# Mathematics

**Part 1**- Whole Numbers
- Integers and Inroduction to Solving Equations
- Solving Equations and Problem Solving
- Fractions
- Decimals
- Lesson 1: Introduction to Decimals
- Lesson 2: Adding and Subtracting Decimals
- Lesson 3: Multiplying Decimals
- Lesson 4: Circumference of a Circle
- Lesson 5: Dividing Decimal Numbers by Integers
- Lesson 6: Dividing a Decimal Number by a Decimal Number
- Lesson 7: Fractions and Decimals
- Lesson 8: Applications of Fractions and Decimals
- Lesson 9: Solving Equations with Decimals
- Lesson 10: Mean, Mode and Median

- 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 Decimals

### Challenge Problem

Polygon Pizza Place caters children's parties with square-shaped pizza. Each pizza is exactly the same size and is divided into equal parts called slices. At Sam's party, each child had 2 out of 10 slices from a single pizza. At Elena's party, each child had 15 out of 100 slices from a single pizza. At which party did each child have more pizza?

How can we solve this problem? What do you notice about the denominators of the given fractions?

### Examples

**Using a calculator, divide each:**

- 7/10
- 4/10
- 11/100
- 72/100
- 1/1000
- 341/1000

### Note

A __decimal__ is used when a whole number is divided by 10, 100, 1000, etc. equal parts.

Decimals are fractions with denominators that are powers of 10.

Decimals, as well as fractions represent parts of a whole.

In the challenge problem we can compare the given fractions or we can convert them to decimals and compare the decimals.

### What's the Connection?

hundred-thousands | 100,000 | Whole Number part |
---|---|---|

ten-thousands | 10,000 | |

thousands | 1,000 | |

hundreds | 100 | |

tens | 10 | |

ones | 1 | |

• | . | Decimal Point read "and" |

tenths | 1/10 or .1 | Fractional/Deciamal part |

hundredths | 1/100 or .01 | |

thousandths | 1/1,000 or .001 | |

ten-thousandsths | 1/10,000 or .0001 | |

hundred-thousandsths | 1/100,000 or .00001 |

The decimal point separates the whole number from the fractional/decimal part. We only need the word “and” when we have both parts.

### Note

- Notice that the ones place is at the center of the place value chart. There is no "oneths" place.
- Each place is 10 times the value of the place to its right.
- Any place to the right of the decimal ends in
, any place to the left of the decimal ends in*ths*.*s* - The number of decimal places in the decimal number is the same as the number of zeros in the denominator of the equivalent fraction.

### Using the decimal form of fractions

Fractions | Decimal | Read as... |
---|---|---|

4/10 | 0.4 | four tenths |

-9/100 | -0.09 | negative nine hundredths |

71/100 | 0.71 | seventy-one hundredths |

8/1000 | 0.008 | eight thousandsths |

-45/1000 | -0.045 | negative forty-five thousandths |

832/1000 | 0.832 | eight hundred thirty-two thousandths |

### Examples

**Write each decimal as a fraction:**

- -0.7
- 0.09
- 0.0678
- 0.877

### Example

**Write the decimal as a fraction or mixed number in simplest form:**

7.89 = 7 + 89/100 = 789/100

### Example

**Write the decimal as a fraction or mixed number in simplest form:**

Can we write 7.89 as 789/100?

789/100 = 7 R89 = 789/100

**Yes, we get the same answer!**

### Examples

**Write the decimal as a fraction or mixed number in simplest form:**

- 0.750
- 18.105
- -4.3

### Examples

In 1 and 2, circle the equivalent decimals.

- 0.058 0.580 0.5800 0.58000
- 43 43.00 0.43 0.043

**Prove your answer by expressing each decimal as fractions in simplest form.**

### Key Information

- For any decimal, writing zeros after the last digit to the right of the decimal point does not change the value of the number.

Ex: 7.6 = 7.60 = 7.600, etc. - When a whole number is written as a decimal, the decimal point is placed to the right of the ones digit.

Ex: 34 = 34.0 = 34.00, etc.

### Writing (or reading) a Decimal in Words

- If there is a whole number part, write it in words.
- Write "and" for the decimal point.
- Write the decimal part in words as though it was a whole number, followed by the place value of the last digit.

**Ex:** Write -451.08 in words

**Ans:** negative four hundred fifty one and eight hundred*ths*.

The eight (or last digit) is in the hundredths place.

### Examples

**Write each decimal in standard form:**

**Note:** Writing a number in __standard form__ means to write the number for the words.

- Negative twelve and seven hundredths.
- Eighty-three ten-thousandths.

### Helpful Hint

When writing a number in standard form, make sure the last digit is in the correct place by inserting 0's if necessary.

### Comparing Decimals

**Procedure to compare decimals:**

- Write each decimal number with the same number of decimal places, you may need to add zeros for place-holders.
- Compare the numbers as if there was no decimal.

### Revisiting the Challenge Problem

You're comparing 2/10 and 15/100

2/10 → .2 one decimal place .20

15/100 → .15 two decimal places .15

**20 > 15**

They had more pizza at Sam's party.

### Examples

**Insert < or > to form a true statement/**

- 26.208 26.28
- 0.12 0.026
- -0.039 -0.0209

### Rounding Decimals

Round 3.065 to the nearest tenth

(asking if that number is closer to 3.0 or 3.1)

3.__0__65

5 or higher add 1 to the 0

**Ans: 3.1**

When rounding decimals the last place in your answer is the place that you are rounding to.

**Procedure to round decimals:**

- Underline the digit in the place that you're rounding to.
- Look at the digit to its immediate right.
- If that digit is a
keep the underlined digit as the last digit.*4 or less* - If the digit is a
add one to the underlined digit and that will be your last digit.*5 or higher*

- If that digit is a

### Examples

**Round to the indicated place:**

- 5.8903; to the nearest hundreth
- 11.0295; to the nearest thousandth
- 0.545; to the nearest tenth