SWBAT describe lines and circles in terms of the properties of 'direction' and 'distance'. Students will understand how the properties of lines and circles can be used to construct them.

How did an ancient culture create drawings so large they can only be seen from the air? By understanding the properties of geometric objects, of course!

10 minutes

**Team Warm-up**

This lesson opener repeats the **Team Warm-up **routine students were introduced to in last lesson. I may still need to provide some coaching for that as I also introduce Learning Journals. The purpose of the lesson opener for this lesson is to get students thinking about the meaning of the terms ‘line’ and ‘circle’.

As students enter the classroom, I ask them to take a Learning Journal handout from the Resource Center and display the Warm Up prompt on the front board. Students record answers in their journals. While the students work, I walk the room checking on how teams work together to complete the task. To help students write High Quality Answers, I might provide sentence stems on the board: “A line is…”, “A circle is…” At the time limit (3 minutes), I randomly select a number 1-4 and the students who have that seat number represent their team by writing the team’s answer on the front board. Students’ seat numbers are on the Management Mats on their desks. Today, I pick the team scribes at random, because I want to emphasize, as I will again and again during the first weeks, that I expect teamwork, and teams cannot succeed if every student does not participate. If the student whose number is picked is not ready to be the team’s scribe (watch them scramble!), the team does not receive full Team Points for the day’s activities. It takes practice to learn to work as a team, and they will have many awesome opportunities to demonstrate teamwork during the remainder of the lesson!

**Be on the Lookout For**

- If students are looking up terms in their textbooks (good use of resources), are they dividing the work with their team-mates? I coach students to share the work when completing team tasks. This will not come easily to some. Dividing up the work helps teams to use class time efficiently and encourages students to use their peers to help them with their own learning.
*Can you really not trust a classmate to look up a word in the glossary?* - Are students writing high-quality answers in their journals?
- Did they remember to date their journal entries?
- Are students copying textbook definitions that they are unlikely to understand, e.g. “an undefined term in geometry”? I encourage them to dig deeper.
*What do you know about lines? Can you describe one property of a circle? If you had to make a circle right now, how would you do it?*

**Goal-Setting**

Following the warm-up, I display the lesson Agenda and Learning Targets. I let the class know that they all know what lines and circles are…but do they really ** know**? In this and the following lessons, we will dig deep to understand the simplest objects in geometry. By understanding the properties of lines and circles, we will be able to use them to answer questions about the real world. This is the goal of modeling with mathematics (

9 minutes

To motivate students about studying properties of geometric objects using constructions I show *Secrets of the Nazca Lines**. I show the first 7 minutes of the episode (up to the scene where the pilot lands the ultra-lite aircraft and congratulates the host of the show on discovering a here-to-for unknown geoglyph. This leads to a class question: "How could the Nazca people have created these drawings?"

As I set up the DVD, I ask if anyone knows about the Nazca Lines of Peru? I call on students and ask them to share what they know and the summarize by displaying:

**How could an ancient culture create accurate drawings on such a large scale that you have to be in an airplane to see them?**

At the conclusion of the video, I show a few of the some of the Nazca Lines (geoglyphs) using Google Earth: http://www.peruvian-tours.com/nazca_lines_google_earth.html using Google Earth and ask students if they have any theories about how the Nazca could have created these huge drawings. Using geometry, of course. (A student will almost certainly call out that answer.)

After hearing any ideas, I announce that we are going to learn how the Nazca *could *have created accurate drawings without being able to see them. We are going to create our own (slightly less giant) geoglyph. The purpose of the activity is not to uncover the secrets of the Nazca, but to uncover the properties of geometrical objects. I encourage the class to pay close attention to how the different parts of the figure are constructed. Students will be doing the constructing.

**Video Information:**

*Volume 3 of *Digging for the Truth – The Complete Season 1*. The History Channel (2006). Before class you would need to obtain a copy of the video (ISBN: 0-7670-8992-8). I use the intra-library loan service of my local library district.

**Additional resource: **

http://en.wikipedia.org/wiki/Geoglyph

30 minutes

In this activity, students learn how the geometric properties of objects can explain an archaeological mystery (**MP4**). They also get to try some tools and techniques which are often used in the real world, but not often found in a classroom. For example, they may use a rope to create a circle, or sight on a distant object to create a line (**MP5**).

The activity is structured as a Team Jigsaw.

I display Constructing a Geoglyph instructions as I explain to the class that they will be working in teams to construct assigned parts of a geoglyph. Detailed instructions are found in the Constructing a Geoglyph activity handout.

The purpose of this activity is to make students think about the properties of lines and circles. The activity is kinesthetic, and I return to it in later lessons when asking students to think about lines and circles in a new way (for most). The procedures students use to create lines and circles are based on the geometric properties of the objects. A line consists of connected points (irrigation flags) which lie in a fixed direction. That direction may be determined by a pair of points (Anchor person and Distant Object). (I do not quibble about the difference between lines, line segments, and rays at this point.) A circle consists of connected points which are equidistant (because the length of the rope did not change) from a central point (Anchor person).

A line can also be defined as the set of points which are equidistant from two fixed points, and so the last line of the figure is constructed by swinging arcs of the same radius from two centers (a perpendicular bisector). However, I don’t want students to focus on this property of lines just yet. In fact, I anticipate that we may not have time to complete the last part of the geoglyph. It is an activity that can fill 20-25 minutes later. If we do complete the line, however... *We swung arcs using a pair of ropes, yet we ended up with points in a line!* It is a mystery to be explained later.

I begin the activity by assigning tasks to each of the cooperative learning teams. The figure can be completed by as few as three or as many as seven teams of 3-4 students. I begin by asking the teams to read their instructions carefully, rehearse their parts (using erasers or other objects), and ask any questions. When all teams have had a chance to ask questions and appear to be ready, we go outside to a level field next to the school.

I carry rope and irrigation marking flags in a gym bag or backpack. Students should take their belongings with them.

As teams are constructing parts of the figure, I stand with the rest of the class in a group. I remind them that their goial in this lesson is to be able to describe lines and circles in terms of their properties. If they watch carefully, they will see those properties highlighted in their construction.

Questioning Strategies (properties of shapes):

- What is this team doing? What shape do they seem to be making? How can you tell?
- How are they making the shape? What is the purpose of (point out an action or a member of the team)?
- What property of the shape guarantees that their method will produce the right shape? (Just describe it in your own words.)

If student answers include terms like “straight”, “direction”, or “distance”, I may ask them to explain what they mean by those terms. However, I do not expect students to be able to define these terms very precisely or accurately at this point, and I refrain from offering my own definitions. (The explanations tend to be circular, which is probably because these are "undefined" terms in geometry. Nevertheless, we need some working definitions for these fundamental concepts.)

- What does it mean for a line to be “straight”? What is a “direction”?
- What does it mean for the flags to be “the same distance” from a point? How can we be sure that they are all the same distance?

When 10 minutes remain before the bell, it is time to clean up. If necessary, students can complete the lesson close activity outside.

I expect that the arc teams (who construct a bisector) may not have time to perform. It takes time for students to learn to function effectively in teams, and I will probably have to step in to help them understand what they are supposed to do.) I tell the arc teams that they will get their chance in the next lesson.

Before class: Acquire materials (see Constructing a Geoglyph activity). Locate a suitable field for the activity outside. Identify a Distant Object. (See the instructions in the activity.) Scout out the location ahead of time, and be sure to recheck the morning of the lesson (lawn sprinklers)!

5 minutes

**Individual Size-Up**

I keep this lesson close simple using our Individual Size-Up Routine. If we have returned to the classroom, then I display the Lesson Close prompt on the front board and students record their response as a journal entry.

**Recognizing Good Work**

While the class is completing the lesson close activity, I invite a student from each team to assign his or her team a score for the lesson.

If necessary, the lesson close can be completed outside. I ask students to take a seat and take out their notebooks, then I simply give the individual size-up prompt verbally.

**Homework**

For homework, I assign problems #6-8 of Homework Set 1 for this unit. The first three homework assignments give students a chance to shake off the rust, since some will not have done any math over their summer vacation. During the second full week of school, I will give students a diagnostic test in order to get a baseline and to identify students who may be in need of additional support.