· Activity Name: Midpoint Connections: Triangles and Squares

· Objectives:

The students will investigate properties of midpoints, congruence, and similarity in triangles and squares. Students will also be introduced to the relationships between linear and 2-dimensional measurements. Depending on the level of students, proofs can also be introduced with this lesson.

· EALR/Standards:

1.2 understand and apply concepts and procedures from measrement.

1.3 understand and apply concepts and procedures from geometric sense.

2.1 investigate situations.

2.3 construct solutions.

3.1 analyze information.

3.3 draw conclusions and verify results.

4.1 gather information.

4.2 organize and interpret information.

· Materials:

Rulers (1 per student)

Protractors (1 per student)

Compasses (1 per student)

Scissors (1 per 2 students)

Scratch paper (at least 2 per student)

· Teacher Notes

o Prerequisites for the learner:

Definition of Midpoint

o Teacher hints for the activity:

This activity will work on any schedule. It takes about 45 minutes for the triangle part, and the rest can be done on a second day or as a continuation. Have students work in groups of four with each person keeping a list of findings.

o Introductory questions:

What is a midpoint?

What happens to a triangle that has its midpoints connected in a circular fashion?

o Wrap-up questions:

What did you discover about the triangle with its midpoints?

Would the same things happen if we used an acute triangle? (Obtuse?)

What did you discover about the square?

Were there any findings that held for both the square and triangle?

What would happen if we used a pentagon or some other polygon?

o Solutions:

Introductory questions: A point exactly halfway between two given points.

The answers here will depend on the level of student and prior

knowledge. Accept all answers and then ask the students to verify

them during the investigation.

Wrap-up questions: 1) Answers will very, but here are some examples:

Forms 4 smaller congruent triangles: 1 is upside down

Area of each is ¼ the original triangle

They are all similar to the original triangle

Forms 3 parallelograms (overlapping)

Area of each is 1/2 the original triangle

Forms 3 trapezoids (overlapping)

Area of each is ¾ the original triangle

Can fold to a regular tetrahedron

2) Not all are the same

3) Answers will very

4) Both create one more shape then number of sides

5) Number of shapes created would be one more than the side number.

o Assessment suggestions:

Verify that students participated and understand the concepts by collecting their “findings” papers. Ask students to repeat this investigation using sketchpad to make it dynamic.

· The Activity:

Have the students divide into groups of four, take out paper and pencil for recording their findings, and place the materials in the middle of each group’s table. Students should all begin by constructing/drawing an equilateral triangle that has side length of 15 cm. Next, have the students locate the midpoints on all three sides. To make it easier for whole class discussions, have the students label the midpoint between vertices A and B as D, between B and C as E, and the last one F. Connect the midpoints to create a polygon.

With this as a starting point, ask the students to discover as many findings as they can. This could include relationships, shapes, locations, etc. Try to not correct “findings,” instead, ask leading questions so that the groups will continue to look for more.

When the groups have slowed down, have the class come together to make a “master” findings list on the board. If a group questions a finding made by a different group, spend some time letting the students discuss the validity of the finding in question. If the discussion does not come to a conclusion, place the finding and a question mark on the board and allow the students to investigate it further at a later time.

When the list is completed, ask the students to identify which statements will hold true for all triangles. Allow time for the students to repeat the process using triangles of their own choosing. When a student discovers an example that goes against a finding, have him/her share it with the class and then mark the finding off the board.

When this process is completed, have the students repeat the investigation with a square.

· Extensions:

Ask the students to repeat this investigation with sketchpad. After they set up their shapes, ask the students to check their findings on convex as well as concave polygons.