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Measuring a Flagpole with a Pocket Mirror

7th grade nonfiction passage about similar triangles: 271 words, six questions, an answer key and vocabulary.

7th grade nonfiction similar triangles 271 words

Measuring a Flagpole with a Pocket Mirror

Climbing a flagpole to measure it would be difficult and unsafe, but a small mirror and a tape measure can do the job from the ground. The trick relies on a simple fact about light. When light bounces off a flat mirror, it leaves at the same angle at which it arrived.

Place the mirror flat on the ground between you and the pole. Then step backward until you can see the very top of the pole in the mirror. At that spot, a ray of light from the top of the pole hits the mirror and bounces up to your eyes. Two triangles have formed. One runs from the top of the pole down to the mirror and along the ground to the pole's base. The other runs from your eyes down to the mirror and along the ground to your feet. Because the light meets the mirror at matching angles, the triangles have the same shape, even though one is much bigger. Mathematicians call them similar triangles.

In similar triangles, matching sides grow by the same ratio. Suppose your eyes are 1.5 meters above the ground, you stand 2 meters from the mirror, and the mirror is 20 meters from the pole. The mirror is 10 times as far from the pole as it is from you, so the pole must be 10 times as tall as your eye height. That makes it 15 meters tall.

Measure carefully, because a small mistake near the mirror grows into a large one at the top of the pole. Surveyors and builders use the same kind of reasoning to find heights and distances that they cannot reach directly.

Questions

  1. What two tools do you need to measure the flagpole from the ground? in the text

  2. Where should the mirror be placed? in the text

  3. What fact about light makes the trick work? in the text

  4. What are triangles called when they have the same shape but different sizes? in the text

  5. How tall is the pole in the example? in the text

  6. Why does a small mistake near the mirror cause trouble? think about it

Words to know

mirror
a smooth surface that shows a picture of what is in front of it
angle
the opening between two lines that meet at a point
similar
having the same shape but not always the same size
ratio
a comparison of two amounts by division
surveyors
people who measure land and distances
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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 · 229 words · Flesch-Kincaid grade 2.9

Climbing a flagpole to measure it is hard and not safe. But you can measure it from the ground. You need a small mirror and a tape measure. The trick uses a fact about light. When light hits a flat mirror, it bounces off at the same angle it came in.

Put the mirror flat on the ground between you and the pole. Step back until you see the top of the pole in the mirror. Now light from the top of the pole hits the mirror and bounces to your eyes. Two triangles are formed. One goes from the top of the pole to the mirror and then to the base of the pole. The other goes from your eyes to the mirror and then to your feet. The angles at the mirror match. So the triangles have the same shape. One is just bigger. They are called similar triangles.

In similar triangles, matching sides grow by the same ratio. Say your eyes are 1.5 meters high. You stand 2 meters from the mirror. The mirror is 20 meters from the pole. The mirror is 10 times as far from the pole as from you. So the pole is 10 times as tall as your eyes. It is 15 meters tall.

Measure with care. A small mistake near the mirror becomes a big mistake at the top. Surveyors and builders think this way too.

▲ This passage · 271 words · Flesch-Kincaid grade 6.3

Climbing a flagpole to measure it would be difficult and unsafe, but a small mirror and a tape measure can do the job from the ground. The trick relies on a simple fact about light. When light bounces off a flat mirror, it leaves at the same angle at which it arrived.

Place the mirror flat on the ground between you and the pole. Then step backward until you can see the very top of the pole in the mirror. At that spot, a ray of light from the top of the pole hits the mirror and bounces up to your eyes. Two triangles have formed. One runs from the top of the pole down to the mirror and along the ground to the pole's base. The other runs from your eyes down to the mirror and along the ground to your feet. Because the light meets the mirror at matching angles, the triangles have the same shape, even though one is much bigger. Mathematicians call them similar triangles.

In similar triangles, matching sides grow by the same ratio. Suppose your eyes are 1.5 meters above the ground, you stand 2 meters from the mirror, and the mirror is 20 meters from the pole. The mirror is 10 times as far from the pole as it is from you, so the pole must be 10 times as tall as your eye height. That makes it 15 meters tall.

Measure carefully, because a small mistake near the mirror grows into a large one at the top of the pole. Surveyors and builders use the same kind of reasoning to find heights and distances that they cannot reach directly.

■ Harder · 299 words · Flesch-Kincaid grade 10.6

Scaling a flagpole in order to measure it would be both difficult and hazardous, yet a small mirror and a tape measure are enough to determine its height without leaving the ground. The method depends on a basic principle of optics: when light strikes a flat mirror, it rebounds at exactly the same angle at which it arrived.

Lay the mirror flat on the ground somewhere between yourself and the pole, and then back away until the very top of the pole appears in its surface. At that position, a ray of light travels from the top of the pole to the mirror and reflects upward into your eyes. Two triangles have been created. One extends from the top of the pole down to the mirror and then along the ground to the base of the pole, while the other extends from your eyes down to the mirror and then along the ground to your feet. Since the light meets the mirror at matching angles, the two triangles share the same shape, although one is far larger than the other. Mathematicians describe such figures as similar triangles.

In similar triangles, corresponding sides are related by the same ratio. Imagine that your eyes are 1.5 meters above the ground, that you are standing 2 meters from the mirror, and that the mirror is 20 meters from the pole. Because the mirror is 10 times as far from the pole as it is from you, the pole must be 10 times as tall as your eye height, which makes it 15 meters high.

Careful measuring is essential, since a slight error close to the mirror is multiplied into a considerable one at the top of the pole. Surveyors and builders rely on this same style of reasoning to calculate heights and distances that they cannot reach directly.

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 mirror, angle, similar, ratio and surveyors 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 3 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 6 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 "Measuring a Flagpole with a Pocket Mirror"?

It was written for 7th grade. It has 271 words and works out to a Flesch-Kincaid grade level of about 6.3. 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. 5 ask for something the text says directly and 1 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 3 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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