Finding areas and perimeters from decimal side lengths
The purpose of this activity is to support students to measure areas and perimeters of rectangle when the side lengths are decimal measurements.
About this resource
New Zealand Curriculum: Level 3 to early Level 4
Learning Progression Frameworks: Measurement sense, Signpost 5 to Signpost 7
These activities are intended for students who understand how to use metric units of measure to find lengths. When working with units, they should understand the following:
- Units relate to the attribute being measured (for example, length is measured with iterations of length).
- Identical units need to be used when measuring.
- Units should be tiled (or iterated) with no gaps or overlaps to create a measure.
- Units can be equally partitioned into smaller units when greater accuracy is needed.
Students should also know how to use a measurement scale, such as a ruler or tape measure. They should be familiar with the most common metric units of length, metres, centimetres, and possibly millimetres, though they may not be able to convert measures (e.g., 45 cm = 450 mm). Students should also have a partial or full grasp of their basic multiplication and division facts.
Finding areas and perimeters from decimal side lengths
Achievement objectives
GM4-3: Use side or edge lengths to find the perimeters and areas of rectangles, parallelograms, and triangles and the volumes of cuboids.
Required materials
- rulers
- 1B5 exercise book
- calculators
See Materials that come with this resource to download:
- 1cm grid paper (.pdf)
1.
Remind students how the area and perimeter of a rectangle can be found using the side lengths. Use a 1B5 exercise book is possible, these usually measure 25.5 cm x 29.5 cm. Present the scenario of the cost of plastic book covers being based on area.
- What is the area of a 1B5 exercise book? (25.5 x 29.5 = 752.25 cm2)
- What is the perimeter of the double page cover (opened out)? (2 x (51 + 29.5) = 169 cm).
2.
Provide the students with a variety of books from the book corner or library. Alternatively, you might investigate the sizes of the different exercise books students use in school.
- Imagine you are making plastic covers for these books. (You might show videos of covering exercise books if students are unfamiliar).
- What area of plastic cover will you need for each book?. Allow for the extra that overlaps the edges. Be as accurate as you can.
3.
Let the students work in pairs to calculate areas and perimeters. Organise students in pairs that will encourage peer scaffolding and extension, as well as productive learning conversations. Roam as they work and look for students to:
- measure decimals in numbers of centimetres. They may use millimetres and convert those measures to centimetres.
- use multiplication with a calculator to find the areas and perimeters.
- express the measures using correct numbers and units, such as 624.6 cm2 for area and 121.8 cm for perimeter.
4.
Throughout the investigation, gather the students to discuss important points, or ask students to discuss the points in pairs. above or discuss the same idea with pairs. Points might include:
- Why do you think book sizes are similar?
- What is the maximum area for a book to be "lap friendly" or "shared book friendly"?
- Does cutting the corners off the rectangle of wrap increase or decrease the perimeter?
You might introduce relevant te reo Māori kupu such as roa (length), tapawhā hāngai (rectangle), paenga (perimeter), and horahanga (area).
1.
Explore maximisation and minimisation problems. For example,
- Henny has a 50 m roll of chicken wire to make a rectangular run for her chickens. Which rectangle has the greatest area?
Consider how digital technology could be used to represent problems.
2.
Students might recognise that while the perimeter cannot exceed 50 m, the side lengths can vary. The area is to be maximised. Some examples of runs with calculated areas are shown below:
Students might organise their findings into a table or spreadsheet to look for patterns:
Length (m) |
Width (m) |
Area (m2) |
|---|---|---|
1 |
24 |
24 |
2 |
23 |
46 |
3 |
22 |
66 |
4 |
21 |
84 |
5 |
20 |
100 |
Students might also graph the relationship between length and area when the perimeter is fixed. First plot order pairs like (1,24), (2,46), (3,66), etc., then look for a pattern by drawing a curve through the points:
Students should notice that the area peaks between lengths 12 and 13. Investigate lengths between 12 and 13 that are decimals, such as 12.3 m x 12.7 m.
Area is maximised when the rectangle is 12.5 m x 12.5 m. Note that 12.52 = 156.25 m2, and √156.25 = 12.5.
3.
Solve problems where the perimeter or area is given but a side length or both side lengths are missing. Make the areas decimal amounts. For example:
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