Ratio 3-2-1 Exit Card – Exposing Misconceptions and Engaging Metacognition

The expectation regarding student depth of understanding of equivalent ratios in middle school  has risen significantly since the advent of the Common Core.   A few years ago, students just scaled up or scaled down ratios to find equivalent ratios.   Now, they are expected to utilize ratio tables, tape diagrams, and double number lines to solve fairly complex problems.   I think, though, that oftentimes students don’t see the connections between these ideas.   I think that a lot of the time, they see them as completely disparate concepts, just one more thing to know about ratios.    As we wrapped up today’s lesson on double number lines (the last of the three representations for my class), I wanted to explore this idea.   Do they see the big picture, that these are all different ways to represent equivalent ratios and that the whole point of the representation is to visualize things so that one can solve problems?    I gave my students this exit ticket to find out a little bit more about their thinking.    img_1772

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Their responses were illuminating, as always.    Student responses to the first question seemed to fall into two groups: those who saw the big picture (like the one shown) and those who responded with three ways to write a ratio.   I think those who were mistaken on this question zeroed in on the word three and went straight for the ways to write a ratio without closely reading the question.   However, I also think that if they had made the connections between the representations, they would not have honed on that word.   So, I have some more work to do.

Student responses to the second and third question showed that most of them are still figuring out how to think about their thinking.    I am reading Making Thinking Visible  and am hoping to gain some insights into how to develop stronger metacognitio in my students.  .

Which one is the “orangiest” – A lesson exploring comparisons full of open-middle questions

“Which one is the most orangey?”   The question referred to a set of four different recipes for orange juice.    Students were to determine which recipe was the most “orangey” and which was the least.   What does it mean to be “orangey”, though?   That was the question that I didn’t anticipate the first time that I taught this lesson.   It seemed so obvious.   It was to me, but it wasn’t to my students.   It was especially not so for my ELL students.

As my students entered class, they were greeted with table laden with stacks of paper cups and four different bottles filled with lemonade.   I told them that each bottle had been made with a different lemonade recipe.  In front of each bottle was a post-it with a letter:   A, B, C, or D.   I asked them to come up to the table by table groups and taste test a sample from each bottle to help me figure out which tasted the “lemoniest”.

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After everyone had completed the taste test,  I asked students to “Vote With Their Feet”.  “Vote With Your Feet” is a high-engagement strategy that utilizes movement to increase levels of engagement.  (It is taken from Marzano’s The Highly Engaged Classroom.)   In the activity, students were directed to go to one corner of the room if they selected lemonade A as the “lemoniest”, another for lemonade B, another for lemonade C, and another one for lemonade D.   Next, they voted on the recipe that was the least “lemony”.   After concluding the vote, the class discussed what it meant to be the “lemoniest” or the least “lemony”.    At the conclusion of the taste test, the vote, and the discussion, everyone in the room understood what it meant to be the “lemoniest”.

After everyone understood the terminology, I introduced the problem context.    (The lesson is a Connected Math investigation.).   Two girls are at camp and are taking their turn helping to prepare breakfast.    They are making orange juice and need to decide which one of the four recipes to use.   Students are asked to analyze the recipes and decide which is the most “orangey” and which is the least “orangey”.  This seems like such a straight-forward question, but it always produces such rich discussion.

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I have students answer these two questions using Kagan’s Numbered Heads Together cooperative learning structure.    In this structure, students work independently.   When they have completed the question, they stand up.    When all of the students at the table group are standing, they discuss their responses.    When they have a consensus, they sit down.    At that point, the class debriefs with a whole class discussion.

I selected which students I called upon to share their thinking very intentionally.    As I listened to students discussing their thinking at table groups, I decided which students I would call upon to share and in what order.   I wanted to be sure to draw out misconceptions and I want to draw out multiple solution paths.   I tried call upon someone who had used  ratio reasoning to share his or thinking first.  Some of them compared the ratios two at a time and reason through which is more or less concentrated to draw their conclusions.   Some of them found equivalent ratios with common “denominators”.      If someone had made the mistake of seeing the ratio of concentrate to water as a fraction instead of as a ratio, they were my first choice.   I wanted to draw out the fact that they can compare with ratios but that there is a significant difference between the ratio and the fraction for a given recipe.   By selecting the ratio pathway as my first presenter, someone who used fractions could catch it if someone using ratios refers to them as fractions.    They could point out that concentrate is not part of the water so it has to be a part to part relationship instead of a part to whole relationship.

After the ratio pathway, I selected someone who used fractions to share their solution.   These students usually seemed to go directly to finding equivalent fractions with a common denominator.   If no one had considered the possibility of converting the fraction to a decimal for the purpose of comparison, I posed the question of whether or not there is another way to compare the fractions.   I wanted to draw out the fact that decimals are also an option without that option being something that I offer.   Finally, I drew out the possibility of using percents.    My intention was for students to see the different pathways and in so doing to help students make connections between ratios, fractions, decimals, and percents.

The following question in the investigation asked students whether recipe B  is 5/9 concentrate of 5/14.   I knew that the authors were posing this question to draw forth the part to part vs part to whole comparisons that are possible.    Since I had spent so much time drawing out different solution paths on the fist part of the investigation, this ended up being something that was addressed in passing because it had been so thoroughly hashed out already.

The final two questions in the investigation focused on the idea of scaling the recipes up or down.

First, students were told that each camper would receive 1/2 cup of juice and that there were 240 campers.    Students were asked to determine how many batches they would need to make  and how many cups of concentrate/water they would need for each of the different recipes.    Students used the Numbered Heads Together cooperative learning structure for this question as well.   I expected to see two different solution paths.   Some of the student would see that they need 120 cups of juice.   They would then divide the 120 cups by the number of cups for a given recipe to find the number of batches.    Some of the students would look at a given recipe and determine how many students it would serve.    They would then divide the 240 students by the number of students served by a recipe to find the number of batches.     I tried to find someone  to speak to each of the two different strategies if I could.   Once students had determined the number of batches, most of them could successfully find the amount of concentrate and the amount of water needed to serve 240 campers.

The final question that student addressed asked them to scale each recipe down to serve only a single cup.

After summarizing the main points of the lesson, I asked students to write a reflection comparing and contrasting the final two questions in the investigation to their experience playing one of the puzzles in the Lure of the Labyrinth.   I wanted students to realize exactly what the math behind this puzzle that they had been playing for a month was, and I wanted them to come to that realization on their own.

I love so many things about this lesson.   The context is real and “relatable” for 11 year olds.   The task is so rich.   It is filled with questions that allow for so many different solution paths and so many connections between ratios, fractions, decimals, and percents.  It is so good at exposing and clarifying misconceptions.