# Let’s Experiment: Discovering Area and Volume Formulas

Mathematics

Level 7Level 8

This learning sequence aims to develop students’ conceptual understanding of perimeter, area and volume. When students are given the chance to develop measurement formulas, they acquire a deeper conceptual understanding of the relationships involved in measuring and they engage with an authentic mathematical process (Siemon, 2015). By investigating and forming relationships, students will see how area and volume formulas are derived and make sense of them in context.

 Big understandings An understanding of the important relationships between various measures, such as area and volume, is the key to extending measurement concepts. Measurement skills are usually used in the context of estimating. Estimating involves reasoning, not guessing. An understanding of how formulas are derived is the foundation for successful use of measurement formulas.

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 five flexible stages, including suggested learning intentions.

Overview of stages

• 1. Quadrilaterals: Investigating Areas of Trapeziums

Suggested Learning Intentions

• To investigate the area formulas of trapeziums

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• 3. Circles: Establishing Pi

Suggested Learning Intentions

• To investigate the ratio of circumference to diameter

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• 5. Prisms: Establishing Volume Formulas

Suggested Learning Intentions

• To build our understanding of the relationship between surface area and volume
• To develop the formula to find the volume of prisms

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• 2. Quadrilaterals: Investigating areas of Kites and Rhombuses

Suggested Learning Intentions

• To investigate the area formulas of rhombuses and kites

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• 4. Circles: Establishing Area Formula

Suggested Learning Intentions

• To visualise and evaluate how the area of a circle is determined

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• Prior knowledge

Before you commence this sequence, it is suggested that you ensure your students are familiar with converting between common metric units of length (such as from cm to mm), calculating the perimeter and area of rectangles and constructing and deconstructing three-dimensional shapes from their nets.

You can find support for building students’ understanding of these concepts in the Mathematics Curriculum Companion:

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

Vocabulary

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

 Formula Kite Estimate Circle Perimeter Circumference Area Radius Surface area Diameter Volume Ratio Prism Pi Parallelogram Irrational Trapezium Sector Rhombus

You can find definitions of some of these terms in the Glossary for the Mathematics Curriculum.

It is recommended that the explicit teaching of vocabulary occur throughout this 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 you 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, usually under ‘Reflect and consolidate’.

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 7.

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 7

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 8, but also addresses the following content descriptions from Level 7:

 Content description Stage Mathematics Establish the formulas for areas of rectangles, triangles and parallelograms and use these in problem solving (VCMMG258) Quadrilaterals: Investigating Areas of Trapeziums Calculate the volume of rectangular prisms (VCMMG259) Prisms: Investigating Volume Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Quadrilaterals: Investigating Areas of Trapeziums Quadrilaterals: Investigating Areas of Kites and Rhombuses Circles: Investigating Pi Circles: Investigating Area Prisms: Investigating Volume

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

• Students use formulas for the area and perimeter of rectangles.
• Students calculate volume of rectangular prisms

Level 8

This sequence addresses content from the Victorian Curriculum in Mathematics and Critical and Creative Thinking. It is primarily designed for Level 8 and addresses the following content descriptions.

 Content description Stage Mathematics Investigate the concept of irrational numbers, including pi (VCMNA275) Circles: Investigating Pi Find perimeters and areas of parallelograms, trapeziums, rhombuses and kites (VCMMG287) Quadrilaterals: Investigating Areas of Trapeziums Quadrilaterals: Investigating Areas of Kites and Rhombuses Investigate the relationship between features of circles such as circumference, area, radius and diameter. Use formulas to solve problems involving determining radius, diameter, circumference and area from each other (VCMMG288) Circles: Investigating Pi Circles: Investigating Area Develop the formulas for volumes of rectangular and triangular prisms and prisms in general. Use formulas to solve problems involving volume (VCMMG289) Prisms: Investigating Volume Critical and Creative Thinking Consider a range of strategies to represent ideas and explain and justify thinking processes to others (VCCCTM040) Quadrilaterals: Investigating Areas of Trapeziums Quadrilaterals: Investigating Areas of Kites and Rhombuses Circles: Investigating Pi Circles: Investigating Area Prisms: Investigating Volume

The sequence can be used to assess student achievement in relation to the following Achievement Standards from the Victorian Curriculum: Mathematics Level 8:

• Students convert between units of measurement for area and for volume.
• Students find the perimeter and area of parallelograms, rhombuses and kites.
• Students name the features of circles and calculate circumference and area.
• Students solve problems relating to the volume of prisms

Learning Progressions

The Numeracy Learning Progressions support teachers to develop a comprehensive view of how literacy 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 7 Level 8 Level 9 Understanding Units of Measurement Converting Units Calculating Measurements Circle Measurements N/A

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