Revised Lesson Plan with Comments

EDUC 531: Term III Assignment Plan, Teach and Analyze, and Represent a Mathematics Lesson

Student teacher(s): Melina Varney and Kelsey Jurewicz Location: Penn Alexander School Grade: First grade

What In this lesson, students will learn to apply their additive reasoning to an unfamiliar problem structure--an equal grouping multiplication problem. The first grade at Penn Alexander uses the Investigations curriculum, which has recently introduced “number stories” in the form of contextualized addition problems. The primary focus of these number stories has been addition within 20. In our placements, we have noticed that some of the students in our classes are doing this fluently and utilizing advanced additive strategies such as expanded algorithms. The goal of this lesson is to challenge these students to extend the use of these additive strategies to multiplicative situations. The task presented during this lesson will be a contextualization of 4 x 11 = 44. Although our students have not yet worked with problems involving more than two addends, we believe that they will recognize the problem situation as repeated addition and apply strategies that fall within the early additive and additive strategy portions of the OGAP Multiplicative Reasoning Framework. Furthermore, we are interested to see if students will use the fact that 11 is one more than 10 to skip count 10, 20, 30 40 and then add the four leftover 1s to get 44.

In addition to the development of early multiplicative reasoning strategies, our secondary goal for this lesson involves the associated mathematical practices, which are twofold. First, students will choose, among a variety of tools, the one that most appropriately represents their thinking. Second, because there will be multiple ways students may go about solving this problem, we want students to participate in a reflective discussion of the various ways that they and their peers approached the problem. This second goal involves both the ability to articulate their own thought process and the practice of listening to and reflecting on alternative strategies.

How The goals of this lesson will be accomplished through the three-phase lesson format: launch, work and explore, debrief and wrap up. Upon being presented with the mathematical task, students will be given ample time to work through the problem independently with their choice of tools. During this time, we will closely observe the strategies students in our respective small groups are using, asking probing questions and providing support as needed. Our hope is that by providing a variety of different tools, students will develop multiple strategies for solving the problem. In the discussion to follow, we will act as facilitators, helping students to articulate how they thought about the problem and how they approached solving the problem. To further evaluate how the whole group debrief and discussion informed students’ approaches to solving equal grouping problems, we would conclude the lesson with an one of two “exit slips,” depending on which we feel is most appropriate given the strategies and discourse we observe taking place throughout the lesson. The first option will ask students to show how one of their peers solved the problem in a different way. This would emphasize the importance of listening to and attempting to understand the approaches taken by their peers. Alternatively, the second exit slip option will ask students to show how they would solve the problem if a 5th group was added to the initial 4 groups of 11. The exit slip will ask students to describe or illustrate how they would solve the problem if there were 5 seed packets with 12 seeds in each packet. This exit slip would further demonstrate to us the ways that students are thinking about equal groups, and possibly demonstrate a student’s ability to apply a more efficient strategy used by one of

Comment [CE1]: This is excellent!

TERM III MATH -- LESSON PLAN ROUGH DRAFT

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their peers, which would provide insight into the effectiveness of the debrief portion of the lesson. Why The task for this lesson was chosen specifically as a means of challenging students to apply their additive reasoning skills to a multiplicative situation. The students we will be including in these groups are students whom we have consistently observed to use advanced addition strategies (i.e. partial-sums algorithms, number decomposition, skip counting) to solve number stories in class. Hiebert et al. discuss that tasks should allow for students to invent and examine strategies for solving problems, but that tasks must also be structured in a way that allows students to use skills and knowledge they already possess. Our hope is that by presenting our students with an unfamiliar problem format, they will extend their additive reasoning strategies to invent solutions to an equal groups multiplication problem.

Although addition involving more than two two-digit numbers is not included in the PA Common Core State Standards until grade 2 (2.NBT.6), we believe that the students selected for this small group lesson are capable of developing solutions to this problem. Our goal is to present them with a task where the solution approach is not immediately evident, thereby forcing them to think creatively and invent strategies to solve the problem. Furthermore, their ability to solve this problem will demonstrate mastery of several grade 1 Common Core Standards, outlined in the standards section below. While our chosen task falls beyond the scope of the Common Core Practice Standards for grade 1, it is supported by a number of grade 1 NCTM Content and Process Standards, identified in our lesson plan below. The applicable NCTM standards include multiple aspects of the task, such as the ability to understand the problem as repeated addition, visually and numerically represent their thinking and choose appropriate tools to model the situation it falls within the Common Core Standards for Mathematical Practice. These include constructing viable arguments and critiquing the reasoning of others, modeling with mathematics, and using appropriate tools strategically. Finally, the task encompasses components of the grade 1 Investigations curriculum used by Penn Alexander, including composing numbers with three addends and the ability to name and compare different strategies used for solving problems, as well as discuss how different tools can be used to model and solve problems. The latter elements will be emphasized directly during the debriefing at the end of the lesson.

Task: Solve a contextualized multiplication problem using existing additive strategies. Based on our knowledge of our students, this task will challenge them to invent solutions to an unfamiliar type of problem, but will remain accessible because they can use strategies that they already know.

Discourse: The debriefing will encourage student-to-student talk, with the student teacher acting merely as a facilitator. The purpose of this discussion is for students to articulate and reflect on their own problem solving approach, as well as listen to and understand the alternative strategies used by their peers.

Tools: Students will be provided with a variety of appropriate and familiar tools to complete the task. A component of our formative assessment for this task will be making note of the tools students select and how they use them.

Norms: Explicit expectations at the beginning of the lesson will remind students that there are multiple “right” and appropriate ways to approach the problem and that each student will be responsible for engaging in discussion around these various strategies. Additionally, students will be expected to explain and justify their method for solving the problem.

Comment [CE2]: I really like this option

Comment [CE3]: Wonderful

Comment [CE4]: agreed

Comment [CE5]: You also probably want to emphasize the norm that you will be expecting them to explain and justify their solution method (that how they solve it and being able to explain that is as important as getting the solution)


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Goals & Objectives Content

Students will invent strategies for solving a contextualized multiplicative problem by extending their existing additive reasoning strategies to an equal grouping multiplication word problem. For example, they could apply repeated addition through counting, the partial-sums algorithm or decomposing the factors into more “friendly” numbers.

Mathematical Practice

1. Students will be able to represent their thinking through the use of appropriate tools and visual models.

2. Students will actively participate in a discussion of various approaches to the problem by explaining their own thinking to the group, as well as listening and responding to the strategies used by their peers.

Standards PA Common Core State Standards 1.OA.2 Represent and solve problems involving addition and subtraction. Solve word problems that call for addition of three whole numbers whose sum is less than or equal to 20, e.g., by using objects, drawings, and equations with a symbol for the unknown number to represent the problem. 2.NBT.6 Use place value understanding and properties of operations to add and subtract. Add up to four two-digit numbers using strategies based on place value and properties of operations.

PA Common Core Standards for Mathematical Practice 3. Construct viable arguments and critique the reasoning of others 4. Model with mathematics 5. Use appropriate tools strategically NCTM Content and Process Standards (Grade 1)

Numbers and Operations 1B: Understand meanings of operations and how they relate to one another.

-Investigate multiplication as repeated addition as it relates to literature. -Use various representations to illustrate understanding of addition and subtraction as it relates to story problems.

1C: Compute fluently and make reasonable estimates -Develop strategies for whole number computations for addition and subtraction (i.e. using manipulatives, counting on, number line). -Write numbers and symbols to represent addition and subtraction. -Use a variety of methods and tools to compute (i.e. objects, mental computation, estimation, paper and pencil, calculators).

Algebra 2B: Represent and analyze math situations and structures using algebraic symbols.

-Use concrete, pictorial and verbal representations to develop an understanding of conventional symbols for addition, subtraction and equals.

2C: Use mathematical models to represent and understand quantitative relationships.

Comment [CE6]: At this age, they probably won’t use the algorithm, but they may decompose 11 into 10 and 1 and add or multiply the tens and ones separately.

Comment [CE7]: Can you connect these directly to the Common Core mathematical practices?

Comment [CE8]: This isn’t necessary… but I’m guessing you were looking at an old lesson plan that was written before we had common core.Your PA common core standards are sufficient here, but I would also add OA 2.2.A.3 on working with equal groups to develop a foundation for multiplication

Comment [CE9]: ?


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-Model and explain situations that involve addition/subtraction of whole numbers using objects, pictures and symbols.

Materials and preparation Instructor Needs

Whiteboard Dry-erase markers with eraser Behavior expectations chart Observation checklist Exit slips (to be handed out after debrief)

Each student needs

Clipboard Worksheet to record thinking with problem written on it Pencil & eraser

Available tools: blank paper, crayons & markers, connecting cubes, hundreds chart, bear counters, colored math tiles, straws with rubber bands Classroom arrangement and management issues Location This lesson will be conducted in one of the small conference rooms at Penn Alexander. The room contains one small circular table with room for about six students. Around the perimeter of the room are several arm chairs. There is also a counter behind the circular table where the whiteboard and materials will be placed. For the launch of the lesson, all students will be sitting at the circular table facing the whiteboard. This small learning room will limit the number of distractions and noise from other students that would occur in the classroom or in the hallway. During the exploratory portion of the lesson, students will be able to move throughout the room to work on the floor or in the armchairs. Then, they’ll come back to the table for the discussion portion to share strategies. Materials During the launch, students will be given the worksheet where the word problem is written. A container with pencils and erasers will be set out on the table after the launch (to ensure that students are listening while directions are given). The other tools (crayons, connecting cubes, hundreds charts, and bear counters), as well as the clipboards, will be set up on the counter beside the whiteboard. There is not very much space in the room, so half the students will be told to line up to choose their tools while the other half re-read the problem and write their names on their papers at the table. Then, they’ll switch. Once everyone has their tools and name written on their paper, they can spread out throughout the room. If students decide they want to use a different tool, they can come back to the counter at any time throughout the assignment. At the end of the exploration, students will be called in groups of two to put their tools away and move to the table. After the lesson, students will put their pencils and erasers back in the container and leave their papers in a pile on the table. Each student will be asked to carry a box of materials back to the classroom.

Comment [CE10]: Good idea


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Management Overall, I don’t anticipate management concerns. Students were chosen because they demonstrated advanced math knowledge. The material should be at the appropriate level for all students. Furthermore, behavior expectations and norms will be explicitly established at the beginning of the lesson. All directions will be given before materials are handed out. The only concern is that the room is small, so not all students can be moving at once. To avoid any management concerns, I’ll only have a few students moving around the room at once and give specific directions for who should be moving when.

Plan

Task (50 min total) Ms. Jurewicz and Ms. Varney want to plant flowers in their garden. They bought 4 packets of seeds. Each packet has 11 seeds in it. How many seeds are Ms. Jurewicz and Ms. Varney planting? Launch (10 min) Expectations and Norms

a. Go over behavior expectations and norms through anchor chart i. Connect to number talk, explain that at the end they need to be able to explain

their thinking to other members of the group and listen to other group members’ thinking

ii. Explain that there is more than one way to solve the problem and more than one tool to use

b. Tell students they’ll be working independently, talk about what that looks like Task introduction

a. Introduce word problem & tools and their appropriate uses. Pass out worksheet, reread word problem asking students to follow along with finger

Introduce problem in narrative form. “Ms. Jurewicz and I wanted to plant flowers in our garden. What do we need to grow flowers? (seeds). The seeds came in these packets (show & open packet). Inside each packet is 11 seeds. Ms. Jurewicz and I need you to help us figure out how many flowers we’re planting because we can’t agree on a number and we need to know how much space to leave in our garden.”

b. Ask students to repeat information such as how many seed packets we purchased, how many seeds are in each packet, and the problem we’re solving

c. Ask students to repeat the directions & ask if anyone has questions d. Place pencils and erasers on table. Call half of the group to choose tools while half re-

read question and write name on paper. Then groups switch. I want to keep the introduction & review short because I want students to bring their own strategies to the problem. I’ll introduce the word problem through narrative form so that students have an additional context. I’ll also show them a packet of seeds and the seeds inside. An introduction to the tools will be the only discussion of prior knowledge. I’ll then review the ways in which students have used the tools in their classroom. The students work frequently with addition word problems in their Investigations book and shouldn’t need a review of how to solve these problems.

Comment [CE11]: I wonder if you want to have a packet to show them what it means to have a packet of seeds? Will the context of seed packets be familiar to them? You could also choose a different context…. e.g, boxes of markers that might be easier to visualize.

Comment [CE12]: Here is where you want to make sure they understand the context of a seed packet. Maybe ask a question like, has anyone every planted seeds? Have you seen how seeds are sold in packets? Here is an example…Or tell it like a story…. when I went to buy the seeds, they came in these packets. Each packet has 11 seeds. I want to know how many I will have in all…


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Work and Explore (20 min) a. Students spread out to solve problem b. Walk around to students & ask questions such as

i. Why did you choose that tool? ii. How are you (counting the items/using a derived fact/using a 100s chart)? iii. How did you keep track when counting? iv. How are you representing this on the paper? v. How will people be able to look at the paper and know how many seeds there

are? vi. Are there other ways to group the items?

c. Adjust the problem for students who need additional support or challenge (see accommodations)

During the work period, my role would be to facilitate students’ thinking. I’ll be checking in with individual students while they work to find out how they’re solving the problem. and why they chose that method. I’ll also try to pose questions that gets them thinking beyond their strategy (e.g., encourage someone who’s counting by 1s to group items, someone who’s skip counting to represent that using a derived fact, etc.) Debrief (20 min) Close task & introduce discussion

a. Call students in pairs to return their tools and sit at the table Call students to return to the table, ask them to keep their tools so they can model their strategy for the rest of the group

b. Reiterate norms of discussion i. Student shares their strategy & representation ii. Student sharing calls on students for feedback and questions -- Model “good”

feedback and possible questions students could ask Discussion

a. Call on each student/group of students to share their work with the group b. Let students provide feedback and ask questions (limit to 2 students due to time) c. Once other students have asked questions, I’ll Ask follow up questions if needed &

provide feedback such as i. When counting, what number did you start with? ii. How did you keep track of the groups? iii. How is this strategy similar to ______’s?

d. If students used the same strategy, ask how we could use (a 100s chart, a derived fact) Close

a. Exit slip: number sentence for how to add one more packet of seeds OR describe someone else’s strategy Use pictures or words to show how you would figure out how many seeds Ms. Jurewicz and Ms. Varney would have if they bought 5 packets of 12 seeds.

i. I will also have exit slips without the numbers filled in so that I can adjust for students who had a difficult time with the initial problem and during the discussion

ii. I will encourage students to think about the strategies their classmates used or if they can find a way to make their own strategy more efficient

b. Students turn in exit slip & worksheet c. Each student brings one container of tools/supply back to the classroom

Comment [CE13]: What will you get out of asking this question?

Comment [CE14]: So these are probably not the exact questions you will ask, but it gets at what you will be looking for when observing.

Comment [CE15]: Probably not necessary—since this is a new problem structure, multiplication, there isn’t a need to push them toward efficiency just yet.

Comment [CE16]: Such as, “how did you keep track of the groups?”

Comment [CE17]: How will you decide?


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I want the discussion to be student led. Often in the classroom, I re-voice students’ thinking to make it accessible to the class. However, I want students to be able to demonstrate and explain their own thinking. I also want to give students the opportunity to respond to their classmates’ thinking. The student sharing the strategy will have the choice of which students to call on. I will have asked clarifying questions during the work period, so students will have already thought out their work.

Assessment of goals/objectives

● Checking in with students as they complete the task ● Listening to how students explain their strategies ● Observations based on follow-up discussion (students re-voicing other students’

strategies, questions posed by students) ● Worksheet with student’s solution and strategy ● Exit slip: Number sentence showing how they would solve the problem if another packet

of seeds was added OR showing a strategy that someone else used Use pictures or words to show how you would figure out how many seeds Ms. Jurewicz and Ms. Varney would have if they bought 5 packets of 12 seeds.

Informal assessment will be conducted throughout the lesson. While students are working, I’ll walk around to individual students to ask clarification questions about their chosen strategies their choice of tool, strategies, and representations. I’ll use the assessment checklist (see below) to record whether students are direct modeling, counting by 1s, skip counting, or using an algorithm to solve the problem. Furthermore, I’ll ask questions about how students are choosing to represent the groups on paper: whether their drawings reflect grouping, include labels, etc. After the explore period, I’ll be listening to how students articulate their thinking to the group. I’ll also be looking for engagement with other people’s strategies through re-voicing, questioning, or providing feedback. Finally, I’ll give one of two exit slips depending on how the lesson goes. If the students seem to understand the problem well, I would ask them to provide the open number sentence that represents the problem if I purchased another package of seeds. If it seems like students struggled, I would ask them to represent someone else’s strategy. Both exit slip would reinforce one of the initial goals. I’ll finish the lesson with an exit slip that asks students to model or explain (without solving) their strategy for a slightly different problem (5 packets of 12 seeds). Anticipating students’ responses and your possible responses Student Strategies Direct modeling This is probably going to be the most frequently used strategy. Response: Help students make a connection between this strategy and repeated addition or skip counting by asking students to compare their strategy with those of other group members’. Ask how students could group items to make counting easier. Ask students to label the groups. Ask what the number sentence is that corresponds to the work they did. Repeated addition Students might understand that it’s a repeated addition problem when they write the number sentence. The students don’t know an algorithm for adding two digit numbers, so they would either invent their own or rely on counting strategies. Response: Suggest students use a 100s chart to see if they can invent their own algorithm or if they count by ones.

Comment [CE18]: I think you want your questions to focus on clarification rather than choice

Comment [CE19]: Yes, and having a checklist or place to jot down notes on what each student is doing could be helpful here, especially if they are not all producing a written solution

Comment [CE20]: If you get a range of strategies, you could ask students to make comparisons. For example, one student uses repeated addition and another direct modeling—how did each represent the groups of 11?


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Skip counting Skip counting will probably be the second most likely strategy. Also, students who counted by ones may group the items when they represent them on the paper. Response: Ask students who represent the items in groups to model how they counted the items. If students counted by ones, ask if they think it would be quicker to skip count. Ask students if they can think of a number sentence to represent their skip counting (e.g., counting by 5s, 5+5+1=11) to encourage they think about derived facts. Multiplication fact/derived fact It’s unlikely that students will know a multiplication fact to solve the problem. However, they may solve the problem using a derived fact. Many of the students chosen for the group use derived facts in class to solve addition problems. 11 was chosen to see if students would use 10+1. Response: Ask students to describe how they know 11 is the same as 10+1. Ask how they could best represent this. Possible points of confusion

● Students have never worked with more than two addends at a time. Response: Help students think about the problem, encourage their use of tools to represent the problem

● Students may see the 4 and 11 and automatically add those two numbers together. Response: ask students to explain what the 4 and 11 represent and what we should be adding. Help the students model one seed packet of 11 seeds.

● Students aren’t used to working with such big numbers, may lose track when counting. Response: ask students if they could group items to make them easier to keep track of. Encourage students to check their work with someone else if they’re having difficulty.

Management issues Scenario 1: Students finish work quickly and are distracting other students Response: Ask them to solve with another tool or give them a more difficult problem to

solve (see accommodations) S2: Students begin to draw on their paper, play with cubes, etc. R: Ask the student to explain their thinking. If the problem is too difficult, help break it into manageable sub-steps (see accommodations). If the problem is too easy, give the student an additional challenge (see accommodations). If the student continues to play with the manipulatives… R: Tell her that the tools are there to help her solve the problem and if the student continues to misuse them, she’ll have to find a different tool Scenario 3: The problem is too difficult for multiple students and they’re doing nothing/talking to each other/coloring R: Sit with the group of struggling kids and ask them to solve an easier problem (e.g., two seeds per packet, see accommodations)

Accommodations

For students who find the material too challenging I would talk to individual students to see if anyone was finding the material too

challenging. If a student was stuck, I would first break the problem into sub-steps by asking something like “Could you show me using one of the tools how many seeds are in one packet?” If the student needed further assistance, I would try to find another student in the group who would be able to explain their thinking. If the problem is still too difficult for a student, I would ask

Comment [CE21]: nice

Comment [CE22]: great ideas


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him how he’d approach it if there were only 2 seeds in each packet. Using his strategy, I would try larger numbers, and finally return to the original problem of 11 seeds. For students who need greater challenge and/or finish early

If students finish the problem quickly, the first question I would ask is if they can find a different tool or way to represent their strategy. For example, if a student used direct modeling, I would ask if she could represent her thinking using a 100s chart, or if a student drew out the objects, I would ask if he could group them in another way. If the problem is not challenging enough for a student, I would ask him what would happen if each of the seed packets had another number of seeds (between 12-20 depending on how much challenge the student needed).


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Name: __________________________________________________________________ Ms. Jurewicz and Ms. Varney want to plant flowers in their garden. They bought 4 packets of seeds. Each packet has 11 seeds in it. How many seeds are Ms. Jurewicz and Ms. Varney planting? Show your work.

Number sentence: ____________________________________________________________


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Name: __________________________________________________________________ Exit slip 1: Show What You Know! Write an open number sentence to say how many seeds Ms. Jurewicz and Ms. Varney would have if they bought another packet of seeds. Use pictures or words to show how you would figure out how many seeds Ms. Jurewicz and Ms. Varney would have if they bought 5 packets of 12 seeds.

Name: __________________________________________________________________ Exit slip 2: Use pictures or words to explain a way that someone else solved the problem that is different from the way you solved it.

Comment [CE23]: Do students know what this means? You could also just say write a number sentence to show how many seeds you would have if you bought 5 packets of seeds.

Comment [CE24]: show


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Assessment checklist: solving an equal grouping multiplication problem

Understanding Strategy Notation Other

Student name

Interprets problem correctly

Represents problem correctly

Direct model / counting

Repeated Addition

Skip counting

Multiplication fact or derived

fact

Writes number

sentence correctly

Comments

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Revised Lesson Plan with Comments