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Mathematics

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:

  1. 7/10
  2. 4/10
  3. 11/100
  4. 72/100
  5. 1/1000
  6. 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-thousands100,000Whole Number part
ten-thousands10,000
thousands1,000
hundreds100
tens10
ones1
.Decimal Point read "and"
tenths1/10 or .1Fractional/Deciamal part
hundredths1/100 or .01
thousandths1/1,000 or .001
ten-thousandsths1/10,000 or .0001
hundred-thousandsths1/100,000 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 ths, any place to the left of the decimal ends in 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

FractionsDecimalRead as...
4/100.4four tenths
-9/100-0.09negative nine hundredths
71/1000.71seventy-one hundredths
8/10000.008eight thousandsths
-45/1000-0.045negative forty-five thousandths
832/10000.832eight hundred thirty-two thousandths

Examples

Write each decimal as a fraction:

  1. -0.7
  2. 0.09
  3. 0.0678
  4. 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:

  1. 0.750
  2. 18.105
  3. -4.3

Examples

In 1 and 2, circle the equivalent decimals.

  1. 0.058    0.580    0.5800    0.58000
  2. 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

  1. If there is a whole number part, write it in words.
  2. Write "and" for the decimal point.
  3. 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 hundredths.

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.

  1. Negative twelve and seven hundredths.
  2. 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:

  1. Write each decimal number with the same number of decimal places, you may need to add zeros for place-holders.
  2. 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/

  1. 26.208   26.28
  2. 0.12   0.026
  3. -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)

3.065
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:

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

Examples

Round to the indicated place:

  1. 5.8903; to the nearest hundreth
  2. 11.0295; to the nearest thousandth
  3. 0.545; to the nearest tenth

Assessment: Introduction to Decimals