Lesson Title: Discovering Primes
Creator: Sheila Pell
Grade Level: 5th
Big Idea: Algebra
Essential question: How are rectangular arrays related to factors of numbers?
CSO:
M.O.5.2.4 Students will model, identify, and describe square, prime, and composite
numbers.
Learning Skills:
21C.O.5-8.1.LS.3
Student presents thoughts, ideas, and conceptual information efficiently, accurately and
in a compelling manner and enhances the oral and written presentation through the use
of technology.
21C.O.5-8. 3.LS2
Student is flexible in approach to solving problems and completing tasks, considers
alternative methods, solutions and perspectives, abandons strategies that do not work,
and reallocates time and resources as priorities change.
Technology Tools:
21C.O.5-8.1.TT.5
Student uses advanced features and utilities of word processing software (e.g.,
bullets, numbering, tables, find and replace, thesaurus, help menus and toolbars).
Launch/Introduction: (Approximately 20 minutes)
 To activate prior knowledge, ask students to talk with a partner about what the
dimensions of a rectangle are. Then have a student explain length and width
to the class.
 Be sure that students see the connection between the length and width of the
rectangle and its area. Ask: How is the area of a rectangle related to its length
and width? (Possible answer: the number of squares in the area is equal to the
product of the length times the width) Stress that area is measured in square
units.
 Remind students that when they write numbers for the length of a rectangle
multiplied by the width, they have a factor pair.
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To begin have students work in pairs. Give each pair of students 25 square tiles.
Demonstrate with a document camera (or overhead projector) acceptable ways
to place square tiles together to form rectangles. One entire side of the tile must
be aligned to an entire side of another tile.
As an example, have students select 4 tiles and ask them to make as many of
the above described rectangles as possible.
Have a pair of students show their responses and explain them using the
document camera. They should have found a rectangle that is 1x4 and one that
is 2x2. If they did not show the square, ask the class if there are other
possibilities. Wait until the square is mentioned. Students must be aware that
squares are also rectangles in order to accurately complete the task. (It
might also be a good time to mention that all rectangles are not squares.)
Discuss the dimensions of these rectangles and how these can be written as
factor pairs.
Show students how to write the factor pairs into a factor set.
1 x 4 and 2 x 2
{1, 2, 4}
Investigation: (Approximately 1 hour over two class periods. Teachers will use
discretion as to a stopping point for day 1. Students will continue on day 2 where they
left off on day 1.)
 Explain that in this lesson students will be finding groups of special numbers that
are created when making rectangles from square tiles.
 Students will use square tiles, colored grid paper, and chart paper. (Other
students may use Microsoft Word to create the chart. Squares to make
rectangles are found under “symbols.” Students set up the chart by inserting a
table. Go to “symbols” to make a square, use the corner of square to make it
very small, then copy and paste many more squares. Click and drag the squares
to put them together to make rectangles. See chart template.)
 Direct students to organize their paper into 5 columns labeled Number,
Rectangle(s), Factor Pairs, Factor Set, and leave the last column blank. (See
sample chart and blank chart.)
 Working with the first one as a whole group, students will begin with 1 tile. They
will record the number 1 in their first column, a drawing of the rectangle it makes
(or a cut-out of the rectangles from grid paper), a list of the dimensions of that
rectangle (factor pairs), and the factor set of the number 1 under the correct
headings.
 Students will then continue to work in pairs with 2 tiles, then 3 tiles, etc.,
recording what they have discovered as they go.
 After students have completed through number 25 (or as far as you feel is
necessary), ask them to list observations from analyzing their chart.
 To differentiate, have all students work through the first 10. Then direct students
who are grasping the concept readily to skip to other larger numbers before
writing their observations.
Summarize/Debrief: (Approximately 40 minutes)
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Ask students what they have discovered or observed. (Possible responses: that
some numbers could only be made one way with the tiles, that these numbers
always had only two factors, that some numbers formed squares, that some
numbers had many ways to make a rectangle)
Ask each student pair to discuss results for two of the numbers as they show
their chart.
Vocabulary: Do you know of a word that describes numbers that can be made
into a square with the tiles? (square numbers) Have you heard of a name for
numbers that have exactly 2 factors, one and the number itself? (prime
numbers) What are numbers called that have more than two factors?
(composite numbers)
Emphasize to students that the number 1, by definition, is neither prime nor
composite. (It does not have exactly two factors. It does not have more than two
factors.)
Now have students go back to their charts and fill in the correct vocabulary
word(s) in column five for each number.
Discuss the connection between the configuration of the tiles, the number of
factors, and the type of number that you have.
Refer to the essential question: How are rectangular arrays related to factors of
numbers? Have students offer answers. (The dimensions of the rectangles
made from square tiles are factor pairs which can be listed in a factor set. If the
factor set has only two factors, the number of tiles needed to make the rectangle
is a prime number. If the factor set has more than two factors, the number
needed to make the rectangle is a composite number.)
Materials: scissors, centimeter squared paper, glue stick, chart paper, markers,
computer(s), printer, document camera or overhead projector
Duration: two 60-minute periods
Teacher Notes:
This task is adapted from a lesson in Scott Foresman’s fifth grade Investigations in
Number, Data, and Space curriculum.
The National Math Panel has stressed that in order for students to be prepared for
algebra, they need to strengthen fraction skills in the upper elementary grades. This
lesson gives students an understanding of prime and composite numbers, as well as
square numbers. These concepts provide a basis for mastering fractions.
Pairs of students in this lesson will work collaboratively with manipulatives to visually
represent rectangles of various sizes. They will create representations of their square
tiles on chart paper or using the computer. The dimensions of the rectangles help
introduce students to prime, composite, and square numbers. Students will be given
opportunities to express their thinking verbally as well as by creating the
representations.
Students will “discover” that some numbers have only one rectangle. This shows that
they have only one set or exactly two factors, 1 and the number itself. This is a prime
number. Composite numbers will have more than two factors. Square numbers will
be represented in the shape of a square by the tiles.