Essential Questions for students (objectives): How can curiosity and perseverance lead to new discoveries in math? How does a number sieve work to help find prime numbers?
Supplies: Sieve of Eratosthenes background page, Sieve chart, graph paper, Center supplies – measuring tape, grapefruits cut in half.
Instructional format: Whole Group OR Small Group/Centers
Vocabulary for Word Wall: Sieve, prime numbers, greatest common factor (GCF), conjecture, multiples
Time needed: 40-80 minutes
Prior Knowledge/Possible Warm-up Activities: Instruction on prime and composite numbers and how to find multiples.
Description of Lesson:
Whole Group Lesson –
1) Read The Librarian Who Measured the Earth by Kevin Hawkes to the class. Ask students to think about what they notice about the main character – Eratosthenes. After the reading, gather their thoughts as a class. Hopefully, they notice that Eratosthenes was inquisitive, methodical, perseverant, and was willing to use others’ research to help him solve problems.
2) Ask students to think about how they would go about finding all of the prime numbers (obviously, they will ask about the infiniteness of the number system here, so you can say that they are going to get a list to 100,000 OR you can think of how a computer program would go about doing it indefinitely). Gather ideas as a class. Without computers or calculators, how would someone go about this daunting task? Have them consider that and come up with possibilities.
3) Introduce the Sieve of Eratosthenes (see background information page) by explaining it from the handout OR you can visit the website on the background information page. After students review, see if they can explain it to a partner. Why would it work for finding prime numbers? Is it similar/different from ideas the class generated?
4) Using the 100’s chart (or a 500 hundred chart), have students try the Sieve of Eratosthenes and see what they notice and what prime numbers they find. This can be a great launch into a discussion on the twin prime conjecture and students can explore prime numbers further. (see this suggested exploration by Ian Byrd: http://www.byrdseed.com/prime-number-explorations/)
Extension into Greatest Common factor GCF:
1) Have students read (or teacher explain) Euclid’s algorithm for finding the greatest common divisor (factor), see background information page.
2) Have students practice using the algorithm on some examples (you can even have them use it for reducing fractions).
3) Compare and contrast Euclid’s algorithm and the Sieve of Eratosthenes. Are they more alike or different?
Small group/centers lessons –
1) Have students look at pages 34-44 in The Librarian Who Measured the Earth and recreate the calculations step by step to see if they get the same answer as in the book. Then, they should go back and explain each step.
2) [Supplied needed – measuring tapes grapefruit cut in half] Students could use the calculations and explanation from pages 34-44 in The Librarian Who Measured the Earth to find the circumference of a grapefruit (similar to page 34).
3) Students can practice using the Sieve of Eratosthenes or Euclid’s algorithm after some whole class discussion, or from reading the background information and trying it on their own. Number charts or graph paper are helpful supplies.
4) Students design their own Sieve for finding a set of numbers (their choice) and explain the Sieve in action. Students may need whole group instruction on Sieves OR read the background information on the Sieve of Eratosthenes.
5) Students look through the book The Librarian Who Measured the Earth and find the questions Eratosthenes asked about his world. Which questions do we have answers to now? Which questions are still unanswered? From reading the book – have students write up some unanswered questions of their own to research at a later date.
6) Students read What’s Your Angle, Pythagoras? By Julie Ellis and The Librarian Who Measured the Earth. Compare/contrast the thinking and mathematics of the two mathematicians. Whose process is easier to understand? In your opinion – which is more useful? [Make sure students read the Afterword in The Librarian Who Measured the Earth.]
1) Social Studies – This book connects perfectly with instruction on the ancient civilizations of Egypt and Greece. Students can research the Library of Alexandria as well as Eratosthenes and other famous philosophers. We still use mathematics developed at that time by Euclid, Eratosthenes and Pythagoras to name a few. What other achievements do we still use from those civilizations. Students can also research one of the most famous female mathematicians – Hypatia.
2) Science – Analyzation of the Earth’s crust and layers would fit in measuring the circumference. There is also a nice tie in to the movement of the planets and shadows as well as angles of light from the sun and the position of the planets.
3) ELA – Students can work on writing procedural (how-to) pieces explaining how to use a Sieve or Euclid’s Algorithm (or possibly their own procedure for doing math).
Assessment (Acceptable Evidence): Formative information can be collected from each of the center work.