2. Perimeter of Rectangles and Squares

Session 2 Learning Plan

Learning objective: Calculate and explain the perimeter of rectangles and squares.

Key vocabulary: length, width, side, opposite sides, rectangle, square

Success criterion: Use \(P=2(l+w)\) or \(P=4s\), show how the formula represents all sides, and reject area distractors.

Concept Foundation

A rectangle has two equal lengths and two equal widths, so its perimeter is
\[ P = 2l + 2w = 2(l + w). \]

Worked example: a rectangle 8 cm by 5 cm has perimeter
\[ P = 2(8 + 5) = 2(13) = 26 \text{ cm}. \]

A square has four equal sides, so its perimeter is just four times one side:
\[ P = 4s. \]
A square with side 6 m has perimeter \(4 \times 6 = 24\) m.

Case study - framing a picture: a 10 in by 8 in photo needs frame molding equal to its perimeter, \(2(10 + 8) = 36\) inches.

Common mistake: doubling only one side. Both the length and the width appear twice, so add them first, then double.

Visual Models

Deep Explanation: Why Rectangle and Square Formulas Work

A rectangle has two pairs of equal opposite sides. If its length is \(l\) and its width is \(w\), walking around the boundary gives
\[ l+w+l+w=2l+2w=2(l+w). \]
These are equivalent forms of the same formula. Use the form that makes the structure easiest to see.

Guided Example: Rectangle

A community notice board is \(14\) feet long and \(6\) feet wide. A metal frame follows all four edges:
\[ P=2(l+w)=2(14+6)=2(20)=40\text{ ft}. \]
The expression inside parentheses represents one length and one width. Doubling accounts for the opposite pair. A useful mental check is that the perimeter must be greater than \(14+6=20\), because that sum covers only half of the trip.

Guided Example: Square

A square has four equal sides, so \(P=4s\). If a square tile display has side length \(11\) inches,
\[ P=4(11)=44\text{ in}. \]
The formula \(s^2\) would calculate area, not perimeter. Notice how the unit supports the formula: four times a length is still a length, whereas multiplying one length by another creates square units.

Reading Diagrams Carefully

A diagram may label only the top and one side of a rectangle. That is enough because opposite sides are equal. Copy the top measurement to the bottom and the left measurement to the right before adding. If a diagram is tilted, the relationship does not change. Orientation never changes the formula.

When algebraic expressions label the sides, preserve the same structure. A rectangle with length \(x+3\) and width \(x\) has
\[ P=2[(x+3)+x]=2(2x+3)=4x+6. \]
Use parentheses so the factor of \(2\) reaches every term.

Compare Without Confusing

Two rectangles can have the same perimeter but different dimensions. A \(9\)-by-\(3\) rectangle and an \(8\)-by-\(4\) rectangle both have perimeter \(24\), yet their areas are \(27\) and \(32\) square units. Perimeter depends on the sum \(l+w\), not on the product \(lw\).

Error Analysis

• \(l+w\): counts only half the boundary.
• \(lw\): calculates area.
• \(2l+w\): doubles only one dimension.
• \(4l\): treats a rectangle as if all sides were equal.

Write the formula, substitute both dimensions, and keep the parentheses visible until the final calculation. That habit makes each missing or duplicated edge easy to detect.

GED® Coaching: Make the Boundary Visible

Before calculating, trace the boundary with a finger or pencil. For a polygon, place a small mark on every side as you include it. For a circle, identify whether the labeled segment is a radius or a diameter before choosing a formula. This simple pause prevents the two most common errors: skipping part of the outside edge and using an area formula for a distance-around question.

Keep units attached throughout the work. Perimeter and circumference are one-dimensional lengths, so the final unit is plain: cm, m, in, ft, or another linear unit. Square units belong to area. If measurements use different units, convert them before adding or substituting.

Use answer choices as a final diagnostic, not as the first strategy. A distractor often represents one recognizable mistake: multiplying dimensions to get area, adding only one length and one width, treating radius as diameter, omitting \(\pi\), or stopping before a cost step. Name the mistake before rejecting the choice.

Session Practice Routine

1. State the exact quantity requested and its expected unit type.
2. Trace or mark the complete required boundary.
3. Label the given dimensions and derive any missing measurement.
4. Write the side sum or formula before substituting values.
5. Complete any cost, packaging, comparison, or repeated-distance step.
6. Estimate, attach a linear unit, and explain why the answer is reasonable.

Exit Reflection

Without looking back, state the session's central relationship, describe one common wrong turn, and write the boundary-checking habit that prevents it.