Project AMP Dr. Antonio Quesada – Director, Project AMP

Author: Karen Balog and Dee Smith

Title: School Pride Week

Subject: General Math, Pre-Algebra, Pre-Geometry

Grades: 7, 8

Strands:

Strand 3: Number and Number Relations

3: Solve and use proportions.

Strand 4: Geometry

6: Validate fundamental geometric theorems using manipulatives.

Strand 5: Algebra

3: Describe problem situations involving ratios and proportions using

algebraic expression.

8: Relate ratio and proportion concepts to various situations, direct and

inverse.

Strand 6: Measurement

1: Select and compute with appropriate standard or metric units to measure

length.

Topics: Similar Triangles, Ratios, and Proportions

Objectives: The student will measure the height of a goal post and its cross bar using properties of similar triangles. The student will also compute the angles of the similar triangles. Application of these solutions to a real life situation will involve the students working in a small group, using technology to make a scale drawing and create a unique design, and writing a paragraph describing the processes which led to their exclusive outcome.

Materials: a meter stick, a mirror, calculator, a Geometry computer program such as Cabri or Geometer's SketchPad, and worksheets

Expected time: Two or three forty-two minute periods

Description: Students will use mirrors, similar triangles, and proportions to find the approximate height of an object not directly measurable.

Keywords: ratios, proportions, similar triangles, equivalent fractions, metric measurement, tangent, co-tangent

Prerequisites: Metric measurement, knowledge of theorems involving similar angles and triangles, algebraic solution of proportions.

Concepts: Students who approach relationship found in their world in mathematical patterns are better prepared for higher order thinking skills. Proportions create a pathway to these logical associations. A is to B as C is to D. The real world demands classroom discoveries be applied to realistic situations. Students need to articulate knowledge in both the written and spoken word. The students, working in small cooperative groups, will use properties of similar triangles to solve a real life situation. The group will make visual scale drawing of their solution using technology and write a paragraph describing the process that led to the outcome.

Assessment: Evaluation will be based upon class participation, peer and self-evaluation forms, and completion of the written assignment.

Sharing: The student will demonstrate a clear understanding of similar triangles, proportions, trigonometric ratios, scale drawings, and measurement in centimeters by creating a unique decorating scheme for a goal post including cross bar. The student will write a paragraph describing the solution’s development. The solution will be presented to the class.

Results: The student will gain an intuitive understanding of Geometric Theorems dealing with similar triangles. The student will practice viewing relationships in proportions, working to solve real life problems algebraically, and represent their solution to problems visually through a scale drawing and in the written and spoken word.

Proficiency Outcomes: OPT 9th and 10 Grade Objectives

3: Solve and use proportions.

4: Validate fundamental geometric theorems using manipulatives.

5: Describe problem situations involving ratios and proportions using algebraic

expression.

6: Select and compute with appropriate standard or metric units to measure

length.

Learning Objectives: The student will use the concepts of measurement, similarity, and proportions to approximate the height of a goal post and its cross bar. The students will calculate the measures of the similar triangles using the trigonometric ratios. Using a Geometry computer program such as Cabri or Geometer's SketchPad,the student will make a scale drawing showing their crepe paper decoration ideas for the goal posts and cross bar. The student will complete the accompanying work sheets, write a paragraph describing the solution, andpresent their ideas to the class.

Task Overview: A group of three students will set up similar triangles using their height, the distance from their feet to a mirror, the distance from the mirror to a goal post, and the unknown (height of the goal post). The process will be repeated with the unknown becoming the height of the cross bar. The students will make a scale drawing of the goal posts and cross bar using a Geometry computer program. The students determine the number of roles of crepe paper needed to decorate the goal post. A paragraph allows the students the opportunity to describe their work through the written word. The students will also present their decoration ideas to the class.

Learning Strategies:

  1. Review theorems on similar triangles, the algebraic solution to proportions, trigonometric ratios, and measurement in centimeters.
  2. Divide the class into groups. Explain that the group willmeasure the distance from each group member’s eyes to the floor and note this on part one of the work sheet.
  3. The class will proceed to the football field where each group will measure the height of the goal post and it’s cross bar by doing the following:
  4. Place a mirror on a level spot on the ground between a group member and the goal post.
  5. Two group members will measure this distance in centimeters.
  6. The stationary group member will back up until the top of the goal post can be seen in the mirror.
  7. The two other group members measure the distance between the mirror and group member’s feet in centimeters.
  8. The work sheet is filled in including the sketch at the bottom of Part I of the work sheet.
  9. Solve the algebraic equation found in the set up of the proportion to approximate the unknown height of the goal post.
  10. Repeat this process to obtain the approximate height the cross bar intersects the goal post. Have a different member of the group be the stationary person.
  1. Make a sketch of the new similar triangles on Part II of the worksheet.
  2. Solve the algebraic equation found in the set up of the proportion to approximate the unknown height of the intersection of cross bar to goal post.
  1. Measure the distance between the legs of the goal post.
  2. Using a calculator, the congruent angles formed in the mirror will be calculated using trigonometric ratios.
  3. Proceed back to the classroom where the groups will brain storm decoration ideas for the goal post.
  4. The amount of crepe paper needed to decorate the goal post will be determined
  5. A scale drawing of the solution will be made using a geometric computer program.
  6. A paragraph describing the solution will accompany the scale drawing.
  7. The groups will present their solutions to the class. The class will vote on the best design, which will be forwarded to the spirit club.

Tools/Resources:

  • Meter stick
  • Work sheet
  • Cabri or Geometer’s SketchPad
  • Calculator
  • Computer Lab
  • Teacher

Classroom Management:

This lesson on similar triangles will be taught to seventh or eighth grade students. Each class is forty-two minutes in length with approximately twenty students in each class. Teaching time is about two to three days but will vary in the length of time needed, depending on the individual students.

School Pride Worksheet

The Spirit Club would like to decorate the goal post for the big football game this weekend. You will be using concepts of measurement, similarity, proportions, and angles that you have learned to measure the approximate height of the goal post and the cross bar. You will make a scale drawing using The Geometer’s SketchPad to show how you would decorate the goal post.

Part I: Work with your group to measure the height of the goal post in centimeters following these steps.

  1. Measure the distance in centimeters from your eyes to the ground. ______
  1. Place a mirror on a level spot on the ground between you and the goal post you are measuring.
  1. Measure the distance from the goal line to the mirror. ______
  1. Move back until you see the top of the goal post in the mirror.
  1. Measure the distance from the mirror to your feet. ______
  1. Label the distances and heights in this sketch.


  1. Make a drawing showing the similar triangles in the sketch and their corresponding parts.
  1. Write a word proportion using the height of the goal post, its distances from the mirror, the distance of your eyes from the ground, and your distance from the mirror.

______= ______

  1. Write a proportion using the correct measurements and find the height of the goal post algebraically.

______= ______

Part II: Work with your group to measure the height of the cross bar in centimeters following these steps.

  1. Place a mirror on a level spot on the ground between you and the cross bar you are measuring.
  1. Measure the distance of the goal line to the mirror. ______
  1. Move back until you see the cross bar in the mirror.
  1. Measure the distance from the mirror to your feet. ______
  1. Make a sketch of this picture and label the distances and heights.
  1. Draw the similar triangles in the sketch and show the corresponding parts.
  1. Write a word proportion using the height of the cross bar, its distances from the mirror, the distance of your eyes from the ground, and your distance from the mirror.

______= ______

  1. Write a proportion using the correct measurements and find the height of the cross bar algebraically.

______= ______

Part III: The two angles formed by the mirror are congruent angles. Calculate these angles using the following steps.

  1. Is the ratio used to calculate the heights of the goal post or the cross bar the sine, cosine, or the tangent?
  1. Write this ratio and its value. ______=
  1. Use your calculator to find the measure of the angle formed by the mirror. (You will be using the arc functions.) ______
  1. Calculate all of the angles of the similar triangles. Make a drawing and label the angles.

Part IV: You will decorate the goal post for the big game on Friday. The crepe paper that you will be using is 202 cm long and 7 cm wide. Follow these steps to decorate the goal post.

  1. Measure the distance between the goal posts to determine the length of the cross bar. ______
  1. Include any other measurements or information that you will need. Show how you calculate any new measurements you need.
  1. Use the information you have gathered to make a scale drawing of the goal post and cross bar. Show how you will decorate it with crepe paper. Use the Geometer’s SketchPad for this scale drawing.
  1. Estimate how much crepe paper you will need to decorate goal posts and cross bar.
  1. Explain how you arrived at your measurements for the goal posts, the crossbar, and your estimate for the crepe paper.