E. Spatial Sense · Angles, Rectangles, Grids and Measurement

Area of Rectangles: Rows, Columns and Multiplying

E2.5 A1About 35 minutes

Learning goals

I can see the squares that cover a rectangle as an array of rows and columns.
I can find the area of a rectangle by multiplying the number of rows by the number in each row.
I can explain why the area of a rectangle is length × width.
I can find areas in square centimetres and square metres.

Words to know

area
The amount of space a flat shape covers, measured in square units.

array
Objects arranged in equal rows and columns.

row
A line of squares going across.

column
A line of squares going up and down.

length and width
The two side lengths of a rectangle. Usually the length is the longer side.

square centimetre (cm²) and square metre (m²)
Square units for area: a square 1 cm on each side, and a square 1 m on each side.

Remember from Grade 3

You measured area by counting squares: square centimetres (cm²) for small things and square metres (m²) for big things. For a rectangle, you counted one row and then added the rows.

This year you'll find a faster way: multiply!

Let's start with a chocolate bar

A chocolate bar is made of little squares in rows. To know how many squares there are, you don't need to count them one by one. If there are 4 rows with 6 squares in each row, there are 4 × 6 = 24 squares.

The squares that cover a rectangle make an array, just like the chocolate bar.

A dark chocolate bar made of 24 squares in 4 rows of 6, on a white plate. — The squares make an array: 4 rows with 6 squares in each row, so 4 × 6 = 24 squares. AI-generated image

A rectangle covered by 1 cm squares: 4 rows with 6 squares in each row. 4 × 6 = 24, so the area is 24 cm².

From array to multiplication

The number of squares in each row is the same as the length of the rectangle (how many 1 cm squares fit along it).
The number of rows is the same as the width (how many 1 cm squares fit up the side).
So: Area = number of rows × squares in each row = length × width.

This works for any rectangle, even when you can't see the squares, because you can always picture the rows.

A 7 cm by 3 cm rectangle. Picture 7 squares along the top and 3 rows. 3 × 7 = 21, so the area is 21 cm².

Worked example

Example 1: A sticky note

A sticky note is a square 8 cm long on each side. What is its area?

Picture the squares: 8 squares in each row, and 8 rows.
8 × 8 = 64.
The units are square centimetres.

Area = 64 cm².

Worked example

Example 2: A bedroom floor

A bedroom floor is a rectangle 4 m long and 3 m wide. What is its area?

Picture 1 m squares: 4 in each row, and 3 rows.
3 × 4 = 12.
The units are square metres.

Area = 12 m².

Turn it, same area

A rectangle that is 3 by 8 has the same area as one that is 8 by 3. It is the same rectangle, turned. 3 × 8 = 8 × 3 = 24. You can multiply in any order.

A rectangle 8 squares long and 3 squares wide, and the same rectangle turned. Both cover 24 square units.

Worked example

Example 3: A bigger rectangle

A garden bed is 12 m long and 5 m wide. What is its area?

We need 5 × 12. Split the 12 into 10 + 2.
10 × 5 = 50 and 2 × 5 = 10.
50 + 10 = 60.

Area = 60 m².

A 12 by 5 rectangle split into a 10 by 5 part (50 squares) and a 2 by 5 part (10 squares). 50 + 10 = 60.

Don't mix up area and perimeter

Area is the space inside, in square units (cm², m²). You multiply length × width.

Perimeter is the distance around, in length units (cm, m). You add all the sides.

A 4 m by 3 m floor has an area of 12 m² and a perimeter of 4 + 3 + 4 + 3 = 14 m.

Math talk

Explain why multiplying works: "There are 3 rows, and each row has 7 squares, so it's 3 groups of 7." Being able to explain why is just as important as getting the answer.

Just for fun

Q: Why did the rectangle stay in to study?

A: It had a big area to cover!

Hands-on activity

Area hunt with a ruler

You need: a ruler, centimetre grid paper (or a sheet where you draw 1 cm squares), a pencil and some rectangles from around the house (a sticky note, a card, a photo, a book cover, a placemat).

Draw a rectangle 5 cm by 3 cm on grid paper. Count the squares, then multiply. Do you get the same answer?
For each household rectangle, measure the length and width to the nearest centimetre.
Estimate which one has the greatest area. Then calculate each area with length × width.
Make a table: object, length, width, area.
With a grown-up, use a measuring tape to find the length and width of a room in metres (round to the nearest metre). Find its area in m².

Math talk: "Which object surprised you? Was your estimate close?"

Your turn

A rectangle has 5 rows with 4 squares in each row. How many squares cover it?

Find the area of a rectangle 6 cm long and 3 cm wide.

Find the area of a square with sides of 9 cm.

A rug is 2 m wide and 5 m long. What is its area?

Find the area of a rectangle 10 cm by 7 cm.

A rectangle is 4 cm by 6 cm. Another is 6 cm by 4 cm. Which has the greater area?

More practice: warm-up

What units do we use for area: cm or cm²?

Area of a rectangle 2 cm by 3 cm?

Area of a rectangle 5 cm by 5 cm?

Area of a rectangle 1 m by 8 m?

Area of a rectangle 10 cm by 2 cm?

More practice: on your own

A patio is 7 m long and 6 m wide. What is its area?

A bookmark is 15 cm long and 4 cm wide. What is its area? Show how you split it.

Find the area and the perimeter of a rectangle 8 m by 2 m.

Ali counted the squares in one row of a rectangle: 9. There are 6 rows. Write a multiplication sentence for the area.

A square tile is 1 m on each side. How many tiles cover a floor 4 m by 4 m?

Challenge

A backyard rink is 13 m long and 6 m wide. What is its area? Show your strategy.

Mia says: "A rectangle 2 cm by 10 cm and a rectangle 4 cm by 5 cm have the same area, so they must be the same shape." Is she right?

An L-shaped garden is made of two rectangles: one is 5 m by 4 m and the other is 3 m by 2 m. What is the total area?

Check your understanding

Quiz

A rectangle has 3 rows of 7 squares. What is its area?

What is the area of a rectangle 9 cm long and 4 cm wide?

Which unit could describe the area of a classroom floor?

Why can we multiply to find the area of a rectangle?

What is the area of a square with 6 m sides?

A rectangle is 11 cm by 3 cm. What is its area?

For parents and teachers

Big idea: The Ontario curriculum (E2.5) asks Grade 4 students to use the row and column structure of an array to measure the areas of rectangles, and to show that the area of any rectangle can be found by multiplying its side lengths. Seeing why length × width works (rows of equal squares) is more important than memorizing a formula.

Common difficulties:

Adding instead of multiplying (3 by 7 gives 10). Draw the grid and count a few rows.
Mixing up area and perimeter. Area is the inside (square units); perimeter is the distance around (length units).
Leaving off the square units. Write cm² or m², not just cm or m.

How to help: Look for arrays everywhere: tiles, muffin pans, egg cartons, chocolate bars, windows with panes. Ask "How many rows? How many in each row? How many altogether?" Measuring real rectangles at home makes the idea stick.