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Exploring Specific Content Area Proportional Reasoning



YDI Math Research Lesson/Lesson Study

Session 2


In Attendance: Renee, Charles, Josh, John, Susan, Gerry, Billy, Moses, Louise, Mark


We started off our second session by reading the notes from our last session and reviewing the decisions and choices we made.


We broke into pairs, and each team focused on one facet of the “Big Picture Goals in math instruction that we want to develop through our research lesson.



During our lesson, 

we want students to:


Why This Goal is Important 




  • have the opportunity to be challenged in a situation that is unfamiliar


  • Students will take the test in likely alien surroundings and have to put the work to use in a place they are not comfortable or familiar with 
  • Students will be measured in an unfamiliar situation, so they need to bring it
  • Adults need to perform in unfamiliar situations all the time, in life as well as in the classroom, and we need practice
  • We want students to be able to comfort and reassure themselves in unfamiliar situation (having access in skills to remain confident)
  • This permeates into the bigger picture in life
  • This is important for us as teachers as well
  • If I have a set way of responding to pre-HSE students, how do I shift what I know to HSE students - it works the same with students, we want them to ask themselves, "What do I know that might be useful? How can I shift that understanding into this new situation?"
  • We want students to be flexible and adaptive and so we have to give them time and experience in situations that require them to be flexible and adapt
  • Students build more confidence when they overcome something challenging  
  • We want students to build up a tolerance of confusion, an expectation of puzzlement, so that when it happens they can experience it, and then move forward. I don't want students to get confused, look around the room, and feel like their confusion is because of them. I want them to understand that math problems require thought and often require pause and creativity. 




  • inventively solve problems finding their own approach, using their own unique toolbox of understanding including (a) concepts and strategies built in math class, and (b) real world knowledge/math from everyday life (not just money)



  • We want students to develop the ability to think and not be afraid to make mistakes - they are so used to math class being “Here’s the formula, now let me do it just like that”  
  • We want students to go through the struggle, maybe not always getting right answer but understanding that they can always learn something from the process
  • It’s like if we all had to get to Times Square - We’d all get there in our own way, by our own avenue
  • Use their own creativity and focus on their own process
  • Louise’s world shifted to high-definition as she was learning to read and seeing words in the world – wanting to read every sign… “I want to see math in the world” 
  • (Josh – Fibonacci sequence, ratios in world)
  • Messages we receive and internalize about math – fixed/growth mindset – society builds up this message of math – who can do it and what it is 





  • feel (a) capable of bravery, (b) comfortable and committed to trying and making mistakes and (c) assertive towards their own sense-making


  • fear and bravery are present in classrooms at all times (even in students 
  • students manage households and do it well – disconnect between household math and classroom math
  • mistakes as learning opportunities (this has come up in all of them) – there are other areas where mistakes are understood as part of the process – but less so in math
  • what can we learn from writing instruction in terms of this allowance/separation within students about first draft and who they are - Can we incorporate writing in math and bring along this idea of revision
  • having a fixed mindset – even if I know the answer I am scared to come forward or be the first one to come forward – As teachers we want to convey to students, "You don’t have to have the answer right, but what do you think"
  • I will make a effort to not use the word “problems” or “opportunities” 
  • Misconception that if you are good at math you are smart – he’s good at math so I’m going to stick in my corner
  • You don’t have to be a math person to be competent in problem-solving in math
  • Stereotype threat – I don’t want to fulfill the stereotype, so I won't try 
  • Affirming and confirming I am not good at math 
  • Students want to make math mathy and overcomplicate things because they thing it should be complicated – they think math should be an uneasy feeling, not a feeling of “this makes sense to me”
  • Fixed mindset versus growth mindset (the work of Carol Dweck) - research has shown that how a person conceptualizes intelligence plays a large role in their ability to learn and take chances. Struggling students with a fixed mindset think "I am just not a math person" as opposed to "I am capable. I have not learned all the math I want and need to, but I can." 




  • be eager to challenge themselves, both with the given work and seeking further problems and challenges. When they get the answer, they will not just say what's next and want to go on to the next thing - they will say, "What else can I learn from this?"



  • The world is more loving and accepting place when we continue to look for more nuance
  • Some things have answers and some things don’t, I want young people to apply this kind of sense-making in other parts of their lives
  • Doing this in math is something
  • I don’t do this in mathematics – but I do it in other realms of life and I want to explore why that is
  • In their academic careers (and life in general), students may find themselves in classroom situations with teachers they do not connect with - having this goal prepares students to takes responsibility and ownership over their own learning. 



We also decided that from this point forward, we would not refer to math problems as “problems” but as “opportunities”.



Then we shifted our conversation to the content focus that our group chose – proportional reasoning, and specifically helping students learn how to recognize when proportional reasoning is appropriate and when it is not.


We talked about the difference between proportions (as traditionally taught and thought of, “this over this equals this over this”) versus proportional reasoning. 



How is proportional reasoning different from a proportion?



  • Proportions can be useful but they can also contribute to the idea that math has to look in a certain prescribed way
  • In real life, like with recipes, we usually use proportional reasoning - in part because we do it mentally.
  • A proportion is more, "This is a formula and this is how you do it"
  • With proportions, once you set it up, it is not proportional thinking anymore. It becomes a multiplication/division problem, solving for x
  • Proportional thinking is about trial and error
  • Proportional reasoning is about shifting recipes and thinking about how it would taste (mixtures, like OJ from concentrate) 
  • Proportional reasoning is about a special and dynamic relationship that can be scaled up and scaled down, but maintains its continuity





Then we reviewed the reading, “Three Balloons for $2”, specifically focused on

the four categories of proportional reasoning “opportunities”.


Part Part Whole


Associated Sets

Well-Known Measures



Mrs. Jones put her students into groups of 5. Each group had

3 girls. If she has 25 students, how many girls and how many

boys does she have in her class?



Ellen, Jim, and Steve bought 3 helium-filled balloons and paid

$2 for all 3 balloons. They decided to go back to the store and

buy enough balloons for everyone in the class. How much did

they pay for 24 balloons?



Dr. Day drove 156 miles and used 6 gallons of gasoline. At

this rate, can he drive 561 miles on a full tank of 21 gallons of




A 6" × 8" photograph was enlarged so that the width changed

from 8" to 12". What is the height of the new photograph?


  • In a PPW problem, a subset of a whole is compared with its complement or with the whole itself


  • Students tend to use informal methods of reasoning


  • Part Part Whole problems lend themselves to counting, matching and building up strategies that don't require advanced proportional reasoning 




  • Associated-set problems relate two quantities, which are not ordinarily associated, through a problem context 


  • Students tend to use a higher level of proportional reasoning with this type of problem


  • The language of ratio is elicited more naturally when students are forced to think about two sets, not typically associated, as a composite that relates one to the other in the context of the problem


  • We decided that the Rose Problem from the last all YDI math training (see below) was an associated sets 




  • Problems involving well-known measures express relationships that are well-known entities or rates (like speed, which is the ratio of distance and time) 


  • For some students, familiarity with well-known measures (like speed and price) may facilitate proportional thinking, but for others the familiar language may allow them to mask their lack of proportional understanding


  • Growth problems express a relationship between two continuous quantities, such as height, length, width, or circumference and involve enlarging/stretching or reducing/shrinking 


  • Often hardest type for students


  • Unlike PPW or Associated Sets, (which involve discrete quantities), growth problems involve continuous quantities, which are more difficult to represent with objects or drawings 



The article also talked about ways/levels of proportional reasoning



Level 0: Non Proportional Reasoning


  • Guesses or uses visual clues
  • Is unable to recognize multiplicative relationships
  • Randomly uses numbers, operations, or strategies
  • Is unable to link the two measures 



Level 1: Informal reasoning about proportional situations


  • Uses pictures, models or manipulatives to make sense of situations
  • Makes qualitative comparisons 











Terry's work on the Rose Problem demonstrates how informal reasoning can help us visualize proportional situations



Level 2: Quantitative Reasoning


  • Unitizes or uses composite units
  • Finds and uses unit rate
  • Identifies or uses scalar factor or table
  • Uses equivalent fractions
  • Builds up both measures 




Our very own Susan's work on the Rose Problem is a great example of Quantitative Reasoning



Level 3: Formal proportional reasoning


  • Sets up proportion using variables and solves using cross-product rule or equivalent fractions
  • Fully understands the invariant and covariant relationships






Recommendations for Teaching Proportional Reasoning from "Three Balloons for Two Dollars"


  1. Instruction should begin with situations that can be modeled or visualized 
  2. Qualitative comparisons should be introduced before numerical comparisons are made or missing values found
  3. Beginning instruction should emphasize informal reasoning with PPW and associated-sets problems
  4. After students can use informal reasoning strategies, quantitative strategies like unit rates and scalar factors can be developed.
  5. Well-known measure and growth problems can be introduced to students who don't need to work with models
  6. Gradually, a full range of quantitative strategies should be encouraged for solving missing-value problems 
  7. Setting up formal proportions using variables and applying the cross-product rule should be delayed until after students have had opportunity to build on their informal reasoning and develop an understanding of the essential components of proportional reasoning 



Next we worked on a few opportunities that required proportional reasoning:

The Water to Rice Activity & The Two-Gear Activity



Water and Rice: Keeping it in Proportion



To start off the conversation, Mark asked who had worked across each row and who had answered all the way down the second column before making they way down the third column. 

Who worked down columns and who worked across? Why?


Renee: I wanted to do the math first, so I went down the 2nd column (To Renee, the 2nd column is the math, the third was not)


Billy has done this exact problem with his students to great effect. He has his students actually make rice


Gerry- I wanted to see 


Moses – Isn't the third column more subjective? – My mother is 87 and she needs needs soft rice - are there right or wrong answers?


We talked about how the third column is not actually asking will the rice taste good or bad - it is asking whether the rice would be too hard or too soft. It is asking students to reason and think deeper upon recognizing that a proportional relationship is not sustained. 


What is the third column – common sense or math, and are those different?


Renee: The second column is computation, the third is reasoning


The second column can be answered using a formal proportion, but it can also be determined using informal or quantitative reasoning


Can you answer the third column without knowing the answers to the second column? Do you know whether it is too hard or too soft if you don’t know whether it is keeping the continuity?


Moses – the third column is also math – you have to compute to get the outcome and then reason from there


It is easier to say no than to say what is bad - the second column is all about whether it keeps the proportional relationship, but for the third column you have to figure out how  


The third column makes you think more – the nature of the question makes you think about why


What about the question of subjectivity? – that actually plays out in the second column 


Billy – one of the things that problem does is it goes from written word into numbers – with students it is important to talk about the original ratio at the very beginning - You can ask questions like, What do you notice about the relationship between the rice and the water? What does that mean, 2 cups of water for every 1 cup of rice? And why would I want to change the recipe at all - Why would I use 4 cups water and 2 cups of rice?


Charles – it accomplishes the real world relativity – they are familiar with the kitchen they will be able to relate quicker – people who have more hands on experience, muscle memory


Louise – imagined herself making the rice – I worked across – At one point, she said to Mark (with some frustration in her voice), "I wish I could just see it" to which I replied, "Can you draw a picture that would allow you to see it?", which she did 


Renee appreciated the fractions – they need to be introduced to it

Josh didn’t like the fractions for the graph – when students have to plot thirds is hard - though this problem only has halves.

One benefit to the fractions is it further encourages students to use reasoning as opposed to setting up a formal proportion.


Follow up Problem: Ask students to “Come up with a few ratios/amounts of rice to water that will result in rice that is too soft. Then come up with some that will result in the rice being too hard)


A few words on the Graph: 


Helps move students away from the yes and no – there are things you can do that get beyond the specific answers you are tasked to find – you produce answers to produce something else. Students work on the chart and then they 


If you do the follow up, have students go back to the graph and plot the too hard points and too soft points. Ask them, "What do you notice?" – This is a way to talk about how graphs can help us see things in different ways and answer different questions in a visual representation. Once thing they can see is that the points that are in proportion form a straight-line (with a slope of 2, but let's not get ahead of ourselves). They amy also notice that the points that don't maintain the continuity are not on the line. They may even notice that all the combinations that will result in mushy rice are above the straight line, and all the hard rice is below the straight line.


Gerry asked, "What if students connect all the dots? (not just the points that are in proportion with 2 cups water/1 cup rice and are on the line)"


John talked about letting one of his students explain something to the rest of the students in a language that John doesn't speak - it gave him the opportunity to let students lead the discussion  – it was really helpful in getting reluctant students to participate in group discussions. We talked about how we have different goals at different times and for different students. In general, we may want to help ELL math students practice expressing their ideas in English, but there are times, as John pointed out, where it is important to take into account other goals we have for them





Proportional Reasoning "Opportunities"





Click here to see some of our work on the Gear Problem (linked above)




Choosing an "Opportunity" for Our Lesson 



Click here to see the criteria we decided to use in choosing the opportunity to serve as the heart of our lesson


Please vote for the opportunity you think should be at the heart of our research lesson, and write a brief explanation of why you chose that opportunity. Please include some predictions – what approaches do you think student might use? What do you think students might struggle with? Are there things you are particularly curious about in terms of how students will approach the opportunity?



The Opportunities The Reasons for Using Each


Kate bought some roses to make some money for a gift she wanted to buy.  She paid $6.00 for every 12 roses she bought.  Later, Kate was charging $6.00 for 8 of them. She sold them all and made a profit of $12.00. How many roses did Kate buy and sell?



I submit this opportunity for 2 reasons.  First,  I have already taught it and would appreciate the opportunity to study it and see how others would teach/improve upon it.  Secondly, I really like that there are many different ways to solve it.  Students who may be intimidated can be asked to draw pictures (a la Terry), tables (a la me) or other brave and creative ways. - Susan



The scale on a road map indicates that 1/4 inch equals 20 miles. How far apart are Dankerville and Baytown if they are 3 1/4 inches apart on the road map.



1.       I chose this opportunity because there are different ways to solve it.  It can be modeled and visualized. It also is a real world skill. Reading and understanding maps are important. – Renee


2.       I like that opportunity. It is cross curricular, which I love, it's visual, it allows for multiple avenues for solving including additive and multiplicative, and it could require them to utilize appropriate tools to measure.  The only problem I have with this opportunity is that I don't know if it would take an entire class period or lesson. Is there a way would could present students with multiple maps/scales and asked them to rank the distances of the places relative to each other? Or maybe as one of the last questions that would be a part of this opportunity ask them to compare this distance with others using the same scale and then different scales? - John 


3.       I personally like the rationale behind the mapping opportunity. I also do agree if there is a way to stretch it out, then other opportunities should be explored as well. - Charles



1.       When you clasp your hands together, which thumb is on top, the left or the right? (include sketch or photo of clasped hands)


2.       Take a survey of your classmates (include survey sheet). How many are “Left-thumbed?” How many are “Right thumbed?”


3.       If one more student walked through the door right now, would you bet that that student would be Left or Right thumbed? How much would you bet? (In other words, how confident are you?)


4.       A survey of a previous class showed that they had 15 Left-thumbers and 22 Right-thumbers. Is your class more or less “Left-thumbed” than the previous class? Give evidence for your answer.


5.       Using the information you have so far, how many Left-thumbed people would you expect to find in Gooberville Arkansas, where the population is 4,180? What are the steps/logic that you used to arrive at your prediction. [Maybe offer some kind of prize to the student who gets closest to the “correct” number of Gooberville L.T.s—a “correct” answer which we’d need to concoct beforehand or calculate on the spot.]



This question—or topic and series of questions—has a few advantages. It allows the students to relate to the question and come up with their own data set (1 and 2), and then to draw a pretty simple conclusion from that data (3). I think (hope) that 1-3 will allow the students to kind of swing into the question/topic in an accessible way.         


Then it asks them to compare one ratio (their class) to another (the “previous class”): these are bound not to match up very well. Not only did I give “previous class” numbers that aren’t “easy,” I also chose a large enough number of Left-thumbers that it’s almost guaranteed to be more L.T.s than my afternoon class will have (since my class will have like 18 students). And yet (according to the brief research I did) the ratio of the “previous class” is heavily-skewed toward R.T.s, and probably my actual class’s ratio will skew toward L.T.s. So, that’ll make the students figure out which is more important, the amount of L.T.s in a room, or the ratio of L.T. to R.T. Then, in (5), they have to speculate on the breakdown of a different, larger, population: to do so, they’ll have to decide on whether to use their own class’s ratio, or the previous class’s ratio, or a combination of the two. [Note again that the numbers will not match up well, so it won’t be a matter of “equivalent fractions, cross multiply then divide.”] Plus, this will introduce the concept of the “whole” in what had previously been referred to only as part-to-part comparisons. - Josh


The thumb crossing and rolling tongue I think will be interactive and fun for the students.  - Susan     



John took a trip to visit his cousin.  He drove 120 miles to reach his cousin’s house and the same distance back home.

a)       It took 3 hours to get halfway to his cousin’s house.  What was his average speed in miles per hour for the first 3 hours of his trip?

b)      John’s average speed for the remainder of the trip to his cousin’s house was 40 mph.  How long, in hours, did it take him to drive the remaining distance?

c)       Traveling home along the same route, John drove at an average rate of 35 miles per hour.  After 2 hours his car broke down.  How many miles was he from home?



I selected this opportunity because I think that it will offer students several opportunities to inventively solve the problem.  Sourcing the data for the opportunity from the students will allow for the creation of an unfamiliar situation from the familiar confines of the classroom.  This often works effectively in Literacy work and I think this is an opportunity to try it out in a Mathematics “opportunity.”  Finally, the scientific content of the material will allow the classroom teacher multiple opportunities to answer the student question of “What can I Learn from this?” - Gerry



Collect info from the class on right or left thumbness and tongue rolling


Ask the students to organize these numbers into ratios – introduce the idea of PPW here (???)


Opportunity # 1 – A national study report that 50% of the 200 people surveyed were right-thumbed.  Which population is more right-thumbed?  Our class or the national study?  If we wanted to move towards the national number how would we have to change our class ratio?


Opportunity #2 – A similar study of tongue rolling reported that 75% of the 480 people surveyed could roll their tongue.  Which population has more tongue rollers?  Move our class ratio towards the national report.


Opportunity #3– Our class grows!  We no longer have ____ students.  80 students show up!  Based on our class ratios:  How many of the 80 would you expect to be right-thumbed?  How many tongue rollers?



I selected this opportunity because I think that it will offer students several opportunities to inventively solve the problem.  Sourcing the data for the opportunity from the students will allow for the creation of an unfamiliar situation from the familiar confines of the classroom.  This often works effectively in Literacy work and I think this is an opportunity to try it out in a Mathematics “opportunity.”  Finally, the scientific content of the material will allow the classroom teacher multiple opportunities to answer the student question of “What can I Learn from this?” - Billy      


The thumb crossing and rolling tongue I think will be interactive and fun for the students.  - Susan 



























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