# Keeping it in Proportion: Fractions and Decimals

Mathematics

Level 4Level 5Level 6

This learning sequence aims to develop student understanding of fractions and decimals. Fractions underpin the development of proportional reasoning and are important for future mathematics study, including that of algebra and probability (Clarke, Roche, and Mitchell, 2011). This sequence aims to equip students with the skills required to compare and order fractions and decimals and to represent fractions in decimal form, and vice versa.

 Big understandings Fractions are numbers that lie between other numbers, including whole numbers and other fractions. Decimals are another way to represent fractions.

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

There is a strong focus within this sequence on supporting students to develop the four mathematical proficiencies set out in the curriculum: Understanding, Fluency, Problem Solving and Reasoning, as well as their capacity for Critical and Creative Thinking.

Overview of stages

• 1. Fractions as Equal Parts of a Whole

Suggested Learning Intentions

• To understand that fractions can represent equal parts of a whole

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• 3. Fractions to Decimals

Suggested Learning Intentions

• To estimate and locate fractions and decimals on a number line
• To understand fractions and decimals as different representations of the same number

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• 5. Colour in Decimats

Suggested Learning Intentions

• To understand place value in decimals
• To understand how place value determines the size of a decimal

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• 2. Chocolate Cake

Suggested Learning Intentions

• To understand fractions result from division
• To compare fractions, noting how the numerator and denominator affect the size of a fraction

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• 4. Colour in Fractions

Suggested Learning Intentions

• To understand two fractions can be the same size but have different names

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with the terms numerator and denominator, common unit fractions such as ¹/₂, ¹/₃, ¹/₄, ¹/₅ and how to partition areas, lengths and sets to illustrate common unit fractions.

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 VCMNA157, VCMNA159, VCMNA187, VCMNA188, VCMNA189, VCMNA190, and VCMNA211 may be particularly useful.

Teaching strategies

The Mathematics Curriculum Companion is an additional resource that you could refer to when you are planning how you might use the sequence in your school. The 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.

In addition, the Literacy in Mathematics section of the Literacy Teaching Toolkit contains an extensive range of strategies for teaching the literacy demands of mathematics.

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
• Modelling
• Guiding
• Orienting

Vocabulary

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

 Fraction Thousandths Decimal Numerator Equivalent Denominator Tenths Number line Hundredths Estimate

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 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 also published annotated student work samples that provide teachers with examples of student learning achievement in each strand of the Mathematics curriculum.

Victorian Curriculum connections

Contents

Level 4

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 5, but also addresses the following content descriptions from Level 4:

 Content description Stage Mathematics Investigate equivalent fractions used in contexts (VCMNA157) Fractions as Equal Parts of a Whole Chocolate Cake Colour in Fractions Recognise that the place value system can be extended to tenths and hundredths. Make connections between fractions and decimal notation (VCMNA159) Fractions to Decimals Colour in Decimats

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum in Mathematics Level 4:

• Students recognise common equivalent fractions in familiar contexts
• Students make connections between fractions and decimal notations up to two decimal places.

Level 5

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

 Content description Stage Mathematics Compare and order common unit fractions and locate and represent them on a number line (VCMNA187) Chocolate Cake Fractions to Decimals Colour in Decimats Investigate strategies to solve problems involving addition and subtraction of fractions with the same denominator (VCMNA188) Chocolate Cake Colour in Fractions Recognise that the place value system can be extended beyond hundredths (VCMNA189) Fractions to Decimals Colour in Decimats Compare, order and represent decimals (VCMNA190) Fractions to Decimals Colour in Decimats Critical and Creative Thinking Investigate thinking processes using visual models and language strategies (VCCCTM029) Fractions as Equal Parts of a Whole Chocolate Cake Fractions to Decimals Colour in Fractions Colour in Decimats

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

• Students order decimals and unit fractions and locate them on a number line
• Students add and subtract fractions with the same denominator.

Level 6

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

 Content description Stage Mathematics Compare fractions with related denominators and locate and represent them on a number line (VCMNA211) Chocolate Cake Fractions to Decimals Colour in Fractions Add and subtract decimals, with and without digital technologies, and use estimation and rounding to check the reasonableness of answers (VCMNA214) Colour in Decimats Make connections between equivalent fractions, decimals and percentages (VCMNA217) Fractions to Decimals Colour in Fractions Colour in Decimats Critical and Creative Thinking Investigate thinking processes using visual models and language strategies (VCCCTM029) Fractions as Equal Parts of a Whole Chocolate Cake Fractions to Decimals Colour in Fractions Colour in Decimats

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

• Students locate fractions and integers on a number line
• Students connect fractions, decimals and percentages as different representations of the same number
• Students solve problems involving the addition and subtraction of related fractions as well as decimals.

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 4 Level 5 Level 6 Quantifying numbers Understanding decimal place value Understanding decimal place value N/A Operating with decimals Understanding positional value of decimals Understanding and estimating relative size of decimals Understanding the effects of multiplication and division with decimals Interpreting fractions Equivalence of fractions Fractions as numbers Using fractions

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