Place Value with Decimals

1. Compare decimals by making use of the structure of the place value system (MP.7), namely that one of any unit is greater than any amount of a smaller unit. Thus, the largest place value in each number contains the most relevant information when comparing numbers. If both numbers have the same number of largest units, the next largest place value should be attended to next, iteratively, until one digit is greater than another in the same unit. 
2. Record the result of comparisons using >, <, or =.
3. Compare numbers written in various forms, including standard, word, and expanded form. 
4. List numbers in ascending or descending order. 

• There are lots of common misconceptions that come up in the context of comparing decimals. These include:
  • Thinking that a longer decimal is a greater decimal (e.g., 0.493 is greater than 0.8)
  • Thinking that a shorter decimal is a greater decimal (e.g., 0.48 is greater than 0.493)
  • Getting tripped up by an internal zero (e.g., in the number 0.07)
  • Thinking decimals between 0 and 1 are less than zero (e.g., thinking 0.48 is less than 0)
  • Reciprocal thinking (e.g., thinking 0.5 is greater than 0.8 because $$\frac{1}{5}$$ is greater than $$\frac{1}{8}$$)
  • Not seeing the equivalence of decimals with zeros at the end (e.g., thinking that 0.48 is not close to 0.493 because they don’t know 0.48 = 0.480)

Keep an eye out for these and other misunderstandings. 

• Before the Problem Set, you could have students play an adapted version of the game “Corn Shucks” from Building Conceptual Understanding and Fluency Through Games by the Public Schools of North Carolina. (Adapt it so that students are writing decimal numbers - perhaps having them using 6 digits to write a number from the hundreds to the thousandths place.)

• Problem Set

• Problem Set Answer Key