Fill it up - flat space

Purpose

This integrated unit combines measurement of area with multiplication, and algebraic thinking.

Achievement Objectives
GM3-2: Find areas of rectangles and volumes of cuboids by applying multiplication.
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
NA3-8: Connect members of sequential patterns with their ordinal position and use tables, graphs, and diagrams to find relationships between successive elements of number and spatial patterns.
Description of Mathematics

Area is an attribute, a characteristic of an object. The attribute of area is the space taken up by part of a flat or curved surface. Usually, we begin by helping students to attend to area as an attribute before formally measuring it. Use contexts in which students compare flat spaces by size such as comparing pancakes or footprints. Note that “biggest” may be perceived in different ways. The most common confusion is between area and perimeter, distance around the outside.

In context students can be focussed on the attribute of area. Suppose some students think that a given pancake is bigger than another because they wrap string around both shapes and find one length longer than the other. “How many bites would it take to eat each pancake?” is an example of an enabling prompt. Partitioning and combining shapes is also a useful way to promote understanding of conservation of area and can lay groundwork for ideas about the areas of triangles, rectangles, trapezia, parallelograms etc. in later years.

Formal measuring of area with units will only make sense to students if they relate their methods to the process in measuring other attributes such as length and mass. Firstly students need to see the need for units and identify the qualities of units that are appropriate. They also need to realise that a number alone does not convey a measure unless the unit is stated as well.

Units require the following properties:

  1. Units are all the same. You can mix units but that makes it harder to be precise and compare measures.
  2. Units fill a space with no gaps or overlaps which explains the convention for using squares that tessellate by equal measure in both dimensions in arrangements of rows and columns.
  3. More smaller units fit into the same space as larger units. Smaller units tend to give a more precise measure. Note that if the smaller units are one quarter of the size of the larger units then four times as many fill the same space.
  4. Units can be partitioned and joined. Note the connection to fractions, e.g. two half units can make a whole unit. The standard units of area most commonly used in real life are the square centimetre (cm2), square metre (m2), hectare (ha.) and square kilometre (km2). While the proportional difference between metres and centimetres is manageable with length, the proportional difference between square centimetres and square metres makes size comparison difficult.

Consider the relationship between square centimetres and square metres. There are 100 x 100 = 10 000 square centimetres in one square metre. That is the same relationship as between square metres and hectares. A hectare is 10 000 m2. Hectares are most commonly used to measure areas of land. Think of a hectare as an area that is 100m by 100m. That means that 10 x 10 = 100 hectares are in one square kilometre. Square kilometres are used to measure large areas of land. For example, Stewart Island has an area of 1 746 km2 or 174 600 hectares.

Specific Teaching Points

Sessions One and Two
A suitable unit for measuring area must have these qualities:

  • Be a piece of area (two dimensional)
  • Units must be the same size
  • Units should fit together with no gaps or overlaps
  • Units should be of a size that give adequate precision (accuracy)

Session Three
The area of a flat shape is conserved (stays constant) as parts of it are moved to different places on the shape. So any shape can be ‘morphed’ into a shape with the same area by ‘giving and taking’.

Session Four
Area is the amount of flat space enclosed by a shape. Perimeter is distance around the outside of a shape. Shapes with the same area can have different perimeters, and shapes with the same perimeter can have different areas.

Session Five
A growing pattern can be structured by looking at how the figures are organised. Noticing structure helps with counting the area of a particular figure and with predicting further figures in the pattern. Identifying sameness and difference in figures can help in creating a rule (generalisation) for all figures in the pattern.

Observations of students during this unit can be used to inform judgments in relation to the Learning Progression Frameworks. Click for tables of guidelines.

Activity

Prior Experience

Students are expected to have some experience with measurement of other attributes, such as length, using informal units. They should also have some knowledge of multiplication facts and understanding of how to apply multiplication to finding the number of items in arrays.

Session One: Three Islands

Play this video introducing Three Islands (mp4, 13MB).

Discuss:

  • Size of an island can be measured by coastline (perimeter) or inside space (area).
  • Measurement requires the use of a unit because the islands cannot be directly compared, i.e. brought together to size match. What units are they going to use?

Put the students into small groups of two or three participants. Each group needs an A3 enlarged version of Copymaster One. Colour is not necessary. Provide the students with a choice of materials. Include items like; string, nursery sticks, dry pasta, beans (different sizes are good, e.g. plastic, lima, red), chick peas, counters, square tiles, transparent grids (Copymaster Two and Copymaster Three) made with Overhead Projector Film. Ask students to record their thinking as they work.

Allow the students plenty of time to compare the islands. Look for the following:

  • Do they distinguish perimeter from area?
  • Do they use a single unit consistently with awareness of iteration (copying with no gaps or overlaps)?
  • Do they use sensible number strategies to count the units?

If possible take digital photographs of the students working and play these images as a few groups share their methods with the class. You might select groups to focus on the bullet points above.

Session Two: Measuring Flat Space

Tell the students that they are thinking about flat space (area) rather than both area and perimeter. You are interested in how they measure the area of an island.

Work through the slides of Powerpoint One. It shows other students working on Three Islands. Ask the students what they notice. Particular points to highlight are:

  • Slide Two: The students are using different units. How will they compare their measures?
  • Slide Three: The students are measuring coastline (perimeter) using pasta. Will that tell them about flat space (area)? 
  • Slide Four: The students are using square tiles. Are squares a good unit to use? Why or Why not?
  • Slide Five: The students have filled one island up and moved the lima beans to the other islands. Is this a good strategy or not? Why?
  • Slide Six: The students have used square tile and pasta. Will that work to compare the flat spaces (areas) of the islands? What could the students do?

Provide the students with copies of Copymaster Four which contains various approaches to measuring two different islands. Ask the students to discuss the measuring strategy that is used. Tell them to think about the questions, What is right? What is wrong?

Look for:

  • Page One: There are gaps and overlaps with the counters. Why are circles hard to use a unit of area?
  • Page Two: There is a mixture of units (square tiles and beans). Could the units be converted to a measure with one unit, e.g. one square for two beans?
  • Page Three: The units are all the same but the Left Island has area missed and Right Island has tiles outside the coastline. How can you allow to missing or outside parts of the area?
  • Page Four: The units are all square tiles but they are different sizes. How many Left Island squares fit into a Right Island square? How could this ratio be used?

After a suitable period of group discussion gather the class to compare their ideas and to decide which island has the most area. Discuss the ’give and take’ of part units combining to a full unit. Record the measures using both number and units, e.g. 46 small squares (Left island) and 11 large squares (Right Island).

What is our problem? (Need the same unit). Converting 4 small squares to one large square results in 11½ large squares being the area of Left Island, making it larger.

How trustworthy is the result given the ‘give and take’ of part units?

This will raise issues of precision. Small squares are more precise than large squares. Why?

Session Three: Megabites

Use PowerPoint Two to tell the story of Yap, the hardworking sheepdog. When the farmer changes Yap’s dog biscuits he gets suspicious that he has been duped. The key ideas being developed are: 

  • Conservation of area – re-arrangement does not alter the internal space.
  • Partial units can be created for more precision and these partial units can be combined

How might Yap check to see that the biscuits are the same size?

Students might suggest overlapping the biscuits to directly compare them. That is a useful suggestion.

Note that by giving and taking, the overlapping triangles can fill the missing space, transforming the trapezium into the square. So the biscuits are the same area.

Students may suggest other strategies involving units. The fourth slide of the PowerPoint Two has an overlay of square units.

Why might Yap use squares? (no gaps or overlaps)

How will he allow for part squares with the Bonza biscuit?

Read the final slide which has a letter from Yap to the Dog Biscuit Company. Of course the challenge is on to create different shaped biscuits that are still 36 squares in area.

Ask the students what shapes they might try for the new biscuits. Make a list of shapes, e.g. rectangle, parallelogram, equilateral triangle, hexagon, octagon, etc. Closed curves such as circle and ellipse will be very challenging but encourage the students to try.

Provide the students with squared paper, e.g. Copymaster Two enlarged onto A3, rulers and scissors. In their teams they need to create at least eight different new biscuit designs that are 30 squares in area. At this stage leave the shapes loose so they can be sorted later. On the back of any biscuits students might indicate how they checked that the biscuit was 36 squares in area.

After a suitable time of exploration gather the class to look at the different biscuit shapes. Sort the biscuit shapes into categories by their common properties. Visually compare the shapes to see if they look to have the same area. Points to bring out include:

  • What rectangles are possible? Rectangles can be recorded systematically as expressions, i.e. 1 x 36, 2 x 18, 3 x 12, 4 x 9, 6 x 6. Ask the students to identify what the factors refer to in each rectangle and why 4 x 9 is really the same biscuit as 9 x 4 by the commutative property.
  • What is the relationship between a triangle and the surrounding rectangle? For example, this diagram shows two different triangles.

    The diagrams show that the triangle is one half the area of the surrounding rectangle. So if a triangle is 36 squares in area then the rectangle must be twice that area, 72 squares.
  • New shapes can be made by starting with a ‘parent’ shape that is 36 squares in area and altering the shape by ‘give and take’. So a rectangular biscuit might be altered to form an interesting shape with the same area.

Session Four: Yap’s Run

Play the video introducing the square metre (mp4, 38MB). Discuss what kinds of areas are measured in square metres, e.g. house floors, driveways, sports fields and courts. Show the students a square metre made from newspaper and tape. You might choose to construct the square metre in front of the class so they see how it is made.

How would we figure out the area of our classroom in square metres? We might want to recarpet.

Invite suggestions. Using an array of columns and rows is more efficient than mapping in the square metres one at a time. Link multiplication with the arrangement of rows and columns, e.g. 6 columns of 4 square metres each has an area of 6 x 4m2 = 24 m2. Explain that m2 means the unit, square metre. For homework students might investigate the cost of recarpeting the classroom on line.

Show PowerPoint Three: Yap’s Run.

Stop on Slide Three to ask the students to check that each design has an area of 54 m2. The students will need to partition two of the designs into smaller areas and combine the measures. Also ask what the perimeter of each enclosure is? Does the perimeter matter?

Move on to Slide Four where the problem is posed. Challenge the students to create an interesting shaped run for Yap that does not exceed a perimeter of 45 metres. Let the students create scale drawings of the enclosures using grids in their mathematics book. Expect them to label each side of the run with appropriate measures and show clearly how the area was calculated. Slide Four of PowerPoint Three shows an example with some measures shown. The perimeter of that run is greater than 45 metres. See if the students can work that measure out.

Give the students time to create their favourite run. Collect the diagrams at the end as work samples for assessment and display. You may like to go outside with some cones and a trundle wheel to mark out the favourite design in real size. Use the paper square metre as a benchmark and ask the students to calculate the area of parts of the run.

Session Five: Farmer Joe’s Garden

In this lesson students apply their understanding of area to a growing pattern. So the task can be used to assess several aspects of mathematics, including multiplicative thinking, measurement, algebraic thinking and equations and expressions.

Show PowerPoint Four: Farmer Joe’s Garden.

Slide Two presents the shape of the garden in Year Four. Ask the students what they notice. Look for them to identify properties of the shape and sections of the garden that will be useful structures for finding area. Ask the students to work in pairs to decide on the area of the Year Four garden. You may need to remind them that each small brown square represents one square metre (1m2).

After a suitable time discuss the various ways they structured the garden to find its area. Highlight the use of multiplication to find the area of arrays within the garden. Slides 3-6 show different ways to find the area of the garden. For each slide discuss how the structure could be recorded using an equation.

Slide 3: 5 x 4 + 2 x 5 + 2 x 4 = 38 m2. Note that brackets are not needed with the order of operations but you might like to record (5 x 4) + (2 x 5) + (2 x 4) = 38 m2. Ask students to identify the connection between each multiplication expression and the diagram on Slide 3.

Slide 4: 7 x 4 + 2 x 5 = 38 m2. How is this equation similar but different to that for Slide 3. Note that 7 x 4 is split into 5 x 4 + 2 x 4 in the equation for Slide 3.

Slide 5: 5 x 6 + 2 x 4 = 38 m2. Compare this equation to that for Slide 3. Note that 5 x 4 and 2 x 5 combine to form 6 x 5 or 5 x 6 using the commutative and distributive properties.

Slide 6: 7 x 6 – 4 x 1 = 38 m2 or just 7 x 6 – 4 = 38 m2.

Slide 7 invites the students to structure successive members of the growing pattern. Encourage the students to represent arrays in each garden using multiplication. For example, the gardens for Years 1-3 might be shown as: 

This structuring is very important if students are to generalise the pattern for later years. Using the same idea Years 5 and 6 would look like this: 

Slide Eight requires students to predict the area of the garden for Year 12. This is a challenging task but students can use table based strategies if they cannot generalise the structure. Here is a table of values for the pattern:

If they look for pattern in the differences students might notice that those differences grow by two each year.  

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