# Operating with Decimals

Mathematics

Level 5Level 6Level 7 This learning sequence aims to develop student proficiency when adding, subtracting, multiplying and dividing with decimals. A well-developed understanding of place value and number sense is critical for students to flexibly and intuitively operate with decimals. This sequence provides opportunities for students to consolidate their understanding of the continuous nature of the base 10 system and deepen their conceptual understanding about the role of estimation and efficient strategies when adding, subtracting, multiplying and dividing with decimals.

 Big understandings Decimals are an extension of the base-10 system. The base-10 system is a continuous pattern that extends both ways from the decimal point. Well-developed number sense allows us to operate with decimals flexibly and intuitively (including using a calculator). Strategies for whole number estimation and computation are helpful when working with decimals.

The sequence has been written by teachers for teachers. It has been designed to provide students with rich, engaging learning experiences that address the Victorian Curriculum. The sequence consists of four flexible stages, including suggested learning intentions.

Overview of stages

• 1. Decimal Number Sense

Suggested Learning Intentions

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

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• 3. Multiplying and Dividing by Powers of 10

Suggested Learning Intentions

• To understand that scientific notation can be used to describe infinitely large and infinitely small numbers
• To learn how to multiply and divide decimals using knowledge of place value in a base-10 system

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• 2. Adding and Subtracting Decimals

Suggested Learning Intentions

• To solve decimal problems using rounding and estimation
• To use my understanding of place value to solve decimal problems

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• 4. Multiplying and Dividing Decimals

Suggested Learning Intentions

• To recognise the appropriate operation when solving decimal problems in context
• To use my number sense when multiplying and dividing decimal problems

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with terms and concepts of place value, tenths and hundredths (VCMNA159) and the connection between fractions and decimals. Students should also have had experience using intuitive and flexible computation strategies for whole number (VCMNA156), which includes rounding and estimation.

It will be helpful for students to have first explored the concepts contained in the ‘Fractions to decimals’ and ‘Colour in decimats’ stages of the Keeping it in Proportion: Fractions and Decimals sequence, that focus on comparing and ordering decimals (VCMNA187; VCMNA190) as well as decimal place value (VCMNA189).

You can find support for building students’ understanding of these concepts in the Mathematics Curriculum Companion. The Teaching Context and Teaching Ideas related to content descriptions VCMNA159VCMNA189 and VCMNA190 may be particularly useful.

Teaching strategies

The Mathematics Curriculum Companion provides teachers with content knowledge, suggested teaching and learning ideas as well as links to other resources. Resources are organised by Mathematics strands and sub-strands and incorporate the proficiencies: Understanding, Fluency, Problem Solving and Reasoning. The Companion is an additional resource that you could refer to when you are planning how you might use the sequence in your school.

The sequence highlights opportunities to apply the High Impact Teaching Strategies (HITS), which are a component of the Victorian Teaching and Learning Model

The sequence highlights the use of a variety of scaffolding practices to help support students in the learning process, in particular:

• Collaborating
• Convince Me
• Noticing
• Extending

Vocabulary

Students should be able to understand and use the following concepts and terms by the end of the learning sequence:

 Place value Rename Decimals Estimate Tenths Round Hundredths Accurate Thousandths Number of decimal places Partition Remainder

You can find definitions of some of these terms in the F-10 Victorian Curriculum Mathematics Glossary. It is recommended that the explicit teaching of vocabulary occur throughout the learning sequence.

The Literacy in Mathematics section of the Literacy Teaching Toolkit provides several teaching strategies with worked examples demonstrating how teachers can use literacy to support student understanding of mathematical language. A further set of strategies demonstrate how teachers can develop students' literacy skills to support their mathematical problem solving.

Assessment

Opportunities for formative and summative assessment are identified at different stages of the learning sequence. Look for the 'Assessment Opportunity' icon. You may want to develop a rubric to assess students’ progress. A range of Formative Assessment resources are available from the Victorian Curriculum and Assessment Authority. This includes a Guide to Formative Assessment Rubrics, a series of modules to support you to develop your own formative assessment rubrics, and sample rubrics across six curriculum areas that demonstrate how you can put formative assessment rubrics into practice in the classroom.

In developing a rubric, you may wish to co-construct assessment criteria with your students. Each stage of the sequence provides sample success criteria for students working at Level 6.

The Victorian Curriculum and Assessment Authority has published annotated work samples that provide teachers with examples of student learning achievement at multiple levels for each strand of the Mathematics curriculum.

Victorian Curriculum connections

Contents

Level 5

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 6, but also addresses the following content descriptions from Level 5:

 Content description Stage Mathematics Recognise that the place value system can be extended beyond hundredths (VCMNA189) Decimal Number Sense Compare, order and represent decimals (VCMNA190) Decimal Number Sense Use estimation and rounding to check the reasonableness of answers to calculations (VCMNA182) Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Use efficient mental and written strategies and apply appropriate digital technologies to solve problems (VCMNA185) Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Creative and Critical Thinking Investigate common reasoning errors including contradiction and inconsistency, and the influence of context (VCCCTR024) Decimal Number Sense Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Consider the importance of giving reasons and evidence and how the strength of these can be evaluated (VCCCTR025) Decimal Number Sense Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 5:

• Students solve simple problems involving the four operations using a range of strategies including digital technology
• Students estimate to check the reasonableness of answers and approximate answers by rounding
• Students order decimals and unit fractions and locate them on a number line
• Students continue patterns by adding or subtracting fractions and decimals.

Level 6

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 6, addressing the following content descriptions:

 Content description Stage Mathematics Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (VCMNA214) Adding and Subtracting Decimals Multiply decimals by whole numbers and perform divisions by non-zero whole numbers where the results are terminating decimals, with and without digital technologies (VCMNA215) Multiplying and Dividing Decimals Multiply and divide decimals by powers of 10 (VCMNA216) Multiplying and Dividing Decimals Select and apply efficient mental and written strategies and appropriate digital technologies to solve problems involving all four operations with whole numbers and make estimates for these computations (VCMNA209) Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Make connections between equivalent fractions, decimals and percentages (VCMNA217) Decimal Number Sense Multiplying and Dividing Decimals Creative and Critical Thinking Investigate common reasoning errors including contradiction and inconsistency, and the influence of context (VCCCTR024) Decimal Number Sense Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Consider the importance of giving reasons and evidence and how the strength of these can be evaluated (VCCCTR025) Decimal Number Sense Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 6:

• Students make connections between the powers of 10 and the multiplication and division of decimals
• Students add, subtract and multiply decimals and divide decimals where the result is rational.

Level 7

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 6, but also addresses the following content descriptions from Level 7:

 Content description Stage Mathematics Apply the associative, commutative and distributive laws to aid mental and written computation and make estimates for these computations (VCMNA240) Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Multiply and divide fractions and decimals using efficient written strategies and digital technologies (VCMNA244) Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals Creative and Critical Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Decimal Number Sense Adding and Subtracting Decimals Multiplying and Dividing by Powers of 10 Multiplying and Dividing Decimals

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 7:

• Students solve problems involving all four operations with fractions, decimals, percentages and their equivalences
• Students make simple estimates to judge the reasonableness of results.

Learning Progressions

The Numeracy Learning Progressions support teachers to develop a comprehensive view of how numeracy develops over time. You can use the numeracy learning progressions to:

• identify the numeracy capability of your students
• plan targeted teaching strategies, especially for students achieving above or below the age-equivalent expected level in the Victorian Curriculum: Mathematics
• provide targeted feedback to students about their learning within and across the progressions.

The sequence is related to the following progressions:

 Learning Progression Level 5 Level 6 Level 7 Quantifying numbers Understanding decimal place value Understanding decimal place value Representing place value Operating with decimals Understanding and estimating relative size of decimals Understanding the effects of multiplication and division with decimals Understanding the effects of multiplication and division with decimals

Click on the Learning Progression to access more detailed descriptions of student learning at each level.