Exploring Prime Numbers is a hands-on, minds-on math lesson that engages students in the discovery of prime numbers through the construction of multiplication arrays.  Students then extend on their background knowledge of multiples and their new understanding of prime and composite numbers to complete an Eratosthenes’ Sieve.

This lesson plan was created by exemplary Alabama Math Teachers through the AMSTI project.

Teacher Materials:

1. Chart paper
2. Markers
3. Sticky notes or index cards

Student Materials:

1. Circle counters or square tiles (1 bag of 20 for each pair of students)
2. Student math journals or notebook paper (if journals are not used in your classroom)
3. Pencil
4. 8 ½ by 11 inch paper
5. Scissors
6. Markers
7. Graph paper

1. Computer with Internet connection
2. LCD Projector

Teacher Background Information: 

A prime number is any natural number with exactly 2 factors, 1 and itself.  Any numbers with three factors or more is a composite number. An Eratosthenes’ Sieve is a method used to filter out prime numbers by eliminating multiples of 2, 3, 5, 7, and 11 on a 100’s chart.  

Student Background Information: 

In order to teach this lesson successfully, students should have an understanding of the terms multiple, factor, array, and understand how to skip count as a way of multiplying. 

Teacher Preparation: 

1. Prior to teaching this lesson, place each student with a working partner.  Make sure that each partner pair has at least one student with a firm understanding of the terms multiple, factor, array, and understands how to skip count as a way of multiplying. 

2. The partner pairs will be working with counters or square tiles.  Create bags of 20 counters/square tiles for each partner pair.  If possible, make all counters/square tiles the same color to keep students from being distracted by arranging the counters/square tiles by color. 

3. Write the following numbers 1-20 (excluding 12) on index cards or sticky notes to distribute during step two.  You will assign pairs of students a number between 1 and 20 by passing out the index cards or sticky notes at the beginning of step two.  Each pair should have a number or two to investigate.  Pairs receiving prime numbers with only one array arrangement will need to get an additional number that is composite.  However, DO NOT make the students aware of the reason for assigning specific numbers. Also, only assign one number to partners receiving a number that will create several arrays, such as 16, and 18.   

4. Create a t-chart labeled “Two Factors or Less” on one side and “More than Two Factors” on the other side.   A sample t-chart is attached.  
5. View the multimedia presentation entitled Prime Numbers Eratosthenes’ Sieve. (attached)
6. Create a sample foldable for Step 5 (see pictures entitled Foldable 1 and Foldable 2)

Engage: 

Pose the following question to the students:

A group of 12 children are performing in a talent contest tomorrow at the mall.  They will be singing together on stage and need to be arranged into a rectangular array.  How many different ways can they arrange themselves for the performance?  

Distribute the counters/square tiles and explain that one counter/square tile will represent each student.   Instruct the students to work with their partner to construct a rectangular array with 12 of the counters/square tiles.

Allow 5 minutes for students to complete the task.  Gain student attention and allow students to begin sharing their findings.  Record their responses on chart paper to show the different arrays created using 12 counters/square tiles.  If the class does not find all of the arrays for the number 12, have them work again to find the missing arrangement.  (All arrays for 12 include 1 row of 12, 2 rows of 6, 3 rows of 4, 4 rows of 3, 6 rows of 2, and 12 rows of 1.)  Now have the students use this information to identify the factors of twelve.  List the factors on the chart paper beside the sketch of the arrays.  Leave this chart posted as a sample of a math journal entry for the next part of the lesson.   A sample of how this chart should look when finished with this part of the investigation is attached.  It is entitled Sample Chart. 

Ongoing Assessment:  As students are working with their partner, monitor student progress by observing their method of creating arrays.  Do students know how to make arrays automatically?  If not, what strategies do they use to create the arrays? 

Explore:  

Assign pairs of students a numbers between 1 and 20 by passing out the index cards or sticky notes prepared before the lesson (see teacher preparation notes about number assignment).  Instruct the students to find as many arrays as possible for the assigned number(s) and to record their findings in their math journals.  If students do not use math journals, have them record their findings on notebook paper.  Tell students to use the posted chart from the previous activity as a model for their own notebook entry. 

Ongoing Assessment:  As students are working with their partner, monitor student progress by observing their method of creating arrays.  Do students know how to make arrays automatically?  If not, what strategies do they use to create the arrays?  Did the students find all the arrays for the assigned numbers?  If students do not find all the arrays, let them know so they can continue investigating the possibilities.  Did students list all of the factors for the assigned number(s)? 

Note:  This is a good stopping point if you plan to break this lesson into 2 sessions.  

Explain:   

Engage students in a discussion about their assigned number(s) by asking questions such as:  Did anyone have a number with several factors?  Who had more than 4 factors?  More than 5, 6, etc.  Did anyone have a number with only a few factors? 

After allowing a few minutes for discussion, post the t-chart constructed prior to the lesson (see teacher preparation notes above).    Have student tell where their assigned numbers fit on the T-chart. 

Introduce the terms prime numbers and composite numbers.  Write these terms on the t-chart above the correct heading

• Prime = Two Factors or Less (1, 2, 3, 5, 7, 11, 13, 17, and 19)
• Composite = More than Two Factors (4, 6, 8, 9, 10, 12, 14, 15, 16, 18, 20)

Guide the students into creating a definition for prime and composite.  Have them write these definitions in their Math Journal or notebook.  If students do not use journals or notebooks, make sure these definitions are posted in a visible place for all to see for the remainder of the lesson. 

Ongoing Assessment:  Make observations during the student discussion to identify students who  

• have a firm understanding of the term factor
• understand how to create arrays for numbers
• make the connection between multiplication and identifying factors
• understand the terms prime and composite 

Extend:

Explain to the students that now they will learn a method for differentiating between prime and composite numbers, without having to build arrays. 

Have the students create a foldable like the one in the attached photo, entitled Foldable 1. 

Show the attached multimedia presentation entitled Prime Numbers Eratosthenes’ Sieve.  As you go through the presentation, have the students answer the questions written on the foldable, by lifting the top tab and writing the answer on the bottom layer.  All questions can be answered from the first seven slides on the presentation.  (See attached photo entitled Foldable 2 for a photo of the inside of the foldable.)  This foldable can be glued into the student math journals/notebooks to be used as a study aid.

The remaining presentation slides will guide the students through an Eratosthenes’ Sieve to determine all prime numbers through 100.  The students will construct a 100’s chart on graph paper.  Then they will mark out all multiples of 2, 3, 5, 7, and 11.  The remaining numbers are all prime numbers. Students should use a pencil so mistakes can be erased. 

Ongoing Assessment:

As students are working on the sieve, monitor progress to insure they are only marking out correct multiples.  Also, look at the student foldables to insure all information recorded on the inside is accurate.

End of Lesson Questioning:  Some of these questions can be answered quickly, others may need to be evaluated and discussed. 

·         What are the prime numbers?  2, 3, 5, 7, 11, 13, 17, 19, 23, 27, 31, 37, 41, 47, 53, 59, 61, 67, 71, 79, 83, 89, and 97. 

·         What are the composite numbers?  All remaining number to 100 not listed above.

·         What are two methods for determining if a numbers is prime or composite?  Creating arrays and conducting a sieve

·         Why did we not have to eliminate multiples of 4 or 6?  They were eliminated as multiples of two, because multiples of 4 and 6 are all even numbers.

·         Are all prime numbers odd?   No, 2 is an even number and it is prime.

·         Are any other even numbers also prime numbers?  No, because  all even numbers (except 2) are divisible by 2 and other numbers.

·         What can we say about the numbers 2, 3, 5, 7, and 11 for which we eleiminated their multiples?  They are all prime numbers. 

Ongoing Assessment ideas are provided throughout the lesson above. 

An additional paper/pencil assessment is attached and entitled as Prime/Composite Assessment.  This assessment also assesses the understanding of arrays and the knowledge of the terms multiple, factor, prime, and composite. A rubric is also attached to assist in grading the assessment.   

Students who finish Step 2 early can explore making arrays on these interactive tools:

http://nlvm.usu.edu/en/navframes_asid_192_g_2_t_1.html?from= category_g_2_t_1.html

http://www.harcourtschool.com/activity/elab2004/gr3/16.html

http://www.ictgames.com/arrayDisplay.html

Students can use the sieve process of eliminating multiples of 2, 3, 5, 7, and 11 to find prime numbers to 400 (or higher). Students can check their work using this interactive prime number sieve of numbers to 400.  

http://www.hbmeyer.de/eratosiv.htm

Students can use colors to code the sieve in order to look for patterns.  Marks all multiples of 2 with red, multiples of 3 with blue, multiples of 5 with yellow, etc.  Then have the students write about the patterns they observe.  

Students who are having a difficult time understanding prime and composite numbers should continue working with arrays.  This will enable them to see how a prime number can only make an array using one row or one column.