Getting Started 
Misconception/Error The student does not have an effective strategy for comparing decimals. 
Examples of Student Work at this Level The student does not reason about the size of the decimals using place value and explains:
 0.69 is greater than 0.81 because the nine in the hundredths place is greater than the one in the hundredths place.Â
 0.38 is greater than 0.7 because 0.38 is longer (has more digits) which makes it the larger decimal.Â
Despite prompting, the student does not use a model to assist with comparing the decimals.

Questions Eliciting Thinking What is the value of the digit three in 0.38? What is the value of the digit seven in 0.7?
Which digit(s) did you look at to determine which number was greater? Are these digits both in the same place value?
How would you represent 0.38 and 0.7 on 10 x 10 grids?
Can you represent both of these decimals on a number line? What strategy would you use to help you know where to place the decimals on a number line?
How would you write 0.69 as a fraction? What about 0.81? Would that help you compare the decimals?
How would you write 0.7 as a fraction? Is equivalent to ? So is 0.7 equivalent to 0.70? 
Instructional Implications Provide direct instruction on the meaning of decimal representations emphasizing place value. Guide the student to visually represent decimals using 10 x 10 grids. Relate the numbers of shaded squares to the way the decimals are written. Then use the less than, greater than, or equal to symbol to compare the decimals. Provide the student with additional pairs of decimals and have the student compare the decimals by first representing each decimal on a 10 x 10 grid.
Review the relationship between decimals and their fraction equivalents. Assist the student in relating the denominator of the equivalent fraction to the number of digits in the decimal representation. Guide the student to scale a number line appropriately in order to graph decimals. Once the student is able to correctly scale a number line, have the student use the number line to graph and compare decimals. Consider using the MFAS task Using Benchmark Fractions On a Number Line (4.NF.3.6).
Consider using the MFAS task Using Models to Compare DecimalsÂ to determine if the student can compare decimals by using visual models. 
Moving Forward 
Misconception/Error The student attempts an effective strategy but makes errors. 
Examples of Student Work at this Level The student correctly determines that 0.81 is greater than 0.69 but struggles to reason about the size of the decimals. In the second problem, the student uses 10 x 10 grids after teacher prompting but makes significant errors and shades in seven squares instead of seventy to represent seven tenths.
The student correctly determines that 0.81 is greater than 0.69 and reasons about the size of the decimals. The student believes that 0.7 is the same as 7%. He or she uses a visual model to represent the decimals but the student makes significant errors in his or her model.
The student struggles to reason about the size of the decimals. He or she attempts to draw a number line to represent the two decimals but makes mistakes when scaling the number line. The student says that 0.38 is greater than 0.7.

Questions Eliciting Thinking What does one square represent on a 10 x 10 grid? What does one column represent?
When we scale a number line, what is something very important we need to remember when we space the decimals on the line?
Where would 0.5 be on the number line? Would 0.38 be placed before or after 0.5? How do you know?
How would you write 0.69 as a fraction? What about 0.81? Would that help you compare the decimals?
How would you write 0.7 as a fraction? Is equivalent to ? So is 0.7 equivalent to 0.70?
Show the student the number 0.7. Then ask, 'How would you read this number aloud? How many hundredths is that equal to?' 
Instructional Implications Review writing decimals in equivalent forms (e.g., explain that 0.7 and 0.70 are equivalent since seven out of 10 is the same as 70 out of 100 when referring to parts of the same whole). Guide the student to compare 0.7 to 0.38 by drawing appropriate models of each. It may also be helpful to relate the decimals to money. For example, when comparing 0.7 and 0.38, the student can say that 0.7 is the same as seventy cents and 0.38 is the same as 38 cents. Guide the student to observe that 0.38 is less than 0.7 since 38 cents is less than 70 cents.
Provide opportunities for the student to use models (10 x 10 grids or base ten blocks) to compare decimal numbers. Reinforce the relationship between decimals and fractions and guide the student to understand and compare decimals by thinking about them as fractions. Model reading decimals appropriately (e.g., read 0.7 as â€śseven tenthsâ€ť rather than as â€śzeropointsevenâ€ť). 
Almost There 
Misconception/Error The student is unable to justify decimal comparisons appropriately using place value or uses inequality symbols incorrectly. 
Examples of Student Work at this Level The student correctly determines that 0.7 is greater than 0.38. However, when recording the comparison, the student writes 0.7 < 0.38. When asked to read his or her comparison aloud, the student says, â€śSeven tenths is greater than thirtyeight hundredths.â€ť
The student correctly determines that 0.7 is greater than 0.38. However, he or she struggles to justify his or her thinking by using place value understanding or does not reason about the size of the decimals. The student says that 0.7 is greater than 0.38 because seven is greater than three. 
Questions Eliciting Thinking You are correct, 0.7 is greater than 0.38. However, look at the symbol you used to compare the decimals. Do you think you used the correct symbol? Why or why not?
Another student said that 0.38 is greater than 0.7 because 0.38 has more digits. Explain why this reasoning is incorrect when we compare 0.7 and 0.38.
How would you scale a number line to show both 0.7 and 0.38 on the number line? Where would 0.5 be on the number line? Would 0.38 be placed before or after 0.5? How do you know? Where would 0.7 be placed?
How many hundredths is seventenths equal to? 
Instructional Implications Provide clear instruction on using the less than, greater than, or equal to symbols correctly. Then provide additional decimals to compare and ask the student to record the comparisons using the less than, greater than, or equal to symbol.
Guide the student to interpret 0.38 as 38 hundredths and to interpret 0.7 in its equivalent form, 70 hundredths. Model explaining that 0.38 is less than 0.7 because 38 hundredths is less than 70 hundredths. Encourage the student to provide place value explanations when comparing decimals.
Partner the student with a â€śMoving Forwardâ€ť student. Have the student model for the â€śMoving Forwardâ€ť student how to compare decimal numbers using place value understanding. 
Got It 
Misconception/Error The student provides complete and correct responses to all components of the task. 
Examples of Student Work at this Level The student determines that 0.81 is greater than 0.69 and justifies his or her reasoning by saying that eight tenths is greater than six tenths or that 81 hundredths is greater than 69 hundredths. In the second problem, the student correctly writes 0.7 > 0.38 and justifies his or her reasoning by saying that 0.7 is equal to 0.70 and 0.70 is greater than 0.38. He or she may also say that 0.7 is greater because seven tenths is greater than three tenths.

Questions Eliciting Thinking How can you use benchmark fractions (0, , , , and 1) to compare 0.49 and 0.8?
How would you compare 0.45 and 0.451? What about 0.045 and 0.0451? 
Instructional Implications Provide opportunities for the student to compare numbers that contain whole numbers and decimals. Begin with comparing two numbers with the same number of digits such as 1.6 and 1.2 and then transition to comparing two numbers with different numbers of digits following the decimal point such as 1.5 and 1.29.
Provide opportunities for the student to compare decimal numbers up to the thousandths place.
Provide the student with four decimal numbers and have the student write the numbers in order from least to greatest.
Provide the student with the following: 0.45, _____, 0.7, ______. Have the student determine numbers that could be written in the blank spaces.
Encourage the student to order three or more decimals from least to greatest and record the comparison using the less than symbol (e.g., 4.877 < 4.989 < 5.882). 