Operating with Decimals

# 1. Decimal Number Sense

Suggested Learning Intentions

• To understand decimal place value to the thousandths
• To understand the relative size of decimals

Sample Success Criteria

• I can describe decimal place value to thousandths
• I can explain and justify my solutions using a variety of manipulatives such as place value blocks, counters or decimats
• I can use my knowledge of place value to compare, order and round decimals
• Decimat: docx PDF
• Decimal comparison template: docx PDF
• Decimal human number line cards: docx PDF
• Linear Arithmetic Blocks (LAB) blocks or Decipipes
• Athletics Day Shot Put worksheet: docx PDF
• Decimal concept cartoon - decimal misconceptions: cartoon 1 cartoon 2

View the Comparing and Ordering Decimals clip in ClickView. Sign into ClickView using your department credentials.

Display a decimal number in the hundredths such as 31.75. Facilitate a number talk about what students know about the number. Record key points of the discussion on an anchor chart.

Use discussion prompts such as:

• Tell me everything you know about…
• How do we read this? For example, do students use language of fractions such as "31 and 75 hundredths", or "31 and 7 tenths and 5 hundredths"?
• How can we describe this number using place value?
• Is there another way to represent this number (e.g., using fractions and percentages)?
• Re-write this number as 31.750 and focus students' attention on the zero. Do we need to include the zero in this number? Why/Why not?
• What connections can we make to fractions and percentages?

Add another digit in the thousandths place. For example, 31.752. Monitor whether students think about thousandths.

Show the students an enlarged decimat or an online version. Revisit decimats with students and talk about how they can be used as a visual representation of tenths, hundredths, and thousandths.

For more information on decimats, please see ‘Colour in Decimats’ in the Level 5 Keeping it in Proportion: Fractions and Decimals sequence. Alternatively, visit AMSI Calculate Decimats.

Decimal Comparison

Display pairs of decimals and invite students to justify which number is larger and why. Encourage students to use the decimal comparison template (available in Materials and Texts) to record their reasoning in words as well as ‘less than' and ‘greater than' notation.

 less than < greater than >

Use pairs of decimals such as:

• 0.3 and 0.45
• 0.04 and 0.104
• 0.3079 and 0.370

Alternatively, students roll two or three 10-sided dice to create two decimal numbers of unequal length (for example, a 2-digit and a 3-digit decimal number) by arranging the digits after the decimal point. Compare the two numbers created.

Encourage students to use the language of fractions and place value when describing decimals by moving to the notion of tenths to hundredths to thousandths, using place value concepts and renaming.

Enable students by using decimats to represent then compare decimal pairs.

Extend students by inviting them to prepare a set of ‘tricky’ decimal pairs for a friend to compare. They may go up to six decimal places. Monitor what aspects of decimal place value students attend to when designing their own sets. Alternatively, have students compare three or four decimal numbers of unequal length.

Task: More than - less than

This task is sourced from the Maths Curriculum Companion Teaching Ideas

Ask students to fill the boxes with the digits 0-9, to make the following statements below true. The digits do not have to be the same and they can be reused.

As this is an open-ended task, students will have many different answers. Encourage students to describe the range of answers in general terms. For example, in the first statement, the digit in the 'ones' place has to be either a 0, 1, or 2 in order for this to be true. Then the next two places can be anything. However, if the student selects a 3 in the 'ones' place, then they need to think carefully about the tenths digit.

Encourage students to represent these statements using decimats, Linear Arithmetic Blocks (LAB) blocks or Decipipes. Further reading about LAB blocks can be found here and Decipipes can be found here.

Comparing and ordering decimals

Distribute the decimal human number line cards to 10 volunteers. Students imagine a number line which starts at zero. Individually, invite students to come up and stand where they think appropriate, in order, starting at 0. Students need to agree/disagree with where the individual is standing.

Next, present the following task to the students, which they attempt on their own before sharing ideas with others.

On Athletics Day, many students participated in shot put. Dante has the school record for shot put with a personal best of 9.120m. In 10th place, was a put of length 6.271m. 1st, 2nd and 3rd came very close. What might have been the results of the students who came 2nd to 9th? Record your solutions using a mixture of numbers with one, two, and three decimal places.

Enable students requiring further support by asking students: "what are four possible numbers that can come between 6.0 and 9.5?"

Extend students by asking them to convince you that their 1st, 2nd and 3rd places are ‘close’. What is the closest they could be? How do you know? Ask students why official field athletics scores are often recorded to two decimals. Should they be recorded to one or three decimal places instead? Why/why not?

Areas for further exploration

The preceding tasks have encouraged students to explore decimal place value to thousandths with a focus on comparing and ordering decimals. To consolidate student understanding you may wish to review these concepts and skills with your students.

1. Compare Decimals

Compare Decimals is an online timed game where students compare decimals, and draw the correct symbol, either greater than or less than, between the two decimal numbers presented. The pairs present common misconceptions and errors.

2. Who’s the tallest

Who's the tallest requires students to measure their height to the nearest millimetre (to 3 decimal places) and write their result on the whiteboard. They will then order these decimals and place them on a number line or piece of string. Have students explain how they determined each position that is, how did they partition the number line?

Discuss which digits determine the approximate height of each individual. Ask questions such as: is your height closer to 1.5m or 1.6m? How do you know? How many students are between the lengths 1.2 and 1.4? Estimate the average height of all students. How might you do this without calculating it? Then calculate it and see how close your estimate was.

3. Decimal misconception concept cartoons

The size of decimals can create some confusion for students particularly when misconceptions are not addressed adequately. The following concept cartoons provide opportunities to address misconceptions in a more contextual way with students. More information about concept cartoons can be accessed here.

Share and discuss responses with the class. This will address the issue of thinking of decimals as whole numbers (also known as “longer is larger”) and ignoring the decimal point when comparing.

AMSI, 2017. Decimats. [Online]
Available at: https://calculate.org.au/2017/12/05/decimats/
[Accessed 15 March 2022].

Clarke, D. & Roche, A., 2014. Engaging Maths: 25 Favourite Lessons. 2nd Edition ed. Melbourne: Mathematics Teaching and Learning Centre.

Davidson, S. & M, A., 2012. Concept Cartoons as a Way to Elicit Understandings and Encourage Reasoning about Decimals in Year 7. In: J. Dindyal, L. P. Cheng & S. F. Ng, eds. Mathematics education: Expanding horizons (Proceedings of the 35th annual conference of the Mathematics Education Research Group of Australasia). Singapore: MERGA.

Moody, B., 2001. Decipies: Helping students to 'get to the point'. Australian Primary Mathematics Classroom, 16(1), pp. 10-15.

New Zealand Government, n.d. Up and Down with Decimals. [Online]
Available at: nzmaths.co.nz/content/and-down-decimals
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Compare Decimals. [Online]
Available at: https://fuse.education.vic.gov.au/Resource/LandingPage?ObjectId=02b50bcd-8f75-4eb0-85dd-aacb49ab3e35
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Maths Curriculum Companion: More than less than. [Online]
Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA190#tab5
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Maths Curriculum Companion: Teaching ideas. [Online]
Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA190
[Accessed 15 March 2022].

State Government of Victoria (Department of Education and Training), n.d. Maths Curriculum Companion: Who's the tallest. [Online]
Available at: https://fuse.education.vic.gov.au/MCC/CurriculumItem?code=VCMNA190#tab2
[Accessed 15 March 2022].

University of Melbourne, 2014. Teaching and learning about decimals: Linear Arithmetic Blocks (LAB). [Online]
Available at: https://extranet.education.unimelb.edu.au/SME/TNMY/Decimals/Decimals/
[Accessed 15 March 2022].