Domain: Ratios and Proportional Relationships Standard Code: RP3ab & d Teacher Name: Julie Peery, Celeste Gledhill, Alicia Rudd, and Jessica Baum

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  Students will use ratio and rate reasoning to solve real-world problems by reasoning about and the lesson? (i.e., what do you want creating tables of equivalent ratios, tape diagrams, double number line diagrams, and students to know and understand about equations. mathematics as a result of this lesson?)  Students will plot pairs of values from a table to a coordinate plane and make reasonable conclusions about the relationships (in the extension).  Students will solve real world problems using unit rate reasoning.  Students will use ratio reasoning to convert measurement units.  What are your expectations for  Students will choose between four tasks involving ratio and proportions and solve using students as they work on and several strategies that make sense to them. They will record their solution method and be complete this task? prepared to discuss and justify their thinking.  What resources or tools will  Materials and tools students have to use in their - If You Hopped Like a Frog by David Schwartz (if not available, you could use a different book from work that will give them the list attached for similar examples) entry into, and help them - instruction sheet reason through, the task? - tape measures and/or ruler  How will the students work— - connecting cubes independently, in small groups, or - calculators in pairs—to explore this task? - measurement conversion chart if necessary  How will students record and  Students will explore independently and then discuss with a partner who chose to report their work? complete the same task (out of the four choices).  Students will record their work on the instruction sheet and report to the class using a document camera (another option would be to have students record their work on a How will you introduce students to the Launch: Read aposter portion to ofshare the withbook the If Youclass Hopped if a document Like a cameraFrog. Explain is unavailable). how amazing many animals activity so as to provide access to all are in their abilities to run, jump, or lift. Their task will be to find out exactly how amazing the animals students while maintaining the are by calculating their ability to do the same task in relation to the specific animal. Students will refer cognitive demands of the task? to the instruction sheet and begin with one of the four tasks. Then pair with a partner to explore, compare, discuss and share strategies. After students have explored the four tasks, and the teacher has selected students to share for the discuss portion of the lesson, a re-launch will be to plot the pairs of values from one of the four tasks on a coordinate grid and then discuss again looking for patterns, and making conclusions. PART 3:2: SHUPAPRORINGTING AND S DTIUSDCUENTS’SSING EX THEPLO TASKRATION OF THE TASK HAsow stu wdillen ytsou w oorrckh iesndtreapteen theden classtly Choose students to share who have demonstrated the following strategies (and doriscussion in small so that gr yooups,u acc omplishwhat  inWhat the order is the listed information based on that moving we know? from concrete, to pattern seeking, to yqouruest mathematicalions will you ask goals? to—  efficientWhat are uses we oftrying writing to find the outequation (the unknown)? or solving the proportion):  Whelphich a gsorouplution get p statahsrt eddo oryou  What areDrawing we comparing a picture in to this draw problem? what information is given and what is wmaankte tpor haogrvesse shared on the d utask?ring  Can youunknown use a table with to clear help labels.organize the relationships?  ftheocus students’ thinking on  What doUsing you manipulativesneed to do to compare to model your the relationshipsheight with that and of comparisons the animal? in the theclass discussion? In what  Do youratios. see any connections between these problem? okrdeyer ma twhematill theical solutions ideas be  Could Makingthis information a table of help equivalent you find ratios out the to find“output” the missing if the “input” value. was ___? pinr estheen ttaesdk?? Why?  How doUsing you knowthe identity these ratiosproperty are toequivalent? find equivalent fractions for the ratios.  Wassheatss s stupecdifeicn ts’quest ions will  Finding the unit rate through diagrams, division, or a table. undyerstaou asknd isong that of students will—  Writing an equation written using symbols for the unknown. 1key. ma maktehemat senseical of theideas,  Showing conversion between the measurement units either before or problemathematicalm- solving stratidease gthaties, after finding the unknown. or theyou r ewparnest ethemntations? to learn?  *Setting up a proportion set up as an equality and perhaps using the  a2dv. aexpance nstud on,den dts’eb ate, and algorithm of cross-multiplication (*only use this if students have a solid understqauestndiniong the solutions understanding of the concepts and can make connections between one How will you ensure that  What twoother different strategy units and arethis wemethod). comparing? students remabeingin shaengaregded? in  How could you write a ratio to represent the information given? 3. make connections among the task?  Could you draw a picture to help you solve this? the different strategies that You may also want to choose students who have made a mistake and used an  What assistance will you additive What approach are we tryingrather tothan find a multiplicativeout? approach and have a discussion giveare or pwrhatese nqtuested?ions 4. look for patterns? about Could how thereyou write are infinite the comparisons numbers of on equivalent a table? ratios based on multiplying or will you ask a dividing by a number. stu5. dbeginent (or to grfoormup) who generalizations? becomes  QuestionsTry solving to ask the to problem engage studentsin another in way.the discussion and reasoning: quickly frustrated and  HowCan do you your explain two strategies the strategy relate that to ______each other? just shared? requests more direction What will you see or hear that lets  WhatHow is thedoes advantage ______’s of strategy using eachrelate of to the what strategies you did? you chose? and guidance in  TryWhat solving connections the problem to you in the see way so far your in thepartner strategies solved shared? his/hers. you knowsol vthating theall stu task?dents in the class  ChooseAre you another able oneto clearly of the seefour the problems information to solve from using the problemtwo different in the  What will you do if a understand the mathematical ideas that representations.solution method? student (or group) finishes  How could we write what ______did to solve the problem as an you intended for them to learn?  There are two ways to write ratios between two comparisons: within and the task almost betweenequation. What that could will work that mean for any and given can younumber? set up the proportion in both EXTENSION: -Have students use the data from one of their problems and make a table of equivalent ratios (if they haven’t already). Use the value pairs to plot at least three sets on a coordinate grid. Draw conclusions about the graph and choose another set of value pairs (not on their table) that would be on the same line on the graph. - Create more tasks or have students create tasks similar to this using books from bibliography list attached. - Here is a clip from the movie “Monsters vs. Aliens” (with permission) that you can show Bibliography of literature to use with ratios and proportions

If You Hopped Like a Frog by David M. Schwartz - Compares action of insect of animal to humans if they had the same abilities; facts listed in back of the book

If You Made a Million by David M. Schwartz - Begins with one penny and continues earnings throughout the book until it reaches a million; uses interest and unknown amounts If Dogs Were Dinosaurs by David M. Schwartz - Compares size of items to form ratios; facts are listed at the back of the book

Jim and the Beanstalk by Raymond Briggs - Jim helps a giant get eyeglasses, dentures and a wig in the right proportion to his measurements

How Strong is It? by Ben Hillman - Shows real life photos and facts about the strength of items around the world and their comparison to other points of reference

How Big is It? by Ben Hillman - Same as above but comparing size

Biggest, Strongest, Fastest by Steve Jenkins - Can be read as a simple picture book about the record holding animals but includes in sidebars how the ratio relates to humans

The Fattest, Tallest, Biggest Snowman Ever by Bettina Ling - Simple book about students measuring snowmen with arm spans, string, paper clips and tree branch; emphasis is on measurement and using standard units but concepts of using small units taking more to measure the same distance and also comparing ratios can be brought out in the book

Longest, Tallest, Heaviest; Amazing Animal Facts by Calvin Irons - Includes facts and questions already embedded into the text to set up ratios and compare facts

How Big is a Foot? by Rolf Myller - A king orders a bed to be made for the queen and uses his feet to measure the dimensions but the tailor uses the same numbers using his own feet to make the bed

How Big Are They? published by Flying Frog Publishing - More facts about animals and structures that can be used to form ratios to compare between two things Those Amazing Creatures

Name:

Directions: Choose one problem to solve. Use pictures, diagrams, tables or other solution methods to find the answer. Then choose another strategy to solve in a different way. Be prepared to justify your thinking.

Amazing hoppers: A frog that is 3 Amazing jumpers: A flea that is 3 mm tall can inches tall can hop 5 feet. If you are 4 jump up 1.8 meters high. If you could jump ft. 6 inches tall and could hop like a like a flea and are 135 centimeters tall, how frog, how far could you jump? many meters high could you jump?

Amazing weightlifters: An ant that Amazing runners: A female house spider can weighs 1/250 oz. can lift something run 99 times its own length in 3 seconds. If that is 1/5 oz. That means it can lift you are 5 ft. tall and could run like a spider, 50 times its own weight. If you were how far could you run in 1 second? as strong as an ant, and you weighed Approximately how long would it take you to 960 oz., how many pounds could you run 100 yards (the length of a football field)? lift?