- Level 5
- Measurement and Geometry
- Using units of measurement
- Using familiar metric units to calculate perimeter, area, volume and capacity

###### Using units of measurement • Level 5

# Using familiar metric units to calculate perimeter, area, volume and capacity

VCMMG196

## Teaching Context

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The standard unit used to measure the area of shapes is the square meter (m^{2}) with sides that are one meter in length. Students must know which units would be most suitable when measuring area. For
example, if measuring the size of the classroom floor they would use m^{2 }or
when measuring the area of a book then cm^{2 }would be more suitable.

Support students in converting between units of area. Students will often memorise rules for converting instead of understanding how and why they work. A way to overcome this is to teach the meaning of of each prefix.

When measuring the area of familiar shapes such as a rectangle, provide students with square centimetres or grid paper to cover the regions. Rather than providing students with the formula, A = L × W, provide opportunities for them to discover this themselves.

Students may confuse the formulas for finding area and perimeter. This is primarily because the formulas are introduced too early and before students have developed a conceptual understanding of each. That is, that area the amount of space covering a region, whilst perimeter is the measure of the distance/length around the outside of a closed shape. It is important to use concrete, pictorial and abstract representations when teaching these concepts.

For example, as students begin to lay the squares in the rectangles they will see familiar images i.e. rows and columns which resemble arrays used in multiplication. At first, they may begin by counting all the squares individually, if this is the case prompt them to use more efficient strategies. This will lead them to skip count. E.g. (vertically) 2, 4, 6, 8, 10 or (horizontally) 5, 10, or use multiplication i.e. multiply the number of rows with the number of columns 2 × 5 = 10.

From here they may be able to generalise a rule for area: A = L × W. The rule for perimeter can also be determined. That is, P = 2L + 2W, which shows that both the length and width are doubled then added together. An alternative version of this rule is P = 2(L + W), where students first add the length and width then multiply the result by two.

These experiences will also make it easier for students to translate information when working with volume i.e. V= Area of base × Height.

### Victorian Curriculum

Calculate the perimeter and area of rectangles and the volume and capacity of prisms using familiar metric units (VCMMG196)

VCAA Sample Program: A set of sample programs covering the Victorian Curriculum Mathematics.

VCAA Mathematics glossary: A glossary compiled from subject-specific terminology found within the content descriptions of the Victorian Curriculum Mathematics.

##### Achievement standards

Students use appropriate units of measurement for length, area, volume, capacity and mass, and calculate perimeter and area of rectangles and volume, and capacity of rectangular prisms. They convert between 12 and 24-hour time.

Students use a grid reference system to locate landmarks. They estimate angles, and use protractors and digital technology to construct and measure angles.

Students connect three-dimensional objects with their two-dimensional representations. They describe transformations of two-dimensional shapes and identify line and rotational symmetry.

## Online Resources

## Teaching ideas

- Human rectangles
- Make a square metre
- Volume of a cubic metre
- Design a box
- Garden bed
- SA Conceptual Narratives
- Learning from home

### Human rectangles

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In this activity from Illuminations students will explore the relationship between area and perimeter.

Place students into groups of 4 and provide each group with rope or string of length of 16 units. Ensure that the floor is covered in grid lines using chalk or tape. You can make each unit square 30 cm x 30 cm. This means that each unit on the string must have 30 cm spaces. See diagram 1 below. Ask the students to each hold a unit on the string and form a rectangle.

Discussion prompts:

- How do you know that you have created a rectangle? Ensure that students explain the properties of the rectangle eg opposite sides are parallel and have the same length or they can also identify that there are four right angles using the grid on the floor
- Can you tell me the dimensions of the rectangle? Students can use a table with the headings: length, width, perimeter and area
- Explain how you can calculate the perimeter of the rectangle
- Explain how you can calculate the area

Have students create as many different rectangles as they can and ask them to record their findings in the table and encourage them to also analyse their results. Students should identify that the perimeter remains constant i.e. a total of 16 units, whilst the area changes.

They can draw these rectangles on centimetre grid paper or recreate their rectangles on geoboards. Ask students to explain:

- If they have found all the rectangles they can possibly make with their rope
- Which rectangle has the largest or smallest area and describe the rectangles features. They can also sort the rectangles in order from smallest to largest.

Extension prompt: If the difference in the area between the two rectangles is 16 cm^{2} draw the two rectangles and determine their dimensions. Ask students to use grid paper for their drawings and to justify and explain their answers.

### Make a square metre

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In this activity from Booker et al. (2014), students work in groups to make a square metre. They will use this to measure various areas such as the classroom floor. For example, if we need to replace the carpet in the classroom what are the key things we need to consider?

- The area of the room
- The price of carpet per square metre
- Budget (for labour cost and materials)

Have groups present their findings. A further investigation could include students rearranging the furniture in the classroom or even comparing the areas of all the classrooms in the school. As an extension, students can cut the square metre into different shapes and explore how the area remains the same. For example, it can be cut into two triangles and transformed into a larger triangle. See diagram 1 below. Alternatively, it can be cut into two rectangles and transformed into a longer and narrower rectangle but the area remains 1 m^{2}. See diagram 2 below.

### Volume of a cubic metre

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This activity involves students making a cubic metre using rolled up newspaper or straws.

Once they have constructed the cube, ask students to determine:

- How many centimetre blocks will fit in a cubic metre?
- How many thousand blocks will fit in a cubic metre?
- How many hundred blocks will fit in a cubic metre?
- How many tens sticks will fit in a cubic metre?
- When would they use the cubic metre to measure volume? Ensure that students first make a reasonable guess, then work towards a solution and efficient strategy.

### Design a box

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In this activity, students are given 60 chocolates the size of a one-centimetre cube. Ask them to design a box that will hold all 60 chocolates, with minimum space left inside. You may wish to advise students they can have as many layers as they like and can create more than one design. Alternatively, you could begin by asking what they might need to know to develop their design.

As an extension, ask students to:

- Find all possible designs
- Organise the chocolates so that the chocolates are not next to the same flavour. Inform students that there are 12 different flavours and 5 of each kind

### Garden bed

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A farmer wants to protect their garden bed by placing a border of tiles
(0.5 m × 0.5 m) around the edge. Each garden bed will have an area of
6 m^{2}. Ask students to work out the total area of one garden bed if it is
protected with tiles. Answers may vary depending on the rectangles
students create. An example of the garden bed:

If the farmer has one-eighth of a hectare of land to use (one hectare =
100 m × 100 m square = 10 000 m^{2}), how many garden beds can be made?

### SA Conceptual Narratives

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This learning sequence, developed by the SA Education Department, outlines the conceptual development of the substrand 'Using units of Measurement' of the 'Measurement and Geometry' Strand of the Australian Curriculum levels 5 to 7.

This sequence covers the **Victorian Curriculum** content descriptors: VCMMG195, VCMMG196, VCMMG222, VCMMG223, VCMMG224, VCMMG225, VCMMG258, VCMMG259.

The sequence connects the content descriptors with the proficiencies.

### Learning from home

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**Activity 1:**

For this activity ask your child to make a cubic metre using rolled-up newspaper or straws or any other material you may have.

Once they have constructed the cube, ask your child to think about the following questions. The diagrams below may be helpful when answering.

- How many centimetre blocks will fit in a cubic metre?
- How many thousand blocks will fit in a cubic metre?
- How many hundred blocks will fit in a cubic metre?
- How many tens sticks will fit in a cubic metre?
- When would they use the cubic metre to measure volume? Ensure that your child first makes a reasonable guess, then works toward a solution and efficient strategy.

**Activity 2:**

In this activity, your child will design a box that will hold 60 chocolates the size of a one-centimetre cube with minimum space left inside.

They may have as many layers as they like and they can create more than one design. If they are having trouble starting, ask them to think about what they might need to know to develop their design.

As an extension, ask your child to:

- Find all possible designs
- Organise the chocolates so that the chocolates are not next to the same flavour if there are 12 different flavours and 5 of each kind.