# Mathematics

**Part 1**- Whole Numbers
- Integers and Inroduction to Solving Equations
- Solving Equations and Problem Solving
- Fractions
- Lesson 1: Introduction to Fractions and Mixed Numbers
- Lesson 2: Factors and Simplest Form
- Lesson 3: Multiplying Fractions
- Lesson 4: Dividing Fractions
- Lesson 5: Adding and Subtracting Fractions
- Lesson 6: Complex Fractions
- Lesson 7: Order of Operations
- Lesson 8: Multiplying and Dividing Mixed Numbers
- Lesson 9: Combining Mixed Numbers
- Lesson 10: Solving Equations Containing Fractions

- Decimals
- Ratio, Proportion and Percent
- Geometry
**Part 2**- Problem-Solving and Inequalities
- Measurement
- Graphing and Systems of Equations
- Data Analysis / Probability / Statistics

## Lesson 1: Introduction to Fractions and Mixed Numbers

### Are You Hungry?

If 16 people try to share an 8 piece pizza, how many pieces does each person get? (Assume everyone wants the same amount of pizza)

### The Need for Another Type of Number

**Definition:** A __fraction__ (or rational number) is a number in the form a/b where a and b are integers and b =/0. A fraction is one way to show parts of a whole.

**Remember: the fraction bar mean division**

a (top number) is called the numerator

b (bottom number) is called the denominator

### Sets of Numbers

Name of Set | Example | Symbol |
---|---|---|

Natural Numbers | {1,2,3,...} | N |

Whole Numbers | {0,1,2,...} | W |

Integers | {...,-3,-2,-1,0,1,2,3,...} | Z |

Rational Numbers | Any number that can be written as a fraction | Q |

### A Closer Look at Fractions

1 part shaded4 equal parts | one-fourth | |

5 part shaded6 equal parts | five-sixths | |

7 part shaded3 equal parts | seven-thirds |

### Showing Understanding

**How can we show** 1/2, 2/3, 1/4, 5/6 ?

- How do the fractions compare to 1?
- What does the top number represent?

The bottom number?

### Examples

Draw a picture with shaded parts to show each fraction:

Given the numbers 245 and 542, answer the following:

- 2/3
- 5/5
- 7/4

**Definition:** If the numerator < the denominator, the fraction is a __proper fraction__.

If the numerator __>__ the denominator, the fraction is an __improper fraction__.

### Examples

**Given the fractions:**

- 2/3
- 5/5
- 7/4

- State the numerator and denominator of each fraction.
- Which fractions are proper fractions?
- Which fractions are improper fractions?
- How does each fraction compare to one?

### FYI

- Proper fractions are less than one.
- Improper fractions are either equal to one or greater than one.

### Examples

Blast From the Past!

**Simpify:**

- 3/1
- -3/-3
- 0/3
- 3/0

### Examples

**Graph each on separate number lines:**

- 3/5
- -2/3
- 1/2

### Challenge Question

If I work 5 whole days and then one third of a day, how many days did I work in total?

### Challenge Question

If I work 5 whole days and then one third of a day, how many days did I work in total?

What do we have to do to get our answer? 5 + 1/3

5 + 1/3 = 5 1/3 ← 5 1/3 is a mixed number

How can we illustrate this?

How can we graph this?

### Key Information

- An improper fraction is greater than 1; this allows us to write it as a mixed number.
- A mixed number has two parts:

whole number**+**proper fraction

A mixed number is the sum of a whole number and a proper fraction.

### Seeing the Connection

5 + 1/3 = 51/3 = 16/3

51/3 = 3 • 5 + 1/3 = 16/3

### Mixed Numbers → Improper Fractions

51/3 = 3 • 5 + 1/3 = 16/3

- Multiply the denominator of the fraction by the whole number.
- Add this product to the numerator of the fraction.
- Use the answer from (2) as the numerator of the fraction, keep the same denominator.

### Examples

**Write each mixed number as an equivalent improper fraction:**

- 72/3
- 34/5
- 51/9

### Note:

Positive Mixed Numbers

51/3 = 5 + 1/3

Negative Mixed Numbers

-51/3 = -5 + (-1/3)

-51/3 = -5 - 1/3

The rule to express negative mixed numbers as improper fractions is the same as the rule for expressing positive mixed numbers as improper fractions.

### Example

Express the following mixed number as an improper fraction in lowest terms.

-51/3

-3 • 5 + 1/3 = **-**16/3

### Example

Draw the following mixed number using shaded regions.

13/5

### Improper Fractions → Mixed Numbers

**13/5** = 5 13 = 2 Remainder 3 = 23/5

### Examples

- 17/5
- 23/4
- 62/3

### Group the Equivalent Fractions Together:

1/2 -1/2 -1/2 1/-2 -1/-2

### Reviewing Concepts

**Evaluate Each:**

- |7/8|
- |-2/3|