1. science
2. computer science
3. cryptography

Objectives To review equivalency concepts for whole numbers;
and to introduce factor strings and prime factorization.
1
materials
Teaching the Lesson
Key Activities
Students use name-collection boxes to review the idea that numbers can be
represented in many different ways. They represent composite numbers as factor
strings; identify the prime factorization of a number; and use factor trees to find the
prime factorization of numbers.
Math Journal 1, pp. 25 and 26
Study Link 1 8
Teaching Master (Math Masters, p. 25)
quarter sheet of paper
Class Data Pad
calculator
Key Concepts and Skills
• Find factor strings for numbers.
[Number and Numeration Goal 3]
See Advance Preparation
• Write the prime factorization for numbers.
[Number and Numeration Goal 3]
• Rename numbers as factor strings or products of exponents.
[Number and Numeration Goal 4]
• Use exponents to rename numbers.
[Number and Numeration Goal 4]
Key Vocabulary name-collection box • factor string • length of factor string
• prime factorization
Ongoing Assessment: Informing Instruction See page 58.
Ongoing Assessment: Recognizing Student Achievement Use journal page 25.
[Number and Numeration Goal 3]
2
Ongoing Learning & Practice
materials
Students practice arithmetic operations and basic facts by playing Name
That Number.
Math Journal 1, p. 27
Student Reference Book, p. 325
Students practice and maintain skills through Math Boxes and Study Link activities.
Study Link Master (Math Masters, p. 24)
4 each of the number cards 0–9; calculator
3
materials
Differentiation Options
READINESS
Students identify the prime numbers
between 1 and 30.
ENRICHMENT
Students explore square numbers that
are also palindromes.
Additional Information
Advance Preparation Prepare a name-collection box for 16 on the Class Data Pad to use in
the Math Message Follow-up.
Teaching Masters (Math Masters,
pp. 26–28)
Technology
Assessment Management System
Journal page 25
See the iTLG.
Lesson 1 9
57
Getting Started
Mental Math and Reflexes
Math Message
Dictate the problems, repeating each problem once.
224
5 6 30
6 9 54
236
7 3 21
8 9 72
3 4 12
7 6 42
9 7 63
4 5 20
8 5 40
8 6 48
This concludes the multiplication facts routine. Plan to give a quiz
once or twice weekly as needed.
8 8 and 4 4 are two names for the number 16.
On a quarter-sheet of paper, write at least five other
names for 16.
Study Link 1 8 Follow-Up
Have partners compare answers and resolve any
differences.
1 Teaching the Lesson
NOTE Some students may benefit from
doing the Readiness activity before you begin
Part 1 of each lesson. See the Readiness
activity in Part 3 for details.
16
Math Message Follow-Up
WHOLE-CLASS
ACTIVITY
Representing a number in many equivalent ways is an important
concept in Everyday Mathematics. Display a name-collection
box for 16 on the Class Data Pad. (See margin.) Ask students to
share equivalent names for 16, comparing each other’s choices
(and possibly thinking of new ones). Record equivalent names in
the name-collection box.
42
兹256
苶
(4 6) º 6 4 º 11
XVI
Ongoing Assessment: Informing Instruction
Watch for students who write only addition or subtraction expressions in
name-collection boxes. Encourage them to use more than one operation in their
expressions, and to use other types of numbers, such as square roots
and exponents.
A typical name-collection box for
16. There are infinite possibilities.
Introducing Factor Strings
WHOLE-CLASS
ACTIVITY
Explain that another form of an equivalent name for a number is
called a factor string. A factor string is a multiplication
expression that has at least two factors that are greater than 1.
In a factor string, the number 1 may not be used as a factor. For
example, a factor string for the number 24 is 2 3 4. To support
English language learners, write factor string and its definition on
the Class Data Pad. Include the example.
We compare factor strings by comparing their lengths. In the
example, the factor string has three factors, so the length of
the factor string is 3. Note that the turn-around rule for
multiplication applies to factor strings: 2 3 4 and 3 4 2
are considered to be the same factor string.
58
Unit 1 Number Theory
Student Page
Ask students to find other factor strings for 24. Remind
students to exclude 1 as a factor. Record all possible factor
strings on the board. (See margin.)
Ask students to find the factor string for 7. Most students will
automatically answer 1 * 7. Ask: What type of number is 7? Prime
number The number 1 may not be used in a factor string, so there
are no factor strings for prime numbers.
Have students find all possible factor strings for other
numbers, such as 30, 50, 54, and 72. Record them in tables on
the board.
Date
Is 2 a factor of 36? Yes; so 36 2 18
●
Is 2 a factor of 18? Yes; so 36 2 2 9
●
Is 2 a factor of 9? No
●
Is 3 a factor of 9? Yes; so 36 2 2 3 3
Factor Strings
19
A factor string is a name for a number written as a product of two or more factors.
In a factor string, 1 may not be used as a factor.
Example:
K I N E S T H E T I C
Factor Strings
Length
2 º 10
2
4º5
2
2º2º5
3
The order of the factors is not important.
For example, 2 10 and 10 2 are the same factor string.
The longest factor string for 20 is 2 2 5.
So the prime factorization of 20 is 2 2 5.
1. Find all the factor strings for each number below.
a.
b.
Number
Factor Strings
Length
Number
Factor Strings
Length
12
3º4
2º6
3º2º2
2
2
3
16
4º4
4º2º2
8º2
2º2º2º2
2
3
2
4
Number
Factor Strings
Length
Number
Factor Strings
Length
16
18
º 42
49 º
6º
32
4º
2º
38
ºº
32
º2
2º2º2º2
22
32
23
4
28
16
7º4
4
4 14
º 2ºº2 2
8º
22
7º
2º
2º2º2º2
2
2
3
3
2
4
c.
d.
25
Math Journal 1, p. 25
Number
Factor String
Length
24
234
3
46
2
2 12
2
38
2
226
3
2223
4
Remind students that divisibility rules can be used to find the prime
factorization of a number. Model using the divisibility rules to find the prime
factorization of 126. 2 3 3 7
Number
20
Adjusting the Activity
A U D I T O R Y
The length of a factor string is equal to the number of factors in the string. The
longest factor string for a number is made up of prime numbers. The longest factor
string for a number is called the prime factorization of that number.
Share students’ strategies for finding the longest factor
strings. For example, for the number 36, you might start with
4 9 and then rename 4 as 2 2 and 9 as 3 3. Another
strategy is to try the prime numbers in order:
●
Time
LESSON
T A C T I L E
V I S U A L
Ask: What kind of numbers are the factors in the longest possible
factor string for any number? Prime numbers The longest factor
string for a number is called the prime factorization of the
number. For example, the prime factorization of 24 is 2 2 2 3.
To support English language learners, write prime factorization
and its definition on the Class Data Pad. Include the example.
Student Page
Date
Time
LESSON
Factor Strings
19
continued
2. Write the prime factorization (the longest factor string) for each number.
Finding Factor Strings
PARTNER
ACTIVITY
and Prime Factorization
(Math Journal 1, pp. 25 and 26)
a. 27 333
b. 40 c. 36 2233
d. 42 237
e. 48 22223
f. 60 2235
2255
g. 100 An exponent is a raised number that shows how many times the number
to its left is used as a factor.
Examples:
Have partners complete both journal pages. Circulate and assist.
Problems 3 and 4 offer additional practice in writing and decoding
exponential notation. To support English language learners,
clarify the meaning of exponent before students complete the
journal page.
2225
52 ∑ exponent
52 means 5 º 5, which is 25.
52 is read as “5 squared” or as “5 to the second power.”
103 ∑ exponent
103 means 10 º 10 º 10, which is 1,000.
103 is read as “10 cubed” or as “10 to the third power.”
24 ∑ exponent
24 means 2 º 2 º 2 º 2, which is 16.
24 is read as “2 to the fourth power.”
3. Rewrite each number written in exponential notation as a product of factors.
Then find the answer.
2
Examples: 23 22 º 9 a. 10 4
2
º
2
º
2
º
10 10 10 10
8
9
36
10,000
335
2
b. 3 º 5 4
2
c. 2 º 10 º
2
45
2 2 2 2 10 10
1,600
4. Rewrite each product using exponents.
53
Examples: 5 º 5 º 5 a. 3 º 3 º 3 º 3 c. 2 º 5 º 5 º 7 4
3
2 52 7
5º5º3º3
b. 4 º 7 º 7 52 º 32
4 72
d. 2 º 2 º 2 º 5 º 5 23 52
26
Math Journal 1, p. 26
Lesson 1 9
59
Teaching Master
Name
Date
Time
Using Factor Trees
LESSON
19
Ongoing Assessment:
Recognizing Student Achievement
Factor Trees
One way to find all the prime factors of a number is to make a factor tree. First write the
number. Then, underneath, write any two factors whose product is that number. Then
write factors of each of these factors. Continue until all the factors are prime numbers.
6
36
º
6
36
3 º 12
9
3 º 3 º 4
3 º 2 º 3 º 2
º
Use journal page 25 to assess students’ facility with finding factors of a
number. Students are making adequate progress if they have correctly written
more than one factor string for each number.
Below are three factor trees for 36.
36
Journal
Page 25
4
[Number and Numeration Goal 3]
3 º 3 º 2 º 2
3 º 3 º 2 º 2
It does not matter which two factors you begin with. You always end with the same prime
factors—for 36, they are 2, 2, 3, and 3. The prime factorization of 36 is 2 º 2 º 3 º 3.
Make a factor tree for each number. Then write the prime factorization for each number.
Sample answers:
24
2 º12
2 º 25
2º 4º3
2 º 5º 5
2º 2 º 2 º3
2º2º2º3
24 WHOLE-CLASS
ACTIVITY
Using Factor Trees
(Math Masters, p. 25)
2º5º5
50 48
100
4 º 12
10 º 10
2 º 2º 3 º 4
2º5º2º 5
2 º 2º 3 º 2 º 2
2º2º2º2º3
48 Finding the Prime Factorization
50
Explain that sometimes it is helpful to organize factors when we
are looking for the longest factor string (the prime factorization).
One way to do this is to make a factor tree. As a class, read the
directions on the teaching master. Students then work together to
find the prime factorization for the numbers 24 and 50. Circulate
and assist.
2º2º5º5
100 Math Masters, p. 25
2 Ongoing Learning & Practice
Playing Name That Number
PARTNER
ACTIVITY
(Student Reference Book, p. 325)
Students practice applying number properties, equivalent
names, arithmetic operations, and basic facts by playing Name
That Number.
Student Page
Date
Time
LESSON
19
1. Circle the numbers that are divisible by 9.
360
252
819
426
651
2. Round 385.27 to the nearest …
a. hundred.
b. whole number.
c. tenth.
4 28–30
249
4. a. Write a 6-digit numeral with
3. Complete the table.
Fraction
Decimal
Percent
1
2
3
8
0.375 37 %
60%
0.6
0.4
40%
0.55
55%
0.08
8%
6
10
2
5
5
5
100
8
100
7 in the thousands place,
5 in the hundredths place,
4 in the tenths place,
3 in the tens place,
and 9s in all other places.
b.
c.
d.
e.
Study Link 1 9
Seven thousand, nine
hundred thirty-nine and
forty-five hundredths
4 30 31
Home Connection Students write numbers with
exponents as factor strings and vice versa. Then they find
the prime factorization of numbers and express the prime
factorization using exponents.
if each bag contains 8 marbles?
56 marbles
(unit)
b. How many marbles are in 700 bags
if each bag contains 8 marbles?
5,600 marbles
(unit)
18
18–20
27
Math Journal 1, p. 27
60
INDEPENDENT
ACTIVITY
(Math Masters, p. 24)
6. a. How many marbles are in 7 bags
300 40 12,000
16,000 80 200
540,000 900 600
80 80
6,400 60
36,000 600 Mixed Practice Math Boxes in this lesson are paired with
Math Boxes in Lessons 1-5 and 1-7. The skills in
Problems 5 and 6 preview Unit 2 content.
7, 9 3 9 . 4 5
b. Write this numeral in words.
80 90
5. Complete.
INDEPENDENT
ACTIVITY
(Math Journal 1, p. 27)
400
385
385.3
11
a.
Math Boxes 1 9
Math Boxes
Unit 1 Number Theory
Study Link Master
Name
3 Differentiation Options
Date
STUDY LINK
19
Using the Sieve of Eratosthenes
Exponents
An exponent is a raised number that shows how many times the number
to its left is used as a factor.
Examples:
READINESS
Time
INDEPENDENT
ACTIVITY
1.
52
103
24
Ò exponent
Ò exponent
Ò exponent
Write each of the following as a factor string. Then find the product.
Example: 23 15–30 Min
(Math Masters, pp. 26 and 27)
3.
a.
104 10º10º10º10
c.
203 20º20º20
3
9
9º9º9
c.
10,000
8,000
112
50 º 50 º 50 º 50 504
Write each of the following as a factor string that does not have any
exponents. Then use your calculator to find the product.
Example: 23 º 3 4.
49
Write each factor string using an exponent.
64
Example: 6 º 6 º 6 º 6 a. 11 º 11 b.
Social Studies Link To explore the concept of prime
numbers, have students use the Sieve of Eratosthenes to
find the prime numbers. Math Masters, page 27 provides a grid
that will allow students to identify all of the prime numbers from
1 to 30. Have students describe patterns they see in the
marked-up grid. Encourage them to use vocabulary words such as
row, column, and diagonal.
2º2º2 8
7º7
72 b.
2.
16
52 means 5 º 5, which is 25.
103 means 10 º 10 º 10, which is 1,000.
24 means 2 º 2 º 2 º 2, which is 16.
a.
2 º 33 º 52 b.
24 º 42 2º2º2º3
2º2º2º2º4º4
24
2º3º3º3º5º5
1,350
256
Write the prime factorization of each number. Then write it using exponents.
2 º 32
2 º3º3
Example: 18 3
2
º
2
º
2
º
5
2
º
5
a. 40 2 º 32 º 5
b. 90 2 º 3 º 3 º 5 Practice
ENRICHMENT
Exploring Palindromic Squares
INDEPENDENT
ACTIVITY
15–30 Min
5,041
50 R4
12
9. 84 7 720
99,140
47,668
701 68 5.
6,383 1,342 6.
48 15 7.
7冄3
苶5
苶4
苶∑
8.
50,314 48,826 10.
Math Masters, p. 24
(Math Masters, p. 28)
To explore the relationship between numbers and their squares,
have students find palindrome numbers by looking at the
arrangement of their digits. Students list 3- and 4-digit
palindrome numbers. Then they square these to find palindromic
squares—the square of a palindrome number that is also a
palindrome number.
Teaching Master
Name
LESSON
19
Date
Teaching Master
Time
The Sieve of Eratosthenes
The mathematician Eratosthenes, born in 276 B.C., developed this method for
finding prime numbers. Follow the directions below for Math Masters, page 27.
When you have finished, you will have crossed out every number from 1 to 30 in
the grid that is not a prime number.
1.
Since 1 is not a prime number, cross it out.
2.
Circle 2 with a colored marker or crayon. Then count by 2, crossing out all
multiples of 2—that is, 4, 6, 8, 10, and so on.
3.
Name
LESSON
19
Skip 4 on the grid because it is already crossed out, and go on to 5. Use a
new color to circle 5 and cross out the multiples of 5.
5.
Continue. Start each time by circling the next number that is not crossed out.
Cross out all multiples of that number. If a number is already crossed out,
make a mark in a corner of the box. If there are no multiples for a number,
start again. Use a different color for each new set of multiples.
6.
Stop when there are no more numbers to be circled or crossed out. The
circled numbers are the prime numbers from 1 to 30.
7.
List the prime numbers from 1 to 30.
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
Time
Palindrome numbers are numbers that read the same forward or backward. A single-digit
number is also a palindrome. The two-digit palindrome numbers are 11, 22, 33, 44, 55, 66,
77, 88, and 99. The table below lists samples of 3-digit and 4-digit palindromes.
1.
Find 3-digit and 4-digit numbers to add to the table.
Sample answers:
Palindrome Numbers
Circle 3 with a color different from Step 2. Cross out every third number after
3 (6, 9, 12, and so on). If a number is already crossed out, make a mark in a
corner of the box. The numbers you have crossed out or marked are multiples
of 3.
4.
Date
Palindromic Squares
3-digit
4-digit
101, 111
1,001; 1,111
202, 222
2,002; 2,222
303, 333
3,003; 3,333
404, 414, 424, 434, 444
454, 464, 474, 484, 494
4,004; 4,114; 4,224; 4,334;
4,444; 4,554; 4,664; 4,774;
4,884; 4,994
Sometimes finding the square of a palindrome number results in a square number that is
also a palindrome number—a palindromic square. For example, 1112 12,321.
1, 2, and 3
2.
Which 3 single-digit numbers have palindromic squares?
3.
Which 2-digit numbers have palindromic squares?
4.
Find the numbers from the table that have a palindromic square and
write the number model.
11 and 22
Example: 1012 10,201
Sample answers: 1112 = 12,321; 2022 =
40,804; 1,0012 = 1,002,001; 1,1112 = 1,234,321
Math Masters, p. 26
Math Masters, p. 28
Lesson 1 9
61