Perimeter and area inquiry

The prompt

Mathematical inquiry processes: Generate examples; test cases; conjecture and generalise; reason. Conceptual field of inquiry: Measurement; perimeter and area; scale factor of enlargement; proportional reasoning.

Samia Henaine designed the inquiry prompt for pupils in grades 4 and 5 (years 5 and 6) to test a statement about perimeters and areas of rectangles. Pupils create examples by doubling the sides of their own rectangles. They work out the perimeters and areas and decide if the statement is always, sometimes, or never true.

Pupils learn about the concepts of perimeter and area through the inquiry. The teacher aims to co-construct an understanding by drawing out the knowledge pupils already hold and then drawing upon it to develop lines of inquiry.

Pupils might require a more structured explanation of the concepts but this would emerge from their responses to the prompt.

Question, notice, and wonder

In the first phase of the inquiry, Samia anticipates pupils making these contributions:

Question

Is the area really double or has it changed by a different amount?
What would happen if we tripled the sides instead of doubling them?
Is the statement always true?
Does it work only for rectangles or would it happen with other shapes too?

Notice

The sides of Rectangle B are double the matching sides of Rectangle A.
The perimeter is double, but the area is not.
Rectangle B has four times as many squares as Rectangle A, not two times.

Wonder

I wonder what happens to the area if we triple the length of the sides.
I wonder if there is a rule connecting how much the sides grow to how much the area grows.

Explaining the prompt

The first part of the statement in the prompt is true. The perimeter of the new rectangle is double that of the original one. If we express the perimeter of the original rectangle as (2 x length) + (2 x width), then the perimeter of the new rectangle is (4 x length) + (4 x width).

Pupils might show their reasoning in a diagram (see below) or start to develop pre-algebraic thinking of the type that leads to P = 2(l + w) for the original rectagle and P = 4(l + w) for the new one.

The second part of the statement in the prompt is false. The area increases by a factor of four. If we express the area of the original rectangle as (length x width), then the area of the new rectangle is (2 x length) x (2 x width), which equals (4 x length x width).

As with the perimeter, pupils might might show their reasoning in a diagram or start to develop pre-algebraic thinking of the type that leads to A = lw and A = 4lw.

Defining terms

In their responses to the prompt, pupils might interpret the 'length' of the rectangle as the longer side and its 'width' as the shorter side. However, that interpretation can lead to confusion when students create their own examples. Even though a 2-by-3 rectangle is different to the 3-by-2 rectangle in the prompt, they would both have a length of three and width of two if the definition is based on longer and shorter sides.

To facilitate precise communication, the teacher could guide the class to label a horizontal side of the rectangle as the 'length' and a vertical side as the 'width' - although the larger the role pupils have in producing the definition, the greater their feeling of ownership over the inquiry.

Modelling inquiry

Samia designed the area and perimeter inquiry as the second of two inquiries. The teacher can use the perimeter inquiry, which is aimed at pupils in grades 2 and 3, as a preliminary step through which to model the inquiry process before giving students more independence in the second inquiry.

September 2026

Samia Henaine

Samia is an internationally-renown PYP educator and instructional coach. Her website Math Bridges contains more inquiry prompts and other resources.

Samia regularly writes articles about classroom inquiry here.

Lines of inquiry

The slides contain lines of inquiry based on three regulatory cards. In a structured inquiry, the teacher directs the class to follow the sequence of actions on the cards in order. Pupils can use the cards to review the course of the inquiry and develop their understanding of mathematical inquiry.

For the meaning of each card in the context of the inquiry, see the descriptions below.

Pupils draw several pairs of rectangles (an original and another with sides double in length) on squared paper. They measure the length of the sides and sum the lengths to find the perimeter of each rectangle. They also count the squares to find the area.

Pupils might record their results in a table and aim to spot patterns. Alternatively, they might analyse the mathematical structure to decide if the claim in the prompt is true or false.

Pupils change the prompt by using the What-if-not? method.

What if the length and width are not doubled? Pupils test other scale factors (tripling, halving, and multiplying by four) systematically. They could use a comparison table (see below) to make a generalisation about the proportional relationship between the scale factor and the perimeter or area.
What if the shape is not a rectangle? Does the relationship hold if we use a triangle or a different quadrilateral (parallelogram or trapezium) instead?

Pupils use a diagram to explain why doubling both sides increases the perimeter by a factor of two but makes the area four times bigger. One way is to see the enlarged rectangle as four copies of the original arranged on a 2-by-2 grid. Pupils work towards a reasoned argument rather than just rely on a numerical pattern.

Teacher scaffolds

If pupils get stuck testing the claim, the teacher can suggest one of three scaffolds:

(1) Direct counting

Pupils count unit squares to find the areas of the rectangles and count or measure each side to find the perimeters before they try to calculate the area and perimeter with multiplication. For example, the area would be the number of squares in a row multiplied by the number of rows.

(2) Overlaying copies

Pupils place four copies of Rectangle A on top of Rectangle B or draw lines dividing Rectangle B into four sections congruent to Rectangle A to see why the area increases by a factor of four rather than two.

(3) A comparison table

Pupils record side lengths, perimeters, and areas for Rectangle A and Rectangle B side by side in a table of results. They extend the table to cases in which the length and width of the second rectangle are half or three times those of the first rectangle. They then use the table to compare the cases and look for a pattern.

Resources

Prompt sheet

PowerPoint

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