# Lesson 1 How Do You Know That? Develop Understanding

## Jump Start

Notice and Wonder

The following students have observed that the sum of two odd numbers seems to always be an even number, but their peers aren’t convinced that this is always the case. Each student has given a different explanation as to why they think this claim is correct.

Write at least two things you notice and one thing you wonder about the following explanations:

Student 1: I heard one of my teachers say that the sum of two odd numbers is always even, and she knows everything about math. | Student 2: I tried a lot of examples, and I found that the sum of two odd numbers is always even. For example, |

Student 3: I know that an odd number of objects can be paired together in groups of 2, with one object left over. If I combine two such groups together, the extra ones will also pair together, so the sum of two odd numbers will always be even. I drew the following diagram to show what I was thinking: | Student 4: Since an odd number is one more than an even number, I can represent odd numbers algebraically as If I add two odd numbers together I get: The last expression, |

## Learning Focus

Examine ways of knowing that the sum of the angles in a triangle is

How do I know something is true? Are there different ways that I know or accept things to be true?

When I notice a pattern in examples or through experimentation, how do I convince myself that my conjecture is always true?

Regardless of the shape or size of a triangle, is there a characteristic that is the same for all triangles in addition to being a three-sided polygon?

## Open Up the Math: Launch, Explore, Discuss

You may know that the sum of the interior angles of any triangle is

We know a lot of things because we accept it on authority—we believe what other people tell us; things such as the distance from the earth to the sun is

**Experiment #1**

Cut out several triangles of different sizes and shapes. For each triangle, tear off the three corners (angles) and arrange the vertices so they meet at a single point, with the edges of the angles (rays) touching each other like pieces of a puzzle.

### 1.

What does this experiment reveal about the sum of the interior angles of the triangles you cut out, and how does it do so?

### 2.

Since you and your classmates have performed this experiment with several different triangles, does it guarantee that we will observe this same result for all triangles? Why or why not?

**Experiment #2**

Perhaps a different experiment will be more convincing. Cut out another triangle and trace it onto a piece of paper. It will be helpful to color-code each vertex angle of the original triangle with a different color. As new images of the triangle are produced during this experiment, color-code the corresponding angles with the same colors.

Locate the midpoints of each side of your cut out triangle by folding the vertices that form the endpoints of each side onto each other.

Rotate your triangle

about the midpoint of one of its sides. Trace the new triangle onto your paper and color-code the angles of this image triangle so that corresponding image/pre-image pairs of angles are the same color. Now rotate the new “image” triangle

about the midpoint of one of the other two sides. Trace the new triangle onto your paper and color-code the angles of this new image triangle so that corresponding image/pre-image pairs of angles are the same color.

### 3.

What does this experiment reveal about the sum of the interior angles of the triangles you cut out, and how does it do so?

### 4.

Do you think you can rotate all triangles in the same way about the midpoints of its sides, and get the same results? Why or why not?

## Ready for More?

**Examining the Diagram**

Experiment #2 produced a sequence of triangles, as illustrated in the diagrams.

Here are some things we might ask about this diagram:

Will the second figure in the sequence always be a parallelogram? Why or why not?

Will the last figure in the sequence always be a trapezoid? Why or why not?

## Takeaways

Today we learned about four “Ways of Knowing”:

These ways of knowing showed up in our work today when we conjectured:

Which I now accept as true because: (responses will vary)

I noticed:

and I thought about

which led me to believe

So, my way of knowing this conjecture is true is

**Types of Reasoning in Geometry:**

Inductive reasoning: This type of reasoning consists of making conjectures based on experimentation across several examples.

Deductive reasoning: This type of reasoning consists of using:

properties,

definitions,

postulates (),

or theorems ().

## Vocabulary

- postulate
- reasoning – deductive/inductive
- theorem
**Bold**terms are new in this lesson.

## Lesson Summary

In this lesson, we explored different ways of knowing if something is true, such as basing our knowledge on being told by someone in authority, versus basing our knowledge on experimentation or reasoning with a diagram. We examined these ways of knowing in the context of justifying how we know that the sum of the angles in a triangle is

### 1.

Use the diagram to determine which of the symbolic statements are correct and which are incorrect. Then explain why.

#### a.

#### b.

#### c.

#### d.

#### e.

#### f.

A straight line measures

Use this information about a linear pair to find the measure of the missing angle.