Grade by Grade
E. Spatial Sense · 3D Objects, Directions and Measurement

Comparing Area and Measuring with Non-Standard Units

E2.7 E2.8 A1About 35 minutes
Learning goals
  • I can compare areas by matching, covering, or cutting and moving pieces.
  • I can show that different shapes can have the same area.
  • I can measure area by covering a shape with units and counting.
  • I can explain how gaps and overlaps make an area measurement wrong.

Words to know

area
How much surface a flat shape covers.
cover
To place units over a shape so the whole shape is hidden, with no gaps and no overlaps.
gap
An empty space between units.
overlap
When units are on top of each other, so part of the shape is counted twice.
decompose and recompose
Cut a shape into pieces, then put the pieces together in a new way.

Remember from Grade 2

In Grade 2 you put shapes together and took them apart, and you found that moving the pieces around doesn't change how much space they cover.

That space is called area. This year you will compare areas and measure them.

Let's start with a tablecloth

You want a tablecloth that covers the whole kitchen table. A tablecloth that's too small leaves part of the table showing. To choose, you need to compare the area of the cloth with the area of the table.

Area is the amount of surface a flat shape covers. Perimeter goes around a shape. Area fills the inside.

Three ways to compare area

  1. Match: put one shape on top of the other. If one fits completely inside the other, it has a smaller area.
  2. Cover: cover each shape with the same units (like sticky notes) and count. More units means more area.
  3. Decompose and recompose: cut a shape into pieces and move them to make a shape that's easier to compare. The area stays the same.
Worked example

Example 1: Match

Mia has two placemats. When she puts the small one on top of the big one, the small one fits completely inside, with space left over. Which has the greater area?

  1. The small placemat fits inside the big one.
  2. Part of the big placemat still shows, so it covers more surface.

The big placemat has the greater area.

Worked example

Example 2: Cover and count

Jamal covers a book with 6 sticky notes and a cutting board with 9 sticky notes (all the same size, no gaps, no overlaps). Which has the greater area?

  1. Both were covered with the same unit, so we can compare the counts.
  2. 9 is more than 6.
  3. 9 − 6 = 3.

The cutting board has a greater area, by 3 sticky notes.

cut andmove →parallelogram: 8 squaresrectangle: 8 squares
Cut the triangle off one end of the parallelogram and move it to the other end. It becomes a rectangle. Both cover 8 squares.
Worked example

Example 3: Decompose and recompose

Is the area of the parallelogram the same as the area of the rectangle in the picture?

  1. Cut along the dashed line. A triangle comes off the left end of the parallelogram.
  2. Slide the triangle to the right end. It fits exactly, and the shape becomes a rectangle 4 squares long and 2 squares tall.
  3. Count the rectangle's squares: 4 + 4 = 8. Nothing was added and nothing was taken away.

Yes. Both shapes have an area of 8 squares.

Different shapes, same area

Shapes can look very different and still cover the same amount of space. Count the squares in each shape below.

ABC
Shape A (2 by 3 rectangle), shape B (a row of 6) and shape C (a different shape) each cover 6 squares.

Shape A, shape B and shape C each cover 6 squares. They have the same area, even though their shapes are different.

Measuring with non-standard units

To measure area, cover the shape with units and count them. You can use sticky notes, playing cards, square crackers or paper squares.

Good measuring rules:

  • Use units that are all the same.
  • No gaps: the units must touch, with no empty spaces.
  • No overlaps: units must not sit on top of each other.
  • Cover the whole shape, right to the edges.
circles:gapssquares:just rightoverlaps:counted twice
Covering the same rectangle three ways. Circles leave gaps. Squares fit just right. Overlapping squares count some parts twice.
  • Gaps leave parts of the shape not counted, so your count is too small.
  • Overlaps count some parts twice, so your count is too big.
  • Units that are round (like coins or cookies) always leave gaps. Units that are squares or rectangles fit together with no gaps.
Worked example

Example 4: Gaps and overlaps

Ben measured a placemat with playing cards. His cards overlapped a lot, and he counted 15 cards. Is the real area more or less than 15 cards?

  1. Overlapping cards cover some parts of the placemat twice.
  2. So Ben counted more cards than he needed.
  3. If he lays the cards edge to edge, he will need fewer.

The real area is less than 15 cards. His count was too big.

Keep going!

Covering a shape neatly takes patience. If units slide around or the count seems wrong, take a breath and start again, or ask a helper to hold the units still. Checking and redoing is what careful measurers do.

Just for fun

What if you measured your bedroom floor with pancakes? The round pancakes would leave gaps, so your count would be too small... and your floor would be very sticky!

Hands-on activity

Cover it!

You need: sticky notes (or playing cards, or squares of paper all the same size), some coins, and flat things to measure, such as a book, a placemat, a cutting board and a tray.

  1. Estimate how many sticky notes will cover the book.
  2. Cover the book with sticky notes: no gaps, no overlaps. Count them. If the last row hangs over the edge, count about how much of each note is on the book (more than half? about half?).
  3. Now try covering the book with coins. What do you notice about the spaces between them?
  4. Measure two more flat things with sticky notes. Order them from smallest area to largest.
  5. Use 6 paper squares to make as many different shapes as you can. Each shape must have the squares touching side to side. They all have an area of 6 squares!

Grown-up note: keep coins away from babies and toddlers.

Math talk: "Why are squares better units than coins for area?"

Your turn

1

What is area?

2

A tray is covered by 12 cards. A book is covered by 8 of the same cards. Which has the greater area? By how much?

3

A small rug fits completely on top of a big rug with space left over. Which has the smaller area?

4

Will gaps between units make your count too big or too small?

5

Will overlaps make your count too big or too small?

6

Which unit is better for measuring area: round cookies or square crackers? Why?

7

You cut a square into two triangles and put them together to make a new shape. Did the area change?

More practice: warm-up

1

Does area measure the space around a shape or inside it?

2

Should your units all be the same size?

3

A shape is covered by 5 squares. What is its area?

4

Name something at home you could use as an area unit.

5

Which has more area: a door or a postage stamp?

More practice: on your own

1

Shape P covers 10 squares. Shape Q is a different shape and covers 10 squares too. Compare their areas.

2

Ava covered a poster with sticky notes but left gaps. She counted 20 notes. Is the real area more or less than 20 notes? Explain.

3

Draw (or describe) two different shapes that each have an area of 4 squares.

4

Mark measured a table with big cards and got 6. Nia measured it with small sticky notes and got 24. Did the table change? Explain.

5

A placemat is covered by 3 rows of 5 sticky notes. How many notes is that?

Challenge

1

A rectangle covers 12 squares. Find 3 different rectangles with this area.

2

Tess cut a rectangle of 10 squares into two pieces and made an L-shape. Leo says the L-shape has more area because it looks bigger. Who is right?

3

A tray is covered by 4 rows of 6 playing cards. A board is covered by 5 rows of 5 of the same cards. Which has the greater area?

Check your understanding

Quiz

1.

What does area measure?

2.

Overlapping units make your count ...

3.

Which unit leaves gaps when you cover a shape?

4.

Shape X covers 7 squares. Shape Y is a different shape and covers 7 squares. Which is true?

5.

You cut a shape into pieces and rearrange them. The area ...

6.

A book is covered by 2 rows of 4 sticky notes. What is its area?

For parents and teachers

Big idea: Children compare areas by matching, covering, or decomposing and recomposing shapes, and see that different shapes can have the same area (Ontario E2.7). They measure area with non-standard units and explain how gaps and overlaps affect accuracy (E2.8). Covering carefully is also a chance to practise patience (strand A1).

Common difficulties:

  • Confusing area and perimeter. Area fills the inside; perimeter goes around the outside.
  • Which way errors go. Gaps: count too small. Overlaps: count too big.
  • Thinking a longer or odd-looking shape has more area. Count the units to check.

Safety: keep coins away from babies and toddlers.

How to help: Ask "How many of our floor tiles does this rug cover?" or cover a book with sticky notes together.