New Mexico Middle School Lesson Study Team Project 2008/09

Team Members:Amanda Machado

Blaine BallSandra Alvarado Sergio AndradeConstance Sherk

George Manriquez

New Mexico Middle School Lesson Study Team Meeting Dates

August 4 Chicago Lesson Study Conference 8:30 – 4:005 Chicago Lesson Study Conference 8:30 – 4:006 Chicago Lesson Study Conference 8:30 – 3:00

Team Meeting 3:15 - 5:309 Chicago Lesson Study Conference 8:30 – 4:0010 Chicago Lesson Study Conference 8:30 – 4:0012 Team Meeting 4:30 – 6:0019 Team Meeting 4:30 – 6:00

Sept 9 Team Meeting 4:30 – 6:00Oct 27 Practice Lesson (CMS) 8:30 – 1:00

28 Practice Lesson (STMS) 1:00 – 3:30

Mathematics Lesson Plan November 7, 2007

New Mexico Middle School Seventh/Eighth Grade

Instructor: Sandra Alvarado Lesson plan developed by: Sandra Alvarado

Sergio Andrade Blaine Ball

Amanda Machado Constance Sherk

1. Title of the Lesson:

Mixing Juice

2. Goals of the Lesson: Students will use ratios and related forms of comparisons to solve problems.

Overarching goal: Our students will work collaboratively and make mathematical connections in order to become independent problem-solvers.

3. Relationship of the Lesson to the Standards

Prior units of study: Bits & Pieces I, II, III, Stretching & Shrinking Standards: 6th GradeStrand: Number and OperationsBenchmark 3: Convert fractions to decimals and percents and use these representations in estimations, computations and applications. Interpret and use ratios in different contextsCompute and perform multiplication and division of fractions and decimals and apply these procedures to solving problems. Strand Algebra Benchmark 1& 2: Solve problems involving proportional relationships.Benchmark 4 Explain and use the relationships among ratios, proportions and percentsBenchmark 4: Solve problems that involve change using proportional relationshipsBenchmark 4: Use rations to predict changes in proportional situations In 6th grade the students were introduced to the concept of equivalent fractions. They had experience with making comparisons by using difference, ratios Exploring and applying fractional number concepts; Percent defined as a ratio to 100 and connected to fractions and decimals Fractions as a part to whole comparison In Stretching and Shrinking they worked with scale factor, ratios and

equivalent ratios

Related post learning standards (topics/objectives) Strand 1: Benchmark 5 (Numbers and Operations): Use a variety of

computational methods to estimate quantities involving real numbers. Strand 1: Benchmark 5 (Numbers and Operations): Select and use appropriate

forms of rational numbers to solve real world problems including those involving proportional relationships.

Strand 2: Benchmark 5 (Algebra): Demonstrate understanding of the relationships between ratios, proportions and percents and solve for missing term in a proportion.

Strand: Algebra, functions, and graphs 9-12 Benchmark: Represent and analyze mathematical situations and

structures using algebraic symbols. Know, explain, and use equivalent representations for the same real number

including: Integers, decimals, percents, ratios…7th grade Lesson Standard & Benchmarks

5-8 Benchmark 2: Understands the meaning of operations and how they relate to one another.2. Convert terminating decimals into reduced fractions.4. Add and subtract fractions with unlike denominators.

5-8 Benchmark 3: Compute fluently and make reasonable estimates2. Convert fractions to decimals and percents and use these representations in estimations, computations, and applications.5. Add and subtract fractions with unlike denominators.5-8 Benchmark 2: Apply appropriate techniques, tools, and formulas to determine measurements.

Solve problems involving scale factors, ratios, and proportions.

4. Unit PlanName of Unit: Comparing and Scaling from Grade 7 The Connected Mathematics Program 2 (2006). Pearson, Prentice Hall: Michigan State University.

Investigation 1: Making Comparisons Students explore ratios and rates Analyzing comparison statements Writing comparison statements Shown three ways to write ratios

RESEARCH LESSONInvestigation 2: Comparing ratios, percents and fractions

Students develop comparison strategies More comparison strategies Scaling ratios

Investigation 3: Comparing and scaling rates Making and using a rate table Finding rates Using rates in equations Two different rates

Investigation 4: Making sense of proportions Setting up and solving proportions Applications of proportions Developing strategies for solving proportions

4. Instruction of the LessonThe rationale for teaching this lesson includes the following: This is the curriculum point at the time we need to present our public lesson. Our team is comprised of both 7th and 8th grade teachers and this topic is in both 7th and 8th grade standards. Ratios and making comparisons are an important topic for our students because they encounter them in everyday life experiences: advertisements, cooking, making purchasing, academic and athletic performance, just to name a few. Comparisons and ratios are introduced in elementary school when students work with equivalent fractions, using factors and multiples; using ratios in fraction form; fractions, ratios and percents are introduced at elementary level, but need to be developed in 7th and 8th grade. Proportional relationships are introduced for the first time in this 7th grade lesson. This concept is not well developed in our 8th grade curriculum, but needs to be because it is in the 8th grade standards, students need more work with this concept than they are getting in 7th grade and it is a huge concept for the high school curriculum. This particular lesson engages students as active problem-solvers, helps them to make mathematical connections between their elementary, middle school and high school work and allows for collaboration. The problem helps them become better problem-solvers by providing them with an experience in which there are multiple entry points and multiple strategies/paths to a solution followed by a mathematical discussion and summary of the mathematics.

District/school data shows that our students are low in fraction skills so they need problem-solving situations where they manipulate fractions in a meaningful context. (Give them a reason to need to know.)

According to the standards and curriculum the students need to:

Based on observations the students have learned so far: How to apply rational number concepts Percent defined as a ratio to 100 and connected to fractions and decimals Fractions are a part to whole comparison Addition, subtraction, multiplication and division with fractions Determining scale factors Working with equivalent fractions

The major focus of this lesson:The major focus of this lesson is making comparisons using ratios. The students will use their prior experiences with strategies for comparing to investigate different orange juice to concentrate mixtures. The lesson will have students discuss the different strategies they use and lead them to part-to-part, part-to-whole ratios for making comparisons. They will use this knowledge to answer question D asking them to scale up.

The above objective will be accomplished by doing the following: Letting students investigate conjectures about most and least orangey, make

mathematically based decisions and justify those decisions. We will raise the issue of comparing statements using ratios from data

presented.

Steps, Learning Activities

Teacher’s Questions and Expected Student ReactionsTeacher’s Support Points of

Evaluation

IntroductionYou used ratios, fractions, percents and differences in investigation 1. Today you’re going to explore strategies for making comparisons when mixing orange juice. At this time the teacher will demonstrate mixing one can of orange juice concentrate with water.

Introduce terms concentrate; idea that combining concentrate with water produces the volume; introduce term batch; mix;Recipe mixing is different from ones you will encounter today.How many of you have made orange juice from a can before? What was involved in making it?(Show can and demonstrate the mixing.)

When you look at your problem today, think about how the different mixes might taste?

What are some of the different ways we made comparisons in Investigation 1? Teacher records these for students

What happened to the volume of the container as we added the concentrate?

Posing the ProblemHave intro (with four mixtures) to problem on pieces of paper for students (one per pair of students) to read and put on overhead and teacher reads aloud.

Julia and Mariah attend summer camp. Everyone at the camp helps with the cooking and cleanup at mealtimes. One morning, Julia and Mariah make orange juice for all the campers. They plan to make the juice by mixing water and frozen orange juice concentrate. To find the mix that tastes best, they decide to test some mixes. MIX A2 cups concentrate3 cups waterMIX B5 cups concentrate9 cups waterMIX C1 cup concentrate2 cups waterMIX D3 cups concentrate5 cups water

PROBLEM:Which mix will make juice that is the most “orangey”? Explain.Which mix will make juice that is the least “orangey”?Which comparison statement is correct? Explain.5/9 of MIX B is concentrate5/14 of MIX B is concentrateAssume that each camper will get ½ cup of juice.For each mix, how many batches are needed to make juice for 240 campers?For each mix, how much concentrate and how much water are needed to make juice for 240 campers?

Poster with mixesPaper at each table with mixes and problem (visuals to help visual learners)

Change the problem to read, “Explain mathematically” instead of just “explain”. (May want to specify calculations, mathematical representations, and words.)

What is the problem asking you? What are some of the different strategies we have been using to make comparisons?Which strategy would your group like to try first?

For C, ask where the numbers came from. (the 5, the 9, the 5, the 14)What does the term comparison statement mean?

Intro question D. Choose one of the batches and ask if each camper gets one FULL cup of mixture, how many campers will this batch serve? But what about if you don’t have enough for everyone to get a full cup and you can only give each camper ½ cup?

Why are you using this method?

How are you using it to make comparisons?

What does the numerator/denominator really represent?

What do the answers to your calculations mean? (decimal, percent, etc.)

Anticipated Student ResponsesFinding differences between concentrate and water. Choosing most or least orangey based on which mix has the largest amount of concentrate or water.Making fractions and finding common denominators.Pictorial representations.Reasoning by using benchmark fractions.Creating part-to-part ratios (concentrate to water and water to concentrate).Creating part-to-whole ratios.Calculating decimals.Calculating percents.Setting up proportions for part D.Finding scale factor and then scaling up for part D. Scaling up using equivalent fractions or ratiosComparing with ratios, percents, decimals, fractions and differences

Scale up mix B so that it is the same mix just with larger numbers and ask the students if making the same recipe multiple times will change the taste of the juice.

What do the calculations within each method represent in relevance to the problem?

BLACKBOARD PLAN1. Finding differences by subtracting: a. concentrate from water b. concentrate

from concentrate c. water from water 2. Pictorial representation without mathematical support3. Using benchmark fractions/ratios without common denominators.4. Compare ratios with common denominators.5. Compare fractions with common denominators.6. Intuitive ordering of fractions to simplify problem.7. Compare percents.8. Compare decimals.Part D Anticipated ResponsesD1. 240 divided by cups concentrate/water/mix240 divided by ½ divided by cups concentrate/water/mix240 divided by 2 divided by cups concentrate/water240 divided by 2 and then divided by cups of mix (yields correct response)240 multiplied by ½ product divided by cups of mix (yields correct response)D2 Batches divided by cups concentrateBatches divided by cups waterBatches divided by the quantity cups concentrate and cups waterBatches multiplied by the quantity cups concentrate and cups waterBatches multiplied cups concentrate (yields correct response)Batches multiplied cups water (yields correct response)

Summing Up (See Blackboard Plan above)

What was today’s situation?

What are the essential things we found out about this situation? We hope the students will come up with the following:

Teacher Led Discussion:Present student responses in order of blackboard plan. Question students through process. Possible questions:

In this lesson we looked at various strategies for comparing mixes of orange juice concentrate and water. (Summarize the various methods used by the students.)

There are various ways to make comparisons and all the different methods are both correct and related to one another. It is important to understand if you are making a part-to-part or part-to-whole comparison.

Pictorial representation-how did the pictures help you decide most, least orangey? Which of the pictorial representations best represents the problem mathematically?(The teacher will relate ratio/fractions to these pictorial representations.)Address differences and whether or not they work with this problem.

Assessment:

The classroom discussions, observations and shared work will serve as assessments of the lesson.

Lesson Study Final Report Gadsden Independent School District Santa Teresa Middle School Santa Teresa, New Mexico 88008

Our team originally was comprised of five members from Santa Teresa Middle School (STMS). Three of the five were relatively inexperienced teachers. Constance Sherk, the Math Process Trainer, was the only team member who had actually participated in a group before, so the team looked to her for leadership and guidance. Within the group, experience really was not a factor. All members had an equal voice, and all contributed equally as we worked together and applied the essential elements of the lesson study methodology:

1. Think carefully about goals of the particular lesson, unit, and subject area.2. Study and improve the best available lessons.3. Deepen our knowledge of subject matter.4. Think deeply about long term goals for students.5. Collaboratively plan lessons.6. Carefully study student learning and behavior.7. Develop powerful instructional knowledge.8. See our own teach through the eyes of colleagues and students.

On our first day in Chicago we decided that our overarching goal would be for our students to work collaboratively and make mathematical connections in order to become independent problem-solvers. We selected our lesson, “Mixing Juice” from the 7th grade Connected Math Program textbook to support this overall goal as well as the narrower goal of using ratios and related forms of comparisons to solve problems.

The work we did in formal sessions at DePaul University was invaluable, but the conversations that we shared as we extended our lesson study activities into the afternoon and evening hours at our hotel were also essential to the total lesson study experience.We did some of our best work at the coffee shop and in the hotel lobby as we shared ideas and pecked away and laptop computers. For all of us, it was the best possible way to start our lesson study quest.

Our team did not want to lose the knowledge and enthusiasm that had been kindled during our conference in Chicago. We met the first week after returning and continued to meet as often as our schedules permitted. Things were complicated a bit by the transfer of two team members to a campus that was quite distant from STMS. We also lost a team member to a neighboring district and were joined by George Manriquez, the new Math Process Trainer at STMS. Through it all we maintained our cohesion and focus on the lesson study objectives.

STMS Lesson Study ChronologyChicago Lesson Study Conference: DePaul University, Chicago, IL

August 4, 2008 8:30-4:00 Initial Session: Established group norms and selected lesson.

August 5, 2008 8:30-4:00 Did the math involved in the lesson and discussed lesson plan.

August 6, 2008 8:30-3:00 Lesson planning. 3:15-6:15 Team worked at hotel.

August 7, 2008 8:30-4:00 Lesson planning.

August 8, 2008 8:30-12:00 Closing session.

August 12, 2008 4:00-5:00 Team meeting at STMS. Worked on lesson plan.

August 26, 2008 4:00-5:00 Team meeting at Chapparal Middle School (CMS). Worked on lesson plan.

September 9, 2008 3:30-5:00 Sandra, George, and Blaine met at STMS and worked on lesson plan.

October 27, 2008 8:30-10:05 Practice lesson at CMS. 10:05-12:00 Team discussion of lesson.

October 28, 2008 1:00-2:15 Practice lesson at STMS. 2:15-3:15 Discussion among team members and district representatives.

November 3, 2008 8:30-9:30 Blaine observed in regular lesson. 10:30-11:30 Sandra observed Blaine in regular lesson.

November 7, 2008 9:00-10:30 Public lesson in STMS library. 10:30-12:00 Debriefing. 1:00-3:00 Team discussion with Tad and Susana.

November 21, 2008 2:00-3:00 Post-public lesion discussion by team members.

April 6, 2009 2:15-3:15 Team met with Susana at STMS to discuss group reflection and final report. 3:40-4:40 continued discussion.

April 29, 2009 8:45-10:00 Blaine observed Sandra in regular lesson.

May 9, 2009 9:00-11:30 Team met to establish responsibilities for finalizing remaining lesson study requirements.

May 15, 2009 2:15-3:00 Team reviewed Power Point final presentation.

May 18, 2009 2:30-3:30 Final presentation at Santa Teresa Middle School.

Santa Teresa Middle School Lesson Study Team Debriefing November 7, 2008

Susana - I want to welcome everyone who came for the public lesson and debriefing. Thank you Sandra for a very interesting lesson. I will be the moderator. I invite everyone to tell us about their observations of the lesson and ask any questions they may have. Let’s start by having Sandra give us her observations.

Sandra - I thought it went really well. I wish that there could have been more time. We didn’t have a chance to discuss all of the solutions and I wanted to talk about the illustrations that groups used to support their calculations. I loved the way the boys talked with each other and exchanged ideas. Basically, the way they interacted was the same as when they are in class. They didn’t let this being a public lesson change the way they did things. They showed their calculations and explained things very well.

Amanda - The conversations were really good. There was a great variety of strategies. They used differences, percents, fractions, fraction strips, scaling, part-to-part, and part-to-whole comparisons. It was awesome that so many ideas came out. If I did the lesson again (Amanda taught the first practice lesson), I would try to figure out how to take less time on the explore phase of the lesson and focus on the share and summarize portion. How can we make the explore phase more efficient?

Sandra - I wanted to get more into the share and summarize portion. There was a lot of math brought out, but I wanted to discuss more about the part-to-whole comparison and the percents that went with it.

Amanda - Some groups did much the same thing. Maybe we could combine some of them when they share at the end of the lesson. If just one group each explained their part-to-part and part-to-whole comparisons, then there would have been more time for the math and illustrations that you wanted to bring out. It might not have been so rich, but all of the math would have been covered.

Sandra - Yes, for example, one group compared water to concentrate and said that it was using differences, but the group really didn’t use subtraction. I would have liked to get into the calculations the group used.

Amanda - I think that was the group that said, if you use a given amount of concentrate, adding more water will make it less orangey. I was intrigued by how the group was approaching the problem.

Sandra - Did you see that the group really wasn’t using differences. It was more a ratio of cups of water to concentrate. It caught me off guard, though, that their approach was to say that adding water to a certain amount of concentrate made it less orangey. They understood the concept, but had trouble with the math.

Amanda - They even said its getting more yellow and less orangey. It was good that we decided to actually demonstrate how the concentrate was added to the water in the launch phase of the lesson.

Susana - In this case, the group focused on color and it took them away from the math. When I have a situation like this in my classes, I ask students how they can express whatever concept they’re discussing in mathematical terms. This usually gets them focused back on the math.

Cheryl - I liked it that group members truly discussed the math of the lesson. They were not afraid to challenge each other. You could tell that they were comfortable working together. One group used a scale factor and Sandra told them to go talk with another group. They shared their ideas and the other group was really trying to grasp the concept.

Blaine - You could tell that these students were used to working in groups and interacting as they did. Its not something that happens overnight or that can be done without practice. It was clear that Sandra’s classroom routines have allowed students to practice these skills on a daily basis.

Bertha - Sandra, you had clear expectations for your students and this was reflected in how they discussed the problem. They understood the problem and, even when one group used a unworkable approach, they knew how you expected them to work together to solve the problem.

Amanda - I am always torn between not wanting to give my students too much information and feeling a need to prompt them without actually putting specific ideas in their heads.

Sandra - Yes, when one group got confused about scaling up or down to get a common denominator, I challenged the students to show me the math of what they were discussing. I tried to help them see that scaling the ratio of ingredients up or down doesn’t change how orangey it is. They were still struggling with it at the end of the explore phase of the lesson.

Amanda - I have trouble deciding what is too little help and what is too much.

George - A group did discuss using common denominators and comparing the ratios to find differences in this sense, but the students had trouble finding a common denominator.

Sandra - When I talked to the group that was using differences, I wanted to discuss with them an example which would demonstrate why differences would not work in the way that they were trying to use them. I wasn’t able to help them see it, so I had to let them go on and try to use the share and summarize phase of the lesson to clear things up for them.

Amanda - We discussed using charts of the various ways of making comparisons at

different phases of the lesson. Some thought that it might be best to use a chart of all the possible approaches during the launch, some didn’t want to use charts at all, and others favored using it to summarize the lesson. I feared using it too soon, but thought that using it at the end would be very helpful.

Susana - Having something already prepared is probably a good idea. Then you have it available if you need it.

Sandra - Discussing part-to-part and part-to-whole comparisons would have been a good transition to Part C of the problem, which we didn’t get to.

Connie - Originally we planned to do the whole problem, Parts A through D, but, after the first practice lesson, we realized this was pretty ambitious. That’s why we decided to do a share/summarize after Part B, and then, if time permitted, go on and address Parts C and D. I was concerned also about the group that used differences. Ultimately, group members decided that differences would not work because two of the mixes had a difference of one in the amount of concentrate compared to water. When the numbers did not work for them, they decided that differences wouldn’t work, but I don’t think they actually realized that differences was a flawed approach to begin with.

Amanda - Yes, when that group wrote the ratio of concentrate to water as 1 to 2, Sandra asked the group to rewrite it as an equivalent fraction. They chose 2 to 4. She helped them see that the difference in the first ratio was one and in the second ratio was two. Sandra then asked them if the mixes represented by the two ratios were equally orangey. The students realized that the mixes were equally orangey and that differences wouldn’t work, but they were unable to solve the problem using a new strategy.

Sandra - Groups kept asking me which was better to use, part-to-part or part-to-whole comparisons. Without actually coming out and telling them the mathematics of the two comparisons, I tried to get them to tell me what the percent really meant for each of the two. I wanted them to see that even though the percents were different, both were valid comparisons.

Amanda - I think in looking back that the part-to-whole comparison was much easier to understand because the percent for each mix reflected the relative amounts of concentrate in each mix.

Blaine - One group did say that a higher fraction of cups concentrate over cups of mix, when converted to a percent, meant a more orangey mix.

Bertha - I’m not sure that all of the groups understood what the percents really meant. Maybe if they could see a visual representation or an illustration, it would have helped them.

Sandra - I wanted to focus on the calculations. Then, if there was time, I wanted them to do illustrations. The calculations were most important to me.

Connie - I didn’t see where any of the students said that the illustrations really helped them solve the problem.

Sandra - I told them that I wanted to see their calculations. If they used illustrations, then the illustrations must speak for themselves. The illustrations shouldn’t just be pictures. I felt that the mathematics would be richer if the illustrations were not emphasized so much.

Blaine - Some went to the visual representation first in order to make sense of the problem. I think you have to leave it open-ended so that they have the option of using illustrations to help solve the problem.

Amanda - One group did use an illustration first to solve the problem, and then did the math to support it. Another group just put one box on top of two boxes to show the ratio of concentrate to water. The illustration was fairly meaningless. I heard the students discussing it. They knew that an illustration was required, so they just drew something. They weren’t concerned that it didn’t actually support the math of the problem.

George - Its also important to understand why different approaches wouldn’t work. The group that used differences understood that it was not a workable approach. It took the teacher and other students, during the share and summarize phase of the lesson, to make it clear to them and everyone else why it was unworkable.

Connie - A lot of time went into illustrations that did not help solve the problem.

Cheryl - Not having a chalkboard or dry-erase board available during the share and summarize portion of the lesson prevented writing down some of the explanations about why certain approaches do or do not work.

Susana - In math its good to have space to write when we need to explain things. Maybe having charts ready that anticipate student responses would expedite discussion at the end of the lesson.

Sandra - The last group did a part-to-part comparison and ordered the percents from greatest to least. I really wanted to get to them when we were sharing, but we ran out of time. I wanted to compare percents for part-to-part (concentrate to water) and part-to-whole (concentrate to mix and water to mix). Students used all three approaches and I wanted to compare and contrast how they did it, but time ran out.

Bertha - I liked how you monitored your groups. You were there for them, but didn’t give them the answers. Maybe at the end it would have helped to have a chart prepared in advance that made the comparisons you wanted to make. There just wasn’t enough time for everything.

Amanda - We should remember that Part C of the problem does address part-to-part and

part-to-whole comparisons. From what we saw in the practice and public lessons, we would need another day to cover Parts C and D. Since we didn’t have the time, maybe it would have been good to have a chart that showed how something like 5 cups of concentrate and 9 cups of water can be written as 5 to 9, 5:9, 5/9 and also as a decimal and a percent.

Blaine - Comparing and contrasting all of the ways to solve the problem definitely made the lesson longer.

George - And then one group went beyond anything we expected by using scale factors. Even though the group wasn’t able to finish the math for this approach, it was a good idea. Students needed more time to share. Maybe it would have been best to just have one group present for each of the similar solutions.

Tad - I liked the lesson plan. It contained more of the mathematics of possible solutions than we often see. It indicates that the teachers were really looking hard at the math of the problem. When I read the original lesson plan, I was concerned that you planned to have students do all of the parts to the problem. I was glad to see that you broke it down into two phases, Parts A and B with a share/summarize to follow, and then the remaining parts of the problem if time permitted. I was impressed with the richness of the conversations and the way that students communicated. It was clear that they habitually did this. The discussion did go too long and there were issues relating to the illustrations. Actually, the discussion at the end was not necessarily too long. Because the explore part of the lesson lasted 40 minutes instead of the allotted 20 minutes, there was less time to discuss and summarize things at the end. Sometimes in math we emphasize multiple representations at the expense of the mathematics involved. You don’t have to use all types of representations all of the time. Let groups select the type that they want to use. Have them solve the problem one way and then, if there is time, expand on it. Twenty minutes for exploring is enough if they are allowed to focus on one way to solve the problem.

Lesson: Mixing Orange Juice (Group Reflection)

Practice Lessons (10/27/08-10/28/08)

The first practice lesson was conducted with a 7th grade class at Chapparal Middle School and the second with an 8th grade class at STMS. Having them on back-to-back days was not the preferred scheduling alternative, but because time was short before the public lesson and numerous other considerations, it was decided to do them on consecutive days. In the first practice lesson, students were given information on four mixes and asked to determine which was the most “orangey”.

Mix A: 2 cups concentrate 3 cups cold waterMix B: 5 cups concentrate 9 cups cold waterMix C: 1 cup concentrate 2 cups cold waterMix D: 3 cups concentrate 5 cups cold water

On the day prior to the first practice lesson, the 7th grade class at Chapparal Middle School had studied differences as a basis for comparison of two quantities which were represented by whole numbers. In the introduction to the first practice lesson, the teacher displayed a chart which listed several ways to make comparisons: differences, ratios, fractions, decimals, and percents. Because they had just studied differences and it was at the top of the list, all of the groups except one (it used percent concentrated in the overall mix) chose differences as their basis for comparison. Since their comparisons on the previous day had involved whole numbers, they compared the number of cups concentrate in each mix to the number of cups water in the same mix. They decided that the ones (Mix A and Mix C) with the least difference must be the most “orangey”.

At the conclusion of the first practice lesson, time also became an issue. None of the students were able to finish all parts of Problem 2.1:

A. Which mix will make juice that is most “orangey”?B. Which mix will make juice that is least “orangey”?C. Which comparison statement is correct? Explain.

1. 5/9 of Mix B is concentrate. 2. 5/14 of Mix B is concentrate.

D. Assume that each camper will get ½ cup of juice.

1. How many batches of Mix A are needed to make juice for 240 campers?2. For Mix A, how much concentrate and how much water are needed to make

juice for 240 campers?

Two significant changes in the second practice lesson. During the Launch phase, the five methods of comparison addressed in the first practice lesson were again introduced, but

each was discussed with an accompanying numerical example of how it might be applied to the problem. The merits and limitations of each approach were also discussed. The group decided too, that it would be best to reduce the number of requirements presented to students during the Launch of the lesson. Only the parts of the problem requiring students to determine the most and least orangey mixes were introduced. If time permitted after the Share and Summarize for Parts A and B, Parts C and D would also be addressed.

The second practice lesson was given to a class of 8th grade Algebra students. The fact that they were advanced 8th grade students, coupled with the more detailed discussion of methods of comparison during the Launch of the lesson, undoubtedly resulted in them using a greater variety of approaches to make their comparisons. One group used the ratio of cups of concentrate to cups of water and then converted the ratio to a percent to make comparisons. Another used cups of concentrate over cups of mix and converted to the fraction to a decimal for each mix. A third group used the benchmark ratio ½ to order the four ratios of concentrate to water and derive the correct answer. Yet another approach was to get a common denominator of 90 for the part to part ratios and then compare them. The fifth and final group did a scale drawing of the mixes in the pitcher, with a part to whole representation of concentrate to mix. All groups were told that they must use mathematical representations to illustrate their answers, so the fifth group did this extremely well. It was noted, however, that the illustrations of the other groups were more pictures than mathematical representations of the problem and its solution. They simply drew illustrations with the approximate relative amounts of juice and water, but they were not to scale and did not accurately represent the mathematical relationships involved.

Public Lesson (11/07/08)

Launch: The teacher for the public lesson wanted to do the launch in a way that would allow the students enough information to analyze the problem, but not give away the solution. It also needed to be engaging. The students needed to connect with a real-world activity. A thawed can of orange juice concentrate was shown to the class, along with two glass containers, one for holding the water and one for making the mixture. The students were asked if anyone had ever mixed orange juice at home or at a family gathering. Most students raised their hands, acknowledging such activity. The teacher opened up the can of concentrate and poured it into the mixing container. She then asked the students what would happen if she filled the empty can of concentrate with water and poured it into the container. Responses included, “it will not be the same color, it will dilute it, the volume will increase, and it will become less orangey.” At this point, she stated lesson objective and pointed to a chart with the four mixes that were to be compared. She advised the students that their task was to figure out which mixture was most “orangey” and which was least “orangey”.

Initially, the group members debated whether, or not to post the various methods that could possibly be used to solve the problem. One of the concerns of not posting the strategies was that the students may not have come up with viable and/or enough strategies within the allotted time frame of 15 minutes.

During the first practice session, the teacher introduced four different methods that could be used to solve the problem: ratios, percentages, fractions and differences. This first group of students exhibited such a substantial lack of understanding of some strategies that the teacher was forced to review during the launch process. This part of the lesson ended up taking more time than was expected. Furthermore, most of the students opted to use the ‘difference’ method. The reason these students opted to use ‘difference’ as a solution may have been partly due to their limited knowledge of fractions, ratios and percentages. Subsequently, when the teacher reviewed the strategies, she may have inadvertently emphasized that taking the difference between the amount of concentrate and the amount of water, was in fact the simplest method, however not necessarily the correct one. In this case, the students simply took the most obvious and easy way out. Most gave no further consideration to alternative methods, or any of the ones introduced. During the second practice session, the teacher also introduced the same four strategies. However, this time questions were asked that allowed students to reflect not only on the various possible solutions, but also on the effectiveness of the methods and whether or not a given strategy would even work, as well as determining what the solution(s) represented.These students were very comfortable with all four methods, so a minimal amount of time was required for the launch process.In the final lesson presentation, the teacher opted not to list the methods and instead asked the students what problem-solving strategies they though would work for this particular problem. Surprisingly enough, the students came up with a list of five ways to solve the problem: ratios, fractions, decimals, percentages and differences.

Explore: When our group discussed the explore phase of the lesson in Chicago, we planned to leave it up to the students to find and investi- gate possible ways of comparing the different mixes. After the first practice lesson, when many of the students used the difference as their only basis for comparison, it was decided to give them more information about possible methods of comparison prior to the explore phase. In the second practice lesson all methods of comparisons were listed and clarified before students started to explore. Students chose their own comparison strategies but, as the teacher observed the groups, some were redirected by the questions asked of the them by the teacher. Other groups, particularly those who finished very quickly, were told by the teacher to expand the depth and/or breadth of their comparison. For the public lesson, students were allowed to struggle and work through the problems themselves. Help was provided in subtle ways. The teacher walked among the students as they worked and helped them with the different strategies, but they were allowed to find their own solutions, even if they were not correct. There was a variety of excellent approaches, students worked within their groups more effectively, and students learned from each other.

Share/Summarize: This portion of the lesson included a “blackboard plan” where student work was organized by problem-solving approach at the front of the classroom. Differences were on the left, then decimals, percents, fractions, and ratios were displayed to the right. Adequate time was allotted for this activity and the mathematical thinking of students was discussed. This was done in a way that allowed students with erroneous solutions to see where they had gone wrong and permitted those with more appropriate approaches and solutions to explain their thinking. This was the best part of the lesson.

The two shortcomings identified in the practice lessons, inadequate time for summarizing/sharing and inappropriate student approaches to the problem, were both largely remedied in the public lessons. This is not to say that all student groups solved the problem in an appropriate manner, but even those who took an incorrect approach understood what they had done wrong and that there was a variety of better ways to compare the mixtures.

Mixing Juice: Individual Reflection (Sandra Alvarado)

The students understood the fact that they all needed to compare the mix’s and find

out which mix was the most and least orangey. The students understood how to use ratios,

percents, decimals but when they begin explaining the difference between getting the

percent when comparing part to whole and part to part. The students will see the percents

and know which is larger than the other, but they would not understand that one percent

belonged to the part to part and the other to the part to whole. When the students are

looking at the percent of the part-to-part, then they are observing the difference of the

concentrate and the water. Thus, when the students look at the percent of the part-to-whole

they are observing the percent of the water to the whole mix or the concentrate to the whole

mix. The students have been applying what they learned during the juice lesson in other

lessons for example, in one of their future lessons the students had to find the scale factor

of some boxes and they started using proportion and fractions. The students scaled up and

down then look at the difference between them. After the public lesson the class followed

up with the last sharing portion that was not completed, and finally the teacher summarized

the lesson. The students took a unit test five days after they finished this lesson. The

teacher noticed that the students cleared the misconception of the difference of a part-to-

part and the part-to-whole percentages. The test also clearly stated a description of what the

percent of that specific answer meant or told them.

Learning on students understanding o f the concepts was difficult to grasp since

one as a new teacher believes that the students understand something when they get the

correct answer. As a new teacher we have learn that paying attention to the students

learning is not simply asking if they understood or if they obtained the answers correct but

verifying it with some sort of assessment. A teacher needs to ask open ended questions so

that the teacher can listen to what the students have learned from this lesson or if they

accomplished their goal. Listening to student’s conversations and listening to students as

they share with each other a teacher can fully understand what concepts a student

understood. In truth every time that fractions and ratios are present teacher start shutting

down their brains because they either did not like fractions or just do not want to deal with

them and convert them into decimals or percents. Starting with this lesson our appreciation

for fractions, ratios, and proportions began since we spent a lengthy period of time working

with them. The team discussed the entire math that was involved and tried to come up with

all the strategies the team thought the students would develop. Since, Mrs. Alvarado was

the one to present the public lesson and she had honors students she knew that her students

would develop some and other new strategies. She tried her best to be prepared for

everything that day, but as she predicted the team did not come up with all the strategies.

Her students brought other new concepts to the lesson and plus brought upon a richer

discussion than the practice lessons that had been observed. After the public lesson the

team discussed some new mathematics and discussed the strengths and weakness in the

lesson. The team learned that the math that was involved might have been brought to a stop

at a certain point in the lesson so that higher math concepts could have been involved. For

example, the teacher could have started making the students to turn all mixes into one unit

to notice the least and most orangey in that manner. As the video was reviewed it seemed

that there were some students whom wanted to develop that strategy, but since the teacher

was not able to redirect them they stopped.

The lesson study program has made me understand how important planning a

lesson is so that the concepts of that day may be beneficial to the students and the teacher.

As a new teacher I was blinded by the thought that as long as a student got his answer

anything else does not matter. Math is hard to understand and if a student does not know

how they got their answer then they do not know how to do the math. The students may

know the process of finding the answer but they will not know what the process is or why

they are doing what they are doing. I was afraid that I was not going to be able to get my

students to realize that the process is as important as the answer. My life as a teacher has

just begun and I will keep the lesson study ongoing in my life so that I can prepare at least

two to three lessons per year. Lesson planning, I never thought this would be so much

interesting and exciting.

Mixing Juice: Individual Reflection (George Manriquez)

On November 7, 2008 a small group of teachers from Santa Teresa Middle School and me as the Math Process Trainer, presented a Public Lesson on one of the CMP investigations involving making orange juice from a can of concentrate. In this lesson, the students are asked to consider four mixes and to determine which mix is the most and least “orangey”. The mathematical goal was for students to become comfortable with ratio and related forms of comparisons to solve problems.

Although the group encountered several obstacles along the way to create and present the final product, such as loosing some members of the original group who moved on to another campus or district, and acquiring me as the new Math Process Trainer replacing the original MPT who started the group and attended the Chicago Conference, I believe we continued with as much optimism as we possibly could, in lieu of these setbacks. I must admit that not having been with the group initially and being told that I would be an active participant to this, was a bit frustrating.

Before the final presentation was held, the group agreed on two practice lessons which were held on October 27th and 28th of 2008. These practice runs went well, although they were held back- to- back. Ideally, we should have spaced them a little further apart to allow the group more time to debrief and modify for the subsequent practice run. However, this was not possible because of other activities that cut our time short and thus we were forced to proceed in this manner. Much was learned from the first practice session and we were able to make some modifications for the next practice session. For example, students in the first session were not as well prepared for these types of problems and could not grasp the concept of the mathematics. The teacher made a good attempt of introducing and leading the students towards a reasonable solution, but their conceptual learning was not there. Most of the students were confused and were unable to use a correct solution. When the group reconvened, slight modifications were used for the sub-sequent practice. For example, the launch was slightly changed so that students could understand that they were being challenged to find different solutions to the problems and the students responded favorably. However, we did notice that their illustrations were less mathematical and may not have illustrated the concept as the group would have liked.

The final Public Lesson was held on November 7, 2008. Members of the group were excited and nervous at the same time. The launch went well, but I feel that it was a bit long and too much detail may have been given. The frozen can of orange juice and the two glass containers were enough to illustrate the problem at hand and proved to be very effective. Once the problem was launched, the students were given a brief moment to “digest” the problem and then were asked to form groups of four. The students expressed understanding and enthusiastically proceeded to work in groups. The teacher walked around answering the students’ questions, as well as asking them to justify their answers, and leading them on to solve the problem in multiple ways. Members of the group and invited guests were encouraged to walk around as well, but not to offer guidance to the students. This phase of the lesson was very interesting as we watched students disseminate the problems. The conversations within the groups on how to approach these problems reminded me of Howard Brenner’s “Multiple-Intelligence” concept. Some students were very visual and immediately started to illustrate the problems and then find a solution. Others were more mathematical and immediately began to discuss proportions in terms of equivalent fractions, ratios, decimals and percents. However, time became a factor and the Explore phase had to be cut short. Some students were unable to complete the entire activity, but interesting enough all the students were able to grasp the concept during the Share/Summarize portion of the lesson. I believe that the Share/Summarize part of the lesson turned out to be the epiphany for several students, particularly those that may have approached the problem erroneously. These students were able to detect their error while presenting and correct it at the same time.

We also ran into a bit of a problem during the Share/Summarize part of the lesson. Again, we were met with time constraints. In part this problem may have been avoided if the teacher had been more selective of the presenting groups. Instead, all the groups presented and this may not have been as effective. I would seriously consider being more selective of the presentations in future similar lesson.

After consulting with the teacher-members of this lesson, they indicated to me that a follow-up to this lesson later on in the school year included the following material: compared and ordered from least to greatest ratios, fractions and decimals. The students used proportion and solved for the unknown. They covered finding the slope of a line using rise/run and the algorithm (y=mx+b). They also used similar geometric figures and related proportions to solve for an unknown variable. These were covered during the remaining course of the school year in other units of study, to which I was privy to witness during my regular class observations as an MPT.

Mixing Juice: Individual Reflection (Blaine Ball)

1. Concept of Comparing Ratios

A. Student Grasp of Concept: This lesson was taught to a group of advanced 8th grade Algebra students. Each of them understood that comparing the ratio of concentrate to water was a part-to-part comparison. They also clearly understood the distinction between this and part-to-whole comparisons of concentrate to mix and water to mix. All were able to convert both the ratios and fractions involved to decimals and percents.

B. Subsequent Application in Class: My students used the concept of comparing ratios on several occasions after this particular lesson. In practicing number operations, they converted between fractions and decimals to compare and order rational numbers. When we studied scale factors, they applied it to scale the ratio of two numbers up and down and calculate equivalent values. They also set ratios equal to each other to form proportions and solved for unknown values. Other applications involved calculating unit rates and finding the slopes of lines.

C. Assessment of Student Understanding: Assessment of how well students understood the concept of comparing ratios took several forms to include:

- Given a list of rational numbers, order them from least to greatest. - Given a scale factor and either the size of the model or actual figure, find the unknown quantity. - Given a scale factor, dilate a geometric figure to obtain a scaled transformation. - Given two equal ratios, one of which has an unknown quantity, solve for the unknown quantity. - Given distances traveled in various periods of time, determine the respective unit rates and order these rates from greatest to least. - Find the slopes of two lines on the same graph and interpret the meanings of these slopes in general and in comparison to each other.

D. When my students were asked to compare ratios, virtually all of them, including even the most capable students, prefer to convert the ratios to decimals and then compare the decimals. They do this because calculators are readily available and because most are

not comfortable doing operations with fractions. Over the course of the year I worked with them to remedy this problem by creating skill practice centers in which they were required to do add, subtract, multiply, and divide fractions and mixed numbers without using calculators. It was very successful and really boosted their confidence in doing fraction operations.

2. My Thoughts About:

A. Students’ Understanding the Concepts: One of the key instructional principlesreinforced by this lesson was the importance using proper vocabulary. If my thinking is imprecise, I use vocabulary which lacks specificity or is incorrect. The same is true for my students. For instance, a ratio is defined as “a comparison of two numbers using division.” A fraction, on the other hand, is “a way of representing part of a whole or part of a group by telling the number of equal parts in the whole and the number of those parts being described.” (DeBold, Rogalski, and Boudreau, Math On Call). In my own mind, I think of a ratio as a part-to-part comparison which may be written in various ways, including as a fraction, while a fraction is a part-to-whole comparison. Being sure to adhere to the definitions of ratio and fractions when we are making comparison is critical if we are to avoid confusing our students. This is also true, of course, for all math vocabulary.

B. My Own Understanding of Concepts: There were several mathematical concepts which this lesson helped clarify for me. Although I had compared ratios by getting common denominators, I had never thought of it as making the denominator a quantity of one. Likewise, converting ratios to percents was not new to me, but I did not think of it as making one quantity a hundred. I even learned from a student group during the public lesson when the students used water to mix comparisons to decide which mix was most orangey. It was something I had not thought about doing.

Perhaps more than the pure mathematics of the lesson, I learned a lot about pedagogy during the lesson preparation, practice lessons, and public lesson. Preparing a plan for questioning students, anticipating student responses, and having a blackboard plan for displaying and discussing student work were all revisions to my lesson planning process brought about as a result of this lesson study. In the practice lessons, I learned that sharing and summarizing take a lot of time and, during the share phase of the lesson, the class performed better if I remained in the back of the room and let them discuss their work. In the public lesson, it became clear to me that allowing some struggle time during the explore phase facilitated learning when students shared their struggles and discoveries at the conclusion of the lesson.

I believe that what I learned about mathematics and pedagogy during this lesson study will make me a much better teacher in the future. I have an increased understanding of how my students’ thinking and, as a result, will be able help them avoid pitfalls such as using differences between the amount of concentrate and water in the mix to determine which is most orangey or attempting to compare ratios without first finding a common denominator. I will also think through both the mathematics of the lesson and how I wish to bring out all

correct mathematical approaches to the problem during the lesson. Perhaps most importantly, I learned that challenging lessons take time to plan and teach.Adequate time must be available for students to explore and share, and for the teacher to summarize. If I apply all that I have learned, my students will perform at an even higher level and with much less difficulty in the future.