Graphing Polygons & Finding Side Lengths

Lesson Overview (1 of 3) Lesson Objective SWBAT graph polygons on a coordinate plane and calculate the side lengths by engaging in the “Polygon Initials” activity. Lesson Description During this lesson, students will review the characteristics and definitions of polygons, other key vocabulary related to polygons and coordinate planes, and how to graph coordinates (if needed). Then, students will participate in a brief guided practice of graphing points to form polygons. Once this launch is complete, students will be introduced to the main activity, “Polygon Initials”, during which they will draw their initials using only horizontal and vertical lines on a coordinate plane. Then, they will label the coordinates of their vertices and copy them on a separate sheet of paper. All students will then pass in those coordinate logs to be shuffled up and chosen at random by the students to graph on a new coordinate plane to figure out whose initials they received, and compare their polygon drawings to that student’s.

Lesson Overview (2 of 3) Lesson Vocabulary Polygon, coordinate plane, coordinate pairs, vertex/vertices, length, quadrant, distance, horizontal, vertical, x-axis, y-axis. Materials 1 “Warm-up” sheet per student, 2 sheets of graph paper per student , 1 “Activity Guide” per student, 1 “Vertex Coordinate Log” per student, 1 “Exit Ticket” per student, pencils & rulers for each student. Scaffolding Students will experience the content of this lesson through a multi-modal approach: using visuals, in writing, and tactile means to reach the objective. In addition to pre-teaching of vocabulary and some guided practice within this presentation, students requiring more help can access online tools via the links provided in “online resources for absent students”. Enrichment Students who complete the lesson activity, and having demonstrated mastery of the objective to the teacher, will: Take it to the next level! Find the perimeter and area of each letter in your initials on the coordinate plane, then explain your method for finding the perimeter and the area. Students will also write a short reflection on why we used only horizontal and vertical lines, and what difficulty would they encounter by using diagonal lines to build polygon initials. Online Resources for Absent Students Review of coordinate plane: http://www.youtube.com/watch?v=xvEyE3_k1C4 Review of graphing coordinates: http://www.mathopenref.com/coordplane.html; Practice: ittc-web.astate.edu/lessonportal/attachments/25-Activity.doc

Lesson Overview (3 of 3) Common Core State Standard 6.G.3: Draw polygons in the coordinate plane given coordinates for the vertices; use coordinates to find the length of a side of joining points with the same first coordinate or same second coordinate. Apply these techniques in the context of solving real-world and mathematical problems. Before and After Students should have familiarity of polygons and their properties from previous instruction (from the Shapes & Designs unit that occurs in Nov-Dec in the scope and sequence), finding area and perimeter (from the Covering & Surrounding unit in Jan-Feb) as well as the basics of graphing and coordinate planes (from the beginning of this 3rd Common Core investigation on Integers and the Coordinate Plane, to occur in May).* The skills exercised in this lesson will carry over to the many 7th grade investigations involving graphing and geometry, and eventually, to the 8th grade content (even the Pythagorean Theorem—the answer to the question “how can we find the length to a polygon’s side on a graph when the side is not perfectly horizontal or vertical?”). Topic Background The word "polygon” comes from the Greek πολύς (polús) "many" and γωνία (gōnía) "corner" or "angle". Today we identify polygons in terms of the number of sides. The idea of polygons is an ancient one. The ancient Greeks understood the regular polygons, and such designs have been dated as early as 7th century B.C.! The Cartesian Plane was invented by French mathematician Rene Descartes in the 17th century, and it played a huge part in the development of higher math, including Calculus! This lesson on graphing polygons and using the coordinates to find the side lengths is an important step in math learning-- not only to further student skill with using Cartesian planes, but to help students make valuable connections between various strategies for finding area and perimeter.

Warm Up OBJECTIVE: SWBAT graph polygons on a coordinate plane and calculate the side lengths by engaging in the “Polygon Initials” activity. On the “6.G.3 Warm-up” sheet, answer the following to get your brain ready for today’s activity: 1) What is a polygon? Explain using mathematical vocabulary and draw an example. 2) a. Make a coordinate plane and graph the points (0, 2), (4, 2), (0, -3), and (4, -3). b. Connect the points to make a shape. c. Is this shape a polygon? Explain why/why not! (5-7 min) 7 mins In-Class Notes Have the Warm-Up sheet ready at student desks with this slide up for them to do as they settle in. Preparation Notes Make copies! Agenda

WARM-UP: Review/Solutions 1) What is a polygon again? How did you describe a polygon? Raise your hand to share out! A polygon is a closed shape that is formed by line segments. (2 min) 7-9 mins In-Class: At this time, hand out the pre-made Vocab sheet to students. Before-class Prep – give either the blank version to be filled in by students (to place in Math Notebooks after) or pre-filled for kids to use as a reference (depending on level of scaffolding needed for the individual students.) **If students are slow to raise their hands and share out how they described a polygon, remind them that there are many ways to describe one, then give more wait time. Ask too: “what shapes did you draw along with your description?” When the class has agreed on an accurate description, prompt them to write it down and draw any example in the space provided on their vocab sheet. Stay on this slide while they write it for a moment.

WARM-UP Review, Cont. 2) X Y Raise your hand if you would like to volunteer to show your work here on the board! How can we find the side lengths of the rectangle we made? (3 mins) 12 mins In-Class Notes: - Have student come up to board to graph his/her work on the empty coordinate plane here PRIOR to clicking the hyperlink for the solution—use only if needed! Prep: - After briefly having the students share their graphs/where they were stuck, click to make the question in yellow appear. Then call on students to come up to the board to label the side lengths of the rectangle and have them explain specifically how they found the side length. (Counting ‘boxes’ along the line segment, using the coordinates). If no one brings it up, ask the class how they can use only the coordinates of the vertices to find the side lengths—and how absolute value comes into play with this!—as the DISTANCE between two points or numbers. Some more helpful questions to guide this discussion: “Which coordinates do we use to calculate the side length of a vertical line in the polygon—the x-coordinates or the y-coordinates? Which coordinates do we use to find the length of a horizontal side of the polygon? Does this make sense? Why?” Solution!

Agenda: 1) Warm Up – (Partners) then review (ALL) 10 mins OBJECTIVE: SWBAT graph polygons on a coordinate plane and calculate the side lengths by engaging in the “Polygon Initials” activity. 1) Warm Up – (Partners) then review (ALL) 10 mins 2) Launch – Key Vocab & Intro (Teacher/ALL), 10-15 mins 3) Explore – “Polygon Initials” (Independent, Pairs), 35-40 mins 4) Summary – Identify your peer & Discussion (ALL), 5 mins 5) Practice – Homework 6.G.3 6) Assessment – Exit Ticket (Independent), 5 mins (1-2 min) 13 mins In-Class Notes Read over briefly Preparation Notes

Launch During today’s activity, you will be graphing polygons and finding the side lengths of the polygons. Then you will use given coordinates to graph objects and decipher what they are. To prepare us better, let’s briefly review some key vocabulary terms! (1 min) 14 min In-Class Notes You may read this aloud to students, or recruit a student to read it aloud. Preparation Notes Agenda

Key Vocabulary Polygon Vertex (s.) / Vertices (pl.) the “corners” of any polygon, where the line segments meet. Coordinate Plane A 2-Dimensional graph with an x-axis and y-axis. Points can be plotted on the plane. Figures or shapes too! *Let’s review how to use coordinate planes: http://www.mathopenref.com/coordplane.html (4 mins) 18 mins This is a vocabulary review for students to make sure they understand and have in their notes. Please remind the students and pause to allow them to continue filling in their vocabulary sheets at this time. **Students should be maintaining Vocabulary Notebooks or Binders in which to keep their vocab sheets! Please direct students through your norm on this and remind them to add their vocab sheet as appropriate.** *When you enter the link for the coordinate plane review, click on “Full screen” on the bottom, then use the “show” options to the right to scaffold as needed for your students with grid lines (recommended) and the pointers (if needed). You may want to “show coordinates” at first, then uncheck “show coordinates” and ask the kids to give the coordinates of wherever you drag point A. When you feel the students are comfortable with this process, move on in their vocabulary sheet, keeping the coordinate plane review site up– what is a Quadrant?—then click to next slide. The link provided here can act as a great refresher on how to correctly graph points and identify coordinates—click on it and show the students! PREP: Decide which of your students, if any, require the “pre-filled” vocab sheet to follow along due to any special needs. Make copies and distribute prior to this slide.

Vocabulary, cont. Quadrant II Quadrant I Quadrant III Quadrant IV Quadrant: One of the FOUR equal sections of a coordinate plane! Y Quadrant II Quadrant I Do you notice a pattern with the signs in each quadrant? (–, +) (+, +) X (3 mins) 21 mins • On this slide, the signs will pop up after the quadrant indicators only by your click—click for the (+, +) to appear, then ask the kids if they can guess the rest of them in order before clicking to show them each one. • Click again for black points to appear, one in each quadrant, for students to volunteer to identify the coordinates. This will allow them, if they need it, to see the patterns of the signs of coordinates based on quadrant. • Once again, pause to allow students to fill in “quadrant” on their vocabulary sheets at this time, then note that for this lesson, we will be using ALL four quadrants! Prior to moving on to the next slide, ask students if they have any questions about anything so far. Quadrant III Quadrant IV (–, –) (+, –)

Explore  ACTIVITY INTRO Graphing “Polygon Initials” (8 seconds) 21 mins In-Class Notes Preparation Notes Agenda

Explore  ACTIVITY SET-UP You will need these tools to graph your “Polygon Initials”: • pencil & eraser • ruler • handouts - 2 sheets graph paper - 1 coordinate log - checklist (1 min) 22 mins In-Class Notes At the end of reviewing this slide & reading it aloud to the kids, distribute these materials in accordance with your classroom routines (table folders, student roles/helpers, etc.) Preparation Notes Have all copies ready! Agenda

 LOOK AT YOUR CHECKLIST to Follow along! (30 secs) 22 mins • Prompt students to take this sheet out of table folders / hand them out now!

Explore: Graphing “Polygon Initials” During this activity, you will… --------------------------------- 1st HALF of Activity ----------------------------------------- 1) Graph your initials as polygons on the coordinate plane using only horizontal and vertical lines. (3 mins) 2) Label the coordinates of each vertex and find all side lengths. (10 mins) 3) In order, write only the coordinates for all vertices in your initials on the Coordinate Log—DO NOT write your name on this sheet! (3 mins) ---------------------------------- 2nd HALF of Activity --------------------------------------- 4) Randomly choose a coordinate log of an unknown peer and graph their initials to reveal their identity! (10 mins) (3 mins) 25 mins In-Class Notes On this slide, you may cold call on students for them to read each step aloud, as you click to make the steps appear. Pass out the checklist for kids to follow along at this point. Preparation Notes 5) Use the graph and coordinates of your peer’s initials to label the LENGTHS of all sides of their initials polygons. (5 mins) Agenda

Explore: KEEP IN MIND as you work… > Use only the initials of your first and last name > Draw each letter so that both fit onto the one coordinate plane on one sheet of graph paper & use all 4 Quadrants > Use ONLY perfectly horizontal and perfectly vertical line segments to form your polygon initials. NO diagonal lines! Example: a student named “Theresa Alvarez” may graph her initials like this  (2 min) 27 mins In-Class Notes Again, you may have students read this aloud, or read it yourself. This is also on their Checklist/Reference sheets. Spend a few moments looking over the example – how all the lines of the polygon initials are perfectly parallel to the x-axis or perfectly parallel to the y-axis – and that we must do this to easily find the side lengths of each initial. Preparation Notes Agenda

Need an exemplar? Take a look! (-4, 4) (4, 4) (3 min + as needed during the Explore activity) 30 mins In-Class Notes If needed, spend some time asking the students to practice identifying all other coordinates of these initials, and finding the length of all line segments on it using the coordinates—write these in on the board, or type them in as the students say them! Remember to stress that the students must find ALL coordinates for their vertices and ALL side lengths during the Explore. You may also leave this slide up as the kids work, so they can have an idea of what their finished product should look like. Preparation Notes (-2, -4) |––––––– 6 un.–––––––| Agenda

Let’s Begin! Do the First 3 Steps On this Sheet! (15 min) 45 mins • With additional exemplars readily available for students who want them, instruct students to begin the first half of the activity: their own polygon initials, and labeling vertices. • Click the stopwatch to set the online stopwatch feature for 15 mins (adjust as needed for you group of kids) and make the kids aware of their time constraints. You can leave the stopwatch up on the corner of your screen with this slide so students may self-manage. • Again, if your kids need the visual, go back one slide to the exemplar and hand out the Additional Exemplar sheet. When the timer sounds, click next slide! **EXTENSION ALERT: For any student who finishes these 3 steps fast and needs another challenge, have then find the area and perimeter of their initials and record each on the back of their worksheet! You may also ask them to record an explanation of how they found the perimeter and area of their initials.

NEXT STEP… On your Coordinate Log  -3 3 0 3 Write each vertex’s coordinates IN ORDER, as you ‘trace’ around one initial, then the next. You may use any vertex as the starting point for each letter. (-3, 3) (0, 3) (3 mins ) 48 mins • Hand out “Coordinate Log” but instruct students NOT to write on it yet—just to follow along on the slide for now. • IMPORTANT—Once the kids get the idea of how to do the “T” in the example (and letters like it!), direct their attention to the “A”.  “What do letters like “A” have that the “T” doesn’t have (a HOLE!) So, how would you (the students) like to represent it on the Coordinate Log without it getting confusing?” Take a moment to let them ponder this, and to let a student(s) chime in their ideas. A good solution could be to write the word “LIFT” beside the coordinate pair that begins a hole, once students have already listed in order the coordinate pairs for the entire outside perimeter of the letter. Whatever it is, have the class quickly agree on a way to designate it and model it for them if needed prior to moving on. • When ready, click on the stopwatch again and set the timer to 3 mins (adjust a min or two if needed) and let the kids know the time limit. Circulate and confer as the kids work to track their understanding and look for any common errors. • When timer sounds, click next slide.

Please pass in your completed “COORDINATE LOGS” FREEZE! Please pass in your completed “COORDINATE LOGS” (2 mins) 50 mins • Collect all Vertex Coordinate Logs, shuffle them up, and redistribute to students. Ask them to double-check that they do not accidentally have their own sheet, then send them off to complete the 2nd half of the activity. (NEXT SLIDE) **If your class is only 50 mins long, this is a good stopping point. Give HW and tell the students that we will be finishing the activity and debriefing it the following day. Create an appropriate Do-Now to get the kids recalling what they did during the 1st half of the activity, review it, then proceed to the next slide!**

LAST STEPS! Do these 2 Steps using a peer’s “Coordinate Log” (15 mins) 65 mins • Hand out the “Classmate’s Polygon Initials” worksheet, instruct the kids to once again use their rulers to draw an X and Y axis that cut the sheet in half horizontally and vertically. • Reiterate the process quickly or ask a student to explain the directions in their own words to the class: Graph the peer’s coordinates in order, connecting the points as you go. “LIFT” where instructed to graph any “holes” the letters may have. Then find and label the side lengths of your peer’s initials. At the end, write your classmate’s name on the bottom of your paper to reveal their identity. • Ask the kids if they have any questions • Click the timer and set it for 15 mins (or whatever your time allows)—your kids may not require all 15 mins. Let them know how much time they have to complete the task. On this Sheet!

Summary: Let’s wrap up and discuss! • If you have not already, please identify whose initials you graphed, and compare your graph and measurements with theirs! Does your work and your peer’s work match? Why or why not? • What did you find difficult during the “Polygon Initials” activity? What was easy to you? • How can you find the side lengths of polygons using only the coordinates of the vertices? Explain and/or show! (5 mins – as needed in class) 70 mins In-Class Notes Show one question at a time; click to reveal the next question. Please note that the third bullet question is a challenge question and may be tough for students. To help orient them, you may need to click back to the exemplar slide and point out the coordinates for one particular line segment. If students continue to struggle to put their observations into words, you may wish to provide a sentence starter like “When the x-values for the line segment are the same, what information do we use to find the length? What about when the y-values are the same in the line segment?” Preparation Notes You may choose to have the kids to write their answers down on the back of their checklist prior to discussion, if time permits. Agenda