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?
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.
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
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.
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.