Decomposing Decimals into Expanded Form
Decomposing Decimals into Expanded Form teaches Grade 6 students to write any decimal number as a sum of its place-value components, where each digit is multiplied by its place value fraction (tenths, hundredths, thousandths, etc.). Covered in Illustrative Mathematics Grade 6, Unit 5: Arithmetic in Base Ten, this skill reinforces the base-ten number system and supports decimal operations, comparison, and understanding. For example, 3.47 = 3 + 4/10 + 7/100.
Key Concepts
Property A decimal number is a number that includes a decimal point, separating the whole number part from the fractional part. The digits to the right of the decimal point represent fractions with denominators that are powers of ten (tenths, hundredths, thousandths, etc.). For example, a number $a.bcd$ can be written in expanded form as: $$a.bcd = a + \frac{b}{10} + \frac{c}{100} + \frac{d}{1000}$$.
Examples The decimal $0.6$ is equivalent to the fraction $\frac{6}{10}$. The decimal $5.42$ can be written as $5 + \frac{4}{10} + \frac{2}{100}$, which is equal to the mixed number $5\frac{42}{100}$. In money, $1.75$ dollars represents one whole dollar and seventy five hundredths of a dollar, or $1 + \frac{75}{100}$ dollars.
Explanation A decimal is a way to write a number that is not whole. The decimal point acts as a separator between the whole part on the left and the fractional part on the right. Each place value to the right of the decimal point is ten times smaller than the place value to its left. Understanding this structure is key to performing arithmetic with decimals, especially in financial contexts like calculating costs and change.
Common Questions
What is expanded form for decimals?
Expanded form writes a decimal as the sum of each digit multiplied by its place value. For example, 3.47 = 3 × 1 + 4 × (1/10) + 7 × (1/100) = 3 + 0.4 + 0.07.
How do you write 0.256 in expanded form?
0.256 = 2 × (1/10) + 5 × (1/100) + 6 × (1/1000) = 0.2 + 0.05 + 0.006.
Why is expanded form useful for decimals?
It makes the place value of each digit explicit, which helps with comparing decimals, performing operations, and understanding the base-ten structure.
Where is decomposing decimals into expanded form in Illustrative Mathematics Grade 6?
This topic is in Unit 5: Arithmetic in Base Ten of Illustrative Mathematics Grade 6.
What place values are to the right of the decimal point?
Moving right from the decimal point: tenths (1/10), hundredths (1/100), thousandths (1/1000), and so on, each ten times smaller than the previous.