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Progression on Area -- a visual overview, page 1

Grade 3

What is area?Howmanyunitsquaresdo weneed

tocover the rectangle without gaps

or overlaps?

1

1

1

1a unit square

1 square unit of area

Area =18 square units because it

takes 18 unit squares to cover.

Area tells us how muchat space the shape takes up.

Whats a quicker way than counting squares one by one?

3 groups of 6 unit squares

3 6 = 18 square units6 groups of 3 unit squares

6 3 = 18 square units

That is why we can multiply side

lengths to nd areas of rectangles.

4

units

5 units

4 5 = 20

square units

14 square units2 units

? units 142 = 7

3

2

3

3

6

5

5 3 = 15

3 3 = 9

15 + 9 = 24

Area = 24 square units

3

2

3

3

6

5

2 3 = 6

5 6 = 30

30 - 6 = 24

Area = 24 square units

3

2

3

3

6

5

2 3 = 6

3 6 = 18

18 + 6 = 24

Area = 24 square units

Decomposing into rectangles to nd area

Area is dierent from perimeter

Same area, dierent perimeter:

Same perimeter, dierent area:

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Progression on Area -- a visual overview, page 2

Grade 4: apply area and

perimeter formulas

L

W

A = L W

P = L + W + L + W or

P = (2L) + (2W) or

P = 2(L + W)

Grade 5: Extend the area formula for rectangles to

cases of fractional side lengths.

12

A =1

61

3

25

3

4

L W =3

4

2

56

20 =L W = 1

3

1

21

6 =

Grade 6: Finding areas

What is the area of the shaded triangle? Most primitive method:

Move and combine small pieces;

count the number of squares.

More advanced methods that will generalize to develop area formulas:

Move a chunk to create a rectangle

of the same area.

Combine two copies to make a

rectangle of twice the area.

3 2 = 6 parts,

so each is 1/6 ofa square unit.

4 5 = 20 parts, so each is 1/20 of a

square unit. 3 2 = 6 parts shaded.So shaded area is 6/20 square units.sameanswer!

same

answer!

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Progression on Area -- a visual overview, page 3

Grade 6: Developing area formulas.

1) Right triangles

b

h

b

h

b

h1

2

bh12

1

2

h12

A = b

A = bh

A =

or A = bh2

1

2A = bh

or A = bh2

A = bh

2) Parallelograms

b

h

b

b

b b

hh

b b

h

b

h h

h h h

a abig rectangles area =

(b + a)h = bh + ah

two extra triangles

area = ah

3) Very oblique parallelograms

parallelograms area = (big rectangles area) - (two extra triangles area)

= (bh + ah) - ah = bh

4) Other triangles: two copies make a parallelogram or rectangle of the same base and height,

so the area of the triangle is half the area of the parallelogram or rectangle.

take away the extra area

two copies make a rectangle

of the same base and height

Strategies:

Relate to a

rectangle

(or parallelogram

by doubling then

halving, moving,

adding then takin

away area.

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rr

2r

r

r

approximately a parallelogrambase: r height: r area: rr = r

Progression on Area -- a visual overview, page 4

Grade 7: Figure for an informal derivation of the relationship between the

circumference and area of the circle.

Grade 8: Figure for a proof of the Pythagorean theorem.

Decomposing and composing to relate areas.

a

b

c

a

b

c