The+Introduction

 =Introduction:= 
 * 1) Observation- What do you notice? Why is it happening?
 * 2) Categorization- Which figures tile? What properties allow them to tile?
 * 3) Generalization –Do you think there are any other figures that will tile? Why?
 * 4) verification –How do you know you are right?
 * 1) Hand out pieces of paper to group students.
 * 2) First, you will work on a problem with regular polygons that you do not know the answer for. However this lesson is not about polygons, the focus is about thinking and reasoning about math using the polygons as tools.
 * 3) Definition: a regular polygon is one in which all sides and angles are equal (sentence strip posted on board - Gema). We show them a picture of a regular polygon and ask why. Show them a picture of a nonregular polygon (Kim) and ask why.
 * 4) In each of these baskets your group is receiving several tools you may or may not use.
 * 5) Tilings have been used throughout history. For example these are tilings from a mosque. For example, in this tiling, what do you notice? (all the same shape) Explain tiling a plane. In this class we are going to talk about tilings that are vertex to vertex.
 * 6) If you use multiple copies of a single regular polygon, which polygons will tile the plane vertex to vertex? Explain how you determined which ones worked and which ones didn't. You may want to record your thinking. (Worksheet with question on it. Question on top, rest empty )
 * 7) This is the task we want you to work on however, we want you to start by thinking by yourself. Please think alone for 2 minutes.
 * 8) Share your thinking with a partner.
 * 9) Working with your partner, use whatever you need to answer this question. (Time: 10-15 min)

=Discussion:= >> __Anticipated Response:__ Student draws, Student fits third piece in and it doesn't work >> > Anticipated Response: There can't be any space. If the the sides are 3,4,6 then they can tile (Response: You only tried up to 10, how do you know others won't work?) If the angle measures of a regular polygon divide 360 then the polygon will tile.
 * 1) Students explain observations in front of room.
 * 2) We record these observations on a chart paper.
 * 3) Questions to ask to prompt to students who are stuck:
 * What do you notice about the figures that tile? – share with partner
 * Try tiling with a hexagon and try tiling with an triangle. What happens? What's going on?
 * <span style="color: rgb(255, 0, 28);">Try tiling with a hexagon and try tiling with an octagon. What happens? What's going on?
 * Try tiling with a triangle and try tiling with an square. What happens? What's going on?
 * 1) <span style="color: rgb(0,0,255);">Work with your partner to ake a conjecture about the properties of the regular polygons that tile. Write that conjecture down.
 * 1) Now we are going to work on how you know for sure. This is called proving.
 * 2) What are the properties of a regular polygon, what allows them to tile?

If a regular polygon tiles a plane vertex to vertex then the angle measures have to divide 360. a. 360 degrees around a point b. The vertices all have to meet at that point because it tiles c.

c. So the interior angles of the regular polygons must all add up to 360 because it tiles d. There must be a whole number of these angles around a point because it tiles e. All the interior angles of a regular polygon have the same measure f. So the interior angle of a regular polygon that tiles, must divide 360 (by part d) g. So the interior angles must have measurements of g. regular polygons have angles of 60, 90, 108, 120

Hook: Show students several pictures of M.C. Escher
 * 1) <span style="color: rgb(0,0,255);">What is the largest possible angle that a polygon that tiles can have? Why? What is the smallest? Why?
 * 2) <span style="color: rgb(0,0,255);">In your pair, what did we just do? Write each step down on a note card.
 * 3) <span style="color: rgb(0,0,255);">Have board in three parts. Teacher categorizes two kinds of moments in shared experience. Tell students to place note cards in areas of board that seem similar. If unsure, put your card in the third space.
 * 4) <span style="color: rgb(0,0,255);">Two specific examples: 1. Played with shapes. 2. Determined maximum angle.

Activity:
 * Question: Will we need to clearly frame the goal of this lesson?** Do we need to be more explicit that it's about **the form of reasoning,** not about tessellations?

idea from that: We're going to look at two different ways that mathematicians use .... ... time sponge?

"We're going to look at tessellations.......but this lesson is not about tessellations."

1. Give the students several envelopes with polygons in them (an evnvelope with triangles, an envelope with square, etc...) 2. Ask them to fill out the following table while primarily focusing on the questions :"What do you notice?," "Why is it happening?" no - students need this scaffolding:** what is it that you are trying to teach? Where is the core lesson? What role does this part play in the lesson? Idea: Somebody has made a table in investigations: Call on that person (try to utilize the results of a student's work - hope for a table!)
 * Shape Name || What do you Notice? || Why is it happening? ||
 * ||  || <span style="border-collapse: collapse; -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"> ||
 * Questions: is this table giving too much?
 * ||  || <span style="border-collapse: collapse; -webkit-border-horizontal-spacing: 2px; -webkit-border-vertical-spacing: 2px;"> ||
 * Questions: is this table giving too much?
 * Questions: is this table giving too much?

3. Now looking back at your chart, what are some generalizations you can make? Anticipated Responses: a. Difference between square, triangle, hexagon compared to everything else...these actually tile. b.

4. Do you believe that there are other shapes that can do this, why or why not? Anticipated Responses:

5. What do you notice about the angles in these tilings? Anticipated Response: Circle overlaying each common vertex

Looking at items # 4-6 in your third list (head-to-head comparisons of tilings with particular polygons), are those intended as fallback, scaffolding-type questions in case students don't come up with relevant conjectures from the more open-ended questions? Or are they intended as part of the core questioning sequence? --Traci
 * Questions at this point: Please put IN RED**
 * Idea: What propoerties about these polygons allow them to "tile?" FIND THAT PROPERTY.

Tiered levels: Easy, Medium, Hard

Hard: Extension: Come up with another shape that tiles the plane. Can you do this with an irregular shape? Can you create a pentagon (irregular) that will tile the plane?