit Fractions and Sums to One
Grade Levels: 5-8
Objective: Students will understand unit fractions, distinct fractions, and how to find combinations of unit fractions that sum to 1. Working in groups, they will explore mathematical reasoning and learn key concepts through Socratic questioning and guided discovery.
Glossary of Important Terms
Unit Fraction: A fraction where the numerator is 1
Distinct Fractions: Fractions that are different from one another
Sum: Result of adding two or more numbers
Part 1: Starting with Three Unit Fractions
Key Question 1: Find the sum 1/2 + 1/3 + 1/6 .
After confirming sum =1, the following is displayed and stated:
We have now expressed 1 as the sum of three distinct unit fractions.
NOTE: New terminology, DISTINCT and UNIT FRACTIONS, is introduced using context clues before formal definition.
Key Question 2: IN YOUR GROUPS DISCUSS WHETHER THESE THREE ARE THE ONLY DISTINCT UNIT FRACTIONS WITH A SUM OF ONE
Instructor Asks: "Can you think of other ways to add three distinct unit fractions to get 1? Why can't we use 1/2,1/4,1/4?
Instructor Clarifies: "Remember, the fractions must sum to exactly 1, and they must be distinct unit fractions.
Encourage testing combinations systematically, guiding students to discover that there is only one way to represent 1 as the sum of three distinct unit fractions
Part 2: Moving to Four Unit Fractions
Key Question 3: How can we write 1 as the sum of four distinct unit fractions?
Instructor Asks: “Discuss in your group why 1/2 + 1/4 + 1/8 + 1/8 does not qualify. Do the same for 1/2 + 1/3 + 1/12 + 1/12
Instructor Asks: "Why can't we use 1/2, 1/3 and 1/5 as three of the four unit fractions? Ans: 1/5 is greater than 1/6 so the sum would already exceed 1
Key Question 4: If two of the four fractions are 1/2 and 1/3 why do we need to look for two distinct unit fractions whose sum is 1/6?
Guide students to test possible pairs:
Anyone want to suggest a third unit fraction we could use? Why do we have to make sure that it is less than 1/6?
Possible starting point: 1/7 + 1/? = 1/6; —> 1/6 - 1/7 = 1/42 —> 1/7 + 1/42 = 1/6
—> 1/2 + 1/3 + 1/7 + 1/42 = 1
Raise your hand if you believe this is the only possible solution starting with 1/2 and 1/3. Life would be easier if it were but in your groups find at least THRREE more solutions.
NOTE: IN FACT THERE ARE FOUR MORE SOLUTIONS. SHARE YOUR THOUGHTS IN COMMENTS OR RESTACK WITH A NOTE
Key Question 5: In your group take a poll of how many believe there could be other solutions if we do not start with both 1/2 and 1/3.
SUMMARIZE KEY CONCEPTS
"Unit fractions are powerful tools in exploring addition and mathematical reasoning."
"The constraints of distinct fractions lead to limited solutions."
Assign a challenge:
"Can you write 1 as the sum of FIVE distinct unit fractions?"
RESEARCH EGYPTIAN FRACTIONS AND THEIR RELATION TO OUR WORK TODAY