In which there gleam the stars. In which there lie the stones. For the plants that live and grow. The animals that feel and move. Where a human and soul behaves. At the level towards the spirit. I look inwards to my soul. That lives within my eating. The spirit of God is eating. In sunlight and in sunlight. In heights of world above. In depths of soul within. Spirit of God indeed. I turn myself in seeking. That strength and grace and skill. For learning and for work. May live and grow in me. So take pride in our love. All day long. All day long. All day long. I'm going to let myself be looked down by the wall. Good afternoon and welcome. On behalf of the Orchard Valley Waldorf School Board of Trustees, I'd like to welcome you friends, family, faculty, administration, Libby, and you, graduating class of 2018. I wasn't going to cry. I'm trying not to. I'd like to briefly share with you all about the first time I met Libby and her class. It was more than six years ago, and a move to Vermont for my family was imminent. We did not have a house to live in, but my children were practically enrolled at Orchard Valley Waldorf School. We came up to visit the school and observe a class. And while I was 99.9% certain that my children would attend Orchard Valley Waldorf School, we had been invited to observe the classroom. If there was the slightest doubt in enrolling, it was suppressed by seeing this class in action. I witnessed a strong, grounded, caring teacher leading this cooperative class of children in a math main lesson. I'll never forget when Libby asked the class to push their desks to the side so they could take up the next part of their lesson. It was so organized. Skipping to their threes, fours, sixes, and so on. She had their full attention. And they were so happy and worked so beautifully together. I still remember some of their names as Libby had called them out that day, and some of what they were wearing, brightly colored shirts and corduroy pants as their long limbs swung about, as their main lesson work settled within them. It has been a pleasure to watch you all grow and emerge into the wonderful young adults you continue to become. As we bid you adieu today, we are confident in knowing you have been gifted with a foundation that will live within you and serve you and your communities for the rest of your lives. Whether you have been at Orchard Valley for one year, nine years, or any number in between, you have undoubtedly had the great privilege to learn and grow on this glorious campus that has greeted you each day. You have been given the support by Miss Case and all our faculty and staff to grow intellectually and socially and to come into your own bodies as you have been ready to do so. This education will forever serve you as you encounter a life's triumphs as well as life's tribulations. Congratulations on this momentous achievement and know that Orchard Valley will always be a place you can come home to. I'd like to introduce Lindsay Benton, the teacher for the rising seventh grade. Well, admittedly, I have a very fun job this morning because I get to introduce my colleague, my mentor, and my friend Libby Case. I met Libby six years ago when I landed at Orchard Valley. Her class was in second grade at the time, and I remember being impressed by her and her class immediately. She was fun, creative, very deliberate, and I knew that she was going to be a colleague from whom I could learn a lot and respect so much. And I can say from firsthand experience that this initial hunch that I had was very true. Those of you who know Libby well know that she is incredibly capable and talented. While simultaneously being very humble and quick to focus on others before even dreaming of shining a spotlight on her own accomplishments. So let me do that for her now. Libby is quite accomplished. Prior to finding her way to Waldorf Education, she spent many, many, many years as an environmental educator training graduate students aboard a bus for Leslie College. She met her husband, Ben, during these formative years, and the two of them had two daughters, Amelia and Anna.  A passionate interest in her daughter's early childhood education led her to the doorsteps of Waldorf Education. And she spent 14 years as an early childhood educator. During this time, she became the lead teacher of the Montpelier Childs Garden, and she helped facilitate the creation of Orchard Valley Waldorf School.  Over the previous eight years, Libby has guided this class through many academic milestones, from learning cursive to writing essays, from mastering their multiplication tables to graphing linear equations, from discerning morals from fables to discussing civil rights and empowering these young people to be advocates for social justice. Each day of her teaching was approached with consideration and incredible commitment. Now, the work of a Waldorf teacher can easily consume one's entire life, but what has impressed me most with Libby is how she never cloistered herself in her classroom studies, but she did quite the opposite. She openly lent her time and energy in service to our school community. I think she served on every committee that our school possibly had over the years here, be it governance council, the finance committee, school life, to being a faculty rep on the board, and it's such a blessing that a person with such an outstanding character has had such an impact on all the many decisions that form this school.  Now lastly, I really want to emphasize here, and I think everybody who knows Libby will agree, that her greatest attribute is her eagerness to be a service to others in ways that aren't necessarily showy or glamorous, but in the ways that are simple and yet profoundly loving. When I was on maternity leave, for instance, Libby would come by my home regularly with treats like Ben and Jerry's ice cream. And she kept me company and checked up on me often, much like a mother would do, which really meant a lot to me because my mom lives eight hours away. Indeed, Ms. Dunlin and I have dubbed Libby our work mom and for good reason. We have come to rely on her experience and kindness and care. I think anyone who has ever spent time with Libby will have similar anecdotes that expose her dedication and love of people. A famous Rudolph Steiner quote reads, the healthy social life is found only when in the mirror of each soul the whole community finds its reflection and when in the whole community the virtue of each one is living. Thank you, Libby, for letting your virtue shine so fiercely. Our community is stronger because of you. And at this time, I would like to welcome a representative of each of the students in this class. All right, from each of our students? No. What if I knew a good idea? In Waldorf education, we learn to hold carefully the natural development of every child and consider even our school as a living, growing being. The impulse of Orchard Valley is about 18 years old now, so well-grown. And for all those years, Libby Case has been at its heart, guiding, energizing, informing, planning, focusing, listening, singing, playing, hiking, cartwheeling, befriending, leaping, calculating, scheduling, in short, motivating and leading. Today, we want to recognize her roots, indeed the roots of this school as an early childhood teacher at the Child's Garden and our farmhouse kindergartens on the founding boards, finance committees, festivals, school life committees, and more. If you have been a parent or a student in any of those classes or committees, would you please stand for a moment? There's more back there than I'm saying. I saw some there. Thank you. In honor of all the service and growth of our community and our children, we present to our school a middle school play structure dedicated to Libby. We'll just bring out the hammers in there. This has initiated this project as our graduating gift, but we are joined by alumni, parents, past members of our class, past colleagues, grandparents, and many fans. In just one week, we are more than three quarters of the way to our $4,000 goal. Libby, we can't wait to see you designing the fun and learning around this expansion of Orchard Valley's outdoor classroom. In recognition of our bird-loving dear teacher, there is a small purple martin plaque just to remind you of this gift that we're offering to the school in your honor. That says on the back, we, the lucky families who have been blessed to have Libby Case as a teacher, present the gift of a new middle school play structure for the Orchard Valley Waldorf School Playground. With this gift, we honor Libby's kindness, generosity, steadfast encouragement, excellence as a teacher, and her tireless devotion to the school. We offer this gift with love and endless gratitude. Thank you, Libby. Oh, my gosh. All right, I think I need a chair up here to stand on. Let's see. Should I use this? Is that good? OK, we've got to find the right place. Oh, my gosh. I'm sort of overwhelmed. Ben's crying. Well, good. Now we can stop talking about me for a little bit. Although, I don't usually talk about myself in a speech, but I did in this one a little bit, so now I'm a little embarrassed because it's been a little overwhelming so far. But I'll do my best to rush through that part and get to you guys as fast as possible. So I want to start with a poem that we used in fifth grade and sixth grade in a few different years. And also, we just heard it on the boat every morning.  In its brief course, lay all the realities and truths of existence, the joy of growth, the splendor of action, the glory of power. For yesterday's but a memory, and tomorrow only a vision. But today, well lived, makes every yesterday a memory of happiness and every tomorrow a vision of hope. Look well, therefore, to this day. And I want to start with that because look at this day. Holy smokes. We have a great day. So hello to everybody.  And we're in a very beautiful place. It's a day of beauty and gratitude. It's a day we're all going to remember well. But we're going to be in it right now, which is so exciting. So it's with great pleasure that I get to stand up here before you and have this opportunity to tell you a little bit about these folks who are down here sitting in their seats, who I have come to know and who I love dearly. And I hope every one of you know that. But first, this is the part I'm in trouble about. Let me tell you that I'm one of the luckier people in the world. My good fortune comes from many places. I have a lovely home. I have food on my table, thanks to many of you. This week is last week especially. My class has provided me with dinner night after night. Thank you so much. I have clothes on my back. I have a wonderful family. I love and I support them. And they love and support me in return. I have long lasting friendships, because I'm getting old. That if weather the years and remains strong, and I have an amazing array of colleagues with which to work, really amazing. But today I want to focus on one particular great fortune that I have in my life. And that's the privilege and the honor of teaching these students over these past eight years. Many of you know that I was one time a leader of expeditions. And I traveled around the country with college and graduate students teaching environmental education. I loved taking expeditions even when I was little, down to the river, anywhere. I was ready to go. I loved leading them when I began working. It was during one such expedition that I first stepped into a Waldorf classroom. And I saw my future. I thought to myself, now here's an expedition I'd love to make. Taking the students from first to eighth grade would be an awesome journey. And I do know how to spell awesome, by the way. Someday I would like to do that, because this Waldorf education makes sense. It's holistic. It's grounded in the very earth we walk on. It's about community. This education takes childhood seriously. It's an education needed to grow children into amazing people who are destined to be, whoever they are destined to be, whether it's going to be an oak or a maple tree. This education takes that little seed and allows it to take its rightful shape and form. It serves not just the intellect of a child, but the very soul. This is the journey I want to take next. So finally, a few years ago, Shannon Hepburn came up to me and said, some of us kind of think it might be nice if you would take this class. I said, all right. I think I'm ready for that. So here is my fortune of this very day. The joy of having been able, thanks to the trust of you, the parents, and the engagement of you, students, to complete this remarkable journey with these remarkable young people. How lucky am I? How lucky. So now that our expedition together is coming to its end, I wanna offer a little reflection on where we've been and how we got there. However, reflecting on this expedition is not an easy task. It's not nearly as easy as I thought it would be, and here's the reason. Eighth grade is really a synthesizing year of all that's come before. It's a time when physics comes together in the practical study of weather. When math comes together in algebra and history makes its way into the present, it's when writing and reading become the tools of new knowledge and poetry and song true means of expression. It's a time of change in childhood when the student who has been living and learning primarily in the feeling realm now begins to live and learn more in the thinking realm. Standing on the cusp of this change bringing all this together this year has been incredibly exciting for me and quite honestly has been such a whirlwind of activity and learning that I feel like I'm still spinning in it. Really, there has never, and I mean never been a dull moment. We were busy from day one, up all the way until yesterday. We hiked with first grade buddies, we researched plastics, interviewed and filmed seniors from the Twin Valley Senior Center. We developed our eighth grade projects, skied Bolton, produced a Shakespeare play. No small feat. Learned about the roots of racism in America and its place in modern history. Studied platonic solids, acids and bases, weather and clouds. In between all that we read some books, wrote some essays, we put Columbus and his men on trial for their crimes against natives. We journeyed down the Colorado River. We learned new songs and poems and famous speeches and we went to Montreal and we even set our sails and headed out to sea. Through it all, each of you students stepped up and played your part. Whether it was in Romeo and Juliet or out on the soccer field in the eighth grade faculty game, the most awesome game in the whole world. Or decorating the earth for a dance or preparing for a festival, working on your eighth grade projects, completing a group project. We've been very busy making our way to this day, the last day of our journey together. What I love about an expedition is what it builds in both the individual and the group. It always involves working together for the common goal with each person on the expedition being critical to its success. Without him or her, there would be no trip. Through this process, a community emerges, one that has very distinct characteristics that define it. This is true for each class that goes through Orchard Valley Walder School and it is what makes each class so unique. For this class, there are two characteristics that I have seen emerge and develop over these years and then have come together during this last leg of our journey to make you all the admired class that you are and believe me, I get nothing but compliments. The first is your energy and enthusiasm for games. You have always been an active group, a group of action and engagement. This has been true since the get-go. First grade, it was essential for me to begin each day with what are called new exercises. These involve rows of students crab walking, wheel barrowing, seal sliding, donkey kicking, their way across a cleared room for at least a half an hour. In second grade, it was baobabalinda circle dancing and poems done to copper rods and bean bags.  Fourth grade, it was hiking, capture the flag, clapping games and contra dancing. You love contra dancing. Fifth grade was tribal groove body rhythm, kick ball, training for the pentathlon. Sixth grade brought squiring, swinging on the swings and soccer. Seventh grade, we did slow down a little bit with yoga, mindful meditation. Didn't work much, but we tried. Finally, eighth grade, a month of morning calisthenics was probably one of our more successful blocks and lots of piling on top of each other on those two chairs. Now down to one. So, my job over these years has been taking these individual balls of energy and helping them form a cohesive cooperative organism. I have to say, I feel quite successful. This joy of playing and doing is growing into a shared sense of fairness and understanding, of need for cooperation, of an ability to work together in all situations. Just a few weeks ago, when we were trying to get a photo of the whole class up on the roof of the school, close your ears, Todd, please. It was amazing to watch as you work together to get each other up on the roof and to get each other down again. No one judged anyone else if they were scared or hesitant. Instead, they jumped in and problem solved. You worked as a team without any designated leader. It might seem like a small thing, but it was pretty impressive. Amongst all this activity was a lot of laughter, a lot of tears, a lot of fun. And this brings me to my second defining characteristic of this class, inclusiveness. Sometimes a group can get a bit insular after being together for a long time. Sometimes it can even get, need I say, exclusive. This is not true with this group.  Amanu in first grade, Adin second, Rain, Bryce, and Caleb in third, Britton fourth, Hattie in fifth, all slipped into our group as if they'd always been part of it. Last year, when we combined with grade six, I was kind of nervous what was gonna happen. We welcomed Rain River, Anna, and Stella into the class and it was as if we'd always been together. When our French visitor Ishmael arrived and spent three months with us, he too was swept under the wings of this gang and instantly a part of it all. These guys even welcome teachers. Miss Baker, Miss Cozar, Mr. Doring, Miss Fettig can all attest to the generosity of spirit that exists in these students. This inclusiveness extended well beyond the classroom. Our visits to the senior center were excellent examples of this. Your openness and warmth were so appreciated by the seniors and helped facilitate the wonderful interviews that you did with them. Parents, you will get to see it someday. You brought your world to them and welcomed them into it. Your dances where you invited back past class members, your camaraderie with the sixth graders, your care and attention to your first grade buddies. Again, these might seem like small things, but believe me, they're not. This is an exceptional quality. This quality that will be so important in not only your future, but in the future of the whole world, really. Expeditions, they're opportunities for growth, for development of knowledge of self and knowledge of others.  This was no accident on my part, but rather a decision made early on in the journey through these grades. To sail a ship of this size, you need more than two individuals. You need cooperation and the focus of many people. Working together to ready her to sail or tuck her in at night, smooth sailing requires teamwork, smooth sailing requires strength and fortitude, cooperation and consideration. Smooth sailing requires attention to detail, awareness of the whole. It requires setting your own needs aside in any given moment for the good of all. That is what I saw on the ship. At 6.45 each morning, the deck was getting swabbed, breakfast cooked, cabins cleaned. When it was time to hoist the sails, people dropped what they were doing and took up a line and pulled together. You had to listen and stop when needed. That was pretty good. And pull when needed. Furling the main and foresail took everyone's attention and cooperation. One person folded as the other pushed the next fold in. We worked like a well-oiled machine. You made room for each other and all the while, you had fun. That's really kind of it. You had fun, don't you think? All along the way. Now that I'm reflecting on all this, there's one other characteristic I have to add and that's bravery.  That was a cold dip. And some of them made it like many times. When I look at this group of young people that I've had the privilege to sail through both calm and troubled waters, I am nothing but proud. In each one of them I see a young person who has a strong centerboard, a mighty inner compass that will help steer a course and will allow him or her to be into consciously, consciously shape his or her own future. I see a group of young people who have together laughed, cried, fought, made up, challenged each other, demanded the best from each other, teased, flattered, welcome new friends into their group, said goodbye to old friends. But most of all, I see a group of students who accept each other for who they are and as a result, they accept themselves for who they are. One thing about expeditions is that they do come to an end as this one now is. This is a good thing. You know why? It allows for the next one to begin. For new growth, new learning, greater understanding of self and of other. So off you go on your next big adventure, approach it with a full sail, let your new friends in with the generosity of spirit you have shown here. You will meet challenges, pressures, even in opposition.  You have the means to meet them. Lay your course steady. Your dreams will take some time to fulfill, but you have what you need to realize them.  So thank you guys. Hey, that's enough for me. Now it's time to hear from them.  And each one will have an opportunity to come up here and share a little bit with you. So let's see. I have them right up here. So what I'm gonna ask you guys to do when you come up and yours is right on, give me on top as you come up. When you take yours out and read it, don't put it back in. Just you can put it off to the side, okay? So that the next person can get theirs. So first we have Nadia McCollister. I'm Nadia. I just wanna say that I've been so lucky to go to this school and be with all these people. I've been with my class for nine years and this has been the best time of my life. We had some good times and some bad times, but that's normal. I'm so grateful for all the specials teachers that have taught me all that I know today. I hope that I will be able to make some of these projects at home. I might try making more pajamas. Or carving another figure, or maybe learn another language. But the one teacher that I wanna thank the most is Miss Case, because she has been with us for all these years and she has had to deal with so much because we have not been the easiest class to control. This case has had some bad days, like maybe we wouldn't listen to her or maybe we'd argue with everything she said. But she's stuck with us and I'm proud for her. My classmates and I are like a family and we have had so many fun adventures and trips in our years. One of my favorite adventures was the Montreal trip. We got to go on it with our amazing French teacher. She is so good at making plans for a big adventure. We went on a zipline and I've always wanted to do that. We even went to a chocolate shop and I was in heaven. I always want my friends to be happy and if they're sad, I wanna be there to give them a hug because friends are like stars and they will always guide you through the hard times. I'm going to miss my friends so much but I hope we can all get together and hang out sometimes. I will never forget them and I hope they will never forget me. Thank you. Eva Goodwin. Hi, my name is Eva and I've been at Orchard Valley for 11 years. In first grade, at the end of the year, Kavita and I were sad that our eighth grade buddies were leaving us so soon. We hatched a plan that we would meet at Kavita's house and ride our bikes down to Montpelier High School, the only other school that we knew of and we thought our buddies were attending. We made a list of things to bring, which included training wheels. We thought we could make it down the hill and burst into the high school and all our buddies would be there. When I was in first grade, the thought of graduating seemed infinitely long. Even at the beginning of eighth grade, it still seemed like a very long time but all of a sudden the year went by. This year has been one of my favorites because of all the fun things we did and trips we went on. The retreat, the Montreal, and then Maine because our class was so close and had a lot of fun being together. When we were little, we used to get in so many fights. Well, of course, we still fight. After all, we are a family but they are not the huge fights we had when we were young. Although my class can be so annoying, they're also special and are always there for me. They are friends that I will always remember. It will be so strange and new not to have all these people by my side every day next year. I love you guys all so much. I'd like to thank my teachers. Thank you, Madame, for not giving up on me although my friend Jackson is terrible and for taking us on amazing trips. Thank you to Miss Case for being the best teacher. She has an enthusiasm of a nine-year-old especially when it comes to making us more stance. She never gave up on us even though we've made her cry, swear, and yell. We've definitely not been an easy class but she was the right teacher for us because of her energy, which is always high. Thank you to Heinz for teaching me how to work with wood and teaching me how to sweep. Thank you to Miss Camilleti for teaching me how to knit, sew, crochet, and many more things that have been so fun. You're the snazziest teacher ever. Thank you to Miss Davis and thank you to Miss Davis for not getting discouraged when our class comes up, when our class groans about the awesome ideas you come up with. And finally, I'd like to thank my parents for driving me to school every day and in the beginning choosing to come to the school. The 11 years that I've spent at Orchard Valley Walder School have been magical. I've been sprinkled in fairy dust, taken long walks in the woods where the leprechauns live. I've played in the restaurant tree and ship tree and I've knitted a gnome. Most kids never get the chance to do that. Now going into a school that does none of that, I'll walk in and think, why are the walls not painted a beautiful color? Where are all the crayons? All these things I've taken for granted will be gone and now I find myself feeling grateful for the opportunity to experience all these things that you can only do at a Walder School. Thank you. My name is Rain. I've been at Orchard Valley for six years. I've had good times and bad times here. The first memory I had was when I ran a rain third grade. I was new, scared and nervous and I didn't know how to fit in. Some of my favorite times were building our shelters in third grade. Our Croco trip with the infamous Nicholas Greab tales. Our fourth grade play and my part is Loki. Fifth grade was a challenging year with lots of fights. The White Mountain trip was really fun even though I'm not a big fan of hiking. In seventh grade, we had a combined class with the sixth graders and I actually loved it. Eighth grade has been a lot of fun. We all became very close and got new privileges we've never had before. We were able to go to the P bench and use the faculty bathroom. I would like to thank, I would like to thank all the teachers who believed in me and helped me. I would like to thank all the teachers who believed in me and helped me along the way. And my parents who realized I needed a change and paid and drove me to the school. Now, here's a more important topic. What's the secret of life? People change people. People change you and shape you into the best version of yourself. That's why I would like to thank each one of my classmates individually. Nadia, you have shown me the power of perseverance. How to keep pushing through when things get tough. Caleb, you have shown me how to throw myself into a social situation and fake it till I make it. Patty, you have shown me the joy of friendship. How to encourage people and support them. Sophia Casey, you have taught me when it's not my story to tell and been my best next door friend. And Manu, you have taught me the joy of listening and being honest and have been there on need to rant or complain. Aida, I love talking about Riverdale and doing funny things with you. We always end up laughing. Lauren, you have taught me how to be myself no matter what any other group was doing. Bryce, you have been the comedic humor I need to get throughout the challenging days. Eva, you have been a helpful, nice friend. You've always taught me something, whether it's drawing lessons in third grade or how to get through my eighth grade presentation. Ellie, you have shown me how to stand up for myself and met me with so much kindness. Sophia C, you have always encouraged me to be me. All of me, not just the happy silly side. Korra, you were the first person I carpooled with. You introduced me to a new part of my life, a challenging change. Bryn, you seem like the class peacemaker. You never are involved in class-related drama and you are kind and caring to your friends. You guys have changed me a lot. Aren't I different than the girl who joined in third grade? I made it, we made it. Thank you, Ms. Case, for sticking with us till the finish line. I am so grateful for the beautiful 55-acre campus I got to grow and play on these six years. I'm going to end with a quote that's helped me get through the past weeks. Don't cry because it's over. Smile because it happened. Hi, I'm Bryn. I haven't been in this class as long as some. I actually joined in fourth grade. I'm the type of kid that when you first meet me, you think I'm quiet and weird, and then when you get to know me more, you just think I'm weird. Like I said, I joined in fourth grade. Before I came to Orchard Valley, I went to Waysford Elementary, which is a totally different school in almost every way. I'm glad I changed schools though because I got to meet all these great people. My classmates and I have been through so many ups and downs in adventures together from our fourth grade camping trip to our eighth grade sailing trip. Now I would like to say a few thank yous. First, I want to thank my parents and my carpool for driving every day of the distance. 45 minutes is not a little amount of driving. Next, I want to thank my teachers for a few things. Number one, for teaching me so many things over the years, and number two, for putting up with me for four years. I'm not the easiest kid to teach. I would also like to thank my first grade buddy, Liam Moore. And finally, I want to thank all my classmates for joining me on this amazing journey and sometimes crazy, but mostly amazing journey. Every one of them is an amazing person and I hope you can stay friends for life. Thank you. Hi, I'm Cora Sturries. Hi, I'm Cora and this was my 11th year at Orchard Valley. I don't remember when I first came. I just remember being here. It was my place. Some of my first memories were in kindergarten and nursery with Miss Peggy. I've always been in love with kindergarten and nursery and everything about them. The smells, the toys, the songs, and all the magic. It's pretty much just a magical blur. I remember on one soup day, I brought in a butternut squash and I got way too attached. I drew a face on it and I also probably named it. I spent a long time cuddling it in the hall that day. Another moment that sticks out is when I lost my first tooth. I remember coming out of the bathroom and being so proud and excited to show everyone. I want to thank Miss Peggy and all of my kindergarten and nursery teachers for making it all so wonderful. Going into first grade was very exciting.  We all soon discovered that we deeply missed having nap time and should have appreciated it while it lasted. As the years flew by, I really got to know my class. We did everything together and we're like a family. I'm so grateful for you guys. You've made all these years a blast. Spending so much time together was hard at times but we've survived and I'm going to miss it very badly. I can't imagine not being with you guys every day. It's going to be a major adjustment. I want to thank all my teachers from throughout the years for making my experience here so wonderful. To Miss Camilletti for always finding a new way for us to make pouches. To Heinz for teaching me how to carve really cool things that will last a lifetime. Thank you to Madame for always believing in my friend abilities and a special thanks to Miss Case for being an amazing teacher throughout all the years and for dealing with us breaking the nice furniture you put in our classroom. Here are some things that people will be confused about in the future because I went to Waldorf School.  I can smell Prussian blue from a mile away. I put borders on everything. I've had the same teacher for nine years and I always expect there to be crayons everywhere. I know a song for every season. I can play the recorder, kind of. It's normal to play soccer on a hill. And last but not least, I've slain the dragon with my school for the past eight years. When I was younger, all I wanted was to go to a normal school. But things have changed. Now all I want to do is knit gnomes and sing and play outside all day long. But that's not really what high school's about so I'm in trouble. Now some advice for anyone who will be in eighth grade someday. Be nice to your teachers because then you'll get extra recess. Thank you. Hattie girl. I'm Hattie. But you can call me whatever you want as long as it's not Hattie. It's happened. I have been at Orchard Valley for four plus years. I joined at the end of fourth grade and was immediately sucked into a pool of wild craziness that is our class. I couldn't have asked for a better class.  It's not just my class who have been important to me during my time here. The amazing teachers have helped me grown to the person I am today. These teachers deserve as many thanks as we can throw at them. Hines, thanks for dealing with all our clowning around but I definitely know how to sleep correctly now. Miss Kemaleti, I will wear my pajamas until they fall apart.  Madame, I always enjoyed your French class because you made it interesting and fun. The two trips you took us on hold some amazing memories. Miss Trouts, teaching the sixth and eighth grades music is an accomplishment. How you got us all to be quiet, I will never know. Miss Davis, you were always so upbeat during On The Land and you even made it sound interesting to stare at goats. Thanks for dealing with all of our pent up energy. Miss Case, I know I sometimes don't show it but I'm really grateful that I had you as a teacher. You really are superwoman. Trying to teach an eighth grade class about clouds and make it semi interesting is the stuff my nightmares are made of. Through all the interruptions, flips, fights and butt cheeks you'd stayed sane which I think is quite an accomplishment. So thank you. I also want to thank my parents who have dealt with many, many ripped up mash eats. I'm so lucky to have had this experience here at Orchard Valley and one day when I'm older I will pull out all my old man lesson books from my old school and OVWS and flip through them, remembering each pen, pencil and crayon stroke every scribble and glued page through the laughter and many tears mostly shed when we all watched Titanic. I have made so many memories to keep with me forever. To end, I'm going to sing part of the song for good for the musical wicked. To my class, I love you all. Class in second grade and I want to say thanks to all the people who have helped me along the way. I appreciate all the time you spent giving and the work you have done to make me who I am today. My classmates have been great companions like I like how we can play games and work together. My friendships made a difference in my childhood. I will remember the times we shared. Ms. Kimmeletti, I like how you made our class fun. Some of the projects you picked were interesting and exciting. Thank you for showing my hands how to make things out of different materials. Hines, the guidance you gave me while working on the eighth grade gift to the school was helpful. The other projects over the years taught me lots of skills. You are a great teacher. Ms. Baker, you have helped me a lot with language arts. I really appreciate all the time and effort you gave me. Thank you. Madame, you planned great trips. The trip to Quebec City was full of exciting adventures. Thank you for including me. Ms. Davis, you are always upbeat and happy all the time. I like the games and activities you picked. I feel lucky that we are able to spend so much time outdoors on our land. Ms. Case, you have shown and taught me how to be patient and kind to everyone. I enjoyed learning from you. You have been a great teacher for me. Thank you for all the time you have spent with me. To my mom and dad, thank you for letting me go here. I appreciate all the time you spent driving me and putting up with me no matter what happens. Orchard Valley is a beautiful place to go to school. I have enjoyed going to school here very much. Thank you. Hi, I'm Bryce. I came to Orchard Valley in third grade. When I came to school, it was a very big change because I came from a public school who did it very differently in the OVWS. For instance, this school is an eighth the size of the other school. And I think the size of the school brings everyone in it closer together. My favorite year was probably this year because of all the cool stuff we did, like the eighth grade projects, the Montreal trip, the sailing trip, and even the school part. My classmates were the best part of my six years here, and they're all awesome. You probably know that in Waldorf, instead of changing classes, we all stay in the same class every year. That's given us the chance to become really good friends throughout all our time here, even the tough times. Our teachers have been also very awesome, weird, and considerate. I want to thank them for coping with us, being good teachers, and not being boring, which is always a good thing. I want to thank Ms. Case for dealing with our class and for being a great teacher, supporter, and friend, even though when we drive we crazy. Another great thing about OVWS are the trips. All the trips have been very well planned and really fun. My favorites were probably the Montreal trip and the sailing trip, and we really missed the bonding experience, so we go to amazing places with people I've known for so long. This year we got to have so much fun with our first grade buddies. I had Asher, who was awesome. We went on a hike, had buddy lunches, and had recess together. I'm lucky that my buddy was Asher, and then I want to really thank Asher for my buddy. The last people I want to thank are my parents. All the other parents for sending their awesome kids to this school to begin my friends, and especially my parents for forcing me to come to the school. This place has been a place that has changed me, and I'm glad it did, because it may be a better person. I love the school, and I would love it a lot. I miss it a lot. Thank you. My name is Aida. I've been over WS for seven years. Each year here has been a new journey for our class to grow and make memories together. I love how closer classes become over the years. We're like one big family now, and I'm very sad their time together has come to an end. But like Dr. Seuss once said, don't cry because it's over. Smile because it happened. Now I want to fully thank my class. Ever since my first day here at OPWS, I felt welcomed, wanted, and loved by my class. No matter where I go, and no matter where you go, I will always love each one of you. I want to thank the school for making an amazing learning experience for me. OPWS has shaped me into who I am today. There were times when I was younger that I wanted to go to public school because I thought it would be so much better. At the end of sixth grade, I begged my parents to send me to Craftsbury Public School, but I am so glad that I stayed at OPWS all the way to the end. The school was very special, and not many kids get this kind of opportunity. I was very lucky. I can never thank my parents enough for always being there when I needed help with either reading over a paper I wrote, helping me with my horrendous spelling, and driving me to OPWS every day. I want to now thank everyone of my super groovy special teachers. Madame, thank you for always making French class such a blast, but either having some yummy French treat for us to make or bringing in a French movie for us to watch. I will always miss and remember you. Miss Camilletti, thank you for everything. You have the best sense of humor ever, and I will always love that about you. Thank you for teaching me how to sew by hand and knit my own socks.  Hines, thank you for teaching me how to make something beautiful out of a piece of wood.  I did enjoy woodworking class. Thank you. Miss Davis, thank you for always having such fun games for us to play on all the land. There was never a dull moment. Thank you. Miss Trouts, thank you for coming in every Monday and Friday to a room full of noisy sixth and eighth graders. The songs you brought to sing were fun and beautiful. Now finally, I want to thank Miss Case. Thank you, Miss Case, for staying with our class all the way and bringing your peppy cheer for presents whenever it was needed. Sorry, I was a blocker. You somehow would patiently teach our class about clouds no matter how many grumbles and complaints you would receive from us. You have been a wonderful teacher and I will miss you forever and always be grateful. To end, I have one last quote from my classmates as we head off from Orchard Valley. Don't worry about failures. Don't worry about the chances you will miss if you don't even try. Jack Andrews, thank you. Sophia, kind of case tellings. Hi, I'm Sophia. I've been at Orchard Valley for eight years now. It's hard to believe my time has come to an end. When I think about the memories, the opportunities, and the people here, there is so much I could talk about. It's hard to know where to begin, but I'm going to start with saying how thankful I am for getting to be a part of this amazing class. Over the years, the support and the friendships have just kept growing. I will never forget the years spent at OVWS, all of the plays we've done and field trips we've been on, knitting socks and carving mice, learning to always be prepared for the weather, singing tons of songs and the celebrations and events that happen every year. And with the help of the rest of the eighth grade class, me and Cora's victory in the box car race at Mayfest this year. I'm so thankful for all of the teachers who have shared their time, talent and knowledge with us. Yes, it's what you guys are supposed to do as teachers, but I'm so thankful for it. For years now, you have taken the time to explain the assignments. Most of the time needing to repeat them because we weren't paying attention and through all of our complaining arguments and not listening, we step by us to make sure we get the help and support we need. Thank you, Ms. Case, for being there for me since kindergarten. I'm also thankful for my parents and grandma for even making it possible for me to go to school here, supporting me through everything and putting up with all my arguing, even though you were only trying to help me. I especially want to thank everyone in the class. From being at OVWS, I have learned a lot more than just school lessons. Personally, I feel like one of the most important things I've learned is what real friends are and that you don't have to change for anyone. The right people change you for the better. It's been a great eight years through our ups and our downs, through our fights and uncontrollable laughter. I feel like I'm so lucky to have been a part of this crazy journey. So thank you, everyone, for all of the help and support you have given me for the past eight years. Thank you. Allagran Godson. Hi, I'm Allagran. I have been part of OVWS since kindergarten. Over the years, there were times when I wanted to leave this school, but every time I really considered going to a different school, I knew I would rather be here. I would like to thank all the teachers in our school. Each of you are very important to the school and me. I definitely think this case deserves recognition for sticking with our class all the way from kindergarten to now. I love everyone in my class, but sometimes, even though I'm part of all the ridiculousness, you guys can get on my nerves and imagining trying to teach all of us to think of my nightmares. To my classmates, you are all amazing people, and I don't know how else to say it without filling up pages and pages of paper. You guys are incredible, and thank you for making everything seem brighter when I was down. Thank you to my parents and grandparents for supporting me all the way through Orchard Valley, all the driving and homework assignments I thought I couldn't complete. Thank you so much for sticking with me. I really have enjoyed my time at OVWS, and I'm going to miss it and everyone here, but I'm also excited about what's coming next. Sort of. I'm not sure high school sounds that great, but new experiences and all is going to be great. Anyways, thank you to everyone, teachers, parents, and friends. Thank you for this experience. Lauren Stevens. Hi, I'm Lauren. I have been attending Orchard Valley for pretty much as long as I can remember. Although paradoxically, I clearly remember my first day.  It was a truly magical time, during which we heard stories of enchanted castles and faraway lands of princesses and princes and frogs. It was a time when anything was possible, when fairies existed, and plants could talk, and goblins hid in the woods. During these three years, if a leprechaun were to pop out of the ground, I would have been completely unfazed. Finally, I moved on to first grade, and I was a big kid, but it turned out that there were even bigger kids. I recall looking up at the eighth graders, and wondering what it was like to be that big, what they learned about in school, and what their view was like from up there. My guess was pretty far off, and both the good and the bad. I certainly underestimated the homework, the stress, and the discovery that grown-ups didn't, in fact, know everything. I also could never have imagined how important everyone to the school had become to me. I could never have conceived how sad it would be leaving all of you. My 11 years at OBWS have shaped who I am today, and I'd like to thank everyone who tolerated us for the last decade. I'll start with Heinz, who taught me that no matter what I'm doing, whether it's rasping, filing, gouging, sawing, chiseling, sanding, or just hitting an intimate object with a hammer out of sheer frustration, you always have to use two hands. Madam Harper, for teaching me that nearly everything is better with glitter and colorful paper cutouts. Miss Davis, for all of the awesome games in team-building exercises that had us banging our heads on the canvas walls of the year, but were still lots of fun. Miss Benton, for your extraordinary patience and ability to explain any mathematical concept. Miss Case, for sticking with us, even when we made you cry or gave you black eyes, or stapled your fingers, even when we wouldn't stop talking or giving you a hard time about your spelling, even though we have never, and will never let you for your extraordinary capability to get us lost.  I'd like to thank Miss Kimmeletti for all the humor, wisdom, and discussions about movies, books, electron microscopes, quantum physics, and philosophy. I would also like to apologize for all of your property of broken, such as those three sewing machine needles, countless pins, and two glasses. I'd like to thank everyone who drove me, Bryn and Bryce, to school, mostly notably Terry, who helped me answer Bryce and Bryn's strange, yet fascinating questions, questions such as, what is uranium? Is it green? No, was Terry alive when the first car was invented? No, could we construct a new planet made entirely of Legos? How many jelly beans would we need were you to replace all of the Earth's oceans with jelly beans? Approximately 984 sextillion, 654 quintillion, and 640 quadrillion. The list, yeah, of course. The list of people goes on and on, but to keep you all awake, I'll skip ahead to my 13 classmates. My life would be very different if I didn't have all of you to laugh and cry with and give me hugs when I'm sad.  I'm never eating my pizza crust first. I can't imagine my life without seeing you guys every morning and dealing with you guys for most of the day, five days a week. I don't know how I'm going to do it, although I guess I'll have to figure it out soon. I really enjoyed these past 11 years with you all, and I want to thank you for making my life as intricate and interesting as it is today.  Okay, so first of all, I lost my voice yesterday and just got it back this morning, so let's see how I do. Okay, hello. My name is Sophia Clark. I've been coming to Orchard Valley for eight years now, and I would like to say that I have had quite the cascade of adventures, good and bad memories, dramas, some pretty embarrassing stories, and many once-in-a-lifetime experiences that I wish I could go back to. A man who we all know as Forrest Gump once quoted, life is like a box of chocolates. You never know what you're gonna get. Coming to Orchard Valley was not a mistake. This quote only hit me after realizing what a sweet chocolate I got out of life's box of sweet delights and surprises. I am so grateful for being able to have had a good time at this school. So thank you to all of my classmates, to all of my teachers, to all of the faculty administrators who keep this school running, and of course, to all of the amazing parents who support the school and help with fundraisers, events, and for simply being there when times get rough. Thank you. I believe that each individual person has their own story to tell and share with the world and the ones they love. Of course, there is a past, a present, and a future. We all have our own paths ahead of us. I hope to be a part of my friend's journeys and of their life's adventures. I am proud to say that I have already had the privilege of being a part of their pasts and for sharing this moment right here and now with them and with you. Everyone has their ups and downs in life. I myself have had a very difficult and challenging past few years. I want to say thank you to all of my friends and Ms. Case for standing by my side and helping me along the way. This past year, a lot of drama has happened. There have been multiple fights, exclusions, unnecessary comments, and many other things. It's a tug of war and I feel that we are growing distant, but we're not. There is something called fear that plays a huge role in everyone's everyday lives, whether it's fear of being unwanted, unloved, ignored, not fitting in, or feeling the whole world is against you. I think that all of these negative actions have been happening because of how scared we are to lose one another. We have become family and sometimes the people we love the most are the people we hurt the most. We're telling ourselves the high school will be better, that the drama will be easier to escape. We are lying to ourselves. Wherever we go, we take ourselves with us. It's up to us as individuals on how we approach life situations. I want this year to be an awakening for all of us. I want to keep creating many more wonderful memories and still hold on to all of the happy memories, so as never to let them go. So please, take my hand. Will you join me? I have so many memories of first grade. Puppet shows held by Sophia and Cora, Lawrence, and Genius Paper TV. The first row call when Cabot said he wasn't there was actually in China, then was sent to the office for talking back to Miss Case. Fighting over clay that we found under the bridge and going on little adventures to the restaurant and ship tree, then walking down to the stream and getting covered in mud and water. I remember hiking with our eighth grade buddies and doing the golden goose as our class play. We were so little, so innocent. Then thinking back on this year, I remember the row ceremony. We were eighth grade buddies. We got to take our little buddies hiking. We got to play with them as well as get tackled by them. Watched their play the same one we did in first grade. It was such a beautiful experience being the eighth grade buddy and undertaking the responsibilities of caring for a little child and seeing your past little self in them. So many great relationships were created and I am grateful for that. Our plays were always fun. Of course there was the first couple of years when I was terrified of acting and being on stage, but then sometime around fourth grade I started getting more confidence and wanting bigger parts. Slowly I built my way up to playing a lead role in our eighth grade production of Romeo and Juliet. This is only one of my many accomplishments. My eighth grade project was another from stick figures and scribbles in first grade to painting my classmates, my family. Every one of us has done and accomplished so much more than we take credit for. I applaud all of us for making it this far, for being there for one another and for doing things we thought impossible. Thinking back now to the past couple of years, we have grown so much and I am so proud of all of us and who we've become as individuals. I'm not sure if my fellow classmates, fellow friends, realize how beautiful, kind, caring, understanding and trustworthy they are. I know people hold grudges. I for one am a pro, but it's time to let those grudges go. It's time to forgive. I forgive any wrong anyone has done to me. I forgive anyone who has tried to hurt me in any way and I hope that you can forgive me too. I realized how much of an impact my time at the school has made on me. Sure it's Waldorf, sure there's a dress code, sure we've had some pretty unfortunate experience with teachers, but it's the memories that count, the relationships we've created together over the years and that will continue to form in the times ahead of us. We are all going our own ways after graduating.  So I guess this is it for now, Richard Valley. Farewell and thank you. So I think we have our faculty song next, yes. Faculty, wanna come up please? Maybe, maybe around the piano.  Yonder come day, day is a break-in. Sun is rising in my soul. Where they get their degrees, it's time. So students, you are gonna be no longer students in a few minutes, but until then you're not in order in which I'm gonna call you up, so just listen for your name. Okay, Ida, you're first, come on up. A few years ago, most of the time when the students were throughout the years, I would write a poem for them each year for their birthday. So I thought I would share one of them that I still think kind of still reflects some of the things about the student, this is Ida's. The mind decipheres right from wrong. It leads the way to knowledge. The heart feels both pain and joy. It only asks for understanding. The hands know their work on earth. They're intolerant of inaction. Together hands, heart and mind lead the soul on its path to wholeness. For me, this poem captures Ida's many strengths and speaks to her strong work ethic. She surely knows, she truly knows and loves her work here on earth. She's an excellent student, highly intelligent and more importantly inquisitive and eager for understanding new ideas. She's always striving to do her best. She's an excellent actor who brought Ariel to life in her lead role in The Little Mermaid, but also enhanced our shows with her many portrayals of minor yet important characters as well. And now we can say that she's a fine quilter as demonstrated by her eighth grade project. She is a passionate human being who feels things deeply, both pain and joy. She faces obstacles and even when frightened makes her way through them as noted by our trip. Because she is so engaged in all she does, she has learned much over these years and has much to celebrate on this day. Congratulations Ida. Bryce, stand up. Fill each day with curiosity and joy, each moment with opportunities and experiences when moving from one wondrous experience to the next, stop on occasion and take a rest, least on life's journey, great lessons might be missed. When Bryce joined our class in third grade, I was so happy. His even temper and good nature were so wonderful to encounter each morning. And the nice thing is it's been that way ever since. He is taller than me now so that's a little challenging to get a good look in his eyes each morning when we shake hands. But I always get a glimpse and what I see is the same joyful, fun-loving boy I had nut back then. As many of you know, Bryce likes to talk. Quite a bit actually. And this is generally a very good trait. For instance, in meeting new people or getting to know shy seniors, fortunately he has a lot to talk about and he is quite interesting because Bryce is a very intelligent young man. He's quick at math, he's a good writer when he puts his mind to it and he's a pretty good joke teller too. Bryce is also an excellent athlete. Whether it is soccer, hip hop or parkour, Bryce is pretty amazing. It's been wonderful to watch not only his physical abilities develop but also his sense of sportsmanship and camaraderie on the field. Congratulations, Bryce. Elegant, can you come up please? With grace and elegance, the deer moves through the forest. Quiet yet alert, cautious, confident, skillful and wise. With grace and elegance, it's true. Who can forget Elegant in that fabulous red dress as Lady Capulet in our play this year? And what about her role as a sassy sister in Roma Amora in grade six or Prince Eric in the Little Mermaid? Oh yeah, elegant. Elegant even as a boy. Like a deer, Elegant is a gentle soul. She is an incredibly attentive student who works carefully and skillfully in both her written and artwork. She's also a deep thinker who considers the questions before answering, always adding great insights to our discussions, particularly when discussing books we've read or historical dilemmas we're exploring. She's also in possession of a sharp wit, often understated, but quite funny. She's a good friend to her classmates and a hard worker in all she does. I know that these gifts will continue to grow in Elegant in the years ahead and I look forward to seeing where you end up, Elegant. Congratulations. Rain. Each day brings great strength to do what is good and right and to leave behind what is not. Each night brings dreams of peace and calm to bring restoration to my soul. Dawn brings love, dusk brings me home and fills my heart with serenity. And sometimes a sound machine helps her at night too. If a class were to elect a class ambassador, it would be rain. She greets everyone, turn and face them with a warm smile, a happy heart and a joyful attitude.  But the truth is it's rain that is special. Rain has brought all kinds of inspiration and enthusiasm to our class. She's not really a kickball or soccer type or actually anything competitive type. So you won't find her out there on the fields, but you will find her almost everywhere else. Ms. Baker's office in particular. She's a social butterfly. She's a geography buff. She's also the master of presentations. I will never forget her fourth grade snake report when she came in with her stuffed snake around her neck, pictures of snakes and rubber snakes everywhere. It was amazing. She set the bar quite high for report presentations and I will never forget her role as Loki that year in our play. She was fantastic. In those early days rain was a busy girl, though not one to sit much. Really, not at all. In fifth grade, something changed. Rain seemed to realize that indeed she could do anything she wanted to do if she put her mind to it, even learn the times table, even sit at her desk for a whole lesson. Ever since then she's been doing just that, setting her mind to learn something and doing it. Congratulations Rain. Imanu Krosi. Come on up Imanu. Courage and confidence, life's great gifts have built a body of strength that lifts inner wisdom clearly discerning determination that strengthens learning. From the moment we started blocking our very first play I knew I had a competent reliable student on my hands. This is a young man who though he spends his time quietly behind the scenes, knows exactly what is going on and everything that is needed to ensure success. Whether it's a play or an experiment, he lets me know what is needed and then he takes care of it. In our last play Romeo and Juliet, we finally got him out of his role with chief stage manager and lighting technician and onto the stage in not one but two roles. He was able to put his technical skills to use in his eighth grade project of learning to fly a drone. The film he made was excellent. Our film of the seniors would not have come together without his extra help in editing. He is highly respected and loved by his classmates, never using an unkind word. That is not to say he is perfect. Behind the scenes can sometimes mean a bit of mischief, a broken window, or a vending machine with a little shaken up. Maybe a bit of prohibited coffee here and there. Congratulations Imanu and thank you. Kayla, when I follow my heart and strive to fulfill my hopes and at the same time maintain the peaceful nature of my soul, I will move through life with right and confident steps. Kaylam is a quiet yet highly intelligent student in the classroom. He's quick to grasp any new math concepts and he's a competent writer. Whenever the class was assigned an oral presentation, Kaylam would work hard practicing his at home and always deliver a strong one. He carried this over into his many varied roles in our plays as well, learning his lines and bringing his character to life. He doesn't really like to be in the spotlight either by the way, so this was pretty good. Shakespeare can be a challenge for even a professional actor and Kaylam took on the role of the long-winded Mercutio, not an easy role for a quiet guy. Working hard with Morgan, he came to deliver a great performance. Out of the classroom on the playground, Kaylam is much less quiet and I would say more of a social spark, actually. His classmates dearly love him. I wish him all the best in the years ahead. Congratulations, Kaylam. Sophia Cates Cummings. Hidden in meadows of spring's first green sprouts, the marshmallow's bright gold sepals stand out, unassuming and gentle, a cheeky sweet flower, happily blooming in the early spring shower. I just want to, you know, all of these students know all of these poems. They had to say them in the morning. Her classmates awarded her the nicest kind of student award this year to Miss Sophia. She is a friend to all, warm and welcoming, always cheery. She's also quiet and really quite calming class, but there's another side to Sophia and that is often seen on the playground where she's loud and boisterous and really sometimes downright physically abusive. If you have only seen the classroom version of Sophia, you would have been surprised when she wanted the role of Lord Capulet, the ferocious father of Juliet in her play this year. Well, she was perfect for her role and we all love seeing her in all her angry glory. The other big challenge Sophia took on this year was her eighth grade project. She went way beyond her comfort zone and put herself in the dark room learning to print photography. What a great job she did. Congratulations, Sophia. Sophia Clark, firmly planted on the earth, holding high a heart of gold, curious, generous and kind, confidence rising, growing bold. When I looked back over my records from the early grades, I found that Sophia was already preparing herself for her artistic career. She was then and still is today a very dedicated student in her academics, but even in second grade, she had high expectations of her block-cran drawings and watercolor paintings, which were really kind of difficult to control. She also had quite a lovely singing voice, which she finds quite difficult to acknowledge. During those years, she did not really like to get up in front of the class to say her morning poem or do a recitation or even be in our place, but around fourth grade, as she mentioned, she began to change and Sophia began to enjoy performance so much so that the reason she was Juliet in the play this year was because she wanted it so badly. This is a huge role with tons of lines to learn. Sophia embraced that role and did a splendid job. And then right after that, she completed her eighth grade project that's large painting of the class with 15 figures done from our class photo. You can see it in the art exhibit afterwards, if you wanna see it. Both of these accomplishments speak to the tenacity and the grit of this young woman. Sophia, congratulations. Cora. Summer skies at night are lit with stars that sparkle as we sit and listen to the stories of old around the fire so lovingly told. But winter stars, more beautiful still, a rye in the hunter towards the bull, inspire our dreams, our hearts, our wit, jewels of the heaven that never quit. This is a student who is interested in just about everything. It is a rare subject that she did not find something fascinating to learn about. Chemistry, physics, biology all piqued her interest. From her first animal report in fourth grade to her report on the aurora borealis, Cora is consistently exploring new territory. They say that 90% of success in life is based on grit. Well, this is a girl with grit. It's not that everything comes easily to her. No, it's that she is curious and strong and determined to satisfy her curiosity. She's an excellent artist and has grown into a gifted actress and singer and she has a keen sense of humor. Her eighth grade project was a perfect synthesis of all these things. She created beautiful objects through the art of chemistry and presented her work with humor in her presentation and a display that demonstrated her strong sense of color and aestheticism. Congratulations, Cora. Bryn, now I get to talk without being interrupted. Find deep thoughts, traits of character which can't be bought. Warm heart, creative hands, strong legs on which to stand. To squander these would be a waste to hone these gifts of future bright to face. When I think of Bryn, I see him with his first grade buddy, Liam, playing in the first grade playground or sitting with him eating lunch. Liam loves Bryn and with good reason, Bryn loves Liam. They are a good pair. For Bryn, though, he would have loved anybody he had and treated him with the same kindness and gentleness I witnessed time and again. He is truly gifted in working with little children. He has a very warm heart. Bryn is also quite creative. From the day he entered our class, he was making something out of whatever he could get his hands on. One year, he showed up at school with rolls of duct tape and God knows what else to make his things. In sixth grade, he channeled all that creativity into a marvelous beeswax rendition of a Roman soldier. Fantastic, really. He is also a gifted athlete whose talents, unfortunately, had to be thwarted while at school due to the role, due by a rule-imposing teacher of his. He is silly, sometimes to the point of irritation, but often I had to hide my amusement so as not to encourage him anymore. Most importantly, though, Bryn, like the rest of these students, is highly intelligent. And when he would get down to considering something in a serious way, he would offer new perspectives and alternative views to what we were debating. A quick mind and deep thoughts lie behind a very silly young man. Nadia, the monarch's journey is long, you see, thousands of miles over land and sea. With trust in her heart, she makes her way to our true home so far away. There will be trials and obstacles to overcome with strength and courage. It's easily done. She shines like the sun with beauty and grace, surrounded by love, the world's warm embrace. Outside of school, Nadia is a beautiful dancer, a horseback rider, a farm hand. At school, she's a dedicated student who has brought her enthusiastic, sunny disposition to our classroom each day. She is kind to everyone she meets. She will often fall into uncontrollable fits of laughter or bouncing spurts of energy. She took excellent care of her first grade buddy and is always willing to lend a hand. She is fully engaged in life, and our adventure on the merry day was an excellent example of that. On the ship, you would find her coiling line, learning knots and helping in the galley. Lastly, she is as brave as they come, whether it's jumping off a boat or tackling an essay. Congratulations, Nadia. Eva is the sun uncertain as it shines upon the earth, are the plants uncertain of their value and worth, are the stars uncertain as they guide birds to their homes, or the poet uncertain as she shares with you her poem. The sun unwavering brings life to birth, the plants determine their wonders on earth, the stars unchanging shelter winged ones beneath their dome, and the poet ever longing allows our imagination room to roam. Eva is a ray of sunshine, who has shown in my classroom since kindergarten so long ago. She is bubbly, kind, thoughtful, and deeply caring for her fellow classmates. If she sees uncertainty in them, she quickly goes to their aid and comfort. She did herself used to cry quite a bit when involved in a tiff. Some might even describe her as a little stubborn back then in the early days of class, but she's grown far beyond tears and stubbornness, and replaced them with strength and clear thinking and reasoning. You can count on Eva to have something insightful to say about a book we have read or a story we have learned. Aside from erupting into fits of laughter on an odd occasion, she really is quite a well-behaved young lady. Congratulations, Eva. Thank you. Lauren. Swallows hoop across the water, Al's final call has begun, the melodic voice of the woodthrush calls to the setting sun, across the pond a creature swims, followed by a silent wake heading to his evening work, his dam and home to make suddenly a splash, a slap, the dive and he's gone, the birds now silent on the roosts await the coming dawn.  Wherever he goes, he passes by lightly without much disturbance around him. He has a keen wit, though, which emerged long before any of us could even understand his dry sense of humor. A lot of jokes in those early years were wasted on our ears.  You see, until this play, Lauren had had a few meaty parts, but his primary role was lighting and sound. I remember our first play at the opera house, Lauren was up in the balcony. He called down to me asking me how to operate the lighting board. Of course, I didn't remember. A half hour later, Lauren had had it up and running. He had read the manual and figured it all out. In seventh grade, he used his musical talents to learn the score for our play, The Little Mermaid, so that the play was primarily for Lauren on the piano. In eighth grade, though, backstage, he had to come back out from it, and he became our leading man. What a great job he did. I believe that Lauren will find high school an exciting adventure, and I just hope his teachers will be able to keep up with him. Between the sun and snow, rain and sleet, between mountain and valley, tall and deep, between earth and star, below and above, between them all, balance, harmony, and love. When Hattie joined our class, we were truly blessed. She quickly made friends, joined in soccer, and was an active participant in all our work. Most importantly, Hattie brought with her the three things I mentioned in her poem, balance, harmony, and love, and her classmates and I were the beneficiaries of it all. That is not to say that there were not moments of troubles along the way, but Hattie always managed to navigate them with grace. She's an excellent student who applies herself diligently to all her studies, even ones that she's not fond of. That's after I send her a new mask sheet, that is. An avid reader, she persuaded the majority of the class to read to kill a mockingbird. I'm not sure everyone agreed with her about how great the book is, nonetheless they read it thanks to Hattie. Thank you, Hattie. Acting has long been one of her passions, and she has worked both in and out of school, improving her skills. This year in a role as a nurse in Romeo and Juliet, all that hard work paid off, and she delivered a stellar performance. She's also a creative genius behind, the creative genius behind Little Red Riding Hood, a film rendition of an ancient fairy tale, and in addition, she's an excellent singer, as you got to hear today. I was lucky enough to travel to and from Camden with Hattie for our sailing trip and her Shrek CD. I too am now a fan. Thank you for that, Hattie. Congratulations, Hattie. And so graduates, are you ready to stand up and sing your song and say your closing verse? Oh, okay. This is normal. They take over whatever it is I'm doing. So when the students do sing, they're going to sing and you look in your program, a song called These Days, which they've been working on with Ms. Trouts. Do you want to get in your order now? And we put them all together and we just had to give the quote to Ms. Kate. I'm not true. I just want to say thank you. We're evened to find my heart scared. Wish we could come back to these days, please. Don't watch the show for it now. When the fire burns out, calling me when I'm sad. Remind me of what I've had when you tried to move on. With each other about these days, these days. Thank you, rest in. Wish we could come back to these days, please. But then I believe the message.  I wasn't ready to act right. I always think I get you back right. You say that things fall apart. We're going to move to Brooklyn. We're going to study art. Love is just a tool to remind us who we are and that we are not alone when we're walking in the dark. I hope someday we'll sit down together and laugh with each other about these days, these days. All our troubles will lay to rest in. Wish we could come back to these days, these days. Back to these days, these days. With heart aflame, aflame with love. I can light the world. Art exhibit inside the kindergarten. If you want to see some of the work the kids did over the years. And also, eat cake.


We have seen previously that if one wants to write a mathematical proof, that's really the same thing as just writing a logical argument. All right, and what is a logical argument? You have some list of premises that you list right here. You have some conclusion and we say that the argument is valid if the premises being true then imply that the conclusion is likewise true. Now, we've also discussed that every logical argument can be rewritten as a single conditional statement where you take as a very large conjunction, the conjunction where i ranges from 1 to n of all of the statements pi, that then implies the statement q. And now this argument is valid if and only if this resulting conditional statement is always true. And as such, providing valid arguments, which are the only out arguments that we will ever want to provide, is really logically equivalent to proving that this statement over here is true. So the most important and the most fundamental of any proof patterns is being able to prove that a conditional statement is true because in the end every proof is proving a conditional and when you look at in terms of logical equivalent. So how does one prove that a conditional is true? Well, let's remember what we've discussed with truth tables, right? So let's say we have two statements p and q and these could be very compounded statements. Don't worry about that. We have some conditional here. Well, the possibilities are you have true, true, true, false, false, true and false, false. Okay. And when your premise and conclusion are true, the conditional is true. When your premise is true and conclusion is false, that's false. When the premise is false and the conclusion is true, that is true. And when the premise is false and the conclusion is false, that's likewise false. Okay. Now these last two cases that are on this table here, when the premise is false, then the conditional is considered true. It's what we call vacuously true. And likewise, when the conclusion is true, regardless of what happens with the premise, the conditional is likewise true, what we call trivially true. And so the only case that really deserves attention is this one right here. We have to basically argue that this possibility never happens. What is this possibility? This is the possibility where the premise is true, but the conclusion is false. And so to prove that a conditional is a true statement, all you have to do is you can assume that the premise is true. And if there's more than one premise, then you list all of them. You assume that all of the premises are true. Then you provide some more statements, more arguments as necessary, and then you conclude that the conclusion is likewise true. This is the method known commonly as direct proof. And the method of direct proof, which is a technique used to prove a conditional statement, you assume the hypothesis, you assume the premise, and then you use that to show that the conclusion is likewise true. As such, because of the method of direct proof, the hypothesis or premise of a conditional is sometimes called the assumption, because we constantly are assuming it to be true. And then we argue that the conclusion follows by the assumptions that we then hold. And so what I want to do in this video then is demonstrate by example how one can write a proof of a conditional statement using this method of direct proof. The formula for writing the proof is always the same. You will start with the assumptions, and then you end with the conclusion. What happens in the middle depends on the argument itself. So consider the following conditional statement. If n is an odd integer, then n squared is likewise odd. Now, the basic structure of this proof, which we will do via direct proof here is you will start with your assumptions. So the first part says if n is an odd integer, so we're going to change that to say something like assume n is an odd integer or suppose n is an odd integer or let n be an odd integer, something like that. That is then the very first sentence of the proof. That's what you write there. Then the last sentence of the proof should be your conclusion. If we have an if, then there should be a then. And what's the then n squared is likewise odd. Now you can change some of the verbiage, right? If you want to drop the word likewise, that's fine. But a direct proof will always start with something like assume the hypothesis, therefore the conclusion. Now, some other things that we typically write when we write a proof is we typically will start the proof with a word like proof, right? We indicate to the reader that we're now beginning the proof of the statement that then preceded it, which is probably marked as a proposition or theorem or something like that. And then at the very end, there's typically going to be a little square of some kind. This is referred to as the QED symbol that this is an acronym for a Latin phrase that basically says that the question has now been demonstrated. That is, we've proven, we've proven the thing that we went out to go improve. And the little square that indicates we're at the end. It's sometimes called a tombstone symbol or the QED symbol. Tombstones a little bit more morbid, it's like our doubt of the truth of the statement is now dead. You can say that if you want to. But those are also just things you're going to see all the time, the QED symbol and proof indicates where you start and stop to prove everything in the middle is then the proof. For the direct proof, we start off with the hypothesis, which we then assumed to be true. Let N be an integer. And then we write the conclusion. What are we going to do next? Well, there's going to be some type of gap right here. Now, if you were writing this as scratch paper, you might want to leave a lot of space for you to write some sentences here, much like other types of writing, proof writing comes in phases. There's going to be some type of prewriting phase where you will outline or think about how is this proof going to go? There might be some organizational phase where we have to change how things are revision is going to take place there as well. We don't start writing a final draft, the final draft is written and revised as we go through it. Now, in this example, you will see it finalize pretty quickly, because I'm practicing the direct proof method and not actually demonstrating all the steps of writing. So because we have an odd integer, and we have to end with an odd integer, it's often a good practice to unravel the definitions. So after you make an assumption, the assumption basically is something of the following form, we have an object that belongs to a set. And so we then remind the reader, what does it mean to belong to that set? And we might also establish notation. So if n is an odd integer, that means there exists some integer k, such that n is equal to 2k plus one. So let's rewrite that first sentence. Let n be an odd integer. If you want to end the sentence right now with a period that's okay, you can then say something like there exists an integer k such that n is odd, right? I actually chose to just make this into one sentence. I think it reads okay in that manner. Let n be an odd integer. That is, this is going to suggest that we're going to clarify what the previous thing just said. Let n be an odd integer. Therefore, that is, there exists some integer k such that n equals 2k plus one. Be aware that in mathematical proofs, it is okay to use symbols. It's also okay to use words. So you could say there exists an integer k and if you spell it out, you're probably not going to put the symbols in there, but you can do the symbols too if that's how it reads. Now you want to be careful not to use too many symbols, but the appropriate use of symbols in a proof can help simplify the language here. Let n be an odd integer. That is, there exists some integer k such that n equals 2k plus one. This is what it means by n being odd. I don't know what n is, but it is odd. So there is some integer k so that this happens. I don't know what k is, but whatever k is, it's an integer. Two times k will give you an even number plus one to an even give you an odd number. This is what it means to be an odd number. So we've unraveled the definition. Maybe that'll help us somehow get from the assumptions to the conclusion. But what does it mean for the conclusion? Sometimes we unravel backwards. We can start with the assumptions of the problem. We can also work with the conclusion and try to come together, right? At the beginning of writing a proof, the logical gaps between them might be so large, it's hard to see how to connect them. But if we work closer and closer to the logical gap in the middle, maybe we can reach a point where we can bridge that gap. And for these easier exercises, that's exactly what we're going to do. So the next thing I might do is I might unravel the definition. What does it mean for n squared to be likewise odd, right? Well, since n is odd, that's what the likewise means. Okay, it just means two things are odd. n squared would be odd if there exists some integer m such that n squared is equal to two m plus one. Now, unlike this one, I didn't necessarily mention that m is an integer. That's probably because I would have discovered that previously. I don't know what m is yet, but there's, if n squared is odd, that means I have to be able to find some integer m so that two times m plus one gives you the n squared. And I bet you somehow or another the numbers m and the numbers k are somehow related to each other, because after all, n and n squared are related to each other. So this is what we're looking for. We have some k, we have to inform the m. That's how we're going to bridge the gap. And we're going to use the relationship between n and n squared here, because after all, n squared is just n times n. But we also have this assumption that n is two k plus one. And so if I replace the n with two k plus one, maybe I could foil this thing out to then get you something that looks like two m plus one. And that's how I'm going to try to bridge the gap here. So let's take n squared, which is the same thing as two k plus one squared by the previous assumption here. So then if you foil it out, I'm not going to go through all the details of the foil for this audience, those type of mechanical algebraic things are things I can expect the reader to be able to do. So if I were to provide the details, we take two k plus one, we take two k plus one, we foil that out, we get two k times two k, which is four k squared, we get two k times one, we get two k plus one, and then we have one times one, right? We could then combine the like terms, we're going to end up with something like this. But like I said, it's reasonable to expect of this audience that they could foil this out and arrive upon this discovery as well. So two k plus one quantity squared is the same thing as four k squared plus four k plus one. Now, I'm looking for something of the form two m plus one, I have the plus one. So if we separate the plus one, you're left with the four k squared plus four k, both of those coefficients are divisible by two, I could factor out the two and get two times two k squared plus two k plus one. This right here feels like my m that I'm looking for. So I'm then going to declare it. Let m equal two k squared plus one, sorry, two k squared plus two k, which is an integer, then n squared is equal to two m plus one, which means it's an odd number. And so look at this together now, our finalized proof. Let n be an odd integer, that is, there exists some integer k such that n equals two k plus one, then n squared is equal to two k plus one squared, which is equal to four k squared plus four k plus one, which is equal to two times in parenthesis, two k squared plus two k close parenthesis plus one. Well, if we let m equal two k squared plus two k, which itself is an integer, then that means n squared is equal to two m plus one, which shows that n squared is an odd integer. So this has all of the ingredients of a direct proof. We assumed the hypothesis, we proved the conclusion, and we provided details in the middle. The basic steps were we made an assumption, we unraveled, we unraveled definitions, we bridged the gap, and we then end with our conclusion. Those are the ingredients of every direct proof, which I confess some of these steps can get more complicated depending on the complexity of the statement we're trying to prove. Let's look at another example of this. This time let's do one with some integer divisibility. Let A, B, C be integers. If A divides B and B divides C, then we want to conclude that A divides C like so. Now we're going to prove this by direct proof. Now this right here, this first sentence, you could take that as an assumption, but really what we're saying is we're trying to describe what's the universe in play. We have three numbers. What kind of numbers are they? They are integers. So those really aren't the assumptions of the problem. The if then statement is right here. This is just adding context to the following statement. So we understand what A, B, and C are. The conditional that we're trying to prove is this one right here. If A divides B and B divides C, then A divides C. So our hypotheses show up right here and they're connected by an AND. It's a compounded statement. So what I'm going to do for my direct proof is I'm going to assume both of the hypotheses. Suppose that A divides B and B divides C. Then the very last sentence of my proof is going to be the conclusion. Therefore, A divides C. So that's how we begin. I might want to unravel some definitions. What does it mean for A to divide B? What does it mean for B to divide C? What does it mean for A to divide C? If we talk about these terms, we can unravel them and get the following. Suppose A divides B and B divides C. Hence, there exist integers M and N such that B is equal to AM and C is equal to BM. The fact that A divides B means there is an integer M such that AM is equal to B. So this is what it means for A to divide B. And likewise, since B divides C, there has to exist some integer such that B times that integer is equal to C. So this statement right here is unraveling the definition of B divides C. Now, I can also do that with the definition of the conclusion. I want to end up with A divides C. So what does it mean for A to divide C? That means there exists some integer K such that AK is equal to C. Now, notice how I phrase this. When you're working on the back, you actually work backwards. The last thing I want is the conclusion A divides C, which means if I unravel the definition, the definition of divisibility will probably be the thing I see immediately before it. So I get something like therefore C equals AK, which oh wait, if C equals AK, that means A divides C. So we've unraveled the definitions. Now we want to try to bridge the gap here. How can we connect these statements of divisibility with this statement of divisibility? Well, divisibility comes down to equations of integers. Can we connect these equations together to get this one? Well, this last equation involves A and C. This one involves A and B. And this one involves C and B. Could I somehow remove the middle man of B and go forward from there? My hope is exactly yes. And I could, because B, I don't need B in the final statement, A divides C has nothing to do with B. So maybe I could substitute this equation into that one, thus removing B from consideration. And maybe I can find my integer K in that process. Let's spell exactly that out. So what I'm going to do is I'm going to take the equation involving C, it's equal to BN. So that was already established by this assumption. Then by the other assumption, B equals AM, I'm going to substitute this in for B. Thus we get that C equals AM times N. Now as this is an equation of integers, I can use all of the algebraic properties I know about integers. For example, I can use the associative property. AM times N is the same thing as A times AM. And so notice here what we have here, C is equal to A times something, that something is an integer. If K is equal to MN, which is an integer, product of integers or integers, that then gives us the final statement that we're looking for. So now let's read the completed proof here. Suppose that A divides B and B divides C. Hence there exist integers M and N such that B is equal to AM and C is equal to BN. Thus C, which is equal to BN, is also equal to AM times N, which is the same thing as A times MN. Now if K is equal to MN, which is an integer, we then have that C equals AK, and that of course implies that A divides C, thus completing the proof. We'll do some more examples of direct proof in lecture 14, because I don't want this video to be too long, but this gives us some good examples of how does one prove a conditional statement. Direct proof is going to be your go-to move for many exercises like this in the future.


Authorities have recovered the body of a federal dam man whose canoe apparently capsized in Leech Lake. Cass County Sheriff Tom Burch says the body of 66-year-old Kenneth Gale was found about eight this morning. Authorities had begun searching after responding to a report of an overturned canoe the evening of October 25th. Numerous local state and tribal agencies were involved in the search. If you've enjoyed this segment of Lakeland News, please consider making a tax-deductible contribution to Lakeland Public Television.