Classification of solids - home.ne.jp

1. The formula for the volume of a rectangular prism
◎ Square
( rectangle)
Math elementary
If you use short paper, the lengths
of sides are 8.5 inches( 21.5 cm) .
◎ Cube
( square)
( fold and open) ( fold and open) ( fold)
( fold) ( fold) ( open)
( fold) ( back side also) ( fold)
( back side also) ( fold and insert) ( fold and open)
( blow )
If you use short paper, the lengths
of sides are 2.1 inches( 5.4 cm) .

( Example of PWA )★ How to derive the formula for the volume of a rectangular prism ★
Thevolumeofafigureistheamountofspace
inside the figure 1
← 1
Volume is measured using cubic units.
We are going to use the cube which we made by 1 1
paper folding.
☆ LET'S TRY TO FIND OUT THE FORMULA FOR THE VOLUME OF A RECTANGULAR PRISM ! ☆
< ACTIVITY>
● Objective ●
To be able to understand the formula for the volume of a rectangular prism
● Procedure ( group activity)
●
1. Make several groups ( 5 ~ 6 pupils for each group ).
2. Each group makes 24 cubes by folding paper.
3. Work in a group
( Q)
Let those cubes be cubic unit.
Form rectangular prism using 24 cubicunitsasmanyaspossible.
24=2 × 2 × 2 × 3 ( Prime Factorization)
↓
We can form following rectangular prisms.
1 24=1 × 1 × 24
1
2 24=1 × 2 × 12
3 24=1 × 3 × 8
2 5
4 24=2 × 2 × 6
4
5 24=2 × 3 × 4
3
v
4. Each group present their ideas.
5. Make the formula for the volume of a rectangular prism.
Since we use 24 cubic units, the volume of these rectangular prisms is all 24.
On the other hand, can we get the number this volume "24" by computation? not by counting.
Which number do you use in the solids? Is there a common rule in order to get the number 24
by computation?
The number of cubic units we arranged lengthways
× The number of cubic units we arranged widthways
× The number of cubic units we piled
↓
We can find out the following formula.
Volume of a rectangular prism = length × width × height
6. ( Exercise)
Find the volume of each figure
1 2 3
8 cm
5 cm
8 cm 8 cm 7 cm
8 cm 12cm 6 cm
5
cm

2. Classification of solids
◎ Regular hexagon
( rectangle)
open side
crease
◎ Regular Tetrahedron
valley folded lines ( turn it over)
valley folded lines
insert
fold and open
blow

( Example of PWA ) ★ Classification of solids ★
There are plenty of ways of observing a solid. If we have several solids ( as shown in the figure
below ), there would be various ways of observing, for example "How many surfaces are there?", "What
shapes of surface does this solid have ?", "if we project it to the wall, what shape can we see?" and so on.
Concrete materials enable pupils to make sure the
characteristics of solids.
On the other hand, textbooks has these kinds of questions:
A
B
D
"Is this figure a prism or a pyramid?" "Which solid is a cylinder?"
C
In these questions pupils only find the answer. It's nothing
more than a knowledge. We can't state it's a comprehension.
The most important thing for math teachers is to make
E
F
pupils find out new concept and characteristics by themselves.
G
H
If we give this question "which solids belong to the samegroupassolidH?",pupilscoulddecidethe
perspective by themselves and could start classifying solids. ( We call this kind of questions
"Open-ended question".)
< ACTIVITY>
● Objective ●
To be able to classify solids by observing its characteristics
● Procedure ●
1. Pupils make Cube and Regular Tetrahedron by folding paper.
2. ( Q)
What is the common denominator, what is the difference between two solids?
( difference) ( common denominator)
・number of surface
・closed solid
・figure of surface
・number of surface gathering round a vertex
・number of vertex
・every surface is plane figure
・ parallel lines etc.
etc.
○ We may say that these two solids belong to the same group in terms of closed figure.
× However, we can't state that these two solids belong to the same group in terms of number of
surface.
( If there in not enough time, resume 3 in next lesson. )
3. Work in a group ( group activity)
A
B
( Q)
Thereare8solids.Checkthesolidswhose
C
D
characteristics is the same as F. Also, write
its characteristics.
E
F
G
H
Characteristics A B C D E F G H
( example) pyramid solid
∨ ∨
( example) base figure is quadrilateral
∨ ∨ ∨
4. Each group presents their ideas.
( Wecanexpectplentyofideas.Wejustcheckwhetheritiscorrectornot. )
Open-ended !!
5. Give an assignment
( Q)
In above question, check the solids whose characteristics is the same as E.
Also, write its characteristics. ( Pupils make a list as well as 3. )
Ateneo De Davao University - Regional Science Teaching Center

Activity Sheet( Classification of solid)
A
B
C
D
( Hints)
・ figure of surface
・ vertex, side, the number of surface
・ parallel, perpendicular line and face
・ volume and area of surface
etc
E
F
G
H
( Q)
Compare solid F with another solids
Characteristics A B C D E F G H
( example) pyramid solid
∨ ∨
( example) base figure is quadrilateral
∨ ∨ ∨
( Assignment)
Compare solid E with another solids
Characteristics A B C D E F G H

Can you visualize solids?
→ If you understand the characteristics of solids, you could draw them easily.
cube
rectangular prism
sphere
cylinder
pentagonal prism pyramid ( quadrangular pyramid)
triangular pyramid
cone