 Prime, Composite, and Square Numbers

Students often work on concepts like prime, composite, and square numbers by manipulating symbols only. Prime numbers are only divisible by themselves and one. Composites have other factors (e.g., 4 has 4 x 1 and 2 x 2). This lesson presents these concepts through the use of arrays, which are a good way for students to visualize factors in a number as well as understand the properties of prime, composite, and squared numbers. The concept of an array is explained in the classroom discussion section below.

Students can explore the concept of primes, composites, and squared numbers by coloring on graph paper or using the spreadsheet for what amounts to "electronic graph paper." Colored arrays are a quick way for students to examine patterns. Students can also gain computer skills by formatting tables and dragging within them.

One limitation of this exercise is that it does not show students all of the factors in some numbers. For example, 16 can be reduced to 2 x 2 x 2 x 2. This cannot be shown visually using arrays the way they are described in this lesson.

 Math Objective Learn to visualize the prime numbers and the factors for composite numbers and square numbers. Skills/Outcomes Math outcomes Identify prime and composite numbers Recognize, describe, extend, and create mathematical patterns in order to represent and solve problems: multiples and diagrams   Spreadsheet Skills build colored arrays drag and rearrange format cells with color and grid Files to Use Download Info/Instructions facprac.doc (student worksheet) factor.xls (teacher spreadsheet) Creating Arrays (math basics lesson) Optional resources graph paper, scissors, colored pens or pencils

Classroom Discussion and Activities

Computer Lab Activities

Classroom Discussion and Activities (Whole Group)

 Teacher Note: You have two options in this lesson. Students can make arrays in small groups using graph paper or you can do it as a large class exercise using a spreadsheet. In the interest of time, exploring arrays on the spreadsheet should probably be a group lesson, and students can take turns creating, coloring, and manipulating the arrays. You will need to know how to adjust cell heights and widths to make the cells on the spreadsheet square, and you will need to know how to use borders and color the cells in your arrays. Whether this is done with graph paper or the spreadsheet, be sure to stress the patterns in prime numbers and squared numbers.

Option 1: Using Graph Paper and Scissors

• Begin the lesson by talking about multiplication facts. Draw a simple fact like 3 x 5 on the board using cells in an array. It would look like the display below. Show how the column of 3 cells is repeated 5 times. While it may be obvious that 3 x 5 = 15, interesting patterns emerge if we explore different numbers using arrays. • Use another example like 8 to show that it is composed or made up of several facts (8 x 1, 1 x 8, 2 x 4, 4 x 2). Show how these can be drawn using arrays.

• Break students into small groups and assign each group a series of numbers. You may want to have students work on a range of different numbers up to 25. In other words, some numbers for the group should be prime and others should be composite. They should be encouraged to double check their work to make sure that they have all of the factors (or multiplication facts) that make up the number.

• Once students have finished their arrays, have them cut them out and arrange them on a table or on the board in order from 1 to 25. Go through each number pointing out its dimensions. Have the students look for patterns. In the end, make sure that students recognize the characteristics of prime numbers (arrays with only one row or column width), composites, and squares (arrays with the same number of rows and columns).

• Duplicate facprac.doc (student worksheet) for extra practice. Students can fill this out with or without drawing arrays.

Computer Lab Activities

• You can begin with a blank spreadsheet or open factor.xls

• If you begin with a blank spreadsheet, you should first format the spreadsheet so that the cells are squares. Help the students create factors for a number. The example below is for 24. Identify two of its factors (4 x 6). Drag over this area, making sure that you are watching the cell dimensions in the upper left corner of the spreadsheet.

• Format the array by adding a border and grid lines

• Color cells to indicate an array

• Type the factors below the array • Now have students draw a grid with the same number of cells but in a different configuration. As an alternative you may want them to click and drag the colored cells to form the new grid. This reinforces that it is really the same number only rearranged. • Continue with this process until students have found all the factors. Point out that a 2 x 6 and 6 x 2 are the same factors just turned 90 degrees.
 Teacher Note: If you have students create arrays using different numbers, you may want to conclude the lesson by going through factor.xls. This is a concise way of showing all of the prime, composite, and squared arrays for the numbers 1 to 25. Again, make sure to discuss the patterns for each number.