1 LEARNING FRACTIONS THROUGH MATHEMATICS-BASED DISCUSSIONS CURRICULUM by Destiny Nancy Owens A project submitted in partial fulfillment of the requirements for the degree of MASTER OF EDUCATION IN CURRICULUM AND INSTRUCTION WEBER STATE UNIVERSITY Ogden, Utah December 14th, 2021 Approved Sheryl J. Rushton, Ph.D. Stephanie Speicher, Ph.D. Jaymi Rogers, MAME 2 Acknowledgments I would like to express my gratitude to Dr. Sheryl J. Rushton, Ph.D. for her countless hours of examining, editing, and giving feedback on various aspects of my project. She would answer emails late into the night and made herself available when I needed to get her advice over Zoom. Even though we have not met in person up to this point due to COVID, I feel as though she truly cares about me and the success of this project. Dr. Stephanie Speicher, Ph.D., and Jaymi Rogers, MAME were crucial to my project’s success as well. I was able to take a few classes from Stephanie and she inspired me to write a curriculum that makes a difference in students’ lives. I was able to glean a lot of advice and knowledge from meeting with Jaymi, for hours, over Zoom. Both of these women have made an impact on my project as well as my teaching practice. Many thanks also have to go to my friends Melissa Port, M.Ed., Karren Pyfer, Erica Sperry, M.Ed., and Sarah Nelson. Melissa helped me go through the rough drafts of my tasks and helped me think outside of the box when coming up with my mathematical tasks for this project. Karren, Erica, and Sarah reviewed my project and gave me exceptional feedback that helped me finish out my project strong. Lastly, I would like to thank my family, specifically my little sister, Winter. My little sister helped me through a rough year of my life and without her I do not know if I would have had the strength and stamina to complete this project. DJ came into my life towards the end of this project but really has pushed and motivated me to keep going the last few months when my life was extremely busy. He has been an amazing support system. 3 Table of Contents NATURE OF THE PROBLEM ……………………………………………………………….. 6 Literature Review …………………………………………………………………….... 7 Problem: Teacher-Centered Lessons ………………………………………….... 8 Teacher-centered lesson not engaging …………………………………. 8 Mathematics topics covered on a surface level ………………………... 8 Not cognitively demanding ……………………………………………. 9 Consequences: Students Unable to Make Deep Mathematical Connections ….. 9 Rote-memorization and low-level thinking ………………………….. 10 Active participation is not required ………………………………….. 10 Students are unable to make connections between topics ……………. 11 Solution: Active Participation in Mathematics-Based Discussions ………….. 11 Discussions develop a higher order of thinking …………………….... 11 Strategies for facilitating mathematics-based discussions ………….... 13 Time to process thinking ……………………………………………... 14 Engages all participants …………………………………………….... 15 Summary …………………………………………………………………….... 16 PURPOSE ……………………………………………………………………………………. 17 METHOD ……………………………………………………………………………………. 19 Intended Audience ………………………………………………………………….... 19 Reviewers and Instrumentation …………………………………………………….... 19 Procedure …………………………………………………………………………….. 20 FEEDBACK AND IMPLICATIONS ……………………………………………………….... 21 4 Feedback ……………………………………………………………………………... 21 Implications ………………………………………………………………………….. 22 Summary ……………………………………………………………………………... 23 REFERENCES ………………………………………………………………………………. 24 APPENDIX A: Mathematics-Based Discussions Curriculum ………………………………. 27 5 Abstract Students should be regularly engaging in discourse in their mathematics classes but many students in the United States today are still being taught through teacher-centered instruction. The teacher-centered lessons are not engaging, only cover surface-level content, and are not cognitively demanding. Because of this, students in the United States are unable to make deep mathematical connections. In classes where rote-memorization and low-level tasks are the norms and where active participation is not required, students cannot connect and apply their mathematical knowledge from one topic to the next. A way to engage students in important mathematical practices is through mathematics-based discussions. When teachers hold well-facilitated mathematics-based discussions students develop a higher order of thinking and can be actively engaged in their learning of various mathematics topics. This curriculum project includes six lessons that focus on 4th graders learning the addition and subtraction of fractions through mathematics-based discussions. After the project was written, it was reviewed by three mathematics and curriculum experts, and revisions were made. This mathematics-based discussions curriculum supports the Common Core State Standards, Standards for Mathematical Practice, as well as the literature for mathematical discourse. 6 NATURE OF THE PROBLEM The traditional teacher-centered mathematics lesson is failing to engage most students and is not leading to long-term retention and learning in mathematics. According to national level assessments, U.S. students are not improving in mathematics (Boston, 2012; Cobb & Jackson, 2011; National Research Council, 2011). The pressure from standardized testing has led U.S. teachers to resort to rote memorization instruction and surface-level coverage of mathematics topics. (Cobb & Jackson, 2011; da Ponte & Quaresma, 2016; Henning et al. 2012; Moschkovich, 2008) These methods of teaching are not cognitively demanding and do not lead to true, vital mathematical understanding (Cobb & Jackson, 2011; Henning et al., 2012; Moschkovich, 2008). The quality of instruction in U.S. mathematics classes needs to improve in order for students to be successful. Students who fail to retain mathematics concepts are unable to make deep connections and apply those concepts in other areas. When teacher-centered instruction is the norm, students are memorizing facts and procedures without knowing why (Cobb & Jackson, 2011; Henning et al., 2012). This kind of knowledge is not lasting and does not require a higher order of thinking (Henning et al., 2012). In these types of lessons, students are not required to be active participants; therefore, students may not gain sufficient knowledge (Henning et al., 2012). Many U.S. students have a surface-level understanding but do not have the capacity to think deeply about mathematics concepts because of the emphasis on procedures in these traditional lessons (Cobb & Jackson, 2011; Henning et al., 2012). Active participation in mathematics-based discussions can lead to greater retention of mathematics concepts and connections between concepts. Mathematics-based discussions allow 7 students to engage in higher orders of thinking which can improve student learning (Bahr & Bahr, 2017; Boston, 2012; da Ponte & Quaresma, 2016; Henning et al., 2012; McCrone, 2005; Walker, 2014). Teachers can employ a variety of strategies to engage students in successful mathematics-based discussions (Bahr & Bahr, 2017; Cengiz, Kline, & Grant, 2011; da Ponte & Quaresma, 2016; Henning et al., 2012; Walker, 2014). Students benefit from time to think and reason about mathematics concepts in small and large group settings (Bahr & Bahr, 2017; Boston, 2012; Henning et al., 2012; McCrone, 2005; Walker, 2014). Teachers can facilitate these discussions in a way that ensures engagement from all participants (Boston, 2012; Henning et al., 2012; Moschkovich, 2008; Walker, 2014). Many professional mathematics organizations are recommending teachers have more rich mathematics-based discussions in order to deepen students’ learning (McCrone, 2005). 8 Problem: Teacher-Centered Lessons In the current mathematics classroom, teacher-centered lessons are the norm. These lessons can be disengaging, surface level, and not cognitively demanding. Teachers should consider changing to the more student-led approach of mathematics-based discussions. Teacher-centered lesson not engaging The traditional teacher-centered mathematics lesson is failing to engage students and does not lead to long-term retention and learning in mathematics. Students in the U.S. are not improving in mathematics. The National Research Council has said according to various standardized test scores, students in the U.S. have not made the progress in mathematics that was hoped for, despite the recent, strong focus placed on mathematics (National Research Council, 2011). This is not only a concern for the U.S., but all around the world, leaders feel pressure to improve the quality of mathematics instruction in their respective countries (Cobb & Jackson, 2011). It is clear that something needs to change in order for mathematics instruction and students' mathematical understanding and skills to improve. Mathematics topics covered on a surface level There are several factors that contribute to less than outstanding mathematics instruction. One reason is the surface-level coverage of mathematics topics and the sole focus on procedures. U.S. instruction in mathematics relies on learning and performing procedures instead of mathematical ideas and relationships (Cobb & Jackson, 2011). Teachers may feel rushed to cover all of the mathematics topics, so they focus on procedure instead of deeper thinking skills. There are many teachers who provide low-quality tasks for students that do not lead to lasting learning (da Ponte & Quaresma, 2016). Most U.S. teachers are made to teach out of textbooks that are full 9 of symbolic representations instead of starting students with concrete examples as a strong foundation then working up to a symbolic representation (Henning et al., 2012). Concepts and procedures are important to develop simultaneously in mathematics, but in U.S. classrooms, both concepts and procedures are rushed and gone over lightly (Moschkovich, 2008) and most students have not participated in a mathematics class where several solutions are presented and valued. (Kooloos et al., 2018). There are more than just procedures to teach in mathematics, teachers need to also help develop deeper thinking skills in their students. Not cognitively demanding The current, popular methods employed in classrooms across the United States are not cognitively demanding. The National Council of Teachers of Mathematics (NCTM) recommends tasks such as discussing with peers, providing evidence, and making connections between concepts (National Council of Teacher of Mathematics, n.d), but our current instructional models are the opposite of these approaches (Cobb & Jackson, 2011). U.S. teachers try to make mathematical tasks as manageable as possible for students without challenging them. True growth and learning do not come without challenge and cognitive demand (Henning et al., 2012; Smith, M. S. et al., 2009). Many students by secondary grades do not possess the necessary skills to problem solve (Napitupulu et al., 2016). Although teaching procedural skills in mathematics is necessary and usually less demanding for students, it should not be a teacher’s only focus. Consequences: Students Unable to Make Deep Mathematical Connections As a result of being taught in a more teacher-centered classroom, many students are subject to rote-memorization instead of a deep, rich understanding of mathematical concepts and making connections between mathematical topics. Another reason students may experience 10 lower levels of thinking in these mathematics classrooms is that they are not required to actively participate. Rote-memorization and low-level thinking Teacher-centered lessons promote rote-memorization and low levels of thinking. Research on improving mathematics instruction has focused on improving mathematics scores; this leads to “teaching to the test” instead of careful, thoughtful, and meaningful instruction (Cobb & Jackson, 2011). Because of the pressure to perform well on standardized tests, teachers rely on teacher-centered lessons that are not engaging and require students to memorize instead of internalizing. Students may perform temporarily well on a test, but not truly and deeply understand a concept (Henning et al., 2012). In a teacher-centered lesson, students’ voices are not equally heard and valued (Dallimore et al., 2010). Student-centered lessons require teachers to shift their classroom norms and practices to make sure all students are engaged and heard (Kooloos et al., 2018). Communicating and reasoning are essential for mathematical learning, and teacher-centered instruction lacks those elements (Vacc, 1993). When students are memorizing instead of internalizing, they are not truly learning mathematics. Active participation is not required Students in traditional mathematics classrooms are not required to actively participate and engage with the learning. The teacher simply lectures and students take notes. In many classrooms, students copy down a problem from a board without truly thinking it through. They are merely practicing a procedure that was just taught (Henning et al., 2012). Teachers sometimes feel uncomfortable utilizing equitable strategies for student participation, like cold-calling. Cold-calling is the act of calling on a student without them volunteering, or without 11 warning them they are going to be specifically called on. It can be uncomfortable for students to get used to being called on when they are not raising their hand, so that may be why teachers do not attempt this strategy (Dallimore et al., 2010). A discussion monitored and facilitated by a teacher that includes cold-calling, wait time, partner talk, among other active-participation strategies, promotes the engagement that needs to be employed more in mathematics classes (Dallimore et al., 2010; Lack et al., 2014). Students are unable to make connections between topics As a result of students not thinking deeply or actively participating in class, most students are unable to make deep mathematics connections between topics or solve more complex problems outside a topic of study. Students do not have the opportunity to participate in engaging pedagogies in most U.S. classrooms (Cobb & Jackson, 2011). In contrast, students in other countries who engage in classroom discussions are able to make deep connections between mathematics concepts and procedures (Henning et al., 2012). Clearly, changes need to happen in most U.S. mathematics classrooms in order for students to think deeply about mathematics concepts, retain what they learn, and apply those concepts across mathematics topics. Solution: Active Participation in Mathematics-Based Discussions In order to combat the lower levels of thinking that teacher-centered lessons produce, active participation in mathematics-based discussions can develop a higher order of thinking when it comes to mathematics. Teachers can employ strategies to ensure the active participation of all students and time for students to process what they are thinking and learning. Discussions develop a higher order of thinking 12 Well-facilitated mathematics-based discussions require and allow students to have a higher order of thinking about the subject at hand. A teacher can tailor the level of thinking they want from students depending on what they ask the students to listen for and think about during the discussion. During a discussion, one should not assume a student is engaged when they are sitting quietly and “listening.” To ensure engagement and the use of higher-order thinking skills, the teacher can assign students what to listen for and how to listen for it (Bahr & Bahr, 2017). Students can deeply engage in the discussion and have more opportunities for higher levels of thinking and reasoning (Boston, 2012; Smith et al., 2009). Mathematics-based discussions can be particularly helpful when done after a student has had a chance to complete a challenging mathematical task. These classroom discussions can challenge students to struggle, reason, and clearly communicate about mathematics topics (Henning et al., 2012; Smith et al., 2009). When students are actively participating in a high-quality discussion they have the chance to use other higher-order thinking skills such as generalizing and justifying (da Ponte & Quaresma, 2016; ). Rich mathematical discussions can include exploring new ideas, testing the validity of solutions, justifying, and reasoning (McCrone, 2005; Smith et al., 2009), which in turn, can help students develop justifying, thinking, and reasoning skills (Walker, 2014). Teachers can assign rich tasks or problems where there are many correct solutions and many different paths to arrive at a solution so students see that many mathematical problems may have a variety of ways to be solved and a variety of solutions (Vacc, 1993). When introduced with a controversial issue (like deciding who’s solution to a problem or approach to the solution is the best), students are using higher-order thinking skills to create viable arguments and to justify their thinking and critique the reasoning of others (Common Core State Standards Initiative [CCSSI], 2020; Vacc, 1993). When students participate in a classroom mathematics-based discussion, they can use and 13 develop higher-order thinking skills which can lead to a deeper understanding of the topic at hand. Strategies for facilitating mathematics-based discussions Classroom and mathematics-based discussions can be facilitated in many different ways; with careful planning and execution, these discussions can be rewarding for both students and teachers. It can be easy for a student to “fly under the radar” by sitting quietly and idly by in a classroom-wide discussion. Teachers can provide listeners in a whole group mathematics-based discussion opportunities to participate by telling students what to listen for, how to engage in a type of listening, calling on listeners to respond after someone has shared, and having other routines when a listener chooses not to share, such as taking notes and critiquing what other are saying and reporting that back to the teacher verbally, or in writing (Bahr & Bahr, 2017; Kooloos et al., 2018). It is important to have an environment that is more student-guided, with teachers asking more open-ended questions. This can increase student talk and participation (Henning et al., 2012; Kooloos et al., 2018). Teachers need to establish a classroom environment where students are comfortable taking risks and participating (Dallimore et al., 2010). A key element in facilitating a rich discussion where students can have multiple solutions and perspectives is providing a challenging mathematical task (Walker, 2014). It is beneficial for students if teachers take an exploratory approach when preparing for a mathematics-based discussion. They need to find a challenging mathematics problem that is still within students’ reach (da Ponte & Quaresma, 2016). Although discussions should involve a great amount of student voice, while students are sharing their thinking in a mathematical discussion, teachers need to be listening carefully in order to address misconceptions and build mathematical connections (Cengiz, Kline, & Grant, 2011; Smith, M. S. et al., 2009). Teachers can use the strategy of “revoicing” to clarify 14 students’ thinking and give students a chance to fully understand what they are trying to say. Revoicing is when a teacher restates what a student has said. This can be particularly helpful for low-performing students who are not as confident with language in mathematics (Chapin, Connor, & Anderson 2003, Lack et al., 2014). In order to make students feel more comfortable, teachers can allow students to sit in a circle, so they can see everyone and everyone is on the same level. Students can discuss their ideas with a partner or small group before sharing them with the whole group, so they feel more comfortable. Discussing ideas with a partner or small group can allow students to process their thinking and make mistakes in a less intimidating setting before sharing in front of the whole group (Vacc, 1993). In addition, teachers need to teach students how to hold a proper discussion, not just what to discuss. In order for mathematics discussions to be meaningful, teachers need to ask for explanations from students, instead of just answers, and teach them how to respond to one another. Teachers can also make clear expectations and classroom norms for discussions (Kooloos et al., 2018). Teachers can plan a rich mathematical discussion with a challenging task, well-thought-out questions and/or prompts, and chances for students to actively engage. Time to process thinking The vast amount of topics in mathematics that need to be covered in such a short time require teachers to race faster and faster to teach all they need to in the span of a school year. This leaves little room for students to practice and process what is being taught. Students need opportunities to reflect on their work, and discussions give them that opportunity (Boston, 2012). As mentioned earlier, by assigning students specific ways of listening, they are able to think about mathematics concepts as other students are discussing their ideas. This furthers individual and class-wide learning (Bahr & Bahr, 2017). Mathematical discussion can progress knowledge 15 by deepening students’ understanding and at the same time, students with sufficient knowledge are able to hold a longer discussion, giving them more time to process their thinking (Henning et al., 2012). Discussion differs from traditional instruction because students reflect more on their learning in order to justify what they know so they can participate in the discussion (McCrone, 2005). Walker (2014) pointed out that when given multiple opportunities within a lesson to discuss their ideas, students are better able to rethink and revise their ideas. When students are asked to repeat and clarify themselves or others, they have more opportunities to process what they have said and what they understand to be true (Chapin, Connor, & Anderson 2003). One strategy, given sufficient time, teachers could employ would be to give students a few days to think about a problem, so they come better prepared for the discussion (Vacc, 1993). Although teachers are usually pressed for time, given the number of standards they must teach in a year, it is important for them to give students some time to think. Giving sufficient wait time for students to think about the problems presented can have a positive effect on low-performing students. These students have had time to process a problem so they can participate more fully, rather than being confused and overwhelmed during a discussion, therefore not participating (Lack et al., 2014). Time to reflect, discuss, and change their ideas is essential to deepening a student’s understanding of mathematics. Engages all participants There are many strategies that a skilled teacher can employ that can engage students in a discussion. When students are allowed and encouraged to discuss their approaches to challenging mathematics tasks, all students can be actively participating and fully engaged. Even students who are not currently speaking in a discussion could be actively processing what is being presented and deciding if they want to change their own strategy and approach to the 16 mathematics topic being discussed (Walker, 2014). Participants have to pay attention so they can contribute to the discussion and build on others’ ideas (Moschkovich, 2008). Teachers can ask students to restate each others’ ideas and can help build everyone’s ideas and the discussion itself (Kooloos et al., 2018; Smith et al., 2009). When a teacher has more knowledge about the different types of mathematics-based discussions they are more likely to have greater participation from their students (Henning et al., 2012; Kooloos et al., 2018). Boston (2012) mentioned that teachers can use one student’s thoughts to not only teach that student but the entire class. When teachers grade students on participation in a discussion and students know they could be called on at any moment, students are more likely to prepare for that discussion, because they know they could be called on at any time (Dallimore et al., 2010). Whether a student is listening or speaking they can be engaged in and learn from a mathematical discussion. Summary Mathematics-based discussions, when planned out thoughtfully, can enhance the learning of all students who participate. Students have multiple opportunities to practice and strengthen their higher-order thinking skills as they actively contribute to the conversation or think to themselves about their own mathematical strategies. In order to have a successful mathematical discussion, teachers need to plan a challenging mathematics task, open-ended questions, and roles to engage both speakers and listeners. During a discussion, teachers should be listening carefully in order to correct misconceptions and build connections between students’ ideas. When a teacher plans and carries out a discussion conscientiously, they can ensure that all students are actively participating and benefitting from the discussion. 17 PURPOSE The National Council of Teachers of Mathematics (NCTM) along with the Standards of Mathematical Practice (SMP) encourage teachers and students to participate in meaningful discourse around mathematics (CCSSI, 2020; NCTM, n.d.). Currently, there is not a mathematics program that heavily involves mathematics-based discussions as a way to encourage student understanding, attend to precision, and construct viable arguments as laid out in the Standards of Mathematical Practice in the Common Core State Standards (CCSS). Students are lacking these necessary skills in mathematics, and units that are centered around rich mathematics-based discussions that can promote these standards. The purpose of creating a curriculum mathematics unit where mathematics-based discussions, such as whole group, small group, and partner discussions, are woven throughout lessons is to enhance the understanding and development of mathematics skills and concepts for fourth-grade students studying fraction addition and subtraction. In the curriculum that has been created, mathematics-based discussions are defined as any purposeful and facilitated discussions set out during the mathematics lesson, as described in the curriculum. Discussions may take place prior to learning, during the lesson, during the debrief of a lesson, whole group, small group, or in partners. The objectives of this project were: 1. Develop meaningful mathematics tasks and activities that promote mathematics-based discussions. 2. Layout best practices for mathematics-based discussions that are easy to follow for mathematics educators. 18 3. Create a fourth-grade fraction addition and subtraction unit that has mathematics-based discussions woven into each lesson. 19 METHOD To address the Standards of Mathematics Practice (SMP) recommendations for a deeper understanding of mathematics, a curriculum on fourth-grade fractions was developed that is centered around mathematics-based discussions (CCSSI, 2020). Intended Audience The mathematics-based discussion lesson plans that have been developed are intended for teachers teaching fourth-grade students studying the addition and subtraction of fractions. This curriculum has also been developed with professional development for teachers in mind. By following this curriculum, teachers will be able to begin to establish a foundation for equitable mathematics-based discussions in their own classrooms. This curriculum provides a deeper understanding for students starting on their journey into understanding and using the addition and subtraction of fractions. Reviewers and Instrumentation The curriculum developed for this project has been reviewed by three mathematics and curriculum specialists. The experts were Karren Pyfer, Sarah Nelson, and Erica Sperry, ME.d. Karren Pyfer is the curriculum director at Voyage Academy Charter School and has many years of experience in curriculum review and education. Sarah Nelson is a fourth-grade math teacher and has taught in third grade for a number of years as well. Erica Sperry has over 16 years of teaching experience and is a teacher and instructional coach. These specialists have the following qualifications: educator, mathematics or curriculum development background, and familiar with The Common Core State Standards-Standards of Mathematical Practice. Reviewers’ comments and notes were reviewed and documented. A section was provided for free responses so participants could put into words what was accomplished in the curriculum and what still needs 20 to change. Changes were made to the lesson plans based on the feedback given by these three experts. Procedure The set of lesson plans were given to the three expert curriculum reviewers. The reviewers were given two weeks to look over the lessons, answer pre-determined questions, and give free response feedback. All of the reviewers submitted their feedback and the curriculum was adjusted according to the reviewers’ feedback. 21 FEEDBACK AND IMPLICATIONS The purpose of this curriculum project was to create lesson plans for learning the addition and subtraction of fractions through mathematics-based discussions in fourth grade. The goal of the curriculum was to engage students in their learning on a cognitively deeper level and provide teachers with strategies to hold well-facilitated discourses in fourth-grade addition and subtraction of fractions in mathematics. Six lessons were created around the addition and subtraction of fractions and were sent to three expert reviewers for their feedback. They were asked to correct grammar and spelling and ensure the instructions given were clear. They were then asked what they would change or add to the curriculum, and for any other feedback. Their feedback and the changes made are discussed below. Feedback When asked to correct grammar, spelling, and unclear instructions it was very useful to get corrections from all three experts. They would find a lot of the same corrections, but oftentimes found different corrections that had been missed. When asked to check for unclear instructions, Karren Pyfer made many useful suggestions. In the curriculum, it was written, “Remember, you want to emphasize…” Karren suggested instead, “Remember, the emphasis should be on using…” Instead of saying “...understand why it is wrong.” Karren suggested, “...understand why it is incorrect.” Although this is a simple suggestion, there is a negative connotation that comes with telling a student their answer is wrong rather than incorrect. The next question the expert reviewers were asked to answer was, “What would you change in order to make this curriculum better?” Sarah Nelson suggested, “Decide how type will look for fractions, and keep the same throughout.” An example of this would be ½ rather than 5/7. The next question the reviewers answered was, “Would you add anything to the curriculum 22 to make it better?” Once again, Sarah Nelson gave some great feedback. She suggested, “Number the lesson at the beginning of each script.” At last, the experts were asked to give, “Any other feedback:” Erica Sperry, ME.d. said, “Really like the student accountability section.” Sarah Nelson said, “Really, the only fixes are to the visual typing. Not to the quality of the content. Great sequence in lesson! Easy to follow since each lesson follows same steps. It is nice to have questions ready to use and possible student answers! Want to come visit my classroom for a few days when fractions come up?” Karren Pyfer said, “I love the monitoring charts and common misconceptions page. This is very helpful!” The implications and changes to the curriculum based on this feedback are discussed below. Implications The grammar and spelling corrections were extremely helpful and all changes suggested throughout the curriculum were made. If a curriculum is to be used by anyone other than the creator, it should have correct spelling and grammar throughout so those mistakes do not get in the way of the content and methodology of the curriculum. As mentioned by Sarah Nelson, there was a mix of the two styles of fractions throughout the curriculum (½ and 5/7), so the change was made to format the fractions in the first style. This style is more readily recognizable for students and teachers alike. This style is more clear especially when it comes to mixed numbers. It is more clear to have 3 ¾ rather than 3 3/4. In the initial curriculum, the script page just said “Script.” The change, “Lesson # Script” was made. This felt important for ease of use for teachers. There were many more subtle changes made throughout the curriculum that made instructions more clear and easier to understand for educators. The changes made have shifted 23 the focus from trying to understand what is being conveyed in a mathematics-based discussion to more of the content and method of successful mathematics-based discussions. Summary The feedback and changes made based on the feedback have made the curriculum more clear so an educator could easily hold a successful mathematics-based discussion. The curriculum follows a sequence that is similar in each lesson so the instructor and students could more easily fall into the discussion routines and deeper levels of thinking. Mathematics-based discussions are meant to deepen students’ understanding of mathematics topics and make connections between different methods. This curriculum facilitates those discussions well for educators and students. 24 REFERENCES Bahr, D. L., & Bahr, K. (2017). Engaging all students in mathematical discussions. Teaching Children Mathematics, 23(6), 350-359. Boston, M. (2012). Assessing instructional quality in mathematics. The Elementary School Journal, 113(1), 76-104. Cengiz, N., Kline, K., & Grant, T. J. (2011). Extending students’ mathematical thinking during whole-group discussions. Journal of Mathematics Teacher Education, 14(5), 355-374. Chapin, S. H., O’Connor, C., & Anderson, N. C. (2003). Classroom discussions using math talk in elementary classrooms. Mathematics Solutions, 11. Cobb, P., & Jackson, K. (2011). Towards an empirically grounded theory of action for improving the quality of mathematics teaching at scale. Mathematics Teacher Education and Development, 13(1), 6-33. Common Core State Standards Initiative (2020). Standards for mathematical practice. Core standards. http://www.corestandards.org/Math/Practice/ Dallimore, E. J., Hertenstein, J. H., & Platt, M. B. (2004). Classroom participation and discussion effectiveness: Student-generated strategies. Communication Education, 53(1). da Ponte, J. P., & Quaresma, M. (2016). Teachers’ professional practice conducting mathematical discussions. Educational Studies in mathematics, 93(1), 51-66. 25 Henning, J. E., McKeny, T., Foley, G., & Balong, M. (2012). Mathematics discussions by design: creating opportunities for purposeful participation. Journal of Mathematics Teacher Education, 15(6), 453-479. Kooloos, C., Oolbekkink-Marchand, H., Kaenders, R., & Heckman, G. (2019). Orchestrating mathematical classroom discourse about various solution methods: case study of a teacher’s development. Journal für Mathematik-Didaktik, 1-33. Lack, B., Swars, S. L., & Meyers, B. (2014). Low-and high-achieving sixth-grade students’ access to participation during mathematics discourse. The Elementary School Journal, 115(1), 97-123. McCrone, S. S. (2005). The development of mathematical discussions: an investigation in a fifth-grade classroom. Mathematical Thinking and Learning, 7(2), 111-133. Moschkovich, J. N. (2008). “I went by twos, he went by one”: multiple interpretations of inscriptions as resources for mathematical discussions. The Journal of The Learning Sciences, 17(4), 551-587 Napitupulu, E. E., Suryadi, D., & Kusumah, Y. S. (2016). Cultivating upper secondary students’ mathematical reasoning-ability and attitude towards mathematics through problem-based learning. Journal on Mathematics Education, 7(2), 117-128. National Council of Teachers of Mathematics (n.d.). Principles and standards for school mathematics. NCTM. https://www.nctm.org/standards/ 26 National Research Council. (2011). Incentives and test-based accountability in public education. National Academies Press. Smith, M. S., Hughes, E. K., Engle, R. A., & Stein, M. K. (2009). Orchestrating discussions. Mathematics Teaching in the Middle School, 14(9), 548-556. Vacc, N. N. (1993). Teaching and learning mathematics through classroom discussion. Arithmetic teacher, 41(4), 225-228. Walker, N. (2014). Improving the effectiveness of the whole class discussion in summary phase of mathematics lessons. Mathematics Education Research Group of Australasia, 629-636. 27 APPENDIX A LEARNING FRACTIONS THROUGH MATHEMATICS-BASED DISCUSSIONS CURRICULUM 28 Table of Contents Standards for Mathematical Practice……….…..….3 Common Core State Standards Addressed……...…4 Student Accountability…………………...……..…5 Previous Knowledge………………………….....…6 Lesson 1………………………………….……...…7 Lesson 2………………………………………..…21 Lesson 3………………………………………......35 Lesson 4….…………………………………….…49 Lesson 5……………….…………………….……62 Lesson 6………….…………………………….…75 29 Standards for Mathematical Practice ● Make sense of problems and persevere in solving them. ● Reason abstractly and quantitatively. ● Construct viable arguments and critique the reasoning of others. ● Model with mathematics. ● Use appropriate tools strategically. ● Attend to precision. ● Look for and make use of structure. ● Look for and express regularity in repeated reasoning. 30 Common Core State Standards Addressed ● 4.NF.B.3 ○ Understand a fraction a/b with a > 1 as a sum of fractions 1/b. ■ 4.NF.B.3a ● Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. ■ 4.NF.B.3b ● Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. ■ 4.NF.B.3c ● Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. ■ 4.NF.B.3d ● Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. 31 Student Accountability Without students being held accountable, a discussion can fall flat and be meaningless. Even if a student is not actively speaking, they need to be actively participating. Here are some strategies to hold students accountable. ● Revoicing ○ A teacher repeats a part of what a student said or asks another student to repeat what a classmate has said. ● Ask students to restate what another student said ○ The rest of the class has to anticipate that they could be called on next. ● Ask a student to apply their reasoning to someone else’s ○ Good for connection. ● Prompt for more participation ○ “Does anyone want to add to that?” ● Wait time ○ Pause a few seconds before asking a question and wait at least 10 seconds after asking a question. ● Guide students then say “I will be back” ○ The student continues to work on the question. ● Come up with partner and whole-group discussion norms ○ Before beginning any discussion in class decide as a class what norms should be followed during a partner, small group, and whole-class discussion. Example: one person speaks at a time. 32 Previous Knowledge 4th-grade students should have already learned the following standards: Extend understanding of fraction equivalence and ordering. ● CCSS.MATH.CONTENT.4.NF.A.1 ○ Explain why a fraction a/b is equivalent to a fraction (n × a)/(n × b) by using visual fraction models, with attention to how the number and size of the parts differ even though the two fractions themselves are the same size. Use this principle to recognize and generate equivalent fractions. ● CCSS.MATH.CONTENT.4.NF.A.2 ○ Compare two fractions with different numerators and different denominators, e.g., by creating common denominators or numerators, or by comparing to a benchmark fraction such as 1/2. Recognize that comparisons are valid only when the two fractions refer to the same whole. Record the results of comparisons with symbols >, =, or <, and justify the conclusions, e.g., by using a visual fraction model. 33 Lesson 1 ● Common Core State Standards ● 4.NF.B.3a ○ Understand addition and subtraction of fractions as joining and separating parts referring to the same whole. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goals: ● Students will recognize that the addition of fractions is joining parts of the same whole. ● Students will recognize that subtraction of fractions is separating parts of the same whole. ● Students will be able to represent the addition and subtraction of fractions in multiple ways. ● Students will be able to add and subtract fractions with like denominators. 34 Lesson 1 Script Before Task Launch (5 minutes) ● Task found on page: 14 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over again. 35 ○ Possible notices: ■ There are 3 different animals ■ 27 students ■ 4th-grade class ○ Write down each notice on the board or under the document camera. ■ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ■ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ■ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 17). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 18) 36 page in order to spot common misconceptions your students may have. ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ○ I can add and subtract fractions with like denominators. ● Tell students to read it in their heads and show a “finished” signal to you when they are done. ● Read aloud the learning target to the students. 37 ● Under the word “add,” write the + symbol. Then ask the students to think of other words they have seen in math that mean the same as “add.” ○ Tell students to turn and talk to their partner and share the words they have seen in math that mean the same as “add.” ○ Cold call on one to three students to share. List those words by the word “add.” ○ Repeat the process with the word “subtract.” ● Ask the students to name a fraction. Cold call on a student to share a fraction. Write this fraction next to the word “fractions.” ● Tell students that “like” means the same. ● Ask students what part of the fraction is the denominator. ○ Tell students to turn and talk to share their thinking. ○ Cold call on a student to answer. ■ Answer: “It is the bottom number of the fraction.” ● Ask the students to say the learning target in their own words. ● Give students wait time, then tell them to turn and talk with their partner, stating the learning target in their own words. ● Cold call on one to three students to answer. ● Say: “By the end of the lesson you should be able to add and subtract fractions with like denominators.” Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? 38 ● Students who are watching may ask questions of the student. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. ○ To include the students who are very eager to answer, you can then ask the rest of the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ How do you know to add? ○ How do you know to subtract? ○ What is the whole? ○ What are the parts? ○ Why do you agree or disagree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? Debrief/Connect (5 minutes) ● The debriefing and connecting can happen during the whole group discussion and right after the whole group sharing is complete. ● After asking a question give students time to think, then share with a partner before cold calling on them to share with the class. ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method. ○ What do we need to remember when adding and subtracting fractions? 39 ○ How can you explain what the addition of fractions is? ○ How can you explain what subtraction of fractions is? Assessment (5 minutes) ● Hand out the assessment (Found on page: 19). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Think: ○ Do the students understand what adding and subtracting fractions with like denominators looks like? ○ Do you need to spend a day on computation to sharpen up their fractional adding and subtracting? ○ Are there students you need to do tier two or three interventions with? ● If the majority of students did not do well, another tier-one lesson needs to be taught with adding and subtracting fractions. 40 Task A survey of favorite animals was taken in a 4th grade class of 27 students. 13 students said that dogs were their favorite animal. 8 students said that snakes were their favorite animal. The rest of the students said that hamsters were their favorite animal. ● What fraction of the students said that dogs and snakes were their favorite animal? How did you figure that out? ● What fraction of the students said that hamsters were their favorite animal? How did you figure that out? 41 Draw a picture or write an equation to solve the task: What is the whole? What are the parts? 42 Explain your partner’s thinking on the task: 43 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 44 Misconception(s) & How to Address Them ● When adding or subtracting fractions, students also add or subtract the denominator. ○ Examples ■ ⅓ + ⅔ = 3⁄6 ■ 5⁄7 - 3⁄7 = 2⁄0 ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we cannot go from 27 students in the class to 54. ■ When subtracting, we cannot go from 27 students in the class to 0. 45 Assessment 1. 16⁄25 + 5⁄25 = 2. 12⁄27 - 8⁄27 = 3. Karren completely filled up her bucket with paint to paint her room. After she painted her room, 3/7 of the paint remained in her bucket. What fraction of the paint did Karren use? 46 Assessment Key 1. 16⁄25 + 5⁄25 = 21⁄25 2. 12⁄27 - 8⁄27 = 4⁄27 3. 4⁄7 47 Lesson 2 ● 4.NF.B.3b ○ Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goals: ● Students will learn what decomposing a fraction is. ● Students will be able to decompose a fraction into a sum of fractions with the same denominator in more than one way. ● Students will be able to write an equation of their decompositions. ● Students will justify their decompositions using a visual fraction model. 48 Lesson 2 Script Before Task Launch (5 minutes) ● Task found on page: 28 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over again. 49 ○ Possible notices: ■ There are 4 people to share the chocolate orange with ■ There are 20 segments of the chocolate orange ■ Each person can have unequal amounts. ■ I wonder what “composition” means. ● Write down each notice on the board or under the document camera. ○ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ○ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ○ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 31). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 32) 50 page in order to spot common misconceptions your students may have. ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” or “Can you explain how you got that answer?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ○ I can decompose a fraction in multiple ways. ● Tell students to read it in their heads and signal to you when they are done. ● Read aloud the learning target to the students. 51 ● Underline the word “decompose.” Underneath the word “decompose” write the words “de-” and “compose.” Explain that “de-” is a prefix that means to separate. Compose means to put something together. Decompose means to break something down. ● Ask the class to turn and talk with their partner: “What are we breaking down today?” ○ Pull a stick or use a random name generator to call on a student for the answer: ■ “We will be breaking down fractions.” ● Say: “Not only will be breaking down fractions, but you should be able to break them down in multiple ways.” ● Ask the students what “multiple” means. Give them time to think, then turn and talk with a partner to share their answer. Cold call on a student to share out: ○ “Multiple means many.” ● Tell the students that by the end of the lesson they should be able to break apart a fraction in many ways. Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, and call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? ● Students who are watching may ask questions of the student. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and 52 differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. ○ To include the students who are very eager to answer, you can then ask the rest of the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ What is the whole? ○ What are the parts? ○ Does everyone agree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? Debrief/Connect (5 minutes) ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method? ○ What do we need to remember when decomposing fractions? ○ How can you explain how (student) decomposed their fraction? ○ What other ways could you decompose this fraction? Assessment (5 minutes) ● Hand out the assessment (Found on page: 33). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Do the students understand how to decompose a fraction? ● Do you need to spend a day on computation to sharpen up their decomposing fractions skills? ● Are there students you need to do tier 2 or 3 interventions with? ● If the majority of students did not do well, another tier 1 lesson needs to be taught with 53 decomposing fractions. 54 Task Taylor has a chocolate candy orange (pictured below). There are 20 segments. She wants to share all the segments among herself and 3 friends. The friends do not need to have an equal amount of pieces. ● Write as many compositions of fractions as possible for the ways Taylor can divide a single chocolate candy orange between herself and 3 friends. ● Represent 3 of your compositions using a visual fraction model. 55 Draw a picture or write an equation to solve the task: 56 Explain your partner’s thinking on the task: 57 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 58 Misconception(s) & How to Address Them ● When decomposing a fraction, the student also takes apart the denominator. ○ Example ■ 3⁄6 =⅓ + ⅔ ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we never change the whole. 59 Assessment 1. Select all the correct ways from the choices below that 4⁄7 could be decomposed. ❏ ½ + ½ + ⅓ ❏ 3⁄7 + 1⁄7 + 3⁄7 ❏ 3⁄3 + 4⁄4 ❏ 3⁄7 + 4⁄7 2. John had ¾ of a cake left. Show one composition in which John and his brother could divide the cake between them? 60 Assessment Key 1. 3⁄7 + 1⁄7 + 3⁄7, 3⁄7 + 4⁄7 2. ¼ + 2⁄4 61 Lesson 3 ● 4.NF.B.3b ○ Decompose a fraction into a sum of fractions with the same denominator in more than one way, recording each decomposition by an equation. Justify decompositions, e.g., by using a visual fraction model. Examples: 3/8 = 1/8 + 1/8 + 1/8 ; 3/8 = 1/8 + 2/8 ; 2 1/8 = 1 + 1 + 1/8 = 8/8 + 8/8 + 1/8. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goals: ● Students will learn what decomposing a mixed number is. ● Students will be able to decompose a mixed number into a sum of fractions with the same denominator in more than one way. ● Students will be able to write an equation of their decompositions. ● Students will justify their decompositions using a visual fraction model. 62 Lesson 3 Script Before Task Launch (5 minutes) ● Task found on page: 42 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over 63 again. ○ Possible notices: ■ There were seven whole apples that were cut up into 8 pieces but three of the pieces were discarded. ■ There are 5 workers that are going to share the apples that are left. ● Write down each notice on the board or under the document camera. ○ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ○ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ○ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 45). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 46) 64 page in order to spot common misconceptions your students may have. ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” or “Can you explain how you got that answer?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ○ I can decompose a mixed number in multiple ways. ● Tell students to read it in their heads and signal to you when they are done. ● Read aloud the learning target to the students. 65 ● Underline the word “decompose.” Ask the students if they remember what decompose means. Give them time to think then ask them to share what they remember it to mean. ○ Decompose: to break something down. ● Ask the class to turn and talk with their partner. Say: “What are we breaking down today?” ○ Pull a stick or use a random name generator to call on a student for the answer: ■ “We will be breaking down mixed numbers.” ● Remind the students of what a mixed number is. Write down 1 ¾. Say: “A mixed number is a whole number with a fractional part.” ● Say: “Not only will be breaking down mixed numbers, but you should be able to break them down in multiple ways.” ● Ask the students what “multiple” means. Give them time to think. Cold call on a student to share out: ○ “Multiple means many.” ● Tell the students that by the end of the lesson they should be able to break apart mixed numbers in many ways. Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, and call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? ● Students who are watching may ask questions of the student. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather 66 than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. ○ To include the students who are very eager to answer, you can then ask the rest of the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ What is the whole? ○ What are the parts? ○ Does everyone agree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? Debrief/Connect (5 minutes) ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method? ○ What do we need to remember when decomposing mixed numbers? ○ How can you explain how (student) decomposed their mixed number? ○ What other ways could you decompose this mixed number? Assessment (5 minutes) ● Hand out the assessment (Found on page: 47). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Do the students understand how to decompose a fraction? ● Do you need to spend a day on computation to sharpen up their decomposing mixed 67 number skills? ● Are there students you need to do tier 2 or 3 interventions with? ● If the majority of students did not do well, another tier 1 lesson needs to be taught with decomposing mixed numbers. 68 Task Amari has seven whole apples for the five people that show up to work. Amari has an apple slicer that cuts the apples up into eight pieces. After Amari cuts up each of the apples with the slicer, they accidentally drop three pieces on the floor, so they throw those pieces away. ● Show three different compositions of mixed fractions of how the apples could be divided among the workers. ● Represent one of your equations with a model. 69 Draw a picture or write an equation to solve the task: 70 Explain your partner’s thinking on the task: 71 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 72 Misconception(s) & How to Address Them ● When adding or subtracting fractions, students also add or subtract the denominator. ○ Examples ■ ⅓ + ⅔ = 3⁄6 ■ 5⁄7 - 3⁄7 = 2⁄0 ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we cannot go from 27 students in the class to 54. ■ When subtracting, we cannot go from 27 students in the class to 0. ● When decomposing a mixed number, the student also takes apart the denominator. ○ Example ■ 1 3⁄9 = 9⁄3 + ⅓ + ⅔ ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, the whole never changes. ● When decomposing a mixed number, the students leave the whole number as a whole number instead of changing it to a fraction. ○ Example ■ 2 ¼= 1 + 1 + ¼ ■ Correct answer: 4⁄4 + 4⁄4 + ¼ ■ Correct answer: 8⁄4 + ¼ ○ How to address this misconception: ■ Remind students that the entire mixed number should be decomposed into fractions. A whole number would represent all parts being present. If I cut a whole pizza into six slices and all 73 six slices are still there, I would have 6⁄6 of a pizza or a whole pizza. Assessment 1. Select all the correct ways from the choices below that 3 5⁄8 could be decomposed. ❏ 8⁄8 + 8⁄8 + 8⁄8 + ⅝ ❏ 3⁄3 + ⅝ ❏ 16⁄8 + 8⁄8 + ⅜ + 2⁄8 ❏ 24⁄8 + ⅝ ❏ ⅛ + ⅛ + ⅛ + ⅝ 2. How could 5 ⅚ bowls of rice be split between three dishes? Write one composition to show how the bowls of rice could be split between three dishes. 74 Assessment Key 1. Select all the correct ways from the choices below that 3 5⁄8 could be decomposed. ❏ 8⁄8 + 8⁄8 + 8⁄8 + ⅝ ❏ 16⁄8 + 8⁄8 + ⅜ + 2⁄8 ❏ 24⁄8 + ⅝ 2. How could 2 ⅚ bowls of rice be split between three dishes? Write one composition to show how the bowls of ricecould be split between three dishes. Possible answer: 6⁄6 + 6⁄6 + 5⁄6 75 Lesson 4 ● 4.NF.B.3c ○ Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goals: ● Students will be able to add mixed numbers with like denominators in multiple ways. 76 Lesson 4 Script Before Task Launch (5 minutes) ● Task found on page: 55 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over 77 again. ○ Possible notices: ■ Some of the times are in hours and some are in minutes ■ Alex read every day ● Write down each notice on the board or under the document camera. ○ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ○ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ○ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 58). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 59) page in order to spot common misconceptions your students may have. 78 ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” or “Can you explain how you got that answer?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ■ I can add mixed numbers. ○ Tell students to read it in their heads and signal to you when they are done. ○ Read aloud the learning target to the students. ○ Underline the words “mixed numbers.” Write down the following fractions and 79 mixed numbers on the board: ½, 1 ½, ¾, 5 ⅞, 8⁄9. ■ Ask the students if they remember which of these examples are mixed numbers. Tell them to give you thumbs up when they have decided. Once everyone has a thumbs up, tell them to turn and talk with their partner which ones are mixed numbers. ○ Cold call on a student to circle one mixed number (1 ½ or 5 ⅞) ○ Cold call on another student to circle the other mixed number. ○ Say: “A mixed number is a whole number with a fractional part.” ○ Say: “We will be putting wholes and parts together to add mixed numbers today. By the end of the lesson, you should have some strategies for adding mixed numbers.” Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, and call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? ● Students who are watching may ask questions to the student presenting. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to 80 share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. ○ To include the students who are very eager to answer, you can then ask the rest of the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ What is the whole? ○ What are the parts? ○ Does everyone agree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? Debrief/Connect (5 minutes) ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method? ○ What do we need to remember when adding mixed numbers? ○ How can you explain how (student) added their mixed number? ○ What other ways could you add this mixed number? Assessment (5 minutes) ● Hand out the assessment (Found on page: 60). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Do the students understand how to decompose a fraction? ● Do you need to spend a day on computation to sharpen up their decomposing mixed number skills? ● Are there students you need to do tier 2 or 3 interventions with? ● If the majority of students did not do well, another tier 1 lesson needs to be taught with decomposing mixed numbers. 81 Task Alex kept track of his reading for the week in the following table: ● How many hours did Alex read on Tuesday and Saturday combined? ● Alex’s teacher asks each student to read 11 ½ hours a week. ● Alex thinks he met that goal. Alex’s teacher thinks Alex did not meet the goal. Who is correct? Justify your answer with a model or equation. Day of the Week Time Read Monday 2 2⁄6 hours Tuesday 80 minutes Wednesday 7⁄6 hours Thursday 50 minutes Friday 2⁄6 of an hour Saturday 3 4⁄6 hours Sunday 1 ⅚ hour 82 Draw a picture or write an equation to solve the task: 83 Explain your partner’s thinking on the task: 84 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 85 Misconception(s) & How to Address Them ● When adding or subtracting fractions, students also add or subtract the denominator. ○ Examples ■ ⅓ + ⅔ = 3⁄6 ■ 5⁄7 - 3⁄7 = 2⁄0 ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we cannot go from 27 students in the class to 54. ■ When subtracting, we cannot go from 27 students in the class to 0. ● When decomposing a mixed number, the student also takes apart the denominator. ○ Example ■ 1 3⁄9 = 9⁄3 + ⅓ + ⅔ ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, the whole never changes. ● When decomposing a mixed number, the students leave the whole number as a whole number instead of changing it to a fraction. ○ Example ■ 2 ¼= 1 + 1 + ¼ ■ Correct answer: 4⁄4 + 4⁄4 + ¼ ■ Correct answer: 8⁄4 + ¼ ○ How to address this misconception: ■ Remind students that the entire mixed number should be decomposed into fractions. A whole number would represent all parts being present. If I cut a whole pizza into six slices and all six slices are still there, I would have 6/6 of a pizza or a whole pizza. 86 Assessment 1. _____=5 ⅜ + 2 2⁄8 2. Susan is making lemonade for her friends. They need 5 quarts of liquid altogether. Susan has 2 ⅗ quarts of lemon juice and 2 ⅕ quarts of water. Will she have enough lemonade? 87 Assessment Key 1. 8 = 5 ⅜ + 2 5⁄8 2. Susan is making lemonade for her friends. They need 5 quarts of liquid altogether. Susan has 2 ⅗ quarts of lemon juice and 2 ⅕ quarts of water. Will she have enough lemonade? Show how you know. 2 ⅗ + 2 ⅕=4 ⅘ Susan will not have enough to make 5 quarts of lemonade. 88 Lesson 5 ● 4.NF.B.3c ○ Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goal: ● Students will be able to subtract mixed numbers with like denominators in multiple ways. 89 Lesson 5 Script Before Task Launch (5 minutes) ● Task found on page: 68 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over again. 90 ○ Possible notices: ■ The first date is missing, we might have to figure that out first to solve the problem. ■ There are 12 inches in one foot. ● Write down each notice on the board or under the document camera. ○ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ○ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ○ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 71). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 72) page in order to spot common misconceptions your students may have. 91 ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” or “Can you explain how you got that answer?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ■ I can subtract mixed numbers. ○ Tell students to read it in their heads and signal to you when they are done. ○ Read aloud the learning target to the students. ○ Ask the students what the difference between today’s and the previous lesson’s 92 learning target. ○ Cold call on a student to answer: ■ “Today we are going to subtract instead of add mixed numbers.” ○ Cold call on 3 students to name a random mixed number to see if they still understand what a mixed number is. ○ Say: “We will be separating wholes and parts from each other to subtract mixed numbers today. By the end of the lesson, you should have some strategies for subtracting mixed numbers.” Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, and call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? ● Students who are watching may ask questions to the student presenting. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. 93 ○ To include the students who are very eager to answer, you can then ask the rest of the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ What is the whole? ○ What are the parts? ○ Does everyone agree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? Debrief/Connect (5 minutes) ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method? ○ What do we need to remember when subtracting mixed numbers? ○ How can you explain how (student) subtracted their mixed number? ○ What other ways could you subtract this mixed number? ○ How do we borrow and regroup from whole numbers when subtracting mixed numbers? Assessment (5 minutes) ● Hand out the assessment (Found on page: 73). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Do the students understand how to decompose a fraction? ● Do you need to spend a day on computation to sharpen up their decomposing mixed number skills? ● Are there students you need to do tier 2 or 3 interventions with? ● If the majority of students did not do well, another tier 1 lesson needs to be taught with decomposing mixed numbers. 94 Task Angel works at a garden center. Angel measured the height of a plant over the span of a month and kept track of the data on the following chart: ● Angel forgot to measure the plant on September 1st. ● If the plant grew 16 inches from September 1st to September 8th, how tall was it on September 1st? ● How much did the plant grow from September 1st to September 29th? Dates Height September 1st ? September 8th 1 10⁄12 feet September 15th 24 inches September 22nd 2 7⁄12 September 29th 3 3⁄12 feet 95 Draw a picture or write an equation to solve the task: 96 Explain your partner’s thinking on the task: 97 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 98 Misconception(s) & How to Address Them ● When adding or subtracting fractions, students also add or subtract the denominator. ○ Examples ■ ⅓ + ⅔ = 3⁄6 ■ 5⁄7 - 3⁄7 = 2⁄0 ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we cannot go from 27 students in the class to 54. ■ When subtracting, we cannot go from 27 students in the class to 0. ● When decomposing a mixed number, the student also takes apart the denominator. ○ Example ■ 1 3⁄9 = 9⁄3 + ⅓ + ⅔ ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, the whole never changes. ● When decomposing a mixed number, the students leave the whole number as a whole number instead of changing it to a fraction. ○ Example ■ 2 ¼= 1 + 1 + ¼ ■ Correct answer: 4⁄4 + 4⁄4 + ¼ ■ Correct answer: 8⁄4 + ¼ ○ How to address this misconception: ■ Remind students that the entire mixed number should be decomposed into fractions. A whole number would represent all parts being present. If I cut a whole pizza into six slices and all six slices are still there, I would have 6⁄6 of a pizza or a whole pizza. 99 Assessment 1. 10 4⁄6 - 3 ⅚=___ 2. James needs to walk 2 3⁄9 miles to his friend’s house. He has already walked 7⁄9 of a mile. How many more miles does James need to walk? 100 Assessment Key 1. 10 4⁄6 - 3 ⅚= 6 ⅚ 2. James needs to walk 2 3⁄9 miles to his friend’s house. He has already walked 7⁄9 of a mile. How many more miles does James need to walk? James needs to walk 1 5⁄9 miles to reach his friend’s house. 101 Lesson 6 ● 4.NF.B.3c ○ Add and subtract mixed numbers with like denominators, e.g., by replacing each mixed number with an equivalent fraction, and/or by using properties of operations and the relationship between addition and subtraction. ● 4.NF.B.3d ○ Solve word problems involving addition and subtraction of fractions referring to the same whole and having like denominators, e.g., by using visual fraction models and equations to represent the problem. Mathematical Goals: ● Students will be able to add and subtract fractions and mixed numbers with like denominators in multiple ways. 102 Lesson 6 Script Before Task Launch (5 minutes) ● Task found on page: 81 ● Display the task big enough so students can see. ○ Ways to display the task: ■ Digitally ■ Document camera ■ Posterboard ■ Whiteboard ● Ask students to read the task to themselves and show a “finished” signal when they are done reading the task. ○ Examples of “finished” signals: ■ Thumbs up ■ Fold arms ■ Eyes on teacher ● Read the task aloud to students. Try not to emphasize any particular part of the task in speech, gesture, or expression. You want to show a “poker face” throughout this lesson, as students often rely on a teacher’s reactions to gauge correctness towards an answer. ● Ask students what they notice about the problem. Emphasize that there really are no wrong answers here, but we are not looking to solve the problem just yet, we are simply making notices and writing them down to get familiar with the problem. ● Give students 15-30 seconds to think about their notices and give a “finished” signal when they have a “notice” to share. ● Wait until most students have shown you they are ready to share. ● Tell students to turn and talk with their partners to share some of their notices. ○ Make sure to tell the students who they are sharing with beforehand. ○ It may be wise to have designated partners or triads. ○ Call attention from the class before the talking dies down. Calling attention from the class before the talking is finished can build engagement and make students want to quickly share in the future. ● Cold call on a student to share a notice. ○ Ways to cold call: ■ Write each student’s name on a popsicle stick and draw a stick ■ Use a digital random name picker ■ Call on students not raising their hand ● This last one can be tricky because it is not truly random. Teachers can have the tendency to call on the same students over and over again. 103 ○ Possible notices: ■ Muhammed is mixing two types of flour ■ He accidentally spills a whole bag of flour into the mixture ■ He has too much flour mixture after spilling the bag ● Write down each notice on the board or under the document camera. ○ Ask students what is mathematical in the “notices.” This can help steer students in the right direction for picking out useful information in the task. ○ Write down their name next to their notice. This will honor their answer and encourage students in the future to share. ○ Writing down notices can activate student thinking and answer any contextual questions before they begin to solve the task. Task Launch (5 minutes) ● Tell the students to start the task (5 minutes). ○ Students need to be completely silent when they work. ○ Remind talking students that they will get to share their thinking once the private think time and solving is finished. ○ Encourage students to use whatever method they need to in order to complete the task. ■ Concrete pictures to abstract algorithms. To have a rich discussion, there should be a variety of ways of thinking shown. ○ Tell students if they get done early to use a different strategy to complete the task. Monitoring (during task launch) ● Refer to the anticipatory/monitoring chart for this part of the lesson (found on page: 84). ● Your main task is to take note of what methods students are using to solve the task. ● Common possible tasks are already listed on the anticipatory/monitoring chart, but additional spaces have been intentionally left blank in order to list other student methods for solving the task. ● Write the student’s names under the “who” column if they used the method listed on the left under “strategy.” You will be asking these students to share their strategies during the whole group discussion. ● It may be helpful to keep track of which students you are choosing to share to ensure all students are sharing equally throughout their time with you, and you are not asking the same students to share over and over again. Answering Questions During Monitoring (during task launch) ● Review the “Likely Misconceptions and How to Address Them” (found on page: 85) page in order to spot common misconceptions your students may have. 104 ● If more than a few students are making the same mistake, this will need to be addressed during the whole group discussion. Make note of this on the monitoring chart under “misconceptions” in order to bring it into the discussion. ○ If only a few students are making the same mistake, the most effective time to address the problem with the students is during a tier two or one intervention. ● Students will likely ask questions to try to solve the problem as they are working independently. ○ Instead of answering their questions, your role is to make their thinking more visible. ■ See the “questions” column on the monitoring chart in order to address their questions. ■ If you ask them an advancing question (italicized), ask the question then walk away so they can think and continue to work independently. ■ If you ask them an assessing question (bold), ask the question and wait around for their response. You can then judge what advancing questions to ask to get them closer to the goal of the lesson. Partner Discussion (3-5 minutes) ● Before the lesson, partner students up intentionally. Avoid partnering students with widely different ability levels, as the discussion may not be productive for either of them. ● Assign one student “partner A” and the other student “partner B,” if you have an odd number of students, a triad (three students) can be formed. ● Tell Partner A to share their thinking on the task first as Partner B listens. ● Partner B then needs to write down an explanation of the thinking of Partner A on the back of their task paper. ● Now switch roles. Partner B will be sharing and Partner A will be writing an explanation of Partner B’s thinking on the back of their task paper. ● If students disagree on the answer, encourage them to probe deeper into each other’s thinking with questions like “What makes you think that?” or “Can you explain how you got that answer?” ● They do not need to come to a consensus on the answer. They are exploring different methods of thinking. Unpacking The Learning Target (3 minutes) ● Display the following learning target: ■ I can add and subtract mixed numbers. ○ Tell students to read it in their heads and signal to you when they are done. ○ Read aloud the learning target to the students. ○ Ask them to think about adding mixed numbers and the strategies they use. Tell 105 them to show a thumbs up when they have thought about the process they use for adding mixed numbers. Tell them to turn and talk with their partner, sharing their strategy. ○ Repeat the process for subtracting mixed numbers. ○ Say: “We will be separating and combining wholes and parts to and from each other to add and subtract mixed numbers today. By the end of the lesson, you should be fluent in adding and subtracting mixed numbers.” Whole Group Discussion (15 minutes) ● If there were any major misconceptions that need to be addressed (more than a handful of students), now is the time to address the misconception(s). ○ If your students are comfortable enough, you may even have a student share an example of the misconception so all students can visually see and understand why is it incorrect. ○ If you do this, make sure to thank the student for sharing and helping the class move their learning forward. ● If there are no major misconceptions, move forward with the lesson. ● Looking at the monitoring chart that you have filled out, and call up the first student you have selected. The chart is ordered in the method that should be shared first, next, and last: from concrete to abstract methods. ● The student should display their work under the document camera in order for all students to see it. ● Ask the student to explain their thinking and their method for solving the task. ● If you find that the student cannot explain their thinking clearly you may ask some of these questions: ○ What was the first thing you did? ○ What makes you think that? ○ What was the next thing you thought? ● Students who are watching may ask questions to the student presenting. ● At this point, you should not confirm if the student got the correct answer or not. Remember, the emphasis should be on using multiple methods and deeper thinking rather than the correct answer. ● Thank the student and ask the next student to display their work. ● Once this student is done sharing, ask the rest of the class to describe the similarities and differences between the two methods. ● Give the students some think time, then tell them to turn and talk with their partner to share the similarities and differences they saw. ● Cold call on two to five students to share their notices on the similarities and differences between the two methods. ○ To include the students who are very eager to answer, you can then ask the rest of 106 the class if they noticed any more similarities and differences and call on students raising their hands at this point. ● Share the last methods in the same manner. ● Questions you can ask during the whole class discussion: ○ What makes you think that? ○ I’m wondering why you need to do that? ○ Is there another way? ○ Can you show us how? ○ What is the whole? ○ What are the parts? ○ Does everyone agree with (student)? ○ Can you show us what you mean? ○ How can we know for sure? ○ Does Muhammed have the right amount of flour? Debrief/Connect (5 minutes) ● Questions to ask to connect: ○ What is the difference between what (student) and (student) did? ○ How is (student)’s method similar to (student)’s method? ○ Does anyone want to add to that? ○ Turn and discuss with your partner if you agree or disagree with this method? ○ What do we need to remember when adding and subtracting mixed numbers? ○ How can you explain how (student) subtracted or added their mixed number? ○ What other ways could you add or subtract this mixed number? ○ How do we borrow and regroup from whole numbers when adding or subtracting mixed numbers? Assessment (5 minutes) ● Hand out the assessment (Found on page: 86). ● Tell the students that they need to complete this assessment by themselves. ● Collect and grade the assessments. ● Do the students understand how to decompose a fraction? ● Do you need to spend a day on computation to sharpen up their decomposing mixed number skills? ● Are there students you need to do tier 2 or 3 interventions with? ● If the majority of students did not do well, another tier 1 lesson needs to be taught with decomposing mixed numbers. 107 Task A recipe for a dessert calls for 5 1⁄4 cups of white flour and 1 ¾ cups of whole wheat flour. Muhammed put in the white flour he had into the bowl, which wasn’t enough. While he was pouring the wheat flour into the bowl, the bag tore and he spilled the entire bag into the bowl. He read on the bag of wheat flour that it contained 5 ¼ cups. He now has 8 2⁄4 cups of the white and wheat flour mixture. ● How much white flour did Muhammed have? ● How much more white flour does Muhammed need? ● How can Muhammed fix the amount of flour for his recipe to make a correct recipe? 108 Draw a picture or write an equation to solve the task: 109 Explain your partner’s thinking on the task: 110 Advancing Question-Walk away Assessing Question-Stay Anticipatory/Monitoring Chart Strategy Questions Who I’m wondering why you need to do that? What is the next step? What was the first thing you thought? How do you know to add? What makes you think that? How do you know to subtract? What is the whole? What are the parts? Can you show me what you mean? What part is confusing you? Misconceptions 111 Misconception(s) & How to Address Them ● When adding or subtracting fractions, students also add or subtract the denominator. ○ Examples ■ ⅓ + ⅔ = 3⁄6 ■ 5⁄7 - 3⁄7 = 2⁄0 ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, we cannot go from 27 students in the class to 54. ■ When subtracting, we cannot go from 27 students in the class to 0. ● When decomposing a mixed number, the student also takes apart the denominator. ○ Example ■ 1 3⁄9 = 9⁄3 + ⅓ + ⅔ ○ How to address this misconception: ■ Remind students that the denominator is the whole, and the whole does not change. ■ When adding, the whole never changes. ● When decomposing a mixed number, the students leave the whole number as a whole number instead of changing it to a fraction. ○ Example ■ 2 ¼= 1 + 1 + ¼ ■ Correct answer: 4⁄4 + 4⁄4 + ¼ ■ Correct answer: 8⁄4 + ¼ ○ How to address this misconception: ■ Remind students that the entire mixed number should be decomposed into fractions. A whole number would represent all parts being present. If I cut a whole pizza into six slices and all six slices are still there, I would have 6⁄6 of a pizza or a whole pizza. 112 Assessment 1. ___=(3 ¼ + 5 ¼) - 6 ¾ 2. Ann needs 10 5⁄7 feet of wood for a project. She has a piece of wood that is 4 2⁄7 feet long and another piece of wood that is 4 6⁄7 feet long. How much more wood will Ann need? 113 Assessment Key 1. 1 ¾ =(3 ¼ + 5 ¼) - 6 ¾ 2. Ann needs 10 5⁄7 feet of wood for a project. She has a piece of wood that is 4 2⁄7 feet long and another piece of wood that is 4 6⁄7 feet long. How much more wood will Ann need? 1 4⁄7 feet