Lesson 12 — Area of Rectangles

Summary

Learners find the area of a rectangle by covering it with square units and counting, then discover the shortcut — area = length × width — and apply the formula to real problems like tiling a floor or measuring a field. Area becomes “how many squares cover it.”

Objectives

• Apply formulas for the area and perimeter of rectangles to solve real-world and mathematical problems. (D05.S3.04.02)

Connection

How many tiles to cover a floor? How much cloth to make a rug? How much land is this field? Each question asks the same thing: how many squares cover a surface? That is area. If you count the squares once, you notice a shortcut — the number of rows times the number in each row, which is length times width. Area is how people plan a garden, a floor, a piece of cloth, or a roof, anywhere in the world.

Materials

• Square tiles or grid paper (per learner)
• Paper or slate and a drawing tool

Preparation

• Give each learner square tiles or grid paper.
• Prepare (or plan to draw) the worked example: a 6-by-4 rectangle.
• Clear space to cover and count.

Facilitator note

This lesson is written to the learner (“you”). Area begins as covering with square units and counting, then generalizes to length × width — the formula is a shortcut for the count, not a new idea (Van de Walle, Karp & Bay-Williams, 2019). Teach the shortcut as a worked example (philosophy §13): tile a 6-by-4 rectangle, count 24, then see that 6 × 4 = 24. Build on Grade 3’s area work. The number of tiles a learner needs to count before seeing the pattern is not a measure of their worth; an area is a fact about a surface, never about a person. Measuring land by its covering squares is a human and cross-cultural practice, from ancient field surveying to modern tile and cloth work. Count in the home language. For learners who are Deaf or hard of hearing, sign the count; for learners who are blind or have low vision, feel the tiles and count rows and columns by touch. Learners with limited movement direct a partner to place tiles. Mastery arrives at its own pace.

Procedure

1. Gather — cover it with squares (5 min). How many squares would cover the top of your hand? That is area: how many squares cover a surface.
2. Worked example — a 6-by-4 rectangle (10 min). Cover a rectangle that is 6 units long and 4 units wide with square tiles. Count: 4 rows of 6 — that is 6 + 6 + 6 + 6 = 24 square units. Now notice the shortcut: length × width = 6 × 4 = 24. The formula is just the count, made quick.
3. Guided practice — count, then multiply (15 min). Cover several rectangles, count the square units, then check with length × width. Do the count and the multiplication always agree? (They should.)
4. Real problems (10 min). Solve: a floor is 8 tiles long and 5 tiles wide — how many tiles? A garden is 9 meters by 4 meters — how many square meters of soil? For each, say the formula and the answer.
5. Check with a partner (3 min). Trade a problem. Do you agree on the area? Count it together, kindly.
6. Close (2 min). Remember: area is how many squares cover a surface, and length × width finds it in one step.

Differentiation

• Support: Count square units only today; add the formula another day.
• Support (blind/low-vision): Use tactile tiles and count rows and columns by touch.
• Extension: Given an area (like 24) and one side (like 6), find the other side and explain how you know.

Assessment

• Formative (observation): Can the learner find the area of a rectangle by counting square units and by length × width, and state the answer in square units?
• Self: Can the learner solve a real area problem and check their answer by counting?

Home connection

At home, measure a real rectangle — a rug, a table, a door — and find its area in your own hand-spans or steps, then tell a grown-up the area.

Resources

• Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2019). Elementary and Middle School Mathematics: Teaching Developmentally (10th ed.). Pearson. (Area as covering with square units, generalizing to length × width.)