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Table of Contents
Number and Operation
Lesson 1
Goldbach’s Conjecture . . . . . . . . . . . . . . . . . . . . . . . . 5
Prime Factorization
Lesson 2
Micro Mites . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
Division with Decimals
Lesson 3The Stock Market . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
Multiply and Divide Integers
Lesson 4
Let’s Sleep on It . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
Percents
Lesson 5
Shopping at Home . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
Working with Percents
Lesson 6Famous Faces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
Divisibility Rules
Lesson 7Deep Blue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
Square Roots
Geometry
Lesson 8Tremendous Triangles . . . . . . . . . . . . . . . . . . . . . . . 47
Pythagorean Theorem
Lesson 9
Twins . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
Missing Sides of Similar Figures
Lesson 10Alone Around the World . . . . . . . . . . . . . . . . . . . . . 59
Angle Pairs
Measurement
Lesson 11
A Capital Idea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
Four-Quadrant Grid
Lesson 12
The Titanic . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71
Scale Drawings
Lesson 13
Papermaking . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
Surface Area
Lesson 14Let’s Get Cereal! . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
Volume of a Pyramid
Algebra
Lesson 15
Rock Stars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89
Evaluate Expressions
Lesson 16The Drive-In . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 95
Two-Step Equations
Lesson 17
Go Figure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101
Inequalities
Lesson 18
Cable Cars . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
Distributive Property
Lesson 19
On the Move . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
Graph Linear Functions
Data Analysis and Probability
Lesson 20
A Helping Hand . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
Histograms
Lesson 21Living Giants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
Box-and-Whisker Plots
Lesson 22The Earl of Sandwich . . . . . . . . . . . . . . . . . . . . . . . . 131
Combinations
Math Tools
Glossary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 137
Order of Operations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
Divisibility Rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 1
Perimeter, Area, and Volume . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
Angle Pairs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Fractions, Decimals, and Percents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
prime
factorization
1
Goldbach’s Conjecture
In mathematics, a conjecture is a belief that something is right
even though it hasn’t been proven. One of the most famous
conjectures in all of mathematics is called Goldbach’s
Conjecture. In 1742, a mathematics professor named Christian
Goldbach wrote to another mathematician, Leonard Euler, that
he believed that every even integer greater than 2 could be
expressed as the sum of two prime numbers. Think about the
even number 8. It can be expressed as the sum of the prime
numbers 3 and 5. To this day, Golbach’s Conjecture has not
been proven, but no one has been able to disprove it either!
Get Started
Ms. Cardona’s class was exploring Goldbach’s Conjecture. Michael
chose the even numbers below to see if the conjecture works.
Complete Michael’s chart. The first one has been done for you.
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even number
greater than 2
12
14
18
24
36
40
44
56
Name two odd numbers that can also
be written as the sum of two primes:
sum of two
primes
57
Remember
The number 1 is
neither prime
nor composite.
and
Lesson 1: Prime Factorization
5
Working with Prime Factorization
You can write any whole number greater than 1 as a product
of different factors. For example, you can write 12 as 2  6 or
3  4 or 2  2  3. Writing a number this way is called
factoring a number.
You can factor the number 24 in many different ways:
24
24
24
2  12
46
38
226
2223
324
2223
3222
When you write a composite number, like 24, as a product
of only prime factors, you are writing the prime factorization
of that number.
Look at the diagram above. The expression 2  2  2  3,
or 3  2  2  2, is the prime factorization of the number 24.
Each shows the number 24 written as a product of only
prime factors.
prime factorization the expression of a number as a product of prime
1. Why is the expression 3  2  4 not a prime factorization of 24?
2.Can you write a prime number, like 19, as the product of only prime numbers?
Why or why not?
6
Level G
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Practice
Write the prime factorization for each number below.
Show your work.
3. 18
4. 32
5. 15
6. 36
7. 20
8. 25
9. 39
10. 80
11. 28
It’s a Fact!
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Solve a Problem
12. Look at Problem 10 above. Show how you
could use exponents to write the prime
factorization of 80.
In 2000, a British publisher offered a prize of one million dollars to anyone who could submit a correct proof of Goldbach’s conjecture by April 2002. The prize was never claimed.
Lesson 1: Prime Factorization
7
Writing Prime Factorizations
You can use a factor tree to help you write the prime
factorization of a number. Follow the steps below to write
the prime factorization of 56.
56
STEP 1 Write the number you are factoring. Then
write this number as the product of any
two factors. Draw lines to each factor as shown.
78
56
STEP 2 Write each composite number as the
product of two factors and draw lines
to each factor.
78
Bring down any prime factors.
724
56
STEP 3 Repeat Step 2 until you are left with
only prime factors.
78
The prime factorization of the number is
shown on the bottom row of your factor tree.
724
7222
STEP 4 Write the prime factors in order from
least to greatest:


Then use exponents to write the prime factorization:
56 
3

Explain how you can check your answer.
8
Level G
© 2005 Options Publishing
I
Show What You Know
Use the steps you learned to write the prime factorization
of each number below. Use exponents to write each prime
factorization. Remember, the factors in a prime factorization
are written in order from least to greatest.
64
1.
42 
64 
90
3.
90 
42
2.
108
4.
108 
On Your Own
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On Your Own
Look at Problem 3 above.
How could you use the prime
factorization of 90 to write the
prime factorization of 180?
Lesson 1: Prime Factorization
9
1
Test Yourself
1. When has the prime factorization of
a number been found?
4. The expression 23  52 is the prime
factorization of which number?
 when all factors are composite
numbers
 10
 40
 when all factors are even numbers
 100
 when all factors are prime numbers
 200
 when all factors are less than 5
2. Which represents the prime
factorization of 48?
5. Find the prime factorization of 60
by making a factor tree. Show your
work below.
 2  24
 2  2  12
 2  2  2  6
 2  2  2  2  3
3. Which shows the prime
factorization of 50?
 5  10
 2  52
 23  5
 22  32
60 
6. Think Back Write the formula for
the area of a circle:
Then find the area of a circle with a
radius of 8 feet. Use π = 3.14.
10
Level G
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