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Why Pentagons Cannot Tile a Bathroom Floor

5th grade nonfiction passage about tiling with shapes: 210 words, six questions, an answer key and vocabulary.

5th grade nonfiction tiling with shapes 210 words

Why Pentagons Cannot Tile a Bathroom Floor

Look at a bathroom floor, and you will probably see squares. Tiles must cover the floor with no gaps and no overlaps, and squares do that job well. Mathematicians call this kind of covering a tessellation. Only a few of the simplest shapes can do it by themselves, and the reason comes down to the corners.

When tiles meet at a corner, their angles must fit together to make a full turn. A full turn is 360 degrees. A square has four corners of 90 degrees each, so four squares fit around one point. 4 times 90 is 360, and nothing is left over.

Triangles work, too, when all three sides are equal. Each corner measures 60 degrees, so six triangles fit around a point. Regular hexagons, which have six equal sides, also work. Each corner is 120 degrees, so three of them meet perfectly. That is why you see hexagon tiles in some floors and why bees build their honeycomb cells in that shape.

Now try a regular pentagon. Each of its corners is 108 degrees. Three pentagons make 324 degrees and leave a gap. Four would need 432 degrees, which is too much, so they would have to overlap. No number of them adds up to exactly 360. A floor made only of regular pentagons would always have holes.

Questions

  1. What two things must tiles avoid when they cover a floor? in the text

  2. How many degrees make a full turn? in the text

  3. How many squares fit around one point? in the text

  4. How many degrees is each corner of a regular hexagon? in the text

  5. Why do three pentagons around a point leave a gap? think about it

  6. Why would a floor of only regular pentagons always have holes? think about it

Words to know

tessellation
a pattern of shapes that cover a surface with no gaps
overlap
to lie partly on top of something else
degrees
the units used to measure the size of an angle
hexagon
a shape with six sides
pentagon
a shape with five sides
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One passage, three levels

The same topic and facts written three ways, so a mixed class can read and talk about one text. Each group’s sheet shows only a shape; the key says which is which.

● Easier · 188 words · Flesch-Kincaid grade 3.6

Look at a bathroom floor. You will likely see squares. Tiles must cover the floor with no gaps. They must not overlap, either. Squares do this well. Math people call this covering a tessellation. Only a few simple shapes can do it alone. The reason is in the corners.

When tiles meet at a corner, their angles must make a full turn. A full turn is 360 degrees. A square has four corners of 90 degrees. Four squares fit around a point. 4 times 90 is 360. Nothing is left over.

Triangles work, too, if all three sides are equal. Each corner is 60 degrees. Six triangles fit around a point. Regular hexagons work, too. A hexagon has six equal sides. Each corner is 120 degrees. Three of them fit just right. That is why you see hexagon tiles on some floors. Bees build their honeycomb in that shape, too.

Now try a regular pentagon. Each corner is 108 degrees. Three pentagons make 324 degrees. That leaves a gap. Four would make 432 degrees. That is too much, so they would overlap. No number of them makes exactly 360. A floor of only regular pentagons would always have holes.

▲ This passage · 210 words · Flesch-Kincaid grade 5.5

Look at a bathroom floor, and you will probably see squares. Tiles must cover the floor with no gaps and no overlaps, and squares do that job well. Mathematicians call this kind of covering a tessellation. Only a few of the simplest shapes can do it by themselves, and the reason comes down to the corners.

When tiles meet at a corner, their angles must fit together to make a full turn. A full turn is 360 degrees. A square has four corners of 90 degrees each, so four squares fit around one point. 4 times 90 is 360, and nothing is left over.

Triangles work, too, when all three sides are equal. Each corner measures 60 degrees, so six triangles fit around a point. Regular hexagons, which have six equal sides, also work. Each corner is 120 degrees, so three of them meet perfectly. That is why you see hexagon tiles in some floors and why bees build their honeycomb cells in that shape.

Now try a regular pentagon. Each of its corners is 108 degrees. Three pentagons make 324 degrees and leave a gap. Four would need 432 degrees, which is too much, so they would have to overlap. No number of them adds up to exactly 360. A floor made only of regular pentagons would always have holes.

■ Harder · 230 words · Flesch-Kincaid grade 8

Glance down at an ordinary bathroom floor and you will probably notice squares. Tiles are supposed to cover a floor completely, with no gaps and no overlapping edges, and squares accomplish that job beautifully. Mathematicians call this kind of covering a tessellation. Only a handful of the simplest shapes can manage it by themselves, and the explanation comes down to what happens at the corners.

Wherever several tiles meet at a corner, their angles must fit together to form one complete turn. A complete turn measures 360 degrees. A square has four corners of 90 degrees each, so four squares can surround a single point. Since 4 times 90 equals 360, nothing is left over.

Triangles succeed as well, provided that all three of their sides are equal. Each corner measures 60 degrees, so six triangles fit around a point. Regular hexagons, which have six equal sides, also work. Each of their corners is 120 degrees, so three hexagons meet perfectly. That is why hexagon tiles appear on some floors, and why bees build their honeycomb cells in that shape.

Now consider a regular pentagon, whose corners each measure 108 degrees. Three pentagons add up to only 324 degrees, which leaves an awkward gap. Four would require 432 degrees, which is too much, so they would be forced to overlap. No whole number of pentagons totals exactly 360. A floor made solely of regular pentagons would always be full of holes.

Download as one PDF

How to use it

  1. Press Read it big (or B) to put the text on the projector, or Full screen (F) to fill the whole screen.
  2. Ask the questions one at a time and use Show answer when the class has decided. A shows or hides every answer.
  3. Print student copy for the class, or Print with answers for yourself.

Ideas for class

  • Before reading, put tessellation, overlap, degrees, hexagon and pentagon on the board with their meanings from the Words to know section and have students predict what the text will be about.
  • Read it aloud once while students follow along, then have them reread it alone or with a partner. At a calm 100 words a minute it takes about 2 minutes to read aloud.
  • Use the Read it big button (or B) to put the text on the projector, and Show answer on one question at a time so the class commits to an answer first.
  • Question 5 asks the reader to infer. Ask students to point to the words in the passage that led them to their answer.
  • Ask students to write one question the passage did not answer, then look for the answer in a book or on a classroom screen.
  • Print the student copy for independent work and keep the answer-key copy for yourself.

Related

Questions

What reading level is "Why Pentagons Cannot Tile a Bathroom Floor"?

It was written for 5th grade. It has 210 words and works out to a Flesch-Kincaid grade level of about 5.5. That is a formula's estimate from sentence length and syllables, and it is not a Lexile or a guided reading level, so check it against your students.

What do the questions ask?

There are six. 4 ask for something the text says directly and 2 ask the reader to work something out from clues. Each has an answer in the key, and the answers are written as a teacher would accept them, not as the only possible wording.

How long does it take to read?

About 2 minutes read aloud at 100 words a minute. For a one-minute fluency check, paste it into the fluency timer and use the word count.

Can I print it?

Yes. Print student copy gives the passage, the questions with lines to write on, and the word meanings. Print with answers gives the same page with the answers filled in. Both are made in your browser.

Is this passage from a published book?

No. It was written for Peabody Teach, so there is no book to buy and no login to read it.

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