Elementary POW, February 17-21, 1997


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Fractured Farmland

While flying over farmland, a pilot notices the rectangular shape of the fields below. She sketches the lines that divide the fields.

When she returns to the airport, she wonders how many different rectangles can be formed by the lines drawn.

Puzzle Picture

HINT: Don't forget that a square is also a rectangle.

(This problem was taken, with permission, from Michael DiSpenzio's book called Critical Thinking Puzzles.)

This week's mentor are from Paul Nass' geometry class at Georgetown Day School in Washington, DC.

Correct Solutions submitted by:

Highlighted Solutions

Many people just sent in a number.  However some did explain their reasoning
and you can see the different ways people thought about this problem.  We
had a good time reading everyone's solutions and writing responses.  Many thanks
to all of you for the experience.  Paul, Geri and GDS 8th grade geometry.

**************

Katie, Lauren, Claire, Daina, Tim, and Steve from Mrs. Brennan's 4th grade
class at the Drexel Hill School of the Holy Child in Pennsylvania shared the
following.

We found 18 different rectangles.  We skectched six of the figures shown of
the field.  We labeled the three triangles 1, 3, and 5 and we labled the
squares 2, 4, and 6.  This gave us 6 rectangles.  Then we noticed that 1
and 2 together gave us 7 rectangles and 3 and 4 together gave us 8
rectangles and 5 and 6 together gave us 9 rectangles.  We skecthed our
third figure and noticed that 1 and 3 together gave us 10 rectangles and 2
and 4 together gave us 11 rectangles and 3 and 5 together gave us 12
rectangles and 4 and 6 together gave us 13 rectangles.  In our fourth
sketch 1, 3, and 5 gave us 14 rectangles and 2, 4, and 6 gave us 15
rectangles.  In our fifther sketch 1, 2, 3, and 4 gave us number 16 and
3,4,5 and 6 gave us 17.  Our last sketch to give us 18 was 1, 2,3,4,5,and
6.

********************
Alex Tuchel-Veyhl, Brian Powers, Emily Shaw, Nathan Strauss, Zach Selke,
Nate Litz, Carlyn Dean-Burris, Sarah Haspiel, Elena Slavin, Jessica Casey,
Ariel Franks from Bob  Coulter's Third Grade Math Class at the Forsyth School
in St Louis, MO made a chart:

We labelled the six sections like this:

a d
b e
c f

We then created all of the rectangles we could:

a,b,c,d,e,f,ab,bc,abc,de,ef,def,ad,be,cf,adbe,becf,abcdef

These 18 are all we could find. We think we covered all of the ones that
are possible, since we worked to find all of the possible combinations of
one, two, three, four, and six rectangles when we made our master list.


************************
>From Stuart Wyss's 4th grade Math Class at Kutz Elementary School in Central
Bucks, Doylestown, PA
found EIGHTEEN squares/rectangles:

Steven Withers, Michael Rose, Derek Denard, Josh Dukeman, PJ Raduta,
Allie Alpers, Philip Gant, Grant Castor, James Ingoldsby, Adam
Stolzenberg, Robbie Sheerr, Forrest Durey, Danielle Rotzell, Chris Coia,
Ben Samuels, David Salantri, Colyn Locker, and Steven Vernaci.

Two of the students, Steven Vernaci and Adam Stolzenberg handed in their
diagrams of how they did it.  They each drew the diagram more than 20
times.  Then with a colored marker, drew a different rectange on each
diagram.  This way they could check if they had a repeat or not!  Other
students did it completely mentally.

Thank you for a neat problem!  (oh, I hope 18 is the correct answer.  I
couldn't find any more either!)
********************
Sara Walcott and Anna Grotberg, period 1 grade 5
Highcrest Middle School Wilmette, Illinois
Mrs. Freeman's class

There is a large rectangle divided into 6 parts. The three rectangles
on the left side we labled 1,2,3. The three squares on the right side
we labled A,B,C. This made it simpiler to solve the problem. These are
the solutions we came up with for how many rectangles are inside this
large rectangle:
1. large rectangle
2. 1
3. 2
4. 3
5. A
6. B
7. C
8. 1,A
9. 2,B
10. 3,C
11. A,B
12.B,C
13.1,2
14.2,3
15.A,B,C
16.1,2,3
17.1,A,2,B
18.2,B,3,C

In conclusion there are a total of 18 rectangles.
********************
Derek Caldwell, Heather Comerci, Andrew Dang, and Sam Henderson, grade 5
Western Salisbury School, (Mrs. Geschel),  Allentown, PA also found 18
rectangles on the field.

SIZE OF RECTANGLES       HOW MANY
     1x1                    3
     2x1                    2
     3x1                    1
     4x1                    3
     5x1                    3
     2x4                    2
     2x5                    2
     3x4                    1
     3x5                    1
                         -------
                           18

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