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Extending Place Value Beyond the Decimal
Place value does not stop at the ones place. It continues to the right of the decimal point, where each place is worth one-tenth of the place to its left. This allows us to represent numbers that are between whole numbers, such as tenths, hundredths, and thousandths. Decimals help us describe parts of a whole, measurements, money, and many quantities that are not whole numbers.
In this topic, we connect decimals to the larger idea of number types, including rational numbers and irrational numbers. Rational numbers can be written as fractions and often have decimal forms that either end or repeat. Irrational numbers, like π or √2, cannot be written as fractions and have decimal forms that go on forever without repeating.
We will use several models to make decimal place value more visible, including decimal squares, base ten blocks, and number lines. These models help show the size of decimal pieces and where decimals belong in relation to whole numbers and fractions. We will also practice comparing decimals by looking carefully at place value from left to right. Instead of comparing decimals by “how many digits” they have, we compare the value of each place until we find a difference.
Student Learning Goals
By the end of this topic, students should be able to:
Explain how place value continues to the right of the decimal point.
Identify tenths, hundredths, thousandths, and beyond.
Represent decimals using decimal squares, base ten blocks, and number lines.
Compare and order decimals using place value.
Explain why a longer decimal is not always a larger number.
Describe the difference between rational and irrational numbers.
Connect decimals to fractions, measurement, and real-world quantities.
Key Vocabulary
Decimal - A number that uses place value to show parts of a whole.
Decimal Point - The symbol that separates whole number places from decimal places.
Tenths - Ten equal parts of one whole.
Hundredths - One hundred equal parts of one whole.
Thousandths - One thousand equal parts of one whole.
Rational Number - A number that can be written as a fraction.
Irrational Number - A number that cannot be written as a fraction and has a decimal that does not end or repeat.
Number Line - A visual model that shows where numbers are located in order.
Common Student Misunderstandings
If a student is thinking that 0.45 is larger than 0.7 because 45 is larger than 7, then the student may be comparing digits instead of place values.
If a student is thinking 0.30 and 0.3 are different values, then the student may not yet understand trailing zeros.
If a student places 0.8 closer to 0 than 1, then the student may need more practice with decimals on a number line.
If a student things all decimals are irrational, then the student may need help connecting decimals and fractions.
If a student reads 0.106 as “one hundred six” without considering place value, then the student may need to connect decimal names to place value positions.
Teacher Connection
Decimals can be challenging because students may try to use whole-number thinking in a new setting. For example, they may think that more digits automatically means a larger number. Visual models are especially helpful because they show that decimal places represent smaller and smaller parts of a whole.
As a future teacher, it is important to help students connect decimals to quantities, not just symbols. Decimal squares can show the size of tenths and hundredths, base ten blocks can model regrouping and place value, and number lines can help students see decimals as numbers with a specific location.
Quick Reflection Question
Why might a student think that 0.125 is larger than 0.5? What model or explanation could you use to help them compare the two numbers correctly?