## How Tall Is That Flag Pole?

### Amelia McCurdySanta Rosa District Schools

#### Description

Students learn that similar triangles have sides that are proportional. They will use this knowledge to determine the height of a flagpole. This method was used by the ancient Egyptians to determine the height of the great pyramids.

#### Objectives

Relates the concepts of measurement to similarity and proportionality in real-world situations.

The student understands the geometric concepts of symmetry, reflections, congruency, similarity, perpendicularity, parallelism, and transformations, including flips, slides, turns, and enlargements.

represents and applies geometric properties and relationships to solve real-world and mathematical problems including ratio, proportion, and properties of right triangle trigonometry.

The student knows the appropriate operations to solve real-world problems involving integers, ratios, rates, proportions, numbers expressed as percents, decimals, and fractions.

The student finds measures of length, weight or mass, and capacity or volume using proportional relationships and properties of similar geometric figures (for example, using shadow measurement and properties of similar triangles to find the height of a flag pole.

The student finds measures of length, weight or mass, and capacity or volume using proportional relationships and properties of similar geometric figures.

The student identifies congruent and similar figures in real-world situations and justifies the identification.

#### Materials

-Geoboards, graph paper or dot paper
-Protractors
-Textbook with similar figures or worksheet
-Flag pole or any tall, straight object on the school’s grounds
-Tape measure (metric or English)
-Straight stick about 1 yard long
--Flag Pole Worksheet- (1 per student)
-Calculators

#### Preparations

1. Make copies of worksheet.
2. Make transparency of -Flagpole Problem.-
3. Gather materials.

#### Procedures

Day One:

1.-Write the word -similar- on the overhead and discuss its everyday meaning. Students should realize that the word means-‘looks the same but different-; -having certain things in common.-

2. Ask, “How are similar figures used in real life?” Possible answers: scale models, photo enlargements, and copy machine reduction and enlargement.

3. Discuss how the word similar is used in math. Geometric shapes can have the same shape but are not necessarily the same size.

4. Explore triangles that are similar in shape but not in size. Give students the coordinates to construct a triangle on the geoboard, graph paper or dot paper. Ask, “How could we construct a similar triangle?” Possible answer is to double each of the sizes. Construct the new triangle.

5. Compare and contrast the first and second triangle. Make a T chart of the similarities and the differences. Students should discover that corresponding angles are congruent. Show students how the corresponding sides of the triangle are proportional.

6. If needed, you may want to repeat the above to convince students that similar figures have corresponding angles which are congruent and corresponding sides are proportional.

7. Work a few example problems and then have students solve several similar figure problems from their textbook or a worksheet that you have provided.

8. Go over problems and clarify any misunderstandings. After students are familiar with how to set up and solve problems involving similar figures, they are ready to determine how tall the school’s flagpole is.

Day Two:

9. Pass out -Flagpole Worksheets.- Explain to the class that they will be using similar triangles to determine the height of the school’s flagpole. The method they will use is called -shadow reckoning.- The Greek mathematician Thales first used it in ancient Egypt. He had visited Egypt and amazed the people when he determined the height of the great pyramids. (see weblink for further information)

10. While showing students the flagpole transparency, explain shadow reckoning. First, they will measure the shadow cast by the flagpole, then hold a stick vertically nearby, and measure the shadow that the stick casts. The height of the flagpole and the stick are a vertical measurement, so they have two right angles. The sun is a fixed point in the sky and so very far away that we can assume its rays are parallel. The sun’s rays create congruent angle at the top of the pole and the top of the stick. Therefore, the two right triangles are similar by AA, and we can write a proportion and find the needed height, x.

11. Take the class outside to measure the flagpole’s shadow and the shadow of the stick. Be sure all measurements are in the same units. Be sure to measure the height of the stick.

12. Once in the classroom, students may complete the -Flagpole Worksheet.-

#### Assessments

Check accuracy of worksheet