A PROBLEM OF
PERIMETER
Getting Ready
What You’ll Need
Base Ten Blocks, 1 or more sets
per group
Children use Base Ten Blocks to create a variety of shapes with a given
value. Then they find and compare the perimeters of their shapes. In this
activity, children have the opportunity to:
Overhead Base Ten Blocks and/or Base
Ten Block Grid Paper transparency
(optional)
◆
◆
strengthen their understanding of the concept of perimeter
◆
discover that shapes with the same value (area) do not necessarily
have the same perimeter
◆
recognize how the compactness of a shape affects its perimeter
Introducing
Since value is synonymous with area,
the word value alone has been chosen
for use here. This is to help avoid the
confusion that children often have in
differentiating between the meanings
of area and perimeter when faced
with these terms in a single context.
18
• Perimeter
• Comparing
Overview
Base Ten Block Grid Paper, several
sheets per group, page 93
The Activity
NUMBER • GEOMETRY • MEASUREMENT
Base Ten Blocks
◆
◆
Make this arrangement with four
Base Ten longs and four units
and have children copy it.
◆
Elicit that the value of the longs is 40 and the value
of the units is 4, so the total value of the shape is 44.
◆
Then push the arrangement of blocks together to form
a single L-shape with no spaces between the blocks.
Have children do the same with their arrangements.
◆
Establish that the perimeter of a shape is the distance around it.
◆
Review the fact that each Base Ten unit measures 1 centimeter
(1 cm) on a side.
◆
Have children find the perimeter of the shape (either in units or in
centimeters). Call on several volunteers to explain how they determined that measurement.
Grades 5–6
© ETA/Cuisenaire®
On Their Own
What can you predict about the perimeter of different shapes that have
the same value?
• Work with a partner. Take any combination of Base 10 Blocks whose total value
equals 258.
• Use all your blocks to form any flat shape. Your shape should be no higher than
1 cm. Your shape should not have empty spaces that are completely surrounded
by blocks. Build your shape to follow this rule:
At least 1 complete unit of each block must touch at least 1 complete
unit of another block.
• Record your shape on grid paper. (You may have to tape some sheets of grid
paper together to do this.) Find and record the perimeter of your shape.
• Now use the same blocks to make a different shape. Predict whether or not the
perimeter of this shape is the same as the perimeter of your first shape.
• Record this shape and find its perimeter. How good was your prediction?
• Make more shapes from these same blocks. Continue predicting the perimeter
of each shape and then recording and finding its perimeter.
• Be ready to talk about your predictions and to tell how to compare the
perimeters of shapes with the same value.
The Bigger Picture
Thinking and Sharing
Have children tell what they discovered about the perimeters of their shapes. Then call for
the recordings of the shapes with the shortest and longest perimeters. Post the one with the
shortest perimeter at the extreme left of the board and the one with the longest perimeter at
the extreme right. Have pairs post their shapes, in between, according to perimeter length.
Use prompts like these to promote class discussion:
◆
How did your shapes differ? How were they alike?
◆
Did the blocks you chose to model 258 affect your results? Explain.
◆
How did you predict the perimeter of each of your shapes? Did you make different
kinds of predictions after you had made a few shapes?
◆
How can you explain the differences in the perimeters of the shapes that have
the same value?
◆
What kinds of shapes have the shortest perimeters? What kinds have the longest?
© ETA/Cuisenaire®
A PROBLEM OF PERIMETER
◆
Base Ten Blocks
◆
Grades 5–6
19
Writing
Have children describe what they learned about why shapes with the same
value can have different perimeters.
Teacher Talk
Where’s the Mathematics?
Any group of blocks that children choose to model the value 258 represents
one of 48 possible combinations. These combinations can be comprised of
flats, longs, and units; longs and units; or units alone.
Six combinations of flats, longs, and units have 2 flats.
Flats
2
2
2
2
2
2
Longs
5
4
3
2
1
0
Units
8
18
28
38
48
58
Sixteen combinations of flats, longs, and units have 1 flat.
Flats
1
1
1
1
1
1
1
1
:
:
1
Longs
15
14
13
12
11
10
9
8
:
:
0
Units
8
18
28
38
48
58
68
78
:
:
158
There are 25 combinations of longs and units.
Flats
0
0
0
0
0
0
0
:
:
0
Longs
25
24
23
22
21
20
19
:
:
1
Units
8
18
28
38
48
58
68
:
:
248
Only one possibility uses units alone.
Flats
0
20
◆
Base Ten Blocks
◆
Grades 5–6
Longs
0
Units
258
© ETA/Cuisenaire®
Extending the Activity
Challenge pairs to make several shapes, each with a perimeter of 258 units
(or 258 cm). Have them find and compare the values of these shapes with
the same perimeter.
Some children may realize that shapes made without flats can have the
longest perimeters. The 1-by-258 rectangle has the longest possible perimeter—518 units. The smallest block group, the one made up of 15 blocks
(2 flats, 5 longs, and 8 units), can be used to make compact shapes with the
shortest possible perimeter—66 units—in more than one way.
Some children may go about finding the perimeter of their shapes
by counting all the unit lengths by ones all the way around.
Others may count the edges of the flats and longs by tens, then
count on by ones. Still others may combine these methods and
then subtract 2 wherever one unit length touches another.
Whatever their methods, children are sure to realize the need for
devising good ways of keeping track of their counts and to recount
in different ways to check for accuracy.
Though some children may start out by thinking that different
shapes always have different perimeters, as the class’s findings are
compiled, it is likely to be revealed that some different shapes can
have the same perimeters.
P = 66 units
Children may see that their perimeters all represent even numbers.
They may be able to explain that this is because Base Ten Blocks
are rectangular and are made up of opposite sides with matching
lengths. Each individual block has an even-number perimeter.
When blocks are pushed together to touch along whole-unit
lengths, the perimeter of each individual block is reduced by some
multiple of 2 wherever it touches another whole-unit length. (This
makes odd-number perimeters impossible.)
P = 66 units
After children have formed several flat shapes, they will notice that some
shapes appear to be compact, or “squarish,” while others appear elongated,
or “stretched out.” Children are likely to point out that the most compact
shapes have the shortest perimeters and the most elongated shapes have the
longest perimeters. They may realize that this is because more unit lengths
touch one another in compact shapes, leaving fewer “exposed” unit lengths
to remain as part of the perimeter.
Exploring the range of perimeters possible with the block combinations of
their choice helps children to understand why shapes with exactly the same
value (or area) can have different perimeters.
© ETA/Cuisenaire®
A PROBLEM OF PERIMETER
◆
Base Ten Blocks
◆
Grades 5–6
21