Comparing Decimal Numbers: 4.0

• Indicator of Progress
• Teaching Strategies
• Further Resouces

Supporting materials

• Related Progression Points
• Developmental Overview of Numeration: Base Ten and Place Value Properties (PDF - 28Kb)

Indicator of Progress

Success depends on students realising that it is the digits with the highest place value that contribute the most to the size of a number. This is especially difficult for decimal numbers.

Young students often think that the size of a decimal number is related to the length of the decimal, because they draw false analogies with whole numbers. For example, they might think that "1576 is larger than 742 as it is contains more digits" and so conclude that "8.1576 is bigger than 8.742 as it is longer". Understanding decimal numbers requires a deeper understanding of place value.

 

Illustration 1: Why comparison is a revealing task

The activities below use the task of comparing decimal numbers of different length to reveal students' misconceptions about place value. Whilst the ability to compare numbers is of importance in itself and is closely connected to number sense, the comparing tasks used here are a way to highlight fundamental aspects of place value and make them an object of class discussion.

The process of comparing decimal numbers is actually exactly the same process for comparing whole numbers, although students might not appreciate this to be the case. Instead of focusing on place value, students might be distracted by the 'length' of the whole numbers, and hence base their perception of the relative size of whole numbers on the relative number of digits involved. For example, they might say that "1576 is larger than 742 as it is contains more digits" or "1576 is bigger than 742 as it is longer". While this strategy will provide correct answers for whole numbers, this lack of focus on place value is likely to lead to similar (but incorrect) strategies for comparing decimal numbers. For example, some students think that long decimals are usually large numbers, and surprisingly, some think that long decimals are usually small numbers.

The fact that these incorrect strategies sometimes result in correct answers leads teachers to over-estimate the student's understanding of the size of decimal numbers. It is important that the teacher provides examples that will reveal a student's incorrect strategy so that this student will appreciate that they do, in fact, have something more to learn.

 

Illustration 2: Longer-is-larger students

Many students beginning to learn about decimals have longer-is-larger misconceptions.

Students who say that "1576 is larger than 742 as it is contains more digits" or "1576 is bigger than 742 as it is longer" are likely to say that "4.11 is larger than 4.9 as it is longer". While this is incorrect, it is a very understandable belief, as they are treating the digits after the decimal point as a separate number. We often use a point to separate a pair of whole numbers. For example, students might note that an Australian Rules football team with 4 goals and 9 behinds, which then scores another 2 behinds, now has 4 goals and 11 behinds. The difficulty is when this is recorded as a pair of whole numbers separated by a dot: 4.9, 4.10, 4.11, which now look like decimal numbers. This clearly supports the idea that 4.11 is larger than 4.9, and hence supports the incorrect generalisation that 'longer numbers are larger'. Inspect any textbook with numbered sections (or an agenda of a long meeting) and you will find that after section 4.9 (perhaps the 9th section in Chapter 4) comes section 4.10 then 4.11, etc. Students without a good understanding of place value are surprised when the teacher states that 4.11 is smaller than 4.9, or that the result of adding 4.9 and 0.2 is 5.1 (when they are expecting 4.11). Research indicates that the majority of students make this error when first encountering decimal notation; the concern is that some students retain this misconception of decimal numbers for many years.

Note that these students are likely to order these numbers correctly: 0.34, 0.35, 0.36, 0.37, as they have the same number of digits after the decimal points. In order to identify these students, we need to ask them to order numbers involving different numbers of digits after the decimal points (called "ragged decimals") as the correct answer cannot be obtained by a student who is treating the digits after the decimal point as another whole number. For example, if we ask them to order this set of numbers: 0.3, 0.4, 0.38, 0.39, 0.312, they might write 0.3, 0.4, 0.38, 0.39, 0.312 (instead of the correct order: 0.3, 0.312, 0.38, 0.39, 0.4).

Counting sequences - here you can see how a longer-is-larger student would place numbers between 1 and 2.

Illustration 3: Shorter-is-larger students

Another incorrect strategy that students use to order decimal numbers, is to choose the 'shorter' number (i.e. fewer digits) when asked to choose the larger number. The students incorrectly believe, for example, that 0.4 is larger than 0.87. There are various reasons. Some of them confuse decimals with fractions and argue that "if we share the cake among 4 people we get more than if we share it among 87", effectively treating 0.4 as 1/4, and 0.87 as 1/87. Note that it is desirable that students do link fractions and decimal numbers, but this has not been done correctly by these students.

Other students also incorrectly believe that 0.4 is larger than 0.87, but for a different reason. They argue that "tenths are bigger than hundredths, so 0.4 is bigger than 0.87". This incorrect statement is based on a partial truth "one tenth is bigger than one hundredth". Such students are also likely to incorrectly order fractions with different denominators like this: 1/8, 2/8, 3/8, 5/8, 1/4, 2/4, 3/4 (i.e. firstly the eighths, then the fourths).

As above, we can find these students by providing 'ragged decimals' to order. They might write 0.312, 0.38, 0.39, 0.3, 0.4 (or possibly 0.312, 0.39, 0.38, 0.4, 0.3) instead of the correct order (0.3, 0.312, 0.38, 0.39, 0.4).

Shorter-is-larger misconception remain prevalent amongst older students, whereas longer-is-larger misconceptions tend to be dispelled by later experiences.

Counting sequences - here you can see how a shorter-is-larger student would place numbers between 1 and 2.

Teaching Strategies

The teaching strategies here are to use a diagnostic test so that instruction can be targetted where it is needed, to use open-ended tasks so that misconceptions can be revealed, discussed and eliminated, and to support instruction on decimals by a concrete model which is easy for students to understand.

Activity 1: Diagnostic test takes students only a few minutes to complete and provides useful information to teachers about likely misconceptions regarding decimal numbers.
Activity 2 and Activity 3 are open-ended tasks that require students to compare decimal numbers with different numbers of digits after the decimal point. They should be encouraged to find more than one solution for each task, because it is in the pattern of solutions that they start to see the importance of place value in determining the size of numbers.
Activity 4: Linear Arithmetic Blocks provides instructions for a simple concrete model. Students need access to a good physical model that will assist them to develop their place value understanding and hence construct meaning for decimal numbers. The model can also be used to explain arithmetic operations on decimals.

Many more activities are available on Teaching and Learning about Decimals (http://extranet.edfac.unimelb.edu.au/DSME/decimals).

 

Activity 1: Diagnostic Test

The test will determine which students are longer-is-larger thinkers and which are shorter-is-larger thinkers. Print the Decimal Comparison Test (PDF - 3Kb) which asks students to choose the larger decimal number from each pair. This test can be completed by students in about 10 minutes. Print the Decimal Classification (PDF - 9Kb) sheet to see if any of your students match the samples provided. Alternatively, enrol to use the SMART::tests diagnostic system (at www.smartvic.com or for teachers at Victorian Catholic schools at https://www.smart-quiz.edu.au/teacher/). The quiz Understanding decimals is an online version of the Decimal Comparison Test.

The test will determine which students are longer-is-larger thinkers and which are shorter-is-larger thinkers. Note that it is the pattern of choices throughout the test (rather than an overall score) that is important for diagnosing students who are using either of the incorrect strategies (longer-is-larger or shorter-is-larger) mentioned above. In other words, we can detect students who do not understand the place value properties of decimal numbers.

Once misconceptions have been identified, teachers can make sure their instruction and examples address the students' problems. The first important idea to note is that the above incorrect strategies reveal that the students do not understand decimal numeration; they are resorting to these incorrect strategies as they do not have a full place value understanding on which to base their choices. They are unable to say, for example: "0.38 is less than 0.4, as 0.38 is 3 tenths and 8 hundredths which is less than 4 tenths". While, at first, it seems reasonable to quickly 'solve the problem' by providing students with a rule or algorithm to follow (eg 'write zeros at the end and then compare as whole numbers', or 'compare from left-to-right'), it is clear that giving students rules to follow is not a substitute for teaching for understanding. Indeed, while there are likely to be gains in the short-term, in the long-term students tend to forget rules that are not supported by understanding.

NOTE: It is possible for students to choose the correct answers on the test if they correctly use a rule. The activities below are useful to determine which of these students who can correctly compare numbers really have a good understanding of place value.

Activity 2: Decimal numbers between

Open-ended task

Ask students to write down 15 numbers between 3.1 and 3.4

Some students will claim that there are only two numbers 3.2 and 3.3, while others will appreciate that 3.18 is between 3.1 and 3.2. Encourage students to share their answers and to explain why their numbers are between 3.1 and 3.4, using models and diagrams to support their explanation. It is essential that students have a model to refer to which will resolve any disputes. Linear Arithmetic Blocks (LAB) is a model which has proven to be very useful in assisting students to improve their conceptual understanding of decimal notation. (Instructions are provided for making this model from washers and tubing which are inexpensive to buy and available from any hardware store.)

Number Between: Classroom Game and Computer Game

Number Between is a good game and is also available as a computer game for individuals and pairs (see references).

• The teacher writes a pair of numbers far apart on the board (smallest on the left) and calls on a student to write a number in between the pair.
• If a correct answer is given (it does not have to be the midpoint), another student is called on to write a number between the new number and one of the earlier endpoints. A dice could be thrown to decide whether it is to be larger or smaller than the new value, or you could have a simple rule - eg alternate between the larger and smaller sides, always go for the smaller etc.
• Continue as the number line is divided into smaller and smaller segments. Stop when interest wains.
• Dividing the students into teams, to take turns writing a number and to challenge the correctness of the answers of the other teams, creates a competition if desired.

 

Activity 3: Less than - more than

Ask students to fill the boxes with some of the digits 0, 1, 2, ...8, 9 to make the following true. The digits do not have to be the same and can be reused.

Task 1

Task 2

These are open-ended tasks with many correct answers. Encourage students to describe the range of answers in general terms. For example, in the first case, if the digit in the ones place (in the number on the left) is 0, 1 or 2, then the digits in the next two places (tenths and hundredths) can be anything (because it is always true that 0.XX < 3.X, and 1.XX < 3.X, and 2.XX < 3.X). However, if the digit in the ones place (in the number on the left) is 3, then the digits in the tenths place need to be considered carefully; the tenths digit on the left needs to be smaller than the tenths digit on the right.

It is important to review the students' answers carefully, as those students using incorrect strategies such as longer-is-larger or shorter-is-larger, need to know that they have something to learn.

 

Activity 4: Linear Arithmetic Blocks

Linear Arithmetic Blocks are a simple model for decimal numbers, which can be purchased commercially or made from simple materials. The size of a decimal number is modelled by length, which is conceptually simpler than other material such as multi-base arithmetic blocks (MAB) which represent size by volume.

Linear Arithmetic Blocks can be used to compare numbers, as well as to demonstrate arithmetic operations in a similar way to MAB.

 

References and Acknowledgements

The descriptions of LAB and the diagnostic test are published with permission from Steinle, V., Stacey, K. & Chambers, D. (2006) Teaching and Learning about Decimals. (Version 3.1 ) Faculty of Education, University of Melbourne . (CD-ROM). This CD contains many lesson ideas for students with identified misconceptions, as well as several interactive computer games for decimals, including Number Between. See Teaching and learning about decimals (http://extranet.edfac.unimelb.edu.au/DSME/decimals)

Tromp, C. (1999) Number Between: making a game of decimal numbers. Australian Primary Mathematics Classroom, 4(3), 9 - 11. Tromp (1999) describes how he used this game in his classroom.

Further Resources

The following resource contains sections that may be useful when designing learning experiences:

Digilearn object *

Hopper Challenge: ultimate (https://www.eduweb.vic.gov.au/dlr/_layouts/dlr/Details.aspx?ID=4877) – Students help a frog to jump along a number line. Students estimate the exact finishing point on a number line, after adding or subtracting multiples of tenths or whole numbers to a starting number. For example, 29.5+(12 x 0.2) = 31.9.

* Note that Digilearn is a secure site; SMART::tests login required.