## Perplexing Perimeters

Purpose

In this unit the students develop a sense of the size of a centimetre and metre as they construct their own “rulers”.  Students come to recognise the exactness of the measurement needed as they calculate the perimeter of objects.

Achievement Objectives
GM3-1: Use linear scales and whole numbers of metric units for length, area, volume and capacity, weight (mass), angle, temperature, and time.
NA3-1: Use a range of additive and simple multiplicative strategies with whole numbers, fractions, decimals, and percentages.
Specific Learning Outcomes
• make appropriate measurements accurately using standard units
• perform addition calculations to find perimeters of objects
Description of Mathematics

Measurement is an important part of maths as a whole, and the primary maths programme in particular.  It is an important area of study because measuring, approximating and estimating are everyday experiences.  In the process of making their own standard metric “ruler” the students begin to develop their own mental image of a centimetre and a metre. This unit also provides opportunities for the students to appreciate that all measurements are approximate and that the selection of the measuring tool for a task depends on the precision required.

Measurement also provides a context for the further development and reinforcement of number skills.    Students can “measure” without the use of number up to the stage of indirect comparison.  However as soon as they repeatedly use a unit to measure an object they need numbers to keep track of the repetitions.

This unit is also designed to allow students to practice their additive strategies as they calculate the perimeters of objects.  Students at the Advanced Additive Stage (Stage 6) of the number framework realise that numbers are abstract units that can be “operated” on as wholes or can be partitioned and recombined.  This is called part-whole thinking.  Students at this stage are learning to choose appropriately from a repertoire of partitioning strategies.

Required Resource Materials

strips of heavy card

centimetre rulers

centimetre cubes

Key Vocabulary

measure, approximate, estimate, accurate, perimeter, centimetre, rectangle, combine, sum, strategies

Activity

#### Getting Started (Session 1)

In this session we make our own centimetre rulers and metre tapes so that we can measure the perimeter of objects in the classroom and playground.

1. Gather the class on the mat and tell them that you would like to know if the perimeter of a book is longer or shorter than the perimeter of a piece of cardboard you have cut out.
Can anyone tell me what a perimeter is?
How could we find out which is longer?

2. Tell the students that they are going to make their own centimetre rulers.

What is a centimetre?
Can you show me with your fingers how big a centimetre is?
Can you find me something that is one centimetre long?
(Centimetre cube, point to markings on a ruler)

3. Give the students a 10 centimetre long strip of card.
How long do you think this strip is?
How could we check? (Use a ruler, line centimetre cubes along the strip.)

4. Discuss how they can use the centimetre cube to make centimetre markings on the strip to create their own ruler.

5. Talk about where the number markings are positioned and why they are at the “end” of each centimetre. Talk about where the “0” would go and how this relates to the “0” on commercially produced rulers.

6. Remind the students of the question that was posed at the start of the session and ask for volunteers to compare the perimeters of the two objects. How can you keep track of the measurement? (Record and then add the length of the sides later.) Do you need to measure all of the sides?

7. When the measurements have been made ask the students to sum the side lengths.  This provides a good opportunity to practice and share strategies for adding numbers.

How did you combine the numbers?

8. Repeat with the second object and then make the comparison.   This provides further opportunities to share strategies for adding and subtracting numbers.

#### Exploring (Session 2)

We begin this session by posing a perimeter problem that is “too large” for our 10 cm rulers.  We discuss the need for a larger measuring instrument and then construct and use a 5 metre tape.

1. Gather the class on the mat and tell them that you would like to know if the perimeter of our classroom is larger or smaller than the perimeter of half a netball court.
How could we find out which is larger? Are our centimetre rulers appropriate for this? Why not? What could we use? (metres) How many of our centimetre rulers make a metre? How do you know?
1. Line 10 of the rulers end to end to show a metre.
2. Place a strip of adding machine tape, or other long paper strip, alongside the rulers and discuss how it can be used to make markings on the tape.
3. Give each pair of students a 3 metre strip of paper and tell them that they are going to use their 10 cm ruler to create a metre measuring tape.
4. Circulate as the students construct their tape. How many centimetres in a metre? How do you write centimetre? What does this line mean on your tape? How many centimetres are there in 2 metres? What objects would you chose to measure with the tape rather than the ruler? How accurate is your tape? Why?
5. Remind the students of the question that was posed at the start of the session and ask for volunteers to compare the perimeters of the two objects.
How can you keep track of the measurement? (Record and then add the length of the sides later)
Do you need to measure all of the sides?
How accurate do you need to be? (As the question posed was one of comparison the level of accuracy will depend on how close the perimeters of the two objects are.)
6. If the measurement requires parts of metres to be used than discuss ways of recording these.
1 m 20 cm             120 cm
7. When the measurements have been made ask the students to sum the side lengths.  This provides a good opportunity to practice and share strategies for adding numbers.

How did you combine the numbers?

1. Repeat with the second object and then make the comparison.   This provides further opportunities to share strategies for adding and subtracting numbers.
2. When the measurements have been made ask the students to sum the side lengths.  This provides a good opportunity to practice and share strategies for adding numbers.

#### Exploring (Session 3)

In this session the students are involved in a Perimeter Hunt. This involves them finding objects whose perimeter is of a certain length (in either metres or centimetres).

1. Give each pair of students a simple folded booklet into which to record the results of their perimeter hunt.

Ask them to select 10 perimeters to find and head up each page with a different perimeter range.

• 0 - 10 cm
• 10 - 20 cm
• 20 - 30 cm
• 30 - 40 cm
• 40 - 50 cm
• 50 – 60 cm
• 60 – 70 cm
• 70 – 80 cm
• 90 – 100 cm
• 1 – 2 metres
• 2 – 3 metres
• 5 – 10 metres
• 10 – 20 metres
• Over 20 metres

2. Pose the perimeter hunt challenge: Find objects that have perimeters which match the measure on each page. The pairs should use their handmade measuring tools and record their findings on each page.

3. Draw the object, labelling the measurement for each side, and calculate its perimeter.

#### Exploring (Session 4)

In this session the students investigate open ended problems involving perimeter. Students may work in pairs to answer the problems.

1. How many different shapes can you draw that have perimeter of 20 centimeters?
2. Explain using words or a diagram why the perimeter of a rectangle can be found by using a combination of addition and multiplication.
3. How could you find the perimeter of a curved shape?
4. If you find the perimeter of a rectangle does it matter which length you increase by one unit to increase the perimeter? Is the same true of the area of the same rectangle? Draw some examples to support your answer.

#### Reflecting

In today’s session we compare the objects we found to match the measure in the Perimeter Hunt and discuss the findings in our investigations.

1. Show the students an object and ask them to estimate which “page” of the book it belongs on.
2. Check the predictions.
What length do you think this side would be?
What do you use to make your estimate?
What about the length of this side? [pointing to another side]
1. Using the agreed estimates ask the students to calculate the perimeter. Share the strategies used for summing the side lengths.
2. Next ask the students to consider which perimeter was the easiest to find objects for.
Why do you think that [perimeter] was the easiest?
3. Share the objects found for that perimeter length.
4. Ask one pair to read out the side measures for their object while the rest of the students check the calculation mentally.Share the strategies used to sum the numbers mentally.
5. Repeat with other perimeter measures.