A Great Mathematics Lesson: From Counting Squares to Understanding Area

How a problem-based lesson helps children discover why mathematics works

Imagine your child comes home and says, “We learned how to find the area of a rectangle.” You might expect them to recite a familiar formula: length × width. That formula is useful—but knowing it is not always the same as understanding it.

What happens when a rectangle is tilted, when its inside squares are not drawn, or when a shape can be split into two rectangles? Can your child still explain what area means?

A well-designed problem-based lesson helps students build the idea before they rely on the shortcut. The following classroom story is an original illustrative example inspired by the teaching approach of Illustrative Mathematics (IM); it is not a reproduction of a specific IM lesson.

START WITH A QUESTION CHILDREN CAN EXPLORE

The teacher shows a rectangle made of square tiles: four rows with six tiles in each row.

“How many square tiles cover this rectangle? Find a way to count without losing track.”

Some children count one square at a time. Others count 6, 12, 18, 24. Another student notices four equal rows and says 4 × 6 = 24.

Instead of announcing the fastest method immediately, the teacher asks: “Can you show me why multiplying works?”

A student traces the rows. Another groups the tiles into columns. Everyone can see that multiplication describes equal groups, and that the area is 24 square units—not simply “24,” and certainly not 20 units around the edge.

This discussion matters because the formula is now connected to the meaning of area.

Original Math School diagram illustrating the example above.

THE TEACHER DOES NOT LEAVE CHILDREN TO FIGURE EVERYTHING OUT ALONE

Problem-based teaching is sometimes misunderstood as giving students difficult questions without support. In a thoughtfully planned lesson, the teacher chooses accessible numbers and diagrams, prepares prompts, notices misconceptions and steers the class toward the mathematical idea.

The teacher might ask: • “How do you know you counted every square?” • “What changes if we turn the rectangle?” • “Is the number of squares inside the same as the distance around it?” • “Can two different arrangements have the same area?”

Students first have time to think, perhaps individually and then with a partner. The teacher selects contrasting strategies to discuss and makes the important mathematical conclusion explicit.

MOVE FROM TILES TO A DRAWING—THEN TO A FORMULA

Now the teacher draws a rectangle on grid paper, but only some squares are visible. The side lengths are labeled 4 and 6 units.

Children use what they learned from the tiled rectangle to imagine the four rows of six squares. The expression 4 × 6 continues to make sense even without every tile being drawn.

Finally, they see a plain rectangle marked 4 units by 6 units. They can calculate area without physically counting tiles because the multiplication summarizes a structure they already understand.

This is a progression from concrete objects to pictures to symbols. It can help students use abstract mathematics with meaning rather than as a list of disconnected rules.

WHAT IF A CHILD MAKES A MISTAKE?

Suppose a student adds 4 + 6 and answers 10. Instead of saying only “wrong,” the teacher might ask what the 4 and the 6 represent. The child may have mixed up side lengths, area and perimeter.

This is a useful mistake. It tells the teacher what concept needs attention. Students can place tiles inside the rectangle, then trace its boundary to compare area with perimeter.

The aim is not to celebrate errors for their own sake but to use them as evidence for more precise teaching.

HOW DOES THE LESSON FINISH?

At the end, the class summarizes: area measures how many square units cover a region, without gaps or overlaps. For a rectangle, the rows and columns form an array, so multiplication gives the number of square units.

A short independent exit question checks whether the understanding transfers. For example: “A rectangle is 3 units wide and 7 units long. Find its area and explain why your calculation works.”

The teacher reads these explanations to decide whether students are ready to move forward or need another model.

WHAT PARENTS CAN TRY AT HOME

You do not need special equipment. Use small paper squares, toy blocks, sticky notes or grid paper. Ask your child to make two different rectangles using 12 squares.

Then ask: “Do both rectangles have the same area? What about their perimeters?” Listen for the reasoning, not only a number.

This kind of conversation turns a familiar formula into an opportunity for exploration.

THE BIGGER LESSON

Great mathematics teaching does not stop at the correct answer. It helps children understand where an idea comes from, connect models to symbols, choose methods thoughtfully, and communicate their reasoning.

Illustrative Mathematics includes a purposeful lesson structure with warm-up, instructional activities, synthesis and a short cool-down. Its Grade 3 materials develop area through square-unit tiling and arrays. These design features informed our example, but the particular rectangle story above was written independently for this article.

At Math School, we believe children should learn not only how to solve a problem, but also how to explain why their solution makes sense.

SOURCES

1. Illustrative Mathematics — Principles for Organizing Mathematical Content. https://illustrativemathematics.org/principles-for-organizing-mathematical-content/
2. Illustrative Mathematics — Grade 3 Scope and Sequence (area and arrays). https://cms-assets.illustrativemathematics.org/b0evstg1rwgmt10wsvgfgdw4lej1
3. Illustrative Mathematics — Grade 3 area standard and tasks (3.MD.C.7). https://tasks.illustrativemathematics.org/content-standards/MD/3/C/7
4. Illustrative Mathematics — K–5 Key Structures. https://accessim.org/k-5/grade-1/course-guide/key-structures?a=teacher&lang=en