Comparing Decimals I

The Comparing Decimals I assessment is designed to elicit information about a common misconception that students have when comparing two decimal numbers:

  • ●   Misconception 1 (M1): Using Whole-Number Thinking / A Focus on "Longer Is Larger"

We strongly recommend that you reference the information below to learn more about this misconception, including how it appears in student work, and how to score pre- and post-assessments once you have given them to students.

Topic Background: Learn about comparing decimals

There are multiple research-based misconceptions related to comparing decimals, but this set of diagnostic assessments focuses on one in particular: overgeneralizing from experiences with whole-number comparisons when comparing the digits to the right of the decimal point. Because students are accustomed to thinking of a number with more digits as the larger number, they extend this rule to decimals; they compare the decimal numbers according to how many digits appear to the right of the decimal point and assume that the decimal number with more digits is larger.


Students typically do not apply this thinking when given numbers with different digits in the ones place, such as comparing 2.36 and 5.1. Instead, they tend to appropriately compare the values of the digits in the ones place, in this case reasoning that since 5 is greater than 2, 5.1 is greater than 2.36.


Connections to Common Core Standards in Mathematics (CCSS)


The CCSS outline specific understandings that students should be able to meet at each grade level.


Grade 4


At grade 4, students should be able to do the following:


4.NF. Understand decimal notation for fractions, and compare decimal fractions.


  • Express a fraction with denominator 10 as an equivalent fraction with denominator 100, and use this technique to add two fractions with respective denominators 10 and 100. For example, express 3/10 as 30/100, and add 3/10 + 4/100 = 34/100.
  • Use decimal notation for fractions with denominators 10 or 100. For example, rewrite 0.62 as 62/100; describe a length as 0.62 meters; locate 0.62 on a number line diagram.
  • Compare two decimals to hundredths by reasoning about their size. Recognize that comparisons are valid only when the two decimals refer to the same whole. Record the results of comparisons with the symbols >, =, or <, and justify the conclusions, e.g., by using a visual model.

  • Grade 5


    At grade 5, students should be able to do the following:


    5.NBT. Understand the place value system.


  • Recognize that in a multi-digit number, a digit in one place represents 10 times as much as it represents in the place to its right and 1/10 of what it represents in the place to its left.
  • Read, write, and compare decimals to thousandths.
  • Read and write decimals to thousandths using base-ten numerals, number names, and expanded form, e.g., 347.392 = 3 × 100 + 4 × 10 + 7 × 1 + 3 × (1/10) + 9 × (1/100) + 2 × (1/1,000).
  • Compare two decimals to thousandths based on meanings of the digits in each place, using >, =, and < symbols to record the results of comparisons.
  • Student Misconceptions: Learn about student misconceptions related to the topic

    When students are developing the understandings described above (see Topic Background), they can develop flawed understanding leading to misconceptions about how to compare decimals.


    The following common misconception when comparing decimals is targeted in the Comparing Decimals I assessments:


    Misconception 1: Using Whole-Number Thinking / A Focus on “Longer Is Larger”


    Students with this misconception consistently compare decimals by comparing the numbers to the right of the decimal point as if they were comparing whole numbers (e.g., they consider 0.34 to be greater than 0.8 because 34 is greater than 8). Because they are accustomed to thinking of numbers with more digits as larger numbers, they overgeneralize from their experiences with whole-number comparisons and extend this rule to decimals.

    Watch a Video explaining Misconception 1


    To see additional examples of student work illustrating this misconception, go to the “Sample Student Responses” tab on this page, and click on the button to download a PDF.


    Resources


    The Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft): 3–5, Number and Operations—Fractions. Retrieved from http://ime.math.arizona.edu/progressions/#products


    The Common Core Standards Writing Team (2011). Progressions for the Common Core State Standards in Mathematics (draft): K–5, Number and Operations in Base Ten. Retrieved from http://ime.math.arizona.edu/progressions/#products


    Irwin, K. (1996). Making sense of decimals. In J. Mulligan & M. Mitchelmore (Eds.), Children’s Number Learning (pp. 243–257). Adelaide, Australia: MERGA & AAMT.


    Siegler, R., Carpenter, T., Fennell, F., Geary, D., Lewis, J., Okamoto, Y., Thompson, L., & Wray, J. (2010). Developing Effective Fractions Instruction for Kindergarten Through 8th Grade: A Practice Guide (NCEE #2010-4039). Washington, DC: National Center for Education Evaluation and Regional Assistance, Institute of Education Sciences, U.S. Department of Education.


    Steinle, V., & Stacey, K. (2004). Persistence of Decimal Misconceptions and Readiness to Move to Expertise. In M. Johnsen Hoines & A. Berit Fuglestad (Eds.), Proceedings of the 28th Conference of the International Group for the Psychology of Mathematics Education—PME 28, 4(1), 225–232. Bergen, Norway: Bergen University College.

    Administering Pre-Assessment: Learn how to introduce the pre-assessment to your students

    About This Assessment


    These EM2 diagnostic, formative pre- and post-assessments are composed of items with specific attributes associated with student conceptions that are specific to comparing decimals. Each item within any EM2 assessment includes a selected response (multiple choice) and an explanation component. Each item has three response choices: Greater than (>), Less than (<), and Equivalent (=). The symbol is provided with the text to provide broader accessibility.


    The learning target for the Comparing Decimals I assessment is as follows:


    The learner will accurately compare decimals to identify which is larger and which is smaller.


    Prior to Giving the Pre-Assessment


    The learner will accurately compare decimals to identify which is larger and which is smaller.


  • Arrange for 15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the pre-assessment is designed to elicit misconceptions before instruction, you do not need to do any special review of this topic before administering the assessment. (See the "Research-Based Misconceptions" tab for information and a video that describes this misconception.)

  • Prior to Giving the Pre-Assessment

  • Inform students about the assessment by reading the following:

  • Today you will complete a short individual activity, which is designed to help me understand how you think about comparing decimals.


  • Distribute the assessment and read the following:

  • The activity includes five problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about comparing decimals, it’s important for you to include some kind of explanation in the space provided. This can be a picture, words, a combination of pictures and words, or something else that shows how you chose your answer.


    You will have about 15 minutes to complete all the problems. When you are finished please place the paper on your desk and quietly [read, work on ____] until everyone is finished.


  • Monitor the students as they work on the assessment, making sure that they understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 15-minute timeframe. Students may have more time if needed. When a few minutes remain, say:

  • You have a few minutes left to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today.


  • Collect the assessments.

  • Click on the button to download a PDF of the Pre-Assessment Administration Process.


    Pre-Assessment Administration Process


    After Administering the Pre-Assessment


    Use the analysis process (found in the Scoring Guide PDF document under the “Scoring Process” tab) to analyze whether your students have this misconception:


  • Misconception 1 (M1): Using Whole-Number Thinking / A Focus on “Longer Is Larger”
  • Scoring: Learn about the scoring process by reviewing the Scoring Guide

    The Comparing Decimals I assessment is composed of five items with specific attributes associated with a misconception that is directly related to comparing decimals. We encourage you to carefully read the Scoring Guide to understand these specific attributes and to find information about analyzing your students’ responses.


    Click on the button to download a PDF of the Scoring Guide.


    Scoring Guide

    Sample Student Responses: Review examples of student responses to assessment items





    To determine the degree of understanding and misunderstanding in the student work, it’s important to consider both the answer to the selected response and the explanation text and representations. The example above is one of many student work samples that provide insight into student thinking about the misconception targeted in these diagnostic assessments (see the “Research-Based Misconceptions” tab for more information and a video about this misconception).


    We encourage you to look at the collection of student work examples provided here. Click on the button to download a PDF.


    Student Work Samples

    Administer Post-Assessment: Learn how to introduce the post-assessment to your students

    If the Comparing Decimals I pre-assessment shows that any of your students have the misconception outlined in the Scoring Guide, plan and implement instructional activities designed to increase students’ understanding. The post-assessment provided here can then be used to determine if the misconception has been addressed.


    Prior to Giving the Post-Assessment


  • Arrange for 15 minutes of class time to complete the administration process, including discussing instructions and student work time. Since the post-assessment is designed to elicit a particular misconception after instruction, you should avoid using or reviewing items from the post-assessment before administering it.

  • Administering the Post-Assessment


  • Inform the students about the assessment by reading the following:

  • Today you will complete a short individual activity, which is designed to help me understand how you think about comparing decimals, a topic we have been working on in class.


  • Distribute the assessment and read the following:

  • This activity includes five problems. For each problem, choose your answer by completely filling in the circle to show which answer you think is correct. Because the goal of the activity is to learn more about how you think about comparing decimals, it’s important for you to include some kind of explanation in the space provided. This can be a picture, words, a combination of pictures and words, or something else that shows how you chose your answer.


    You will have about 15 minutes to complete all the problems. When you are finished, please place the paper on your desk and quietly [read, work on ____] until everyone is finished.


  • Monitor the students as they work on the assessment, making sure that they understand the directions. Although this is not a strictly timed assessment, it is designed to be completed within a 15-minute timeframe. Students may have more time if needed. When a few minutes remain, say:

  • You have a few minutes to finish the activity. Please use this time to make sure that all of your answers are as complete as possible. When you are done, please place the paper face down on your desk. Thank you for working on this activity today.


  • Collect the assessments.

  • Post-Assessment Administration Process


    After Administering the Post-Assessment


    Use the analysis process (found in the Scoring Guide PDF document under the “Scoring Process” tab) to analyze whether your students have this misconception:


  • Misconception 1 (M1): Using Whole-Number Thinking / A Focus on “Longer Is Larger”

  • Some students who previously had the misconception will no longer have it—the ideal case. Consider your instructional next steps for those students who still show evidence of the misconception.