Repeated Addition, Arrays, and Equal Groups Building a Foundation for Multiplication

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1 Repeated Addition, Arrays, and Equal Groups Building a Foundation for Multiplication I. Repeated Addition and Equal Groups: learn about equal groups and begin to build a foundation for multiplication and division. They will learn how to solve addition problems through problem solving, drawing pictures, and making arrays. extensively explore addition and will be able to write their own addition stories. also be able to go from a picture or array to a addition number sentence or from a addition number sentence to a picture or an array. II. Essential Questions: explore the essential questions, how do we solve addition problems? and how do we with equal groups to solve addition problems?. The main idea of this unit is to have the students explore addition in many different ways, so that they build a solid foundation for multiplication and division in the future. with addition in many different ways including word problems, skip counting, number sentences, and arrays. The idea is for students to become experts in addition so that when they learn multiplication next year they will have the knowledge they need to understand equal groupings. III. Learning Goals: A. Development of content understanding (key concepts and ideas): understand the concept of equal groupings. They will be able to solve addition number sentences and identify the array that goes with the addition number sentence, and vice versa. also be able to use problem solving to complete addition word problems, and will be able to write their own addition word problems. In addition, students will be able to talk about arrays in terms of equal groups. B. Development of habits of mind and, including habits of independent or collaborative thinking and doing typical of readers, writers, speakers, creators, researchers, and thinkers in the discipline (ways of knowing): come to know about addition through a variety of independent, group, and class. They will learn to explain how to solve addition problems and identify equal groups. Students will engage in a variety of different activities, ranging from identifying arrays in a book scavenger hunt, completing envision mats, creating their own word problems, and playing addition games. In addition, students will explore arrays through problem solving. Before introducing the topic, students will discover how to solve addition and create arrays through open ended word problems. They will with groups to explain their reasoning and analyze their solutions as a class. C. Literacy development, including capabilities of proficient readers, writers, and speakers: Although this is a math unit, students will be using their literacy skills to demonstrate their understanding. Many lessons will include class

2 IV. discussions which will allow the students to practice their speaking skills. In addition, the two word problems students will be ing on will be depicted on a poster that they will solve in a group. Then, the groups will have to present by explaining how they solved their problem. This will help them practice writing in a group, as well as presenting in a group. It will also help them to begin to explain, in words, how they are thinking about mathematical concepts. also begin to write their own word problems with addition. This will help students practice a different kind of writing than they may be used to and make their writing skills more versatile. They need to begin to learn how to use writing to explain their mathematical thinking, as well as to create their own problems. D. Development of the classroom as a learning community: Each lesson will involve some sort of partner or group. This will allow the students to get to know each other as people and as mathematical thinkers. It will allow them to talk in an informal way about their mathematical thinking, and will give them more confidence when they present or share with the whole class. be asked to complete two open-ended word problems to explore and discover the mathematical concepts of arrays. These problems, although new to them, can be solved through a variety of different methods. be able to on them independently for a short period of time to get their ideas on paper, and then they will in a group before they present. Working in a group will allow them to talk through their ideas and mathematical thinking and gain more confidence in their mathematical abilities and problem solving skills. be encouraged to solve the problems in whatever way they are comfortable with, and their different strategies will be represented on a chart, to include all students and enforce the idea that there is more than one right way to solve a problem. In addition, students will use their knowledge of arrays to on a quilt patch, which in the end will be put all together and hung up in the classroom. This gives meaning to their, and demonstrates the importance of collaboration in learning. also be starting math centers in this unit, which they have not done before. This shows the students that the teacher trusts them to be accountable for their behavior and their learning while they are not in a small group with the teacher. In addition, by teaching the students new concepts in small groups, it creates a low-stakes environment and allows the students to feel more comfortable in asking questions about the mathematical concepts. Rationale: A. Learning Goals: begin to build a foundation for multiplication and division by learning how to solve addition number sentences. explore addition using their problem solving skills to solve an open ended word problem about M&Ms (something they love!), and will use practice explaining their mathematical thinking by ing in groups and presenting their solution. be asked to talk about

3 what they notice about the groups strategies, and they will also be asked to discuss and identify the different problems and solutions in terms of equal groups. Students have already been encouraged to draw pictures to help them solve addition problems, and now they will have to use this picture drawing practice to discover what arrays are. After they discover how to solve the word problem (involving brownies and how we can get them into equal pieces so that everyone in the class gets one), the will have to use this real life problem to think about what their picture looks like, and come to the conclusion that there are an equal number of brownies in each row. Lastly, students will practice solving addition word problems, and use the strategy of UNPACK the question to help them. This strategy can be used for any word problem, but is especially helpful for addition because they have to really think about what the question is asking (underlining the word each, for example, is very helpful to understanding a addition word problem). In addition, this strategy will be helpful for the students later on in math as well, because it is adaptable to all word problems. Furthermore, students will learn how to write their own addition word problems, because if they can write their own problems, then they can also solve them. The students have had a lot of trouble in the past writing their own word problems, so I wanted to take the time in this unit to go through how to write a word problem. If the students are comfortable writing their own word problems and know how they should be worded and what components go into it, then they will undoubtedly understand how to solve a word problem. B. Curriculum Standards: CCSS.Math.Content.2.OA.C.4 Use addition to find the total number of objects arranged in rectangular arrays with up to 5 rows and up to 5 columns; write an equation to express the total as a sum of equal addends. create their own arrays, write number sentences from arrays, solve addition number sentences and word problems. They will play several games to practice they will practice skip counting to reinforce the idea of addition. Furthermore, they will make their own quilt patch with a decorated array built from a number sentence. C. Students Background and Readiness: Students have already learned how to add three numbers together, and have had a lot of practice with word problems. Although I have a lot of different learners in my classroom, they will all be excited to solve the first introductory problem to addition because it involves M&Ms, and my students are very interested in anything food related. They will be able to use the M&Ms as manipulatives, if they need to, and if not they will get them as a snack after to make them very interested. All students know how to use the strategy of drawing a picture to help them solve a problem, so they should all be able to solve the open ended

4 word problems I give them. In addition, students have spent the last few months ing on different addition strategies, and they practice skip counting every morning during morning meeting, so they already have enough background knowledge to solve addition number sentences. D. Students Needs: I have a variety of students in my classroom, so I needed to create a unit that would be applicable for all types of learners. The two word problems I gave the students to discover arrays and addition have multiple entry points and may be solved in a variety of ways. Therefore, all students, even students on IEPs and ELLs, should be able to solve them; however, many of my students on IEPs and my ELLs will have the word problems read to them to help with the literacy portion. also have a lot of time to on these problems, and will on them in several different ways. First they will individually so they can start to think about the problem on their own. Then, they will in their group and explain how they solved the problem. Lastly, they will share with the class. I decided to have these three steps so that students with learning disabilities, or students who just take a longer time to process information have the extra time to reread the question and get their ideas down before they need to talk about them. In addition, students who might not have a lot of confidence in math will be able to talk in a small group first before sharing with the class. For the new concepts that are being taught, I will pull small groups, and my students on IEPs will all be in the same group to assure that they get the scaffolding that they need. In addition, all of the centers are accessible to all students, and will be modeled to the whole class. E. Research and Evidence-Based Best Practice Ideas: This unit involves learning about addition through many different methods. Before learning the term addition and array, students will discover how to solve addition problems and how to create an array through open ended word problems (Zemelman, Daniels, & Hyde, 2005). By allowing students to explore and discover these mathematical concepts, they are acting and thinking like mathematicians and are active members of their learning communities. This holds students accountable for their own actions, because they are guiding their own learning. It holds the students to high standards, and it lets them know that the teacher believes in their abilities and potential (Smith, 2004). No matter what learning preference a student has, they will be able to solve the open-ended problems given, because there are so many different ways to think about solving them. In addition, because there are so many ELLs and students on IEPs in my classroom, students will be read the word problems, and will have a lot of time to talk about their mathematical thinking in their groups. This will allow these students to talk about formal academic thinking in an informal setting, making them more comfortable and allowing them to think through their ideas before they discuss it with the whole class or with the teacher. Furthermore, these students will benefit from small group instruction, as well as math games, and other low stakes,

5 informal assessments to help them practice their addition knowledge (Gibbons, 2009). V. Assessments: A. Explain your main assessments and how they are appropriate for your learning goals: be assessed many times throughout this math unit, as to check for student understanding and misconceptions. This topic is especially difficult for students to understand right away because it is the basis of multiplication and division, which they have not been exposed to before. Therefore, throughout the unit there are several places where student misconceptions can be noticed. In both introductory activities (for arrays), students will be given a sheet with an open ended word problem on it, on which they are asked to draw a picture and create a number sentence. A variety of drawings or strategies would be accepted for this sheet; however, because this is the part that students are ing on individually, it will be helpful for the teacher to see what the students original thoughts are. Then, the students will share their mathematical thinking with a group, and the group will present and explain to the class how they solved the problem. In partners, and as a class, the students will then discuss what they notice about the different strategies and the different problems. The group, discussions, and presentations will be an informal assessment to see how students are thinking about these mathematical concepts and how well they are able to explain their thinking in words. In the math centers I have created for this unit, students will also be assessed appropriately. All the centers will have some sort of a sheet, whether it be for the scavenger hunt, the coverall game, the quilt patches, or the mat, the students will be producing something in their center. This is a way to make sure the students are ing in their centers, and that they are understanding the material, even if it is just with a dice game. Furthermore, when students meet in small groups they will begin to on writing their own addition word problems. This will allow the teacher to talk with a small group of students about the different components of a word problem and how to solve them and write them. Through discussion with the students, and the students drafts and final products they will be able to demonstrate their understanding of the material. Finally, because this unit is based on a topic in the envisions math program, students will have to take several topic quizzes and a final unit test. This will quantify their knowledge of arrays and demonstrate whether or not they understood these mathematical concepts. B. How will students know what to expect and the criteria for good : At this point in the year, students have had a lot of practice ing in groups, and have presented in groups more than once. They know what the expectation is for ing in a group, and understand that everyone must participate and cooperate. Students also know the expectations for presenting, including allowing everyone in the group a chance to speak, and speaking loud and

6 proud. I think the students will do very well with the open ended word problem activities because they are used to this type of. In addition, although math centers are new for the kids, they do Daily 5 every day, and most of them have been doing Daily 5 since kindergarten. They know that in Daily 5 they must maximize their time on learning in the center, by ing the whole time. Therefore, I will tell the students that our new math centers will have the same expectations as our Daily 5 centers. Students also know that I will be collecting their from every center, so they know a center spent not maximizing their time on learning will not go unnoticed. In addition, I will model each center for the students and clearly write out the instructions for each center, so the students have a reminder of what they are expected to be doing. C. Culminating assignment and corresponding assessment criteria/rubric: The students will have two final assessments. The first assessment will have two parts, and is student centered; the students will write four of their own word problems, and then switch with a partner to solve them. They must demonstrate their solution through a drawing and a number sentence. In addition, the students will have to take a unit test, which will assess their understanding of addition, arrays, and addition word problems. VI. Unit Calendar: Lesson Strategies Materials Assessments Lesson 1: What is solve an open Differentiated Word Problems Individual sheet addition? ended word M&Ms Group poster problem based on Construction Paper Presentation addition. Markers individually, then in groups, and then present to the whole class. We will then discuss how they discovered equal groups. Chart Paper Lesson 2: Continuing to with engage in partner envisions interactive book envisions Center sheets envisions quick

7 addition. How do we UNPACK word problems? Lesson 3: What is an array? Lesson 4: Applying our knowledge of arrays. with whole class word problem examples. then break into different math centers to on different addition activities. in small groups with the teacher to with word problems. Anchor chart, whole group discussion and examples. then in math centers to practice arrays. Students will also continue to in small groups with the teacher to on word problems. on an openended word problem involving arrays. individually, then in a group, and then will present. mat Coverall game (sheet and dice) Skip counting cards Differentiated addition problems Memory game cards Coverall game Scavenger hunt sheet The Grapes of Math Amanda Bean s Amazing Dream Differentiated math problems Workmats Quilt patches with number sentence on back Word problem Brownies in a rectangular pan Posters Markers check quiz Small group Worksheets from centers Quilt patches Small group Individual sheets Group posters Presentations Quick check quiz

8 Lesson 5: Practicing arrays. Lesson 6 (2 days): Word problems and review of arrays. continue to in centers to practice arrays. In small groups with the teacher, students will begin to write their own word problems. complete their word problems (writing and solving a partner s) and quilt patches. engage in a review game for their test. Word problem graphic organizers Math notebooks Coverall game Memory game Scavenger hunt sheet Amanda Bean s Amazing Dream The Grapes of Math envisions mat envision s test Center materials (same as above for kids who finish early) Jeopardy questions Computer (to type finish word problems) Math notebooks Graphic organizers Center sheets Small group Word problems envisions test Jeopardy game VII. Resources: A. How will you to actively involve parents in their child s academic activities and performance, and communicate clearly with them? All student envision s will be sent home to their parents. In addition, students will have home every night that they will be encouraged to complete with their parents. B. Resources used: Van de Walle, J. A., Karp, K. S., & Bay-Williams, J. M. (2012). Elementary and middle school mathematics: Teaching developmentally. Pearson. Zemelman, S., Daniels, H., & Hyde, A. (2005). Best practice, today's standards for teaching and learning in America's schools. Heinemann. Common Core Standards (2012). Standards for mathematical practice. Retrieved from Neuschwander, C. & Burns, M. (1998). Amanda bean s amazing dream. Scholastic Press. Tang, G. (2004). The grapes of math. Scholastic Paperbacks.

9 C. Legal and ethical issues involved in Internet use and other resources: I made sure that all of my resources were reliable and valid. All of the books I used I borrowed from the curriculum lab in the education department at Clark University. In addition, all of the ideas and inspiration I got for my unit came from scholarly math textbooks and teaching and learning texts, also acquired through Clark University. Furthermore, the mathematical concept I obtained from the envision s math program, which is aligned with the Common Core standards.