This unit begins with having students use their bodies to measure the area of different shapes drawn on the floor. Understanding of area is developed through using non-standard units (e.g. beans, counters and blocks) to measure the area of objects in the classroom.
- Use comparisons and knowledge of non-standard units to estimate area measurements.
- Cover a shape with non-standard area units and count the number used.
- Compare and order areas of shapes using non-standard area units.
Non-standard units are objects which are used because they are known to students and are readily available, for example, paces for length, books for area, and cups for volume. Non-standard units introduce the students to the idea that units are repeated and counted in order to provide a measure of an attribute of an object. Therefore, students should be provided with many opportunities to measure using these kinds of non-standard units. For example, the width of the desk is 4 handspans.
Non-standard units introduce most of the principles associated with measurement:
- Measures are expressed by counting the total number of units used.
- During a measurement activity, the unit must not change.
- Units are repeated across the attribute being measured with no gaps or overlapping of the units.
- Sometimes fractions of a unit (such as half and quarter) need to be used in order to get a more accurate measure.
- Units of measure are not absolute but are chosen for appropriateness. For example, the length of the room could be measured by handspans but a pace is more appropriate.
Prior to introducing standard units, students need to realise that non-standard units tend to be personal and are not the most suitable for communication. For example, one student's hands will be smaller than another's, so measuring using hand span is not always useful or accurate.
Covering surfaces with a single unit should lead to discussion about which shapes that tessellate, and are therefore useful for covering surfaces. For example, rectangles and squares tessellate the plane, whilst circles do not. Tessellating with non-standard units establishes the need to cover surfaces without leaving gaps and without overlapping. It also demonstrates the advantages of using arrays that can be readily counted by using multiplication, for example, 3 rows of 6 tiles gives an area of 18 tiles.
From the earliest of these experiences, students should be encouraged to estimate. Initially these estimates may be no more than guesses, but estimating involves the students in developing a sense of the size of the unit. As everyday life involves estimating at least as frequently as finding exact measures, the skill of estimating is important.
At this stage students can also be introduced to the appropriateness of units of measure. For example, a hand is more appropriate than a finger tip for measuring the area of a desktop.
The learning opportunities in this unit can be differentiated by providing or removing support to students and by varying the task requirements. Ways to differentiate include:
- modelling and scaffolding the measurement of objects using different non-standard units
- varying the complexity and size of shapes and non-standard units students are asked to measure with
- using a digital camera to record students' responses and reduce the required writing time
- providing additional 'challenges' which require students to choose their own unit of measure
- exploring a digital model of area and tessellation
- providing opportunities for students to work in pairs and small groups in order to encourage peer learning, scaffolding, extension, and the sharing and questioning of ideas
- working alongside individual students (or groups of students) who require further support with specific area of knowledge or activities.
The context for this unit can be adapted to suit the interests and experiences of your students. For example, in session 1 you could draw outlines of characters from a book you are reading the class. Similarly, for other activities, both the shapes being measured and the units being used to measure them could be chosen to appeal to your students.
Te reo Māori kupu such as ine (measure), āhua (shape), and tatau (count) could be introduced in this unit and used throughout other mathematical learning. You could also encourage students, who speak a language other than English at home, to share the words related to measurement that they use at home.
- Items to be used as non-standard units: beans, shoes, counters, shape tiles etc.
Today we introduce non-standard measures and use our bodies as measuring tools. Prior to this lesson, use tape or chalk to mark out several large areas of different shapes on the floor.
- Explain to the students that they are going to cover the shapes on the floor by lying in them. Looking at one shape at a time estimate how many students will be able to lie in each, then check. Record these estimates on a class chart. Discuss the need for all available space to be used up, and for there to be no gaps or overlaps. Record the number of students that can fit into each space on the class chart.
How many students do you think will be able to lie in this space?
Will this space fit more or less students than the last one?
Which space will fit the most / least students?
- Explain to the students that they are now going to draw their own shapes in the playground with chalk. As they draw a shape they are to estimate how many students will be able to lie in it and record that number inside the shape. You might discuss the meaning of "estimate" (e.g. to make a reasonable guess) and practise estimating, and then checking, the height order of students in the class. Model drawing a shape and estimating, as a whole class, the number of students that would fit inside the shape.
- Provide time for students to draw their own shapes and make estimations. Roam and use a sticky note or mini whiteboard to record students' estimations (or have students do this).
- Have each student pick one shape to be measured.
- As a class, measure the area of the chosen shapes. For each shape, have the student who created the shape share their estimate and reasoning (e.g. I think the X number of people will fit in this shape because...) before measuring the number of students that fit in the space. Emphasise that, in measuring, you need to be able to fully cover the object in order to measure its area (no gaps or overlaps).
How many students do you think will fit into this space?
Can you find a space that will fit the same number of students as this one?
Can you find an area that is smaller / larger than this one?
These sessions explore the use of non-standard measures and compare areas using non-standard measuring units.
Over the following days set up non-standard measuring tasks. For each task have students estimate, measure and then compare their results with others to order areas. You could explore each of these tasks as a whole class, or set them up as stations to be explored by groups of students across a few sessions. Consider starting and ending each session with a review of the tasks completed at the stations (e.g. what did you do? What was hard to measure? What was most enjoyable? Did you find any measurements surprising?). Roam and identify any misconceptions demonstrated in learning, and use these as the base for planning your review and/or targeted whole-class/small-group sessions. During this part of the unit, you should also consider any links that could be made between tasks and students' interests, cultural backgrounds, and learning in other curriculum areas (e.g. we have been reading about the stars of Matariki, can we use stars to cover this area? People fly kites at Matariki, can we use kites to cover this area?).
If you choose to use these tasks as stations, provide sufficient modelling of the task and instructions to ensure students can fully participate in the learning at each station. You could place a large graphic organiser (e.g. A3) at each station for groups of students to fill out.
Measuring tasks you could use include:
- Covering sheets of newspaper with shoes. How many muddy shoes can fit onto a sheet of newspaper while they dry?
- Covering a sheet of newspaper with memo cube paper.
- Covering book covers with mosaic shapes (triangles, hexagons or squares)
- Cutting shapes out of paper and covering them with beans.
- How many books laid flat does it take to cover the mat?
- Measuring the area of 3 different size chair seats (a staff chair, a senior student’s chair and a junior student's chair) using blocks.
In this session students use their measuring skills to compare the areas of their feet and hands and find out which is larger.
- Have students look at their hands and feet. Get them to estimate which of them has the greatest area.
Look at your hands and feet. Do you think your feet or hands take up the most space?
- Students then draw around one hand and cut out the outline, and draw around one foot and cut out the outline. Ask the students for their ideas about how they could compare the area of their hands and feet, for example, counters, tiles, blocks, direct comparison.
- Students record whether their hand or foot is larger and explain how they found that out. Look for students to compare the foot and hand cut-outs and/or to use non-standard units accurately (i.e covering the whole area without gaps or overlaps).
- Discuss what they have found out.
Whose foot is larger than their hand?
How did you work that out?
How do you know that you measured the shape accurately?
What else could you have used to measure the shape?
Was your measurement close to your estimate?
- Tell the students that you want them to put themselves in order according to the area of their feet. Discuss ideas for doing this. Select one of the ideas and ask the students to use it to measure their feet.
- Put the feet in order on a chart and display.