RIC-0563 Developing algebraic thinking

PENTOMINO PUZZLES
Teachers notes
Introduction
Looking at
the algebra
Geometrically, a polyomino is a set of squares connected side-byside.
Dominoes are examples of polyominoes. A pentomino is a set of
five squares connected side-by-side. There are twelve pentominoes;
however, only eight pentominoes lend themselves to problem-solving
involving algebra.
Although the pentominoes are all different, the algebra involved is
the same for certain groups.
Group 1
a
b c d e
a
c b d e
For these two pentominoes, we have a + b = b + c + d + e, or a = c + d + e.
The result shows that the digit representing b
can be any digit 0 to 9.
The greatest possible sum for a + b is 17, which comes from 8 + 9 or 9
+ 8. The smallest possible sum for either pentomino can be obtained
with b = 0, a = 6 = 1 + 2 + 3. Therefore c, d and e are equal to 1, 2 and 3
in any order.
Here are some solutions:
7
8
6
9
3 0 2 5
1 9 3 4
1 7 2 3
3 1 4 2
Group 2
A second group contains three pentominoes.
d
b c a b a c
b a c
d
d
e
e
e
For each, a + b + c = a + d + e, or b + c = d + e.
The greatest sum of 22 occurs when a = 9.
The greatest possible sum for two pairs of digits is 13 = 8 + 5 = 7 + 6.
The least possible sum of 5 occurs when a = 0; therefore, the least
possible sum of the two pairs is 5 = 1 + 4 = 2 + 3
Here are some solutions:
5
3
9
1 7 0 6
4
1
7
8
8
6
4
5
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