5.5B - Patterns, relationships, and algebraic thinking. The student
makes generalizations based on observed patterns and relationships.
The student is expected to identify prime and composite numbers using
concrete objects, pictorial models, and patterns in factor pairs.
Suggested Activities/Lessons:
• 5^^ Sense Lesson
• Prime and Composite Lesson
• Be a Prime Number Hunter
TAKS Objective 2:
The student will demonstrate an understanding of
patterns, relationships, and algebraic reasoning.
5-
{
patterns, relationships, and algebraic reasoning.
Student Expectation 5.5
The student is expected
!
concrete models and patterns in factor pairs.
Overview:
Materials:
Vocabula 17:
Lesson:
Lesson 5.5K)
This lesson will help students with the conceptual understanding of prime
and composite numbers.
1.
2.
3.
4.
Small paper plates or coffee filters
Counters
Journal or paper to record on
Sample space visual for room
Prime, composite, factor pair, factor, prime factors
I 1. Review the terms in the vocabulary section.
2. Begin with the number of people in the group.
How many candies wouid we need for eacii person to iiave an
equai siiare?
3. IModel this with counters.
4. Begin with two counters and asl< how many equal shares can be mademodel with plates. Model using one plate and putting 2 candies on it and
use 2 plates with one candy on each.
5. Record the factor pairs. (Ex. 2 candies for 1 plate, 1 candy for 2 plates.)
2 x 1 / 1 x 2 (one factor pair)>I/-e tiiere anymore possibie equai
shares? {no)
6. Repeat with 3, 4, 5, 6, 7, 8, 9,10, 11,12. Students record in their
journals.
7. Reinforce the term factor pair. What do you thinlc is a factor pair?
Did you notice that some numbers oniy had two equal
shares?(yes)
1
•
What number is aiways present in these factor pairs? (1) When
a numberhas oniy two factors^ the number one anditseif, it is
icnown as a prime number. An exampie would be the number
three. You could use three plates and one candy each or one
plate with ail three candies.
• What about four? You couid have one plate with four candies,^
two plates with two candies each orfourpiates with one candy
per plate. How many equal share possibilities did you Und? {3)
Write them in yourjournal.
Identify the entire prime and identify all the composite numbers starting
with two and ending with twelve.
8. Ask What do you think wiil be the next prime number? Why?
What about the number one?
(One is neither prime nor composite.)
Another way to Und prime and composite numbers is making
arrays. Model the number four for the students, (ex. Bingo markers).
Students create arrays with counters to justify if a number is prime or
composite Now have the students model, 6, 7, and 9.
SIX (6)
OO
OO
OOO
ooo
2 groups of 3
2x3
3 groups of 2
3x2
OOOOOO
1 group of 5
1x6
6 is composite
SEVEN r71
OOOOOOO
1 group of 7
1x7
7 is prime
NINE (9^
ooooooooo
1 group of 9
1x9
OOO
ooo
ooo
3 groups of 3
3x3
9 IS composite
Lesson 5.5(^
2
*AdditionalTEKS5.16B
Which of these numbers is a prime number?
16, 21, 17, 42
List the factors of each number. A prime numberhas oniy two
factors, one and itseif. A composite numberhas more three or more
factors.
FACTOR PAIRS
16 = 4 X 4, 8 X 2, 1 X 16
21 = 3 X 7, 21 X 1
17=17x1
42 = 6 X 7, 21 X 2, 3 X 14, 42 x 1
Debriefing
Questions:
FACTORS
1, 2, 4, 8, 16
1, 3, 7, 21
1,17
1, 2, 3, 6, 7, 14, 21, 42
Looic at the factors of 21. Which factors are a prime number factor
ofthe composits number 21. (3 and 7)
What do you notice about the factor pairs?
Which numbers have more than one factor pair?
Which numbers are composite? Why?
Which numbers are prime? Why?
Guided
Practice:
Looic at the numbers beiow.
18, 11, 9, 23
1.
2.
3.
4.
5.
6.
7.
Assessment:
Lesson 5.5(5^)
5
Find the factors of 18
f 1,2.3.6,9,18')
Find the factors of 11
fl.ll)
Find the factors of 9
ri,3,9)
Find the factors of 23
a,23)
What does prime mean? A number with onlv two factors, one and itseif.
What does composite mean? A number that has three or more factors.
Using the given numbers, the prime numbers are_{lland 23}_ and the
composite numbers are 19 and 18>.
1. Which is a prime number factor ofthe composite number 24?
*A
3
B
4
C
6
D
9
3
2. The list below contains the factors of which number?
15, 3, 5, 9, 45, 1
F
15
G
27
*H
45
J
54
3 Which number pair is a factor pair of 64?
A
6,11
B
3,21
C
4,14
*D
2,32
Guided Practice
Objective! TEKS 5.5
Name:
Date:
Look at the numbers below.
18,11, 9, 23
1. Find the factors of 18
2. Find the factors of 11,
3. Find the factors of 9
4. Find the factors of 23
5. What does prime mean?
6. What does composite mean?
7. Using the given numbers, the prime numbers are
numbers are
Lesson 5.5Cg)
and the composite
.
5
Name:
Date:
1. Which is a prime number factor of the composite number 24?
A
3
B
4
C
6
D
9
2. The list below contains the factors of which number?
15, 3, 5, 9, 45, 1
F
15
G
27
H
45
J
54
3. Which number pair is a factor pair of 64?
A
6,11
B
3,21
C
4, 14
D
2,32
Lesson 5.5(^
6
5.50Sample Space
Visual for Room
mmmmmm
le spaces
Set of all
possible outcomes
. Ex: Find all the
possible outcomes
ofthe 2 spinners
Table
^ 1
;''' V M p y e ; i o ^ j S g a K d s ; ' ' '
•towardC
backv/ard
M o t « T o w a r d ;rT^' I spaots;
: sMovc. fonvards-l fjiacbs • '
' • ''Mojfb hkck^
space
' j. AioA'c liiWJcwani;, 3 - spaces
Lesson 5.5(K)
6
space
A: ' y .
•
. ,•
',
• 3 spaces..
1 space.
•• •• 3'spaces
7
Prime and Composite Lesson (TEKS 5.5B)
Summary. Students will use manipulatives t o learn about composite and prime numbers.
Materials:
For class:
• We are Prime/ Composite Numbers posters
• Sieve (or picture of a sieve)
For each student:
• Graph paper
• Centimeter cubes
• Sieve of Eratosthenes worksheet
• Prove or Disprove worksheet
• Colored pencils
Background For Teachers:
Students learned t o classify whole numbers f r o m 2 to 20 as composite or prime, and 0
and 1 as neither prime nor composite, in f i f t h grade.
Every whole number greater than 1 is either a prime or composite number. A prime
number has exactly two f a c t o r s , 1 and itself. A composite number has more than two
factors.
Commutative property is t h e f a c t t h a t changing t h e order of addends or f a c t o r s does
not change t h e sum or product (e.g., 4 x 7 = 28 and 7 x 4 = 28).
Rules of divisibility
o
A whole number is divisible by 2 if and only if t h e ones digit is even,
o
A whole number is divisible by 3 if and only if t h e sum of its digits is
divisible by 3.
o
A whole number is divisible by 5 if and only if t h e ones digit is 5 or 0.
o
A whole number is divisible by 6 i f and only if it is divisible by 2 and 3.
o
A whole number is divisible by 9 if and only if t h e sum of its digits are
divisible by 9 (or 3).
o
A whole number is divisible by 10 if and only if t h e ones digit is 0.
Instructional Procedures:
Pass out 12 centimeter cubes t o each student. Have students make d i f f e r e n t shapes
with t h e cubes. Share examples and point out examples and nonexamples of arrays.
W r i t e a definition o f '"array" as a class.
Instructional Procedures
bayl
Use t h e Manipulative Master t o create an array for numbers f r o m 3 t o 20. For example,
say t o t h e class, "Using your manipulatives, show me what 4 looks like. How many ways
can you make a complete rectangle using only f o u r squares?"
Continue through all numbers, 3-20.
Transfer
arrays onto graph paper and label each rectangle.
3 x 4 = 12
2 x 6 = 12
Circle each f a c t o r using one color. Put a circle around each product using a d i f f e r e n t
color.
(3)x(4)= 12
(2)x(6;= 12
Point out t h e commutative property—the f a c t t h a t changing t h e order of addends or
f a c t o r s does not change t h e sum or product. Example: 3 x 4 = 12 and 4 x 3 = 12 f a c t o r s
1,2,3,4,6,12
Connect two f a c t o r s using a curved line t h a t resembles a rainbow. Draw a square around
numbers t h a t are t h e square root.
1 2 4
1 5
1 2
4
8
Numbers w i t h exactly 2 f a c t o r s are prime numbers.
Numbers w i t h more than two f a c t o r s are composite numbers.
The numbers 0 and 1 are neither prime nor composite. 1 is a unique counting number. 0 is
not a counting number.
Day 2
Before class, place t h e We A r e Prime (Composite) Numbers posters in d i f f e r e n t corners
of t h e room.
Before class, prepare a sheet of graph paper f o r each student in your class by writing a
d i f f e r e n t number at t h e top of each sheet. Begin with t h e number two and continue in
order until you have enough f o r each student (e.g.. I f you have 36 students in your class,
you would use t h e numbers 2-27 f o r this activity.).
Pass out a sheet t o each student. I n s t r u c t students t o complete t h e i r sheets as follows:
• Draw all of t h e possible arrays f o r t h e number.
• Label t h e length and width of each array.
• W r i t e multiplication sentences f o r each array and circle f a c t o r s in red and
products in blue.
Draw a f a c t o r rainbow f o r t h e number.
As t h e students finish, begin sending them t o d i f f e r e n t corners of t h e room depending
on whether t h e y have a prime or composite number. Have t h e students take t h e i r
completed papers with them.
Have t h e students in each corner compare t h e i r numbers with other members of t h e i r
group and look f o r things t h a t t h e i r numbers have in common. I f t h e students are having
a d i f f i c u l t t i m e seeing similarities, prompt them t o look at t h e number of f a c t o r s and
arrays. Discuss t h e findings as a class.
Have one member of each group hold up t h e W e A r e Prime (Composite) Numbers
posters.
Have students r e t u r n t o t h e i r seats and w r i t e definitions of prime and composite
numbers in math journals.
Gather papers f r o m students and randomly pass them out again. A f t e r looking at t h e i r
new number, have students move t o t h e c o r r e c t corner of t h e room again. Continue t o do
this until all students can correctly move it either t h e prime or composite corners of
t h e room.
I f you have been hanging t h e Number Posters up around your classroom, draw students'
attention to them at this time. Look at t h e posters and discuss how t h e number one is
d i f f e r e n t than t h e other numbers. W r i t e a definition f o r unique number in math
journals.
Day 3
Show a sieve. W h a t is it? What is it used f o r ? How does i t work?
We can s o r t numbers j u s t like a sieve separates and sorts material.
Using t h e Sieve of Eratosthenes, complete t h e hundreds chart as follows:
• As a class, cross out t h e 1 on t h e chart.
• Putting your finger on 2 count by twos and color t h e top l e f t corner of each number
square yellow. Teacher Note: As students begin coloring, they will most likely
begin t o see a pattern. Explain t h e rule of divisibility f o r 2—A whole number is
divisible by 2 if t h e ones digit is even.
• Put your finger on 3. Counting by threes color in t h e top right corner of each
number square orange. Teacher Note: As students begin coloring, they will most
likely begin t o see a pattern. Explain t h e rule of divisibility f o r 3—A whole number
is divisible by 3 if t h e sum of its digits is divisible by 3.
• Place centimeter cubes on all multiples of 6. Discuss patterns and divisibility f o r 6.
• Put your finger on 5. Count by fives and color t h e b o t t o m l e f t corner of each
number square green. Teacher Note: As students begin coloring, they will most
likely begin t o see a pattern. Explain t h e rules of divisibility f o r 5 and 10—A whole
number is divisible by 5 if t h e ones digit is 5 or 0. A whole number is divisible by 10
if t h e ones digit is 0.
• Put your f i n g e r on number 7. Count by sevens and color t h e b o t t o m right corner of
each number square blue.
• The boxes with nothing colored are prime numbers. Color these boxes red.
Day 4
How can you t e l l people apart? How can you tell people apart when you cannot see them?
(fingerprint)
J u s t like every person has a unique f i n g e r p r i n t , each number has a unique " f a c t o r print."
This f a c t o r p r i n t is t h e number's prime f a c t o r i z a t i o n . I t is t h e set of prime numbers
whose product equals t h e number. Factor Tree:
36
36
/ \
/\
2
18
3
12
/ \
.1
'
/ \
6
/
2
3
\
/
3
2
\
2
Extensions:
Have students work in pairs or small groups t o develop divisibility rules. Discuss
as a class and complete rules as listed in t h e Background f o r Teachers section.
Prove o r Disprove worksheet.
H a t f i e l d , M., Edwards, N., B i t t e r , G. <& Morrow, J . (2000). Mathematics Methods f o r
Elementary and Middle School Teachers. New York, New York. John Wiley &. Sons Inc.
Academy Handbook Sixth Grade
We Aire PrimefComposite Numbers Posters
3-16b
Hementary CORE Academy 2005
Math Standard I-I S: 2—Activities
Name
Sieve of Eratosthenes
1. Put yourfingeron number 2. Count by twos and color the top left corner of each number
square yellow.
2. Put yourfingeron numbet 3. Count by threes and color the top right comer of each number
square orange.
3. Put yourfingeron number 5. Count byfivesand color tlie bottom left comer of each
number square green.
4. Put yourfingeron number 7. Count by sevens and color the bottom right comer of each
number square blue.
5. Color the remaining boxes red.
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
Utah State Office of Education/Utah State University
3-17
Math Standard I-l & 2—Activities
Name
Prove or Disprove
Prove or disprove these statements:
1. Only three of the numbers below are prime numbers.
123
635
587
247
198
233
267
307
7101
2. There are 25 prime numbers between 1 and 100 and 25 prime numbers between
100 and 200.
3. Each of the whole numbers between 1 and 50 can be written as a product of two or more
prime numbers.
Example: 20 = 2 x 2 x 5
4. Every number from 1 to 100 can be written as the sum of two prime numbers.
Example: 18 = 7 + II
Create three more prove or disprove statements of your own.
Utah State Office of EducationAJtah State Univetsity
3-19
Be a Prime Number Hunter (TEKS 5.5B)
by Jane Oh
Wondering how to help your child get a handle on prime numbers? Why
not turn her into a prime number hunter?
What You Need:
• Colored markers
• Printable hundreds chart
Review: A prime number is a whole number greater than zero that has
exactly two different factors, one and itself. For example, the number 3 is
a prime number because its only factors are 1 and 3. In contrast, a
composite number is a whole number greater than zero that has more
than two different factors. The number 6 is a composite number because
its factors are 1, 6, 2, and 3.
It's important to note that the number 1 is neither prime nor composite.
It is not prime because it does not have exactly two different factors. And
it is not composite because it does not have more than two factors. 1 is a
special number.
What You Do:
1.
2.
3.
Print out a copy of a hundreds chart. Cut or fold the hundreds chart
in half if you only want to focus on the prime numbers through 50.
Take a moment to review what makes a prime number. Then let
the game begin!
To play, tell your child that you will be competing to cross out all
the composite (non prime) numbers, and circle all the prime
numbers. Designate one color marker for the prime numbers, and
4.
5.
anotlier to cross out composite numbers.
Each player will take turns either crossing out a composite number
(1 point), circling a prime number (3 points), or "passing." The
game will get easier as more number are crossed and circled, but
the bigger numbers may present more of a challenge to your child.
You may need to take your child through the definition of prime
numbers a few times as you look at different numbers.
The player with the most points at the end wins!
After you have played the game, check your answers. The prime
numbers through 50 revealed from this activity are: 2, 3, 5, 7,11,13,17,
19, 23, 29, 31, 37, 41, 43, and 47Jane Oh has taught third and fourth grades for 8 years. She has worked
with many diverse groups of students. Most recently, she has written
teacher textbook guides.
© Cop5Tight 2006-2010 Education.com All Rights Reserved.
100 Chart
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
AI
3
education,