Download Section A Number Theory 4-1 Divisibility 4

Section A
Number Theory
4-1 Divisibility
4-2 Factors and Prime Factorization
4-3 Greatest Common Factor
Section A Quiz
Section B
Understanding Fractions
4-4 Decimals and Fractions
4-5 Equivalent Fractions
4-6 Mixed Numbers and Improper Fractions
Section B Quiz
Section C
Introduction to Fraction Operations
4-7 Comparing and Ordering Fractions
4-8 Adding and Subtracting Fractions with Like Denominators
4-9 Estimating Fractions Sums and Differences
Section C Quiz
Number Theory & Fraction Unit Test
4-1 Divisibility
Vocabulary
Divisible ______________________________________________________________
______________________________________________________________
Composite Number _____________________________________________________
______________________________________________________________
Prime Number: ________________________________________________________
______________________________________________________________
Divisibility Rules
A number is divisible by
2 if the last digit is even (0, 2, 4, 6, 8).
3 if the sum of the digits is divisible by 3.
4 if the last two digits form a number divisible by 4.
5 if the last digit is a 0 or 5.
6 if the number is divisible by both 2 and 3.
9 if the sum of the digits is divisible by 9.
10 if the last digit is 0.
Example 1
Checking Divisibility
A) Tell whether 610 is divisible by 2, 3, 4, and 5.
Divisible or Not Divisible
Explain how you know
2
3
4
5
So 610 is divisible by ________________.
Divisible Not Divisible
B) Tell whether 387 is divisible by 6, 9, 10.
Divisible or Not Divisible
Explain how you know
6
9
10
So 387 is divisible by _________________.
Example 2
Identifying Prime and Composite Numbers
Tell whether each number is prime or composite.
A) 45
Divisible by: 1, 3, 5, 9, 15, 45
Prime or Composite
Divisible by: ____________________
Prime or Composite
Divisible by: ____________________
Prime or Composite
D) 49
Divisible by: ____________________
Prime or Composite
B) 13
C) 19
Lightly shade in all the prime numbers.
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
Think and Discuss
1. Tell which whole numbers are divisible by 1.
2. Explain how you know that 87 is a composite number.
3. Tell how the divisibility rules help you identify composite numbers.
4-2 Factors and Prime Factorization
Vocabulary
Factor _______________________________________________________________
Prime Factorization ____________________________________________________
_______________________________________________________________
Example 1
Finding Factors
List all of the factors of each number.
A)
18
Begin listing factors in pairs
18 = 1  18
18 = 2  9
18 = 3  6
18 = 6  3
1 is a factor
2 is a factor
3 is a factor
4 & 5 are not factors
6 & 3 have already been listed
The factors of 18 are 1, 2, 3, 6, 9, 18.
B)
20
The factors of 20 are ___________________.
C)
13
The factors of 13 are __________________.
Example 2
Writing Prime Factorizations
Write the prime factorization of each number.
A) 36  Choose any two factors to begin. Keep finding factors until each branch ends
at a prime factor.
36
36
B) 54  Choose any two factors to begin. Keep finding factors until each branch ends
at a prime factor.
54
54
Think and Discuss
1. Tell how you know when you have found all of the factors of a number.
2. Tell how you know when you have found the prime factorizations of a number.
3. Explain the difference between factors of an umber and prime factors of a number.
4-3 Greatest Common Factor
Vocabulary
Greatest Common Factor (GCF) ___________________________________________
_____________________________________________________________________
Example 1
Finding the GCF
Find the GCF of each set of numbers.
A)
16 and 24
Method 1: list the factors
Factors of 16: ___________________________
Factors of 24: ___________________________
The GCF of 16 and 24 is _______.
B)
28 and 42
Method 1: list the factors
Factors of 28: ___________________________
Factors of 42: ___________________________
The GCF of 28 and 42 is _______.
C)
12, 24, and 32
Method 2: Use prime factorization
12 =
24 =
32 =
The GCF of 12, 24 and 32 is _______.
Example 2
Problem Solving Application
There are 12 boys and 18 girls in Ms. Ruiz’s science class. The students must form lab
groups. Each group must have the same number of boys and girls. What is the
greatest number of groups Ms. Ruiz can make if every student must be in a group?
Jenna has 16 red flowers and 24 yellow flowers. She wants to make bouquets with the
same number of each color flower in each bouquet. What is the greatest number of
bouquets she can make?
Think and Discuss
1. Explain what the GCF of two prime numbers is.
2. Tell what the least common factor of a group of numbers would be.
4-4 Decimals and Fractions
Vocabulary
Mixed Number _________________________________________________________
________________________________________________________________
Terminating Decimal ____________________________________________________
________________________________________________________________
Repeating Decimal _____________________________________________________
________________________________________________________________
Example 1
Writing Decimals as Fractions or Mixed Numbers
Write each decimals as a fraction or mixed number.
A)
0.23
B)
0.67
C)
1.7
D)
5.9
Example 2
Writing Fractions as Decimals
Write each fraction or mixed number as a decimal and circle terminating or repeating.
A)
3
4
terminating or repeating
B)
5
2
3
terminating or repeating
C)
3
20
Example 3
terminating or repeating
D)
6
1
3
terminating or repeating
Comparing and Ordering Fractions and Decimals
Order the fractions and decimals from least to greatest.
0.5,
1
, 0.37
5
First rewrite the fraction as a decimal.
1
= 0.2
5
3
7
, 0.8,
10
4
First rewrite the fraction as a _______________.
3
= ___________
4
7
= ____________
10
Think and Discuss
1. Tell how reading the decimal 6.9 as “six and nine tenths” helps you write 6.9 as a
mixed number.
2. Look at the decimal 0.121122111222……. If the pattern continues, is this a
repeating decimal? Why or why not?
4-5 Equivalent Fractions
Vocabulary
Equivalent Fractions ___________________________________________________
_______________________________________________________________
Simplest Form _______________________________________________________
___________________________________________________________________
Example 1
Finding Equivalent Fractions
6
Find two equivalent fractions for .
8
_________
=
__________
=
_________
The same area is shaded when the rectangle is divided into 8 parts, 12 parts, and 4
parts.
Example 2
Multiplying and Dividing to Find Equivalent Fractions
Find the missing number that makes the fractions equivalent.
A)
2

3 18
B)
3

5 20
C)
70 7

100
D)
4 80

5
Example 3
Writing Fractions in Simplest Form
Write each fraction in simples form.
A)
18
24
B)
20
48
C)
8
9
D)
10
35
Think and Discuss
1. Explain whether a fraction is equivalent to itself.
2. Tell which of the following fractions are in simplest form:
3. Explain how you know that
7
is in simplest form.
16
9 20 5
,
,
. Explain.
21 25 13
4-6 Mixed Numbers and Improper Fractions
Vocabulary
Improper Fraction ______________________________________________________
________________________________________________________________
Proper Fraction ________________________________________________________
________________________________________________________________
Improper and Proper Fractions
Improper Fractions
3
=1
 Numerator equals denominators  fraction is
3
equal to 1
9
>1
 Numerator greater than denominator  fraction is
5
greater than 1
Proper Fractions
 Numerator less than denominator  fraction is
2
<1
less than 1
5
Example 1
102
=1
102
13
>1
1
102
<1
351
Astronomy Application
The longest total solar eclipse in the next 200 years will take place in 2186. It will
15
15
last about
minutes. Write
as a mixed number.
2
2
METHOD 1: Use a model.
Draw squares divided into half sections. Shade 15 of the half sections.
1
2
1
2
1
1
2
1
2
2
1
2
1
2
3
1
2
1
2
4
1
2
1
2
5
1
2
1
2
6
There are ____ whole squares and ___ half square, or 7
1
2
1
2
7
1
2
1
2
1
squares, shaded.
2
1
2
METHOD 2: Use division
18
=
5
2 15 =
Example 2
Writing Mixed Numbers as Improper Fractions
A)
Write 2
1
as an improper fraction. Use multiplication and addition.
5
B)
Write 3
2
as an improper fraction. Use multiplication and addition.
3
Think and Discuss
10 25 31
,
,
.
7 9 16
2. Tell whether each fraction is less than 1, equal to 1, or greater than 1:
21 54 9 7
,
,
,
.
21 103 11 3
1. Read each improper fraction
3. Explain why any mixed number written as a fraction will be improper.
4-7 Comparing and Ordering Fractions
Vocabulary
Like Fractions _________________________________________________________
________________________________________________________________
Unlike Fractions _______________________________________________________
________________________________________________________________
Common Denominator __________________________________________________
________________________________________________________________
Example 1
Comparing Fractions
Compare. Write <, >, =.
A)
1
8
B)
7
10
Example 2
A)
5
8
1
2
Cooking Application
Ray has
2
3
cup of nuts. He needs cup to make cookies. Does he have
3
4
enough nuts for the recipe?
Compare
3
2
and .
3
4
B)
Rachel and Hannah have 1
2
1
cups of cabbage. They need 1 cups to make
3
2
potstickers. Do they have enough for the recipe?
Compare 1
Example 3
A)
Order
3
7
2
1
and 1 .
3
2
Ordering Fractions
3 3
1
, , and from least to greatest.
7 4
4
= ___
3
4
= ___
1
= ___ Rename with like denominators
4
The fractions in order from least to greatest are _____________________________.
B)
Order
4
5
4 2
1
, , and from least to greatest.
5 3
3
= ___
2
3
= ___
1
3
= ___ Rename with like denominators
The fractions in order from least to greatest are _____________________________.
Think and Discuss
1. Tell whether the values of the fractions change when you rename two fractions so
that they have common denominators.
2. Explain how to compare
2
4
and .
5
5
4-8 Adding and Subtracting with Like Denominators
Example 1
A)
Life Science Application
Sophie plants a young oak tree in her backyard. The distance around the trunk
1
grows at a rate of inch per month. Use pictures to model how much this
8
distance will increase in two months, then write your answer in simplest form.
=

1
1
+
8
8
1
1 2
+ =
8
8
8
=
1
4
Add the numerators. Keep the same denominators
Write your answer in simplest form.
The distance around the trunk will increase by _____ inch.
B)
1
inch per hour. How much snow fell after two
4
hours? Write your answer in simplest form.
Snow was falling at a rate of
After two hours _______ inch of snow fell.
Example 2
Subtracting Like Fractions and Mixed Numbers
Subtract. Write each answer in simplest form.
2
3
A) 1 -
___ - ___ = ___
Check:
B)
1-
3
5
___ - ___ = ___
Subtract. Write each answer in simplest form.
C)
3
7
1
- 1
12
12
3
7
1
- 1
12
12
2
6
12
2
Check:
Subtract the fractions. Then subtract the whole numbers.
1
2
Write your answer in lowest terms.
D)
5
5
12
Example 3
- 2
1
12
Evaluation Expressions with Fractions
Evaluate each expression for x =
A)
3
. Write each answer in simplest form.
8
5
- x
8
5
- x
8
Write the expression
5
3
2
=
8
8
8
Substitute
3
8
for x and subtract the numerators. Keep the
same denominator.
=
B)
1
x + 1
8
C)
x +
7
8
1
4
Write your answer in simplest form.
Evaluate each expression for x =
A)
5
- x
9
B)
x + 2
2
. Write each answer in simplest form.
9
4
9
Think and Discuss
1. Explain how to add or subtract like fractions.
2. Tell why the sum of
1
3
4
and is not
. Give the correct sum.
5
5
10
3. Describe how you would add 2
3
1
1
3
and 1 . How would you subtract 1 from 2 ?
8
8
8
8
4-9 Estimating Fraction Sums and Differences
Example 1
Estimating Fractions
Estimate each sum or difference by rounding to 0,
A)
1
, or 1.
2
8
2
+
9 11
8
2
+
9 11
Think:
8
2
rounds to 1 and
rounds to 0.
9
11
Think:
8
7
rounds to ____ and
rounds to ____.
15
12
1 + 0 =1
8
2
+
is about 1.
9 11
B)
8
7
12 15
8
7
12 15
__ - __ = ___
8
7
is about ____.
12 15
C)
6 3
+
7 8
6 3
+
7 8
Think:
3
6
rounds to ____ and rounds to ____.
8
7
__ + __ = ___
6 3
+
is about _____.
7 8
D)
9
7
10 8
9
7
10 8
Think:
9
7
rounds to ____ and rounds to ____.
10
8
__ - ___ = ____
9
7
is about _____.
10 8
Example 2
A)
Sports Application
Nature Club’s
Biking Distances
About how far did the Nature Club ride
on Monday and Tuesday?
Day
Monday
10
Tuesday
9
Wednesday
B)
About how much farther did the Nature
Club ride on Wednesday than Thursday?
Distances (mi)
Thursday
3
10
3
4
1
12
4
7
4
10
C)
Estimate the total distance that the Nature Club rode on Monday, Tuesday,
and Wednesday.
Think and Discuss
1. Tell whether each fraction rounds to 0,
1
, or 1:
2
5 2
7
,
,
.
6 15 13
2. Explain how to round mixed numbers to the nearest whole number.
3. Determine whether the Nature Club met their goal to ride at least 35 total miles.