Thinking Through a Lesson Protocol (TTLP) Template s8

Domain: Geometry Standard Code: 2.G.3 Teacher Name: Amy Bowden, Rachael Bibeault, Cathe Runyan

Adapted from: Smith, Margaret Schwan, Victoria Bill, and Elizabeth K. Hughes. “Thinking Through a Lesson Protocol: Successfully Implementing High-Level Tasks.” Mathematics Teaching in the Middle School 14 (October 2008): 132-138.

PART 1: SELECTING AND SETTING UP A MATHEMATICAL TASK What are your mathematical goals for 2.G.3 Geometry the lesson? (i.e., what do you want Reason with shapes and their attributes students to know and understand about Content Objective: Students will partition circles and rectangles into two, three, or four equal shares. mathematics as a result of this lesson?) Recognize that equal shares of identical wholes need not have the same shape. Language Objective: Students will describe the shares using the words halves, thirds, half of, a third of, etc., and describe the whole as two halves, three thirds, four fourths.

 What are your expectations for Expectations: students as they work on and  Students will make sense of the problem and persevere in solving it. complete this task?  Students will create representations to show their reasoning.  What resources or tools will  Students will reason abstractly and quantitatively students have to use in their  Students will construct viable arguments and critique the reasoning of others. work that will give them  Students will model with mathematics. entry into, and help them  Students will work as a team and use mathematical discourse in solving the task. reason through, the task? Resources:  How will the students work—  Patty paper or cut square paper independently, in small groups, or  Math journal with grids or dot paper in pairs—to explore this task?  Glue sticks  How will students record and  Writing implements report their work?  Visual representations of unusual equal partitioning Grouping:  Students will work independently and share in small groups and finally whole group Recording/Reporting  Record thinking in math journals How will you introduce students to the Hand out patty/square papers and direct students to fold the square into halves. (Don’t give directions activity so as to provide access to all on how to fold) Direct everyone to hold up their folded paper. How many different ways did we fold; students while maintaining the can we think of any other ways to fold halves? Instruct students to open their folded paper and share cognitive demands of the task? observations on what they see. (Both halves are equal shares, same shape, same size – congruent, etc.) Task #2 Geometry

PART 3: SHARING AND DISCUSSING THE TASK How will you orchestrate the class Teacher will observe students while working. Make a note of which students discussion so that you accomplish should present their solutions. your mathematical goals? PART 2:W ShUPichP soORlutionTING p aStThsU DdENTS’o you EXPWhatLORA doTI youON OwantF THE students TASK to share? As studwenatsnt wtoo hark vine dsharedepend ednutlyring  WhatDrawings do you notice about your partitioning?  Reasoning or inthe small groups, what  Can you think of any other ways to show equal shares?  Justification that the shares are equal questionsclass w illd yiscussion?ou ask to— In what  Do all the shares have to be the same shape? What order do you want students to present?  helporder a g wroupill theget st solutionsarted or be  What are you thinking? mapreske nptreodgr? essWhy on? the task?  Students with predictable simple drawings share first.  HowStudents are you with going more to showunusual your drawings. reasoning?  fWochuats stuspecdeifnicts’ q uestthinkingions wonill  ExplainStudents your who process; show howpartitions do you of know same yoursize butshares different are equal? shapes. theyou ask so that students will—  DoesExtensions everyone in your group concur? k1ey. ma maktehemat senseical of theideas Discussion Questions in themathematical task? ideas that  If the shapes within the rectangle are different, can the size (area) still be  asseyssou stu wadnetn tts’hem to learn? 2. expand on, debate, and the same? Are they still halves (thirds, fourths)? understanding of  Are the partitions (halves, thirds, fourths) in separate identical rectangles keyq muestathemation theical solutions ideas, being shared? the same size if the shapes are different? problem- solving strategies,  Can one-half ever be smaller than one-fourth? or3. the ma rkeep conresennectionstations? among the different strategies that  What things are similar? What things are different?  advance students’ are presented? How will you know students have achieved the learning outcome? understanding How will4. y loouo ensk foru rpea tthatterns? If students Students are stuck, will assess use wherethe terms the halves,frustration thirds, is, andfourths, refer etc. back to above questions students5. r emabeginin teonga forgmed in  Students are engaged in the task. the task?generalizations? If students Students’ finish early, drawings add extension will represent their thinking.  What assistance will you give or what questions Extensions: What wwillill y youou s eeask or a hear that lets  Challenge students to look for other ways to partition the shapes. you knowstu dthatent al(orl stu grodup)ents w inho the  Present visual representations of more unusual fraction partitioning if not class becomes previously shared, i.e. the yin-yang symbol, the Google symbol, etc. (some understaquicndk thely f rmausttrhematated aincald ideas that visuals are linked in the wiki). you interenqduestsed for m themore d tirecto learn?ion  Challenge students to work with other shapes (triangles, parallelograms). and guidance is  Silly joke you can use if you choose: Give students a small circle or rectangle solving the task? and say if they can tear it exactly in half, you’ll give them a quarter. If they  What will you do if a succeed in tearing it neatly in half, tear one of the halves in half to give them student (or group) finishes (a quarter). Ha ha. Reason with shapes and their attributes 2.G.3

Part A: Draw circles and divide them in half as many different ways as you can think of.

Share your drawings with your group. How many different ideas did you come up with? Prepare to share some examples with the whole group. Now partition the circles into thirds and share. Finally partition the circles into fourths and share.

Discuss your observations of your results. How many equal shares do you have? What do you call each of these shares.

Part B: Draw rectangles and divide them in half as many different ways as you can think of.

Share your drawings with your group. How many different ideas did you come up with? Prepare to share some examples with the whole group.

Now partition the rectangles into thirds and share. Finally partition the rectangles into fourths and share.

Discuss your observations of your results. How many equal shares do you have? What do you call each of these shares?

Part C:  Using two congruent rectangles, partitioned differently, prove that one fourth of one rectangle is the same size (has the same area) as one fourth of the other (recognizing that equal shares of identical wholes need not have the same shape).  Prove that shares within the same rectangle, which are different shapes from each other, are equal in area.  Discuss and justify, can one-half ever be smaller than one-fourth?