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 decimals using place value and explains 0.34 is greater than 0.4 because it is longer (has more digits).
Despite prompting, the student is not able to use a model to assist with comparing the decimals. |
| Questions Eliciting Thinking How would you write 0.9 as a fraction? How many hundredths are in nine tenths? Is 9/10 equivalent to 90/100? So how does 0.9 compare with 0.90?
What is the value of the digit three in 0.34? What is the value of the digit four in 0.4?
How could you align the numbers to help you compare?
Which digit(s) did you look at to determine which number was greater? Are these digits both in the same place value?
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 represent 0.34 and 0.4 on 10 x 10 grids? |
| 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.
Consider using the MFAS task Using Models to Compare Decimals to determine if the student can compare decimals by using visual models. |
Moving Forward |
| Misconception/Error The student can only correctly compare decimals that contain the same number of digits. |
| Examples of Student Work at this Level The student correctly compares 0.53 and 0.52. However, he or she has difficulty comparing when one number only has digits in the tenths place and the other number has digits through the hundredths place because he or she is applying whole number concepts to decimal concepts. The student says that 0.9 < 0.90.
The student initially says that 0.34 > 0.4, 0.9 < 0.90, 0.07 > 0.6, and 0.53 > 0.52. With minimal prompting, the student changes his or her answers to 0.9 = 0.90 and 0.07 < 0.6. However, when asked if he or she correctly compared 0.34 and 0.4, the student says, “Yes” and does not change his or her thinking. |
| Questions Eliciting Thinking What is your strategy for comparing the decimal numbers?
How would we write 0.9 as a fraction? How many hundredths are in nine tenths? How does 0.90 compare with 0.9?
What does one square represent on a 10 x 10 grid? What does one column represent?
Are 0.34 and 0.4 comparing the same size whole?
Can you locate the decimals on a number line? |
| Instructional Implications Review writing decimals in equivalent forms (e.g., explain that 0.6 and 0.60 are equivalent since six out of 10 is the same as 60 out of 100 when referring to parts of the same whole). Guide the student to compare 0.6 to 0.07 by drawing appropriate models of each. It may also be helpful to relate the decimals to money. For example, when comparing 0.6 and 0.07, the student can say that 0.6 is the same as sixty cents and 0.07 is the same as seven cents. Guide the student to observe that 0.07 is less than 0.6 since seven cents is less than sixty 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 “seven tenths” rather than as “zero-point-seven”). |
Almost There |
| Misconception/Error The student is unable to justify decimal comparisons appropriately using place value. |
| Examples of Student Work at this Level The student says that 0.34 < 0.4 because four is greater than three (instead of saying four tenths is greater than three tenths). |
| Questions Eliciting Thinking What is the value of the three in 0.34?
How can a decimal written in tenths be written as an equivalent decimal in hundredths?
How would you represent 0.34 and 0.4 on 10 x 10 grids? What does one square represent? What does one column represent?
Do 0.34 and 0.4 refer to the same size whole?
How many hundredths is 0.4Â equal to? |
| Instructional Implications Guide the student to interpret 0.34 as 34 hundredths and to interpret 0.4 in its equivalent form, 40 hundredths. Model explaining that 0.34 is less than 0.4 because 34 hundredths is less than 40 hundredths. Encourage the student to provide place value explanations when comparing decimals.
Partner the student with a “Moving Forward” student. Have the student model for the “Moving Forward” 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 correctly makes each comparison and explains his or her reasoning. The student writes:
- 0.34 < 0.4 and explains four tenths is greater than three tenths or 40 hundredths is greater than 34 hundredths.
- 0.9 = 0.90 and explains
 is equivalent to  .
- 0.07 < 0.6 and explains seven hundredths is less than sixty hundredths.
- 0.53 > 0.52 and explains 53 hundredths is greater than 52 hundredths.Â
|
| 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 transition to comparing two decimals 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).
Consider using the MFAS task Comparing Decimals in Context. |
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