1. Number Theory and Fractions
Chapter 4
Number Theory and Fractions

2. 4-1: Exponents
IWBAT write and evaluate exponential expressions

3. 3 3 3∙3∙3 27 Vocabulary Power – How many times to multiply
Exponent – Tells how many times a repeated factor is to be multiplied.
3 3
Power – How many times to multiply
Base – The factor to be multiplied
3∙3∙3 27

4. Writing Exponents
The factor is the base. The number of times the factor appears is the exponent.
2×2×2
2 3
0.8∙0.8∙0.8∙0.8
0.8 4
𝑏∙𝑏∙𝑏∙𝑏∙𝑏∙𝑏
𝑏 6

5. Evaluating Exponents 4 3 1. Write out what the exponent represents
4∙4∙4
2. Multiply. Write out each multiplication step
16∙4 64

6. Examples 1) 2) 3) 4)
7∙7
6∙6∙6
0.5∙0.5∙0.5∙0.5 4 49
36∙6
0.25∙0.25
216
0.0625

7. Evaluating with Substituting
Evaluate when n = 5
𝑛 2 −2
1) Rewrite the problem with the substitution.
𝑛 2 − 𝑛=5
2) Substitute for the variable.
5 2 −2
3) Evaluate using order of operations.
25−2 23

8. Evaluate each expression when n = 5.
Examples
Evaluate each expression when n = 5.
1) (𝑛−3) 4
2) 2 𝑛 2
(5−3) 4
2∙ 5 2
2 4
2∙25
2∙2∙2∙2 50
4∙4 16

9. Word Problem
A certain cell doubles every hour. If you begin with one cell, at the end of 1 hour there are 2 cells, at the end of 2 hours, there are 𝟐 𝟐 or 4 cells, and so on. After 6 hours, how many cells will there be? Hour 1 = 2 Hour 2 = 𝟐 𝟐 = 4 Hour 3 = 𝟐 𝟑 = 8 Hour 6 = 𝟐 𝟔 =𝟐∙𝟐∙𝟐∙𝟐∙𝟐∙𝟐 𝟒∙𝟒∙𝟒 𝟏𝟔∙𝟒 𝟔𝟒 64 cells in 6 hours.

10. 4-2: Prime Factorization
IWBAT use divisibility rules to check for divisibility and write the prime factorization of numbers in exponential form.

11. Divisibility Rules A number is divisible by . . . .
2 if the last digit is even (ends in 0, 2, 4, 6, 8)
3 if the sum of the digits is divisible by 3
4 if the number formed by the last two digits are 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 a 0

12. Tell if 3,742 is divisible by 2, 3, 4, 5, 6, 9, or 10.
Examples
Tell if 3,742 is divisible by 2, 3, 4, 5, 6, 9, or 10.
1) Test the 2s. Is the number even?
yes
2) Test the 3s. Find the digital root.
= 16
Is 16 divisible by 3? No
3) Test the 4s. Look at the last two numbers.
Is 42 divisible by 4? No
4) Test the 5s. Does it end in 0 or 5?
No.
5) Test the 6s. Can it be divisible by 2 and 3?
Just 2 and not 3. No
6) Test the 9s. Find the digital root.
= 16 = 1+6 = 7 No

13. 7) Test the 10s. Does it end in 0?No
3,742 is only divisible by 2

14. Examples
Tell if each number is divisible by 2, 3, 4, 5, 6, 9, 10 or none of these
1) 3,742 2 3 6
2) 5,310 2 5 9 10
3) 47,388 2 3 6
4) 9,999 3 9

15. Vocabulary
Prime Number – A whole number with only two factors, 1 and itself
Composite Number – A whole number with more than two factors.

16. Prime Factorization
Breaking down a number so that it is represented as a product of prime numbers Use the factor tree method to find the prime factorization.

17. 4-3: Reasonable Answer
IWBAT understand when the answer to a problem is reasonable

18. 4-4: Greatest Common Factor
IWBAT find the Greatest Common Factor of two or three numbers

19. Greatest Common Factor (GCF)
Two methods to finding the GCF
Factor Rainbow
Using Prime Factorizing

20. 4-5: Using Logical Reasoning
IWBAT use logical reasoning to draw conclusions and solve problems

21. 4-6: Least Common Multiple
IWBAT find the least common factor of two or three numbers

22. Least Common Multiple (LCM)
Use the same methods of finding the GCF to find the LCM, with one major difference.

23. 4-7: Equivalent Fractions
IWBAT find equivalent fractions and write fractions in simplest form

24. Finding Equivalent Fractions
Multiplying
Dividing
Multiply both the numerator and the denominator 𝟐 𝟑 𝟐 𝟑 ×𝟒 ×𝟒 𝟖 𝟏𝟐
Divide both the numerator and the denominator by the GCF 𝟏𝟐 𝟏𝟓 𝟏𝟐 𝟏𝟓 ÷𝟑 ÷𝟑 𝟒 𝟓

25. Simplest form is when the GCF is 1 (you can’t divide anymore)
Write in simplest form
1) Find the GCF of 12 and 42
12 –
42 –
GCF = 6
2) Divide the numerator and denominator
12 42
÷6 ÷6
= 2 7

26. Examples Write each fraction in simplest form 1) 12 22 2) 14 281) 2)
12 –
14 –
22 –
28 –
GCF = 2
GCF = 14
12 22
÷2 ÷2
= 6 11
14 28
÷14 ÷14
= 1 2

27. Finding the Least Common Denominator
Least Common Denominator (LCD) – The common denominator (or multiple) of two or more fractions. The LCD is the same thing as the LCM but now used with fractions.

28. Rewrite 3 4 and 5 6 as Fractions with the same denominator
1) Find the LCM of the denominator
4 –
6 –
LCM = 12
3 4 = 12
2) Use the LCM as the new denominator for each fraction
5 6 = 12
3) “What you do to the bottom, you do to the top”
3 4
×3 ×3
= 9 12
5 6
×2 ×2
= 10 12
9 12 and

29. Use the LCD to write each set of fractions with the same denominator.
Examples
Use the LCD to write each set of fractions with the same denominator.
1) , 5 20
2) , 4 30
3) , 5 8 , 7 9
20 –
9 –
LCM: 60
LCM: 30
LCM: 72
5 6 = 60
5 20 = 60
6 10 = 30
4 30 = 30
4 3 = 72
5 8 = 72
7 9 = 72
50 60
15 60
18 30
4 30
96 72 , , 56 72

30. 4-8: comparing and Ordering Fractions and Mixed Numbers
IWBAT compare and order fractions and mixed numbers

31. Vocabulary
Improper Fraction – A fraction where the numerator is larger than the denominator. They can be rewritten as mixed numbers. Ex: 𝟒 𝟑 𝟏𝟎𝟎 𝟏𝟎 𝟏𝟕 𝟔 Mixed Number – A number that is made of a whole number and a fraction. Ex: 1 𝟏 𝟑 𝟐 𝟓 𝟔 𝟏𝟕 𝟏𝟑 𝟏𝟓

32. Improper Fractions and Mixed Numbers
Improper Fraction to Mixed Number
Mixed Numbers to Improper Fractions
Divide the denominator into the numerator. The quotient is the whole number. The remainder is the new numerator. The divisor is the new denominator
Multiply the whole number and the denominator. Add the numerator. The sum is the new numerator. Use the same denominator.

33. Improper fraction to Mixed Number1) 2) 3)
7÷3=2 𝑟1
22÷3=7 𝑟1
17÷8=2 𝑟1
2 1 3
7 1 3
2 1 8
MIXED NUMER TO IMPROPER FRACTION 1) 2) 3)
7×9=63
5×10=50
1×4=4
63+4=67
50+3=53
4+3=7
67 7
53 5
7 4

34. Compare. Write <, >, or =.
Comparing Fractions
Compare whole numbers first.
Find the LCD for all fractions.
Compare or order from least to greatest.
Compare. Write <, >, or =. 1) 2) 3)
LCD: 8
LCD: 9
LCD: 20
3 4 = 6 8
7 9
2 3 = 6 9
3 4 = 15 20
5 8
4 5 = 16 20
5 8 < 6 8
7 9 > 6 9
>

35. Ordering Fractions
Rewrite all mixed numbers as improper fractions
Find the LCD for all fractions
Compare
Order from least to greatest
Rewrite using original fractions and mixed numbers
Place the fractions on a number line
Arrange in order from least to greatest and place them on a number line.
1) , , 3 2 , 7 8
7 8 , 3 2 , , 17 8
1 3 4 = 7 4
LCD: 8
7 4 = = 12 8
7 8 , , , 17 8

36. Arrange in order from least to greatest.
EXAMPLES
Arrange in order from least to greatest.
1) , 3 8 , 1 2
2) , 1 3 , 5 6
3) , , 6
LCD: 56
LCD: 18
LCD: 16
5 7 = 40 56
3 8 = 21 56
1 2 = 28 56
4 9 = 8 18
1 3 = 6 18
5 6 = 15 18
1 4 = 4 16
, , 6
5 1 4 , , 6
21 56 , ,
6 18 , ,
3 8 , 1 2 , 5 7
1 3 , 4 9 , 5 6

37. 4-9: Relating Fractions and Decimals
IWBAT write fractions as decimals and decimals as fractions

38. Converting Fractions and Decimals
Fraction to Decimal
Decimal to Fraction
Divide the numerator by the denominator
If there is a whole number, place the whole number in front of the decimal
If the decimal repeats, round to the nearest hundredths
Place the decimal digits over the place value
Rewrite in simplest form
If there is a whole number, place the whole number in front of the fraction

39. Examples Write each fraction as a decimal. 1) 3 4 2) 2 3 3) 4 3 201) 2) 3)
3÷4=0.75
2÷3=0.6666
3÷20=0.15
0.67
4.15
0.75
Write each decimal as a fraction in simplest form.
1) 0.6 2) 3)
Tenths place = 10
Hundredths place = 100
Hundredths place = 100
6 10
= 3 5
8 100
= 2 25
11 100
2 25
3 5

40. 4-10: Using Fractions and Decimals
IWBAT use fractions and decimals to solve problems