Decimals

Here you will learn about decimal numbers including how to read and write them, understand their place value, and strategies to compare them.

Students will first learn about decimals in 4th grade math as part of their work in number and operations with fractions and extend their knowledge as they work through number systems in middle school.

What are decimals?

Decimals are numbers that have a whole part and a fractional part that are separated by a decimal point. Our decimal system splits whole numbers into tenths, hundredths, thousandths, and so on.

This decimal place value chart can help you understand decimal place value.

Step-by-step guide: Decimal place value

Step-by-step guide: Decimal places

You can convert fractions to decimals and decimals to fractions.

Fractions as decimals \hspace{1.7cm}

Fraction → \cfrac{3}{10} Read as three tenths Decimal → 0.3 The 3 is placed in the tenths column. Since there are no digits in the hundredths and thousandths place it can be left blank or 0’ s can be added as placeholders.	Fraction → \cfrac{3}{100} Read as three hundredths Decimal → 0.03 The 3 is placed in the hundredths column and a 0 in the tenths column. Since there are no digits in the thousandths place it can be left blank or a 0 can be added as a placeholder.	Fraction → \cfrac{3}{1000} Read as three thousandths Decimal → 0.003 The 3 is placed in the thousandths column and 0’ s are placed in both the tenths and the hundredths columns.

Decimals as fractions \hspace{1.7cm}

Decimal → 0.43 Read as forty-three hundredths. 43 is the numerator of the fraction and 100 is the denominator because the last digit of the decimal is in the hundredths column of the place value chart, 0.43 =\cfrac{43}{100}	Decimal → 0.009 Read as nine thousandths. 9 is the numerator of the fraction and 1000 is the denominator because the 9 is in the thousandths column of the place value chart, 0.009=\cfrac{9}{1000}

You can compare decimal numbers using a place value chart or converting them to fractions.

Step by step guide: Comparing decimals

You can place decimals on a number line to help visualize the value of decimals.

Decimal on a number line

A decimal number line is a number line that includes decimal increments and can help you to visualize decimal numbers. Here is an example of a decimal number line that is in increments of tenths from 0 to 1. If you zoom in on the number line between 0.4 and 0.5 that goes up in steps of 0.01 you can position the decimal 0.472 more accurately. The position has been estimated between 0.47 and 0.48, but closer to 0.47.

Step by step guide: Decimal number line

You can perform calculations on decimal numbers like you would with multi-digit numbers.

Adding and subtracting decimals \hspace{1.7cm}

Add: 3.87 + 4.21 = \, ? ● Line up the decimal points ● Add 0' s as placeholders where necessary ● Add in the same way as adding multi- digit numbers	Subtract: 5.6- 3.79 = \, ? ● Line up the decimal points ● Add 0' s as placeholders where necessary ● Subtract in the same way as subtracting multi-digit numbers

Step by step guide: Adding decimals

Step by step guide: Subtracting decimals

Step by step guide: Adding and subtracting decimals

Multiplying and dividing decimals \hspace{1.7cm}

Multiply: 0.34\times 3 = \, ? ● Put the number with the most amount of digits on top ● Multiply in the same way as multiplying multi-digit numbers. ● Count up the total digits to the right of the decimal point	Divide: 1.10\div 2 = \, ? ● Line up the decimal point of the dividend with the decimal point of the quotient ● Divide using long division

Step by step guide: Multiplying decimals

Step by step guide: Dividing decimals

Step by step guide: Multiplying and dividing decimals

What are decimals?

Common Core State Standards

How does this relate to 4th grade math, 5th grade math, and 6th grade math?

• Grade 4 – Number and Operations Fractions (4.NF.C.7)
  Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or < , and justify the conclusions, example, by using a visual model.

• Grade 5 – Number and Operations Base Ten (5.NBT.A.3)
  Read, write, and compare decimals to thousandths.

• Grade 5 – Number and Operations in Base Ten (5.NBT.A.1)
  Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and \frac{1}{10} of what it represents in the place to its left.

• Grade 5 – Number and Operations in Base Ten (5.NBT.B)
  Perform operations with multi-digit whole numbers and with decimals to hundredths.

• Grade 6: The Number System (6.NS.B.3)
  Fluently add, subtract, multiply, and divide multi-digit decimals using the standard algorithm for each operation.

How to write decimal numbers

In order to write a fraction as a decimal:

1. Write the fraction in words.
2. Place the numerator of the fraction on the place value chart.
3. Write the decimal number.

In order to write a decimal as a fraction:

1. Write the decimal in words.
2. The digits of the decimal is the numerator, and the denominator is the column of the last digit on the place value chart.
3. Write the fraction and simplify if possible.

Decimals examples

Example 1: write the fraction as a decimal

Change \cfrac{47}{100} \, to a decimal number.

1. Write the fraction in words.

\cfrac{47}{100} \, is 47 hundredths.

2Place the numerator of the fraction in the correct spaces on the place value chart.

3Write the decimal number.

\cfrac{47}{100} \, as a decimal is 0.47.

Example 2: writing a decimal as a fraction

Write 0.017 as a fraction.

The digits of the decimal is the numerator, and the denominator is the column of the last digit on the place value chart.

0 is in the tenths column, 1 is in the hundredths column and 7 is in the thousandths column.

17 is the numerator, and 1000 will be the denominator.

Example 3: comparing decimals using a place value chart

Use the place value chart to list these decimal numbers in order from least to greatest.

0.05, 0.02, 0.205, 0.052, 0.502

Put the number on the place value chart.

Add zeros as place holders as necessary.

Start with the tenths place to compare numbers.

From the chart, you can see that 0.502 is the greatest value decimal number, and then 0.205.

0.050, 0.020, and 0.052 all have 0 ’s in the tenths place which means they are lesser in value than 0.205 and 0.502.

Next, move on to the hundredths place and then the thousandths place.

Moving to the hundredths place on the chart, you can see that 0.020 is less than 0.050 and 0.052.

Moving to the thousandths place on the chart, you can see that 0.050 is less than 0.052.

Example 4: comparing decimals using equivalent fractions

List the decimals in order from least to greatest, using equivalent fractions.

0.4, \, 0.004, \, 0.44, \, 0.04

Example 5: add decimal numbers

Add the decimal numbers,

5.4 + 1.27

Stack the numbers and line up the decimal points (and place value).

Add each column from right to left and regroup the numbers when necessary.

Example 6: multiply decimal numbers

Multiply the decimal numbers,

3.2\times 4.03

Stack the number with the most digits on top.

Multiply as if the numbers were multi-digit whole numbers, regrouping when necessary.

Count the number of digits after the decimal point for each factor.

Put the same number of digits after the decimal point for the product.

Teaching tips for decimals

• Using visual models such as number lines or place value representation helps students build number sense as they learn decimal place values.

• Although decimal worksheets have their place in the classroom, utilizing online platforms for game playing helps to motivate students and also provides time for them to practice.

• Reinforcing placeholder 0 ’s makes the decimal numbers have the same digits so that students can easily compare them. It also builds understanding for decimal operations.

Easy mistakes to make

• Confusing decimal numbers
  For example,
  When comparing 0.50 and 0.05 and thinking they are both 5 hundredths or 5 tenths.

• Forgetting to use to placeholder \bf{0} ’s when comparing decimals
  

• Adding \bf{0} ’s placeholders in the wrong place
  For example,
  Adding the 0 ’s as placeholders before the 9 in the decimal number 0.9 instead of after the 9.

The next lessons are

• Percent
• Converting fractions, decimals, and percentages
• Exponents
• Algebraic expression

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