Isosceles Introduction
Primary Market
Education, Secondary Ed
Character Focus
Cooperation, Teamwork
Items Needed
1 Toobeez set per group, 1 meter stick, 1 notebook per student, pen or pencil
The Activity Time
50 - 60 minutes

Objectives

  • Introduce and define the isosceles triangle
  • Define and model the basic method to determine the area of isosceles triangles
  • Introduce and practice more complex approaches in determining the area of isosceles triangles
  • Work with others in a cooperative learning style
Character Focus
Teamwork/Cooperation
The Challenge

Students will be able to calculate the area of isosceles triangles without directly measuring the height.

Preparation

Setup Time: 5 minutes

Mini-Lesson Time: 25 minutes

Materials

  • 1 Toobeez set per group
  • 1 meter stick
  • 1 notebook per student
  • pen or pencil
Activity Plan

Time: 50 – 60 minutes

Instruction: Whole class, groups and individuals

Space: Medium

Activity Setup

1. Teachers should construct a model of an isosceles triangle. Two sides of this triangle should be built using two 24” Toobeez tubes. The third side should consist of two 16” Toobeez tubes with a sphere in the center connecting to the other two tubes in a triangular arrangement.

Activity Mini-Lesson

1. The class should be led in a discussion to determine the unique qualities of the model. Teacher Note: Students should realize that two sides of the triangle are equal in length.

2. After the class arrives at the appropriate observations, teachers should present the definition of an isosceles triangle on the board. Students should copy this definition in their notebooks.

3. Ask the class to recall any math strategies used in studying triangles. Teacher Note: The desired answer is the Pythagorean theorem. If this has not previously been studied, precede this lesson with Activity #2 “Pythagorean Puzzler,” which offers an excellent introduction.

4. After the class recalls the Pythagorean theorem, ask the class whether that theorem can be directly applied to the model of the isosceles triangle.
Teacher Note: Students should realize the model is not a right triangle and the Pythagorean theorem does not apply.

5. Teachers should introduce the area equation for triangles:

Area = ³ base x height

6. Ask the class the strategy for determining the base length.

7. After students realize the base can be directly measured, have a volunteer measure the model’s base length with a meter stick and report the result to the class.

8. Ask the class: “Can any side of this model be measured to determine the height of the triangle?” Allow a discussion so ideas can be introduced.

9. Once it is agreed upon that no side can be directly measured to determine height, the teacher should insert a 16” Toobeez tube extending from the center sphere of the base to the sphere at the apex of the triangle.

Helpful Hints

  • Be sure to review these tips prior to beginning the activity, and if necessary, share reminders with the group during the activity.
  • Teachers should have students review their notes on the Pythagorean theorem
  • Although there are isosceles right triangles, remind students that this lesson is focused on non-right isosceles triangles
  • Reinforce and remind students of the equation used to determine the area of a triangle

Here are available Training Options!

Activity Instructions

1. Divide the students into groups.

2. Read aloud the following Activity Challenge Box to the group.

Challenge: Students will be able to calculate the area of isosceles triangles without directly measuring the height.

3. Teachers should instruct students that the same area value arrived at in the setup can be attained without measuring the height.

4. As an introductory y clue, groups should be instructed to try and use the Pythagorean equation to solve their problem. Allow the groups ample time to arrive at a solution.

5. In the event that no group offers a solution, present the strategy on the board by slowly introducing individual steps and providing time between each step for the groups to brainstorm.

6. The solution:

  • Demonstrate the Toobeez model and assign it the variable h for height. Point out that two right triangles have been created by this tube.
  • Additionally, point out that the base, represented by a,has been divided in half by h tube. Each half can be represented by 1J2a. The hypotenuse, represented by c, remains unchanged.
  • If we plug these three variables into the Pythagorean theorem, the result is:
  • (½a^2)+h^2=c^2
  • Now rearrange the equation to solve for h
  • h^2=c^2-¼ a^2
  • Substitute the values for a and c measured during the setup and solve the equation for h
  • Plug the calculated value for h into the triangle area equation and solve for the area
  • After the activity, gather the class together and pose the following question: “How do you find the area of an isosceles triangle?”
  • Finally, move to the”Activity Discussion and Processing” section of the activity.

Assessment

  • Have students draw non-right isosceles triangles and determine the area by direct measurement and solving for the value of h. Compare your results.
  • Ask the students to explain why the Pythagorean theorem cannot be directly applied to determine side length in isosceles triangles.

Activity Discussion and Processing

To close the lesson, end with a group discussion about what was learned during the activity. Circle up the group and work through the following questions. If possible, record the group’s responses on flip chart paper so all comments are displayed.

  • What makes a triangle isosceles?
  • What ot her features, besides two equivalent sides,define an isosceles triangle?
  • How did this activity require you to demonstrate trust and test your communication skills?

Here are some additional topics for discussion:

  • Calculating the area of a triangle
  • The qualities that define an isosceles triangle
  • The use of the Pythagorean theorem to determine the height of an isosceles triangle
  • The team effort used in attempting to manipulate the Pythagorean equation  to solve for the height of an isosceles triangle

Activity Variations

1. Additional qualities.

Have the students draw various isosceles triangles and measure the angles. Ask them to determine additional qualities ofthe isosceles triangle.

2. Extension/Follow up.

Provide students with the height and base length of an isosceles triangle. Have the students solve for the length of the two additional sides of the triangle.

Albert J. Reyes, MA and B. Michael McCarver, JD are the principals of Lingua Medica LLC, a partnership of writers, researchers and analysts specializing in science, mathematics and medical education. The goal of Lingua Medica is to create successful educational materials by fusing quality writing with effective presentation formats.
All Activities of Albert J. and B. Michael

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