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Lesson 2 Do You See What I See? Develop Understanding

Learning Focus

Examine features of diagrams to determine the story of how the diagrams were built.

Write a paragraph to prove a conjecture that surfaces when analyzing a diagram.

What can I learn by analyzing the features of a geometric diagram?

What assumptions am I making when I interpret the features of a diagram?

Can I tell the story of a diagram: what feature of the diagram was drawn first, because everything else depends on its existence, and what features were drawn later?

How can I write a concise description justifying something new that I noticed while analyzing a diagram?

Open Up the Math: Launch, Explore, Discuss

In the previous task, How Do You Know That?, we saw how the following diagram could be constructed by rotating a triangle about the midpoint of two of its sides. The final diagram suggests that the sum of the three angles of a triangle is .

Triangle ABC with Angle A with a green arc, angle B with a blue arc, and angle C with a red arc. Triangle A'BC with Angle A' with green arc, angle B with red arc and angle C with blue arc. Triangle ABC and Triangle A'BC share side BC. Triangle A'CC" with Angle A' with blue arc, angle C with green arc, and angle C" with red arc. Triangle A'CC" and Triangle A'BC share side A'C.

This diagram “tells a story” because you saw how it was constructed through a sequence of steps. You may have even carried out those steps yourself.

Sometimes we are asked to draw a conclusion from a diagram when we are given the last diagram in a sequence of steps. We may have to mentally reconstruct the steps that got us to this last diagram, so we can believe in the claim the diagram wants us to see.

a venn diagram with the triangle ABC in the middle
a venn diagram with the triangle ABC in the middle

1.

For example, given that and are the centers of the two circles that intersect at point , what can you say about the triangle in this diagram?

2.

What convinces you that you can make this claim? What assumptions, if any, are you making about the other figures in the diagram?

3.

What is the sequence of steps that led to this final diagram?

4.

What can you say about the triangles, quadrilateral, or diagonals of the quadrilateral that appear in the diagram? List several conjectures that you believe are true.

Given:

a venn diagram with the triangles ABC and ABD in the middle

5.

Here are some possible conjectures you might have made as you thought about the construction of the previously given diagram. Select one of the following statements and write a paragraph that would convince your partner that the statement is true. Your partner should work on the other statement. Then work together to revise and refine each of your proofs.

a.

and are congruent, isosceles triangles.

b.

Quadrilateral is a rhombus.

6.

Together with your partner, select one of the following conjectures and write a paragraph convincing someone else that the conjecture is true.

  1. , , and are congruent triangles

  2. The base angles of the isosceles triangles in the diagram are congruent

  3. The diagonals of the rhombus bisect each other

  4. The diagonals of the rhombus are perpendicular

  5. The diagonals of the rhombus bisect the vertex angles

Work individually on your first draft of the proof for 5 minutes. Think about the sequence of statements you need to make to tell your story in a way that someone else can follow the steps and construct the images you want them to see. Then work together to revise and refine your proof using the given prompts—you may be asked to share your proof with your peers.

Note: Since some of your peers will prove statement 6A, you may use it as a true statement in proving any of the other statements, 6B-6E.

Ready for More?

Now pick a second claim from 6A–E and write a paragraph convincing someone else that this claim is true. You can refer to your previous proof, if you think it supports the new story you are trying to tell.

Takeaways

Working with diagrams is central to geometric thinking.

Like any other text, a diagram is written by an author and has to be read by a reader, since every diagram is intended to tell a story.

That is, a diagram:

Today, we used diagrams to prove the following theorems about a specific isosceles triangle and the diagonals of a rhombus formed by two intersecting circles that shared a common radius: (These theorems will be revisited for more general cases in future lessons).

Lesson Summary

In this lesson, we learned that diagrams are built in consecutive steps, each step depending upon previous steps. When we can tell the story of how the diagram was built, we can also identify additional features that might be true about the diagram and gain insight into how to prove these new conjectures. Today we wrote paragraph proofs about the things we noticed in a diagram to verify that our observations were true.

Retrieval

Fill in the following statements in a way that makes sense. Be prepared to justify your answer.

1.

If a triangle is a right triangle, then the theorem can be applied to find the side lengths.

2.

If a rigid motion transformation (translation, rotation, reflection) occurs then the corresponding parts of the pre-image and image will be .

3.

If triangles are congruent then every corresponding part is .

Perform the indicated transformations.

4.

Translate

A coordinate plane with x- and y-axis with 1-unit increments. A quadrilateral is located at points (4,2) (2,4) (4,6) (8,4). x–5–5–5555y–5–5–5555000

5.

Rotate the quadrilateral counterclockwise around the point

A coordinate plane with x- and y-axis with 1-unit increments. A quadrilateral is located at points (1,1) (-2,2) (-3,6) (0,5) and a point located at (3,1). x–5–5–5555y–5–5–5555000