lesson plan format grade

Lesson Plan Format Date: October 14, 2013 Subject: Mathematics Grade: 3rd Grade

Lesson Topic: Single Digit Division Class/Group Size: 25

New Jersey Common Core State Standards for Content: CCSS.Math.Content.3.OA.A.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ? CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8.

Learning Objective(s): What are the important concepts that this lesson teaches? Students will explore the concept of single- digit division through the use of manipulatives, array modeling, and other mathematical tools and representations. This introductory lesson to single- digit division will focus on division by grouping (Measurement Division) and sharing (Partitive Division). Single-digit mMultiplication represented viawith array modelss will be used to build students’the understanding of howthat the total amount (the area of the array) and one factor (one side of the array) are trying tocan be used to find the other factor. What assessment will you use to assess whether students have learned this objective? Informal assessment will be collected throughout the lesson by observation of students at work. Students will be asked to solve problems by acting out the situation by either grouping or sharing strategies. More advanced sStudents will be asked to recall their multiplication facts to quickly assess ‘how many’ groups can you make of the ‘total amount.’ Students will frequently be asked to think about the largest amount they can put into each group at one time for assessment of the sharing strategy.

I. Core and Supplemental Materials

I Need: Manipulatives

x Unifix cubes or counters x Paper plates x Counting beans

Practice Worksheets with arrays Chalk board and chalk Overhead Transparencies

x Hundreds Chart

x Multiplication Table

Students Need: Manipulatives


Worksheet with arrays Writing utensil Hundreds Chart Multiplication Table

II. Context for Learning

a. Organization of Students: - How do you want students to work? - Students will work individually during the introduction where they will work on the multiplication problem. Then

they will work in partners for the first division problem. Since they are all learning a new concept, they all will have some sort of difficulty when doing the first division problem. Therefore, students will be paired strategically with another partner based on ability. This will allow both partners to work together to figure out the problem.

b. Prerequisite Knowledge:

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Comment [LAL1]: Well-written learning objectives.

Comment [LAL2]: What about formal assessments? Will any of the student work be collected and graded for the class?

Comment [LAL3]: Nice job. I would highly encourage you to re-read your assessment strategies here as it will definitely help you in developing questions to pose to your students. This will address my notes below regarding questioning.

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Comment [LAL4]: What about in terms of students’ basic number facts with multiplication?

- What should students know before they start this lesson? - Students need to be proficient with single digit multiplication. They should also understand how multiplication

works. For example, 10x3= 30. Show students that 10+10+ 10= 30 and pointing out that 10 was added 3 times to get 30.

- Are students familiar with solving problems and sharing their strategies? If not, how will you support them from the very beginning of the lesson?

- Students are familiar with solving multiplication problems. We will go over a single digit multiplication problem in the beginning of class using the array method in order to activate prior knowledge. However, if another strategy is presented, they will be encouraged to share it with the class.

c. Key vocabulary and terms: - What mathematical vocabulary will you introduce and develop meaning for throughout the lesson?

Dividends Divisor Quotient

d. Describe potential difficulties students may experience with the content: - How do you predict students will solve the problem? What difficulties might students have with the problem? - Students will solve the problems using manipulatives. They will be encouraged to use them because it will allow

them to understand the process of division and what it means to divide. Also, when learning the 2 strategies of sharing and grouping, it should be easy for them to see the difference between the two strategies. However, students might have difficulty when it comes to using off decade numbers.

- Will they have difficulty with the numbers? - Students might have difficulty when it comes to using off decade numbers. Therefore, we will begin teaching

division using decade numbers since these are the most familiar to them. - How will you help students record their thinking? - During the first part of class, students will be required to write down the problems and the answers once they

get the answer. This is so that they can become familiar with the symbols. They will be recording their strategies during the second half of class on their worksheet. Since they are using manipulatives to solve the problems, they will be required to write down the problems and the answer that they get.

- How can you accommodate students who lack prerequisite skills while still meeting the learning objectives? o Number Hundreds chart o multiplication table o manipulatives

III. Instruction: Problem Posing

Description of the lesson: Introduction

- How will you activate prior knowledge? - Students will activate prior knowledge by using arrays, which were used to conceptualize multiplication in the

previous lesson. - Will you solve one problem as a whole class to model the problem solving process? - I will pose one multiplication problem to solve with an array to make sure the students are using it correctly, or

else they will be unable to proceed to division with arrays. Ex: make 1 array for 12, such as 3x4 (3 columns of 4). - How will you introduce the problem to students? Can you pose it in a way that they give you different

strategies? - Now that we remember how to correctly use an array, we can move on to some different types of problems that

can also be solved with an array. Students will work in pairs to create different arrays with the number provided to them, which is 40. Then, students will make up their own problems using the arrays they created. Tell students an example would be that they have 40 cookies to work with, and they have to give them out to a certain amount of friends. The problems that they make will follow this model, because they will have to examine the different number of cookies depending on how many people there are, based on the columns and rows of the array. Students will be using division without even realizing they’re doing so! This problem will help the teacher gauge whether students are using the grouping or sharing strategy.

- Do you want to read instructions from the board or have it on a worksheet or both?

Comment [LAL5]: What sorts of problem-solving strategies for multiplication will your students have experienced before the single-division lesson?

Comment [LAL6]: Students should be able to build on their understanding of multiplication number sentences like these in terms of group language. More specifically, 10 groups of 3 items results in a total of 30 items.

Comment [LAL7]: Keep in mind that repeated addition is not the same as multiplication, as discussed in class. This is actually a mid-level strategy for solving multiplication problems that moves beyond students directly modeling. The main focus of students’ interpretation of multiplication should be in terms of grouping language.

Comment [LAL8]: I am not sure how you are making the connection between students’ prior experience with single-digit multiplication and the array model. Did the students use the array model when solving single-digit multiplication problems? It seems as though the array model will more ...

Comment [LAL9]: Instead of using arrays, I would recommend building on students’ prior knowledge of multiplication in terms of groups by ...

Comment [LAL10]: What other strategies would you be expecting your students to provide ...

Comment [LAL11]: It is great that you are encouraging all of your students to use the manipulatives. Unifix cubes will work just fine. ...

Comment [LAL12]: Nice work. Grouping and sharing are basically the directly modeling strategies that will be presented in our class when students ...

Comment [LAL13]: What sorts of questions will you pose to students to help them identify when one problem-solving strategy is more appropriate ...

Comment [LAL14]: You will not have this issue since students will be working on single-digit division problems. However, what sorts of single- ...

Comment [LAL15]: What do you mean by thus? Are you having students construct number sentences? If so, this will be another way to show ...

Comment [LAL16]: What are some recording strategies that you expect your students to employ? Think about our class conversations regarding ...

Comment [LAL17]: These are definitely great resources to provide students with support in building their understanding. However, you need to ...

Comment [LAL18]: See my notes above regarding the recommendation of holding off on ...

Comment [LAL19]: Excellent work in starting from students’ prior understanding of multiplication. I would still use this multiplication ...

Comment [LAL20]: Besides the use of an array, this is an excellent way to pose the Partitive Division problem for your students. As teachers, you would ...

Comment [LAL21]: Wow! I really like this planned activity. Nicely done. I would recommend holding off on this until after students get some ...

Comment [LAL22]: How so? Can you be more specific about that here?

- Students will have instructions on the board, and either a small white board or paper to work on it with their partners.

- Do you want to give students the choice of numbers or choose for them? What numbers will you choose? Why?

- We will give the students the numbers to work with. We are using single digit division without remainders, so we want to make sure the numbers divide evenly with no remainders, so that the array will work. Students will make all the arrays for the number 40. This way, there can be a 5x8 array, a 4x10 array, a 2x20 array, and a 1x40 array, so this is a great number to work with to show all the numbers that 40 can be divided by.

- Do you want to pose multiple problems to students? How will you assign the problems to students? - Students will work in pairs to create division story problems from the arrays that they just made. This will be the

same problem posed to all students as they work with their pairs, to examine the different division story problems that can be configured by using the same numbers.

- What will you do when students finish quickly? Do you want them to use alternative numbers? Do you want them to solve a different problem? Do you want them to solve it another way? Do you want them to do another activity?

- Students will make the arrays and story problems with their partners in a pair- share situation, so that students can individually attempt the problem, and then work with a partner to share what they have made. Then, those pairs will share with another pair if they finish quickly. If there is only one pair finished much earlier, students will be instructed to come up with more story problems with the same number 40. Supporting Students

- What will you do if some students run into problems while solving the problem? - We will proceed with the strategy that we have noticed the majority of the students tended to use, with the

least difficulty. This is only an introductory activity, so it’s ok for students to be unsure, because we are just beginning to explore what it means to divide. To avoid any major conflicts, students will be strategically paired so that there will be minimized distractions and more focus on the problem.

- How will you avoid telling them an operation or keyword? - The nature of this problem is not to use the proper terms of “division” yet, so that they can see that they can

divide by using concepts that they are already comfortable with. With these problems that the students are forming, they will not have to name the operation used, and can use the multiplication from the array to devise a division problem.

- How will you redirect student attention back to the problem? - If there is too much confusion, or if everyone seems to be finishing, this is a good time to redirect attention to

explaining 1 specific strategy to use for these types of problems (division problems). This way, we can use what was just discovered in their activity to explain how to continue with division.

- What strategies can you use to help students move past difficulties (without giving them a strategy)? What questions can you ask?

- To move past difficulties during the activity, students can look at other students’ examples for support, or look at another number array if that helps to put the concept into perspective. Questioning and Student Sharing

- What should students do while students are sharing or recording their strategy? - Students are asked to present one of their story problems to the class. Then, they are instructed to tell the class

how they made that problem, and which parts of the array they looked at to make that problem. - What questions will you ask to engage students that are not sharing? - Students will be asked to think about the question, and if the numbers worked based on the arrays that they

made (since they should have made the same arrays). - What other techniques can you use to capture student attention? - Students can come to the board to draw out and model their story problem. - What questions will you ask to help students who are sharing make their thinking explicit for the rest of the

class? - How did you make this problem? - Did the array help? - How will you respond to a student that shares a mistake or a misconception? - Ask the other students if anyone can tell the student why he/she made a mistake first. If no one can, address the

Comment [LAL23]: Excellent teacher decision!

Comment [LAL24]: See my note above about using a different number for the activity.

Comment [LAL25]: One recommendation would be to have students construct number sentences for the corresponding arrays that they create. I would also be specific about having the first factor in the number sentence represent the array’s length (vertical number of units) and width (horizontal number of units) to be consistent.

Comment [LAL26]: Very nice activity idea. The students can use the three story problems (including the 2 division problems) to use as reference when developing their story problems.

Comment [LAL27]: What other number could be used that has various single-digit divisors? Since students will already hear multiple other “40” story problems, it might be more mathematically challenging to have those students work with a different number instead.

Comment [LAL28]: I am not entirely sure what you mean by this. Even though sharing and grouping are two different division problem-solving strategies, Measurement Division problems will lend themselves more to the “grouping” technique while Partitive Division problems are more appropriately solved using the “fair sharing” tactic. This all direct modeling though. How might you intervene if you feel as though a student is struggling with an advanced strategy (i.e. skip-counting on fingers, deriving a multiplication fact)? In what ways could you modify the array activity at the end to support students struggling with finding the second factor for a particular number?

Comment [LAL29]: Excellent!

Comment [LAL30]: Between presenting the three introductory examples and the array activity, I would advise having students engage in some independent practice with the two division problem types (Measurement and Division). This can simply the original Salamander Problem (Partitive Division) and a variation with the number of friends being unknown in the problem (Measurement Division).

Comment [LAL31]: This should be a whole-class debriefing on the Salamander Problem’s two variations. What are the different strategies that ...

Comment [LAL32]: See my notes above about helping students move onto more advanced problem-solving approaches (i.e. counting or derived facts). What questions might be useful in ...

Comment [LAL33]: What about during the opening and independent practice activities of the lesson?

Comment [LAL34]: What are the questions? Please see various notes above that ask about how you will follow up on students’ thinking with questions and strategy probing.

Comment [LAL35]: Again, these questions should also be considered for the opening and independent practice activities during the lesson.

Comment [LAL36]: These questions need to be a bit more specific and tied back to the division story problems in the sense of talking about groups. For the array model activity, these questions need ...

problem, and move on to another problem if this one seems flawed, such as the subsequent worksheets. - How will you choose students to share? - Students will be chosen to share by having raised their hands first, and then we may call on certain students who

haven’t to get them involved. - How will you guide students towards the critical mathematics to learn? - Only about 2 pairs will be sharing, because hopefully by this point there have been pair shares that have

completed another pair share. This activity is only an introduction, so should only take about 20 mins. - What will you do if the critical mathematics is not embedded in a student strategy that is shared? - We want to utilize the most used strategy at first, so it is fine if a certain strategy is not demonstrated, because

we will get to that later, or in the next lesson. Guided Practice/Independent Practice

- How will you have students independently practice the critical mathematics from this lesson? - Now we have demonstrated the grouping vs. the sharing methods of division to the students. Grouping

demonstrates more conceptual and factual division knowledge, and the sharing method is more investigative as it is individual division. Students will work on another activity using manipulatives and other mathematical tools including the hundreds chart and multiplication table, and will demonstrate knowledge of one of the methods.

- Will you provide an additional problem or two? A different activity? - We will present another activity to further practice these methods. Problem: Dr. Salamander shares his 12 eggs

among 3 friends. We want the students to demonstrate this division problem using paper plate to represent the friends, and the beans to represent the eggs. We can see students’ method use by watching how they divide out the beans, and whether or not they have grouped or shared. By using the word share, we expect most students to give out 1 bean to each plate until they have run out of beans, resulting in 4 beans per plate. However, students who know their multiplication facts more fluently may want to group and just immediately put 4 beans per plate, which is also acceptable. Students will complete this activity individually, and when finished will work individually on more of the same types of division story problems, and can use small counters, such as the ones of the base 10 blocks.

- Time Table:

Clock reading

during the

lesson

“Title of the

activity” Students doing Me doing

10-10:20 (20 min long)

Arrays for Division

1. Make arrays of 40

2. Use arrays to make division story problems

Walking around the room to see which strategy most used (Grouping vs. Sharing)

10:20-11 (40 min long)

Grouping Vs. Sharing

1. use manipulatives to answer salamander problem

2. work individually using manipluatives to answer worksheet

Explain difference between grouping and sharing.

Comment [LAL37]: What do you mean by “flawed”? It is important to use these opportunities to capture where a student went in a mathematically incorrect direction based on the division strategies. Moving away from an error or faulty story problem will help the student in realizing and addressing his/her mathematical struggle.

Comment [LAL38]: There are usually 2-3 students who will always have their hands raised first. These are typically the more advanced students in the class. Think about how you might vary your student sharing selections based on students' level of problem-solving development and connections between the stratgies observed while walking around the classroom.

Comment [LAL39]: I am confused. What are some of the critical mathematical concepts that serve as the content background for your activities? Thinking about division (problem types and strategies), multiplication facts, etc.

Comment [LAL40]: This does not seem to answer the question. The question is asking what will you do if a student volunteers a strategy and he/she does not apply an appropriate division problem-solving strategy. How will you address that and support the student?

Comment [LAL41]: Not sure what you mean by this here. Please see my notes above about what sharing and grouping mean in relation to Partitive and Measurement Division problems respectively.

Comment [LAL42]: Manipulatives only refers to the Unifix cubes listed in your Resources Need section.

Comment [LAL43]: Excellent problem. It might be more appropriate to use the two division story problems involving Madison and the lollipops (see my notes above) as the independent practice problems instead. That way, students are using the beans and paper plans for the opening Salamander activity and the mathematical tools (e.g. Unifix cubes, multiplication table) for the Madison/lollipop activity. You may need to change the numbers since 12 is being used in both story problems.

Comment [LAL44]: This is the only possible direct modeling strategy that will be used since it is Partitive Division. Having the number of friends be the unknown in the problem will lead to students’ use of the grouping strategy.

Comment [LAL45]: How might you get students to reach this more advanced, efficient approach?

Comment [LAL46]: Please provide some examples here of these division problems.

Comment [LAL47]: You will not need these with single-digit division problems.

Comment [LAL48]: Please see my notes above to further partition your time table into smaller time intervals. Please remember that you are having students complete 4 main activities – opening (Salamander), guided/independent practice (Madison/lollipops), the array model activity, and closure with constructing equal groups from 24. Please adjust this table accordingly using the notes throughout.

problems.

IV: Closure

How will you end the lesson? To end the lesson we will ask the students to find as many different ways to divide 24 into equal groups. They will work individually for the first 5 minutes or so then they will have the chance to talk to their partners to see if they found any different ways. We will them ask them to share their ways and will write their responses on the board. Counters will be available to them if needed. The purpose of this exercise is to check the students’ understanding of the process of division and to make sure they can work out the problems on their own.

V. Notes for Modifications/Accommodations

How will you differentiate instruction for: 1. English language learners?

Even if the students understand the process of division, they will need to understand the terms in order to move forward. Before the lesson we will review the terms “array, difference, factor, product” and once the students perform the sharing/ grouping of cookies, we will introduce “division, remainder, dividend, divisor, quotient”. We will then ask them to divide 12 counters equally then explain what they did either in writing or verbally depending on how advanced the students are in English. We will then write out the problem (ex: 12/4= 3) and ask them to label each part.

2. Students with LD or low achievement? If they are having difficulty understanding the meaning of division, we will enforce division as sharing. We will give a number of problems one at a time and ask them to figure it out with unifix cubes and plastic bags. For example we will ask them to share 14 blocks between themselves and a partner. Then ask them to share them between themselves and 3 or four classmates (depending on the group size). Then we will review what we did by asking them “How many cubes did we start with? How many were in you group? How many counters did each of you end up with?”

3. Student with autism or moderate disability? We will enforce the use of unifix cubes in order to help the students visualize the mathematics taking place. We will use our sharing strategy in order to divide the counters equally. We will give them a number such as 10 and ask them to spilt the cubes equally and gradually moving up to more difficult numbers (ex: 15, 21One of the teachers will give the students a problem and ask them to model what is happening. The students will be allowed to work in partners or groups depending on their preference.

Instructor Comments (via E-mail):

Your group did a very fabulous job in incorporating the array model activity to provide a visual approach in meaningfully building students' mathematical understanding of division. If the students did not previously see the array model in the preceding single-digit multiplication lesson, I would highly recommend giving them a brief introduction of the array model for single-multiplication and then delving into your described activity. The lesson's learning objectives are well-written and aligned with the CCSS. Your

Comment [LAL49]: Excellent closure activity! This is a very nice preview into multidgit multiplication/division to be introduced in an upcoming class session.

Comment [LAL50]: Good number choice.

Comment [LAL51]: Will you also provide students with other tools and the array model sheets? It will be nice to use the array model to visually capture the number sentences 2 x 12 = 2 x (10 + 2) = 24 and 12 x 2 = (10 + 2) x 2 = 24.

Comment [LAL52]: Think about some of the strategies discussed in class about question frames, student/small-group support, structured class discussion, etc. How are you incorporating these ELL modification strategies into your single-digit division lesson plan?

Comment [LAL53]: It is not so much that they will need to know the terms to “move forward,” but instead that they should be able to communicate the mathematical language.

Comment [LAL54]: How will you do this?

Comment [LAL55]: What is this?

Comment [LAL56]: Front-loading vocabulary – presenting key terms and definitions at the beginning of the class – should be avoided. How are you incorporating these terms meaningfully into the lesson? Think about your array modeling activity. How will you embed these terms into the discussions of the activity and student strategies?

Comment [LAL57]: When are you doing this? This does not need to be done separately from the rest of the lesson activities.

Comment [LAL58]: For this lesson, students should not be introduced to this notation. They should be working solely based on the multiplication number sentences with varying unknowns.

Comment [LAL59]: Think about how you might modify the independent practice and array model activities for these students. Some possible considerations here include opportunities for students’ think alouds and applications of their prior knowledge of single-digit multiplication.

Comment [LAL60]: How might you support these students in grasping the grouping strategy?

Comment [LAL61]: Don’t forget your use of the array model as an additional visual representation for this student population.

Comment [LAL62]: Very nice scaffolding approach!

Comment [LAL63]: What sorts of questions might you pose to gauge these students’ understanding and verbalize their thinking aloud? Also, Unifix cubes tap into the students' tactic modality of learning. What other tools and techniques might you implement to tap into other learning modalities for these students?

Comment [LAL64]: Excellent use of student groups here considering these students’ communication and social difficulties in the classroom.

guided/independent practice activity does a nice job in activating students' prior knowledge of single-digit multiplication to approach the single-digit division story problems and array model activity. The closure activity involving the number 24 is a great way to get students' feet wet with mutidigit multiplication/division -- one of the proceeding lessons. One recommendation would be to have the Mr. Salamander story problems (involving the beans and plates) as the introductory problems for the lesson with the first problem posed as a grouping multiplication (3 x 4 = ___) task. I would then proceed to the guided/independent practice problems (Madison and the lollipops) followed by the array modeling activity. In the lesson plan, there is mention of "sharing and grouping" strategies. These are directly aligned with the direct modeling strategies for Partitive Division and Measurement Division problems that we saw in yesterday's class. Your lesson plan, therefore, should also include mention of how you as the classroom teacher want to support students in advancing to more efficient, higher-order strategies such as counting and derived facts as well. In addition, the array model should be used as a visual representation of the single-digit division number sentences reserved ONLY for the lesson's third class activity. (Think of this more of an extension activity.) Your students' problem-solving strategy development with single-digit division, thus, should not be solely premised on arrays but instead focuses on having them transition from directly modeling (i.e. counters, Unifix cubes, tally marks) to skip-counting and applying multiplication facts. Yes, I am more than happy to send an additional round of comments later this week before your group presents the lesson plan in class. If your groups feels confident with presenting the lesson next Monday (October 28th) after reading both rounds of feedback, please let me know and I will keep you on the schedule for presenting that day. (For the presentation, I would strongly recommended focusing on either the Salamander or Madison problems and then giving us a taste for the array model activity.) In light of the technical mishap with Sakai, I can always push back your presentation to the following week -- November 4th. The only downside to that would be that you will not be able to present in the GSE Lecture Hall (as opposed to our regular classroom location) if that is your group's presentation preference. Please let me know what you decide and feel more comfortable doing. Overall, I think your group has a pretty strong single-digit division lesson plan and can be further strengthened by addressing its activity sequence, considerations for student support, and inclusion of specific teacher questions and probes. I look forward to reviewing your group's second draft and observing your lesson plan presentation. Please feel free to e-mail me back with any questions or concerns. - Luis

Instructor Feedback (via E-mail):

The introductory Do Now activity is very effective in building on students' prior knowledge of single-digit multiplication presented in terms of constructing equal-sized groups. One major recommendation would be to dedicate more class time to the guided/independent practice activity modified with larger number choice (see notes in document). The array activity could be conducted during the final portion of the class period as the closure activity that shows how division can assist in constructing the same arrays from the single-digit multiplication lesson. I would recommend focusing more on the students using the number sentences to help them build the array models rather than building the array models to solve the division problems. (I provide specifics on that in the document.) As you will notice, I included detailed information on how to incorporate the base-10 blocks as supportive mathematical tools during the class activities especially the independent/guided practice problems. It is important that the students walk away with a being able to flexibly represent the given dividends (e.g. 56, 36) using the base-10 blocks in preparation for the Partial Quotients algorithm used in multidigit division. What I mean by this is having students realize that they can trade one 10-rod for ten individual cubes in order to be able to evenly distribute a given total number of items in a division story problem. For example: Dividing 56 cookies among 4 friends. The 56 can be represented using five 10-rods and 6 individual cubes where four of the five rods can be evenly distributed. This leaves a student with 1 rod and 6 individual cubes to distribute. A teacher should guide a student to realize that she/he cannot evenly distribute as is so she/he must trade the one 10-rod for 10 individual ones and evenly distribute the remaining 16 cubes among the 4 groups in the problem. This should be modeled when solving the MD and PD problems introduced during the Do Now activity. An additional area of improvement for your lesson plan's final draft is being more specific about your use of teacher questioning. it will be helpful for you to actually cite questions and teacher dialogue in the "Question and Student Sharing" section of the plan. There are some great support strategies listed throughout your plan, but it would help to be more specific about what the teacher should be saying as guiding questions to build more advanced strategies, address student misconceptions, etc. I provided a few examples as well as some thoughts to consider in the Track Changes. In terms of the presentation for Monday's class, I would recommend focusing on the Do Now (including the MD and PD problems) and maybe 1 of the independent practice problems. I will bring the Unifix cubes and base-10 blocks with me to class on Monday. If you need anything else for preparation, please let me know. You are able to use the overhead project and chalkboard as well. Please note that since you presenting your lesson plan on Monday, I re-scheduled your final lesson plan draft submission to Monday, November 11th so you have an extra week to use the feedback from the class presentation. If you have any questions over the weekend, please feel free to e-mail me back. Hope this feedback helps. I look forward to observing your presentation on Monday morning. Have a wonderful weekend! - Luis

Lesson Plan Format Date: October 27, 13 Subject: Mathematics Grade: 3rd Grade Lesson Topic: Single Digit Division Class/Group Size: 25 Common Core State Standards for Content: CCSS.Math.Content.3.OA.A.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ? CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Learning Objective(s): Students will explore the concept of single-digit division through the use of manipulatives, array modeling, and other mathematical tools and representations. This introductory lesson to single-digit division will focus on division by grouping (Measurement Division) and sharing (Partitive Division). Students’ prior knowledge of sSingle-digit multiplication as creating equal groups will represented via array models will be used to develop build students’their

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Comment [LAL1]: Good job!

understanding of howsingle-digit division problems with the number of items in each equal-sized group (Measurement) and number of groups (Partitive) as the unknown. the total amount (the area of the array) and one factor (one side of the array) can be used to find the other factor. What assessment will you use to assess whether students have learned this objective? Formative assessment will be collected throughout the lesson by observation of students at work. While the students are working in pairs the teacher will walk around to individual pairs, asking students to explain their work and support their thinking by modeling, writing and verbalizing their answers. More advanced students will be asked to recall their multiplication facts to quickly assess ‘how many’ groups can you make of the ‘total amount.’ Students will frequently be asked to think about the largest amount they can put into each group at one time for assessment of the sharing strategy.


I Need: 1. Manipulatives


2. Practice Worksheets with arrays 3. Chalk board and chalk 4. Overhead Transparencies

x Hundreds chart x Multiplication Table

Students Need: 1. Manipulatives

x Unifix cubes or counters x Base-Ten Blocks x Paper plates x Counting beans

2. Worksheet with arrays 3.2. Writing utensil 4.3. Hundreds chart 5.4. Multiplication table


a. Organization of Students: - Students will work individually during the introduction where they will work on the multiplication problem. - Then they will work in partners for the first division problem. Since they are learning a new concept, they will all

have some sort of difficulty learning this new concept. Students will be paired strategically with another partner based on ability. This will allow both partners to work together to collaborate and come to a solution.

- The teacher will walk around during the problem solving stage and ask students to model their thinking and show how each pair came to find their answer.

b. Prerequisite Knowledge: - Students will be proficient working with whole unknown (multiplication), size unknown (partitive division), and

number of groups unknown (measurement division) types of division problems. - Students know how to both make sense and create an array.

o 10 rows with 3 columns will yield a total array with 30 parts. - Students are proficient with multiplication of base 10. - Students are familiar with solving multiplication problems. We will go over a single digit multiplication problem

in the beginning of class using size unknown (partitive division) in order to scaffold them into the division lesson using prerequisite knowledge.

- Students should have prior place-value knowledge for flexibly modeling two-digit dividends with base-10 blocks (i.e. trading one 10-rod for 10 individual cubes).

c. Key vocabulary and terms: - Dividends - Divisor - Quotient d. Describe potential difficulties students may experience with the content: - Students will solve the problems using manipulatives. They will be encouraged to use them because it will allow

them to understand the process of division and what it means to divide. Also, when learning the two strategies of sharing and grouping, it should be easy for them to see the difference between the two strategies.

- Some students may be more advanced, and may not need the use of manipulatives. Some students may take advantage of the hundreds chart. The difficulty may come in how to count, should I (the student) count in ones,

Comment [LAL2]: Students will already be familiar with array/area modeling from learning single-digit multiplication. Thus, it is important to highlight that this would be no different from the modeling completed with single-digit multiplication and how this maps to the divident, divisor, and quotient for a single-digit division number sentence. In other words, finding the missing side length is the same as dividing the area (dividend) by the given side length (divisor). See notes below.

Comment [LAL3]: Formative assessment is the same thing as an informal assessment of student understanding such as in-class observations, teacher questioning, etc. Are there going to be any formal modes of assessment (i.e. homework assignment, graded classroom activity)?

Comment [LAL4]: The teacher should not be doing these things for the student. Students should be encouraged to write and verbalize their strategies – not the teacher. It appears as though you are trying to convey that the teacher is scaffolding the students’ understanding of single-digit division. (You will need to provide more detail on this in the Student Supports section.)

Comment [LAL5]: A detailed list, indeed!

Formatted: Font: Bold

Comment [LAL6]: Please see my notes below on using base-ten blocks below.

Comment [LAL7]: Nicely written.

Comment [LAL8]: Be careful with your use of the word “model.” Students should be demonstrating their thinking either in written words, action/directly modeling, or thinking aloud.

Comment [LAL9]: I am a little confused. This is what the students are learning during your lesson for single-digit division. The only one that the students should have experience with solving is whole unknown (multiplication). Remember this is the Prerequisite Knowledge section so everything listed here should be content that the students learned BEFORE the taught lesson.

Comment [LAL10]: Yes, this will be covered in the single-digit multiplication lesson.

Comment [LAL11]: The use of 10 was just an abitrary number selected for my example in my last round of feedback. What I meant in my TrackChange comment was that student should be able to discuss multiplication and dividision number sentence using “group” language. For example, if students are solving the division problem for a partitive division (sharing) situation 24 ÷ 4 = 6, the students will be able to translate it as “If there are 24 items that need to be evenly divided into 4 groups, then there will be 6 items in each group.”

Comment [LAL12]: Excellent job!

tens or use skip counting. - The most advanced students will remember their multiplication facts, and will not need to make use of either

manipulatives or the hundreds chart. They will use derived facts to come to the answer. - For all students, we will challenge them to think of at least two ways to find the solution to the problem. This

will help both advanced learners and students who still need to use manipulatives come up with alternative ways to solve one problem.

- Student’s single digit multiplication facts are strong. To support students who cannot remember the few missing facts, we can use other derived facts to help them reach the unknown answer. For example, if a student struggles to solve 7x6, we can ask them to recall 6x6, and knowing that it must be one more group of 6, they will be able to solve the problem.

- The teacher will support students in recording their thinking and applies strategies with the single-digit division problems. For example, sStudents will record the beginning do now as written on the board. 3 X _ = 12.

- To answer the problem, they find the number in each group. They will then be instructed to construct a sentence explaining the entire multiplication problems, correctly using groups and number in each group.

o IE. Johnny has 3 baskets of apples. If he has 12 total apples, he can put 4 apples into each basket. - They will be recording their strategies during the second half of the class on their worksheet. Since they are

using manipulatives to solve problems, they will be required to write down the problems, the answer, and a sentence to demonstrate understanding of total/# of groups/# in each group

- In order to accommodate students who lack prerequisite skills we will supply students with support if they need to check their multiplication facts, or if they need to use the hundreds chart if they are applying a skip counting method.

o Hundreds chart o Multiplication table o Manipulatives



- How will you activate prior knowledge? - The ‘Do Now’ will activate student’s prior knowledge by using single-digit multiplication story problems to

conceptualize multiplication from the previous lessons and changing their unknowns to generate Measurement and Partitive Division problems.

- We will first display (and read aloud as a whole class) a measurement multiplication story problem for 3 x 4 = 12 o Kate has 3 boxes of lollipops. If each box contains 4 lollipops, how many lollipops does Kate have in

total? o We will be looking for students to write out the problem equation (3x4=___) and then to use their

manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to model their thinking.

o Students will work independently on this problem. - The ‘Do Now’ continues with another story problem to practice (measurement division) multiplication where

the number of groups in the equation is unknown. o We will first display (and read aloud as a whole class) the story problem. Bill has 21 cookies. If he can

place 7 cookies in each box, how many boxes will he need? o We will be looking for students to write out the problem equation (___x7=21) and then to use their

manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to model their thinking.

o Students will be allowed to work with a partner to solve this problem. - The ‘Do Now’ wraps up with a story problem to practice size unknown (partitive division) multiplication where

the number in each group is the unknown part of the equation. o We will first display (and read aloud) the story problem. You have 15 cupcakes and 3 boxes. You want to

put the same number of cupcakes in each box. How many cupcakes can you put in each box? o We will be looking for students to write out the problem equation (3x___=15) and then to use their

manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to

Comment [LAL13]: Very nice way of thinking ahead here on students’ approaches to the problems. It might help to given example of the student struggles and use of derived facts here to be a bit more explicit.

Comment [LAL14]: It may be helpful to specify which division problem the students are solving where this multiplication fact knowledge will come in handly. More specifically, the students could be solving a sotry problem represented as 42 ÷ 7 = 6 ij a number sentence.

Comment [LAL15]: See notes above about your use of the word “model.”

Comment [LAL16]: Along with the Unifix cubes/counters, I would also provide the students with base-10 blocks so they can begin practicing working flexibly with 10s and 1s. For example, the students will directly model “21 cookies” with 2 rods (20) and 1 cube. In order to divide the 21 cookies to have an equal number in each of the 7 boxes, the students will have to trade the 2 rods for 20 individual cubes. This is good preparation for when students begin learning multidigit division and use the Partial Quotient algorithm (to be discussed in our next class session).

model their thinking. o Students will be allowed to work with a partner to solve this problem.

- Once students have solved the last problem, we will discuss all three problems and the types of strategies that students are using.

- Array Activity - I will pose one multiplication problem to represent and solve with an array/area model to make sure the

students are using it correctly, or else they will be unable to proceed to visually represent division with arrays. Ex: make 1 array for 6, (ie, 1x6, 2x3, 3x2, 6x1)

- Now that we remember how to correctly use an array, we can move on to some different types of problems that can also be solved with an array. Students will work in pairs to create different arrays with the number provided to them, which is 18. Then, students will make up their own problems using the arrays they created. Tell students an example would be that they have 18 cookies to work with, and they have to give them out to a certain amount of friends. The problems that they make will follow this model, because they will have to examine the different number of cookies depending on how many people there are, based on the columns and rows of the array. Students will be using division without even realizing they’re doing so! This problem will help the teacher gauge whether students are using the grouping or sharing strategy.

o Grouping – Students will make the array of 18 using (1x18, 2x9, 3x6, 6x3, 9x2, and 18x1). If students are grouping they will look at the columns.

o Sharing - Students will make the array of 18 using (1x18, 2x9, 3x6, 6x3, 9x2, and 18x1). If students are sharing they will look at the rows.

- If they have 18 cookies, and they are to give the cookies out to friends we are looking for the students to know that the number of friends (row) x number of cookies (column) = total array.

- Students will have instructions on the board, and either a small white board or paper to work on it with their partners.

- We will give the students the numbers to work with. We are using single digit division without remainders, so we want to make sure the numbers divide evenly with no remainders, so that the array will work. Students will make all the arrays for the number 18. This way there can be multiple arrays using (1x18, 2x9, 3x6, 6x3, 9x2, and 18x1), so this is a great number to work with to show how many cookies (rows) each of the students’ friends (columns) will get if the student has 18 cookies to start with.

- To check for assessment, we will have students construct number sentences to model how many cookies each friend will get if they have 18 total cookies. We will make it clear to students that they will NOT be including themselves in the count of friends. They will do this for more than one array model.

- Students will work in pairs to create division story problems from the arrays that they just made. This will be the same problem posed to all students as they work with their pairs, to examine the different division story problems that can be configured by using the same numbers.

o (***This could possibly be where we are gauging whether or not students are using the grouping or sharing strategy from a few lines above***)

o If we see students struggling with the problems, we can point them in the direction of the ‘Do Now’ story problems where all three problem types were modeled.

- Students will make the arrays and story problems with their partners in a shade situation, so that students can

individually attempt the problem, and then work with a partner to share what they have made. Then, those pairs will share with another pair if they finish quickly. If there is only one pair finished much earlier, students will be instructed to come up with more story problems with additional numbers.

o 24 – 1x24, 2x12, 3x8, 4x6, 6x4, 12x2, 1x24 o 16 – 1x16, 2x8, 4x4, 8x2, 16x1

Supporting Students

- If a student is struggling with moving onto an advanced strategy (ie, skip-counting on fingers, deriving a

Comment [LAL17]: Fabulous job in generating these two different single-digit division problem types following the Do Now multiplication task. Since you are introducing this concept for the first time, it might be conducive to have the teacher debrief the students’ strategies after each problem is presented and completed. The concluding debrief described here should focus on making connections across the the three problems on how they differed based on what was unknown in the context of the story problems (e.g. the total number of lollipops, the number of boxes).

Comment [LAL18]: Before starting the Array Activity, I would pose two division problems (one Measurement and one Partitive) as the guided/independent practice activity for the class. The two problems that you have listed under Guided/Independent Practice would work just fine. My only recommendation would be to use some larger numbers such as 56 (instead of 24 for the balloon problem) and 36 (instead of 16 for the brownie problem). With this larger number choice (for Partitive balloon problem in particular), you will encourage students to flexibly trade 10-rods for ones so they can even distribute in their direct modeling approach. [More specifically, the student will see that he/she can give one 10-red per group and then need to trade the remaining 10-rod into 10 ...

Comment [LAL19]: This will be recalling students’ prior engagement with area models when tackling single-digit multiplication problems.

Comment [LAL20]: It will be helpful to point out how the arrays created represent the same area when the number is directly modeled using the base-10 blocks. For example, you can show how

Comment [LAL21]: Before mapping the different number of friends and the different number of cookies received by each friend to the array model, I would recommend writing out ALL of the different poissibilities on the board as students volunteer them and then listing the corresponding ...

Comment [LAL22]: You can model and encourage students to label each side length as “Cookies for Each Friend” and “Friends.”

Comment [LAL23]: I would give out the same grid paper that we used during the in-class presentation for single-digit multiplication during our llast class session. I have attached an electronic copy for you.

Comment [LAL24]: Remember that students are going to be volunteering their predictions for the differnet number of friends so it is possible that someone will raise a number that does not divide 18 evenly. What are some examples of teacher queestioning that will help this student see that this ...

Comment [LAL25]: I would just stick with presenting the initial Partitive Division example of dividing cookies among friend. The students can keep using this same story problem to generate their array models. When they are all done with building the arrays and if there is enough time left, ...

Comment [LAL26]: Excellent reference point for student support.

Comment [LAL27]: You probably will run out of class time after students have completed the array modeling exercise and debriefed. I think it is safe to just delete this part. Good advance planning!

multiplication fact) we can first determine what types of problems are giving them difficulty. - With measurement division, a more advanced student will understand what he or she needs to skip count by,

and they will count until they reach the total. - However, with partition division, the student will understand how many times they need to skip count but they

do not know what they are counting by, so they will start to make guesses as to how many would be in each group.

o Student guessing demonstrates that a student may not be ready to work without manipulatives for partitive division problems. Support students with another partitive division problem.

- We will proceed with the strategy that we have noticed the majority of the students tended to use, with the least difficulty. This is only an introductory activity, so it’s ok for students to be unsure, because we are just beginning to explore what it means to divide. To avoid any major conflicts, students will be strategically paired so that there will be minimized distractions and more focus on the problem.

- The nature of this problem is not to use the proper terms of “division” yet, so that they can see that they can divide by using concepts that they are already comfortable with. With these problems that the students are forming, they will not have to name the operation used, and can use the form equal groups from single-digit multiplication from the array to devise the differenta single-digit division problems.

- If there is too much confusion, or if everyone seems to be finishing, this is a good time to redirect attention to explaining 1 specific strategy to use for these types of problems (division problems). This way, we can use what was just discovered in their activity to explain how to continue with division.

- To move past difficulties during the activity, students can look at other students’ examples for support, or look at another number array if that helps to put the concept into perspective.

Questioning and Student Sharing

- During the opening activity, all students should have come to the same answer. Their strategies may have been different. If their strategies were different, students are to make note of the other strategies in their notebooks.

o How did you solve the problem? o Did anybody approach the problem in a different way? -o What was my number of groups in this problem? Did we already know how many items each group

received? - Students are asked to present one of their story problems array models to the class. Then, they are instructed to

tell the class how they made that problemconstructed their volunteered array modeling, and which parts of the array they looked at to make that a different division story problem.

- Students will be asked to think about the question, and if the numbers worked based on the arrays that they made (since they should have made the same arrays).

- Students sharing their solutions to problems will be asked how they made their array, and to share their number

sentences with the class. - We will check for understanding by assessing whether or not students use of rows vs columns match the correct

amount of friends vs cookies.

- If a student gives an incorrect answer we will address it immediately. We will ask if another student can tell why the student made a mistake. If no one can address the mistake, we will address the mistake.

- - Students will be chosen to share by having raised their hands first. At each part of the lesson, based on hands

raised, we will chose students who have not participated to involve the most amounts of students. If we notice that the same 2-3 students are the only students to participate, we will begin to call on students based on teacher observations.

- During the lesson, as we have walked around to students and pairs, we are observing student work. We will make note of different and correct modeling techniques and chose students to share.

Comment [LAL28]: How might students’ prior knowledge of single-digit multiplication facts help here? In what ways are the base-10 blocks useful for problems involving 2-digit dividends like those in the independent/guided practice activities? How will the teacher be using the language of creating equal groups to connect the division problems back to single-digit multiplication? See notes from above.

Comment [LAL29]: Not necessarily. Student guessing initially will help the student in making decisions about whether the selected is too small or too large. The manipulatives simply help the student to visualize if the selected number is too big or too large for the partitive division problems.

Comment [LAL30]: It is important to monitor the strategies that students feel most comfortable using. At the same time, it is crucial to encourage them to move onto a more advanced strategy only if they are indeed ready for it. You can make these determination using your discussion above about assessing students’ approach to MD and PD tasks.

Comment [LAL31]: I am not sure what this means.

Comment [LAL32]: This section does a really nice job in outlining class moments when students are sharing with one another. However, there should also be a list of teacher questioning during these instances of student sharing and support. I have include some examples for you, but please be sure to include additional ones as noted.

Comment [LAL33]: Please include examples of teacher questioning for the independent practice section. Thinking about how students will be flexibly modeling with the base-10 blocks during these exercises. “Are we able to evenly distribute 5 rods to the 4 groups? What can we do in order for us to be able to evenly distribute?” etc.

Formatted

Comment [LAL34]: What are some questions that can be raised when asking students to consider how many friends will allow the cookies to be evely distributed?

Comment [LAL35]: Great questioning here.

Comment [LAL36]: Nice way to check back.

Comment [LAL37]: Can you given a specific example of that here? What is a possible mistake that a student can make? Provide dialogue about how this would be addressed.

Comment [LAL38]: What observations? Keep in mind that you will want to showcase as many different strategies as possible based on student selections in the debriefs.

Guided Practice/Independent Practice

- Now we have demonstrated the grouping vs. sharing methods of division to the students. o Grouping demonstrates measurement division. o Sharing demonstrates partitive division

- Students will work on another activity using manipulatives and other mathematical tools including the hundreds chart and multiplication table, and will demonstrate knowledge of one of the methods.

- We will present another activity to further practice these methods. Students will work independently.

- Problem1: - Mike shares his 24 balloons among 4 friends. How many balloons will each friend get? We want the students to

demonstrate this division problem using paper plates to represent the four friends, and the beans to represent the balloons. We can see students’ method use by watching how they divide out the beans, and whether or not they have grouped or shared. By using the word share, we expect most students to give out 1 bean to each plate until they have run out of beans, resulting in 6 beans per plate. However, students who know their multiplication facts more fluently may want to group and just immediately put 6 beans per plate, which is also acceptable.

- Problem2: - There are 16 brownies. If you are to give 2 brownies to each friend. How friends will get brownies? - We want the students to demonstrate this division problem by realizing that they can count out groups of 2

“brownies” until they reach 16 “brownies.” When they have finished counting, the total number of groups will represent the number of friends.

- How will you have students independently practice the critical mathematics from this lesson? - Will you provide an additional problem or two? A different activity? - Time Table:

Clock reading during the

lesson

“Title of the


10-10:10 (10 min long)

Do Now

1. Solving 3x4=___ and writing number sentence

2. ___x7=21 and writing number sentence

3. 3x___=15 and writing number sentence

Walking around the room to see which strategy most used

10:10-10:50 (40 min long)

Array Activity

Creating array for 18 cookies and number sentence (# Friends vs # cookies)

Monitoring progress, explaining Friends=groups & cookies=rows

Comment [LAL39]: See my notes above about presenting these two problems with larger number choice before the concluding Array activity.

Comment [LAL40]: After students complete each of these activities, it is important for the teacher to debrief the students’ different solution strategies and draw connections across them. The teacher should have already modeled trading 10-rods for 10 individual cubes in the introductory Do Now section. That way, the students will attempt to use the base-10 blocks when tackling these two independent exercises in class.

Comment [LAL41]: Please adjust the contents of the Time Table according the notes/suggestions listed throughout the lesson plan.

Comment [LAL42]: Since you are presenting and modeling the MD and PD problem types, this will probably take you 20 minutes.

Comment [LAL43]: I would spend an additional 20-25 minutes on the Independent/Guided Practice activity. The last 15-20 minutes of the class should be reserved for the Array Modeling activity with 18. Please adjust these timings accordingly.

10:50-11 (10 min long)

Additional Problem Solving

1. 4x___=24

2. ___x2=16


IV: Closure

How will you end the lesson? - To end the lesson we will ask the students to find as many different ways to divide 24 into equal groups. They

will work individually for the first 5 minutes or so then they will have the chance to talk to their partners to see if they found any different ways. We will them ask them to share their ways and will write their responses on the board. Counters will be available if needed.

o Students will be encouraged to use the array model to visually capture the number sentences. o 1x24, 2x12, 3x8, 4x6, 6x4, 8x3, 12x2, 1x24

- The purpose of this activity is to check the students’ understanding of the process of division and to make sure they can work out the problems on their own.


How will you differentiate instruction for: 1. English language learners? - These students will be paired with students who are bilingual in order to assist the students during the parts of

the lesson where they are instructed to write sentences. - If there are multiple English Language Learners in the classroom, translations of the ‘Do Now’ will be done ahead

of time, and presented along with the English versions. 2. Students with LD or low achievement? - If they are having difficulty understanding the meaning of division, we will enforce division as sharing among

groups of equal size. We will give a number of problems one at a time and ask them to figure it out with unifix cubes and plastic bags. For example we will ask them to share 14 blocks between themselves and a partner. Then ask them to share them between themselves and 3 or four classmates (depending on the group size). Then we will review what we did by asking them “How many cubes did we start with? How many were in you group? How many counters did each of you end up with?”

- Using an array model, we will have the students remember the multiplication parts of the array. o # of groups (rows) times # in each group (column) = the total amount of the array

- For partitive division o They will be able to easily see the total amount of the array. o We will tell them that in order to find the missing piece; we can look for the number of columns.

- For measurement division o They will be able to easily see the total amount of the array. o We will tell them that in order to find the missing piece; we can look for the number of rows.

3. Student with autism or moderate disability? - We will enforce the use of unifix cubes in order to help the students visualize the mathematics taking place. We

will use our sharing strategy in order to divide the counters equally. We will give them a number such as 10 and ask them to spilt the cubes equally and gradually moving up to more difficult numbers (ex: 15, 21). One of the teachers will give the students a problem and ask them to model what is happening. The students will be allowed to work in partners or groups depending on their preference.

Comment [LAL44]: Please see my notes from above about sticking with the array activity involving 18 as the closure activity. There will probably not be enough time to consider another number like 24 with the Do Now, independent practice, and debriefing the arrays/story problems for 18. You can simply transfer that information down here.

Comment [LAL45]: What about the teacher’s use of sentence frames with the multiplication/division number sentences? Example: “If there are ___ total items to be evenly distributed among ___ groups, how many items will each group contain?” (for PD problems).

Comment [LAL46]: When are they writing sentences? Besides writing, how else can this partnering support ELL students? Think about our discussions regarding “turn to a partner” prompts.

Comment [LAL47]: Remember that you should not be “telling” students. This is something that the students should be producing on their own. The arrays and cubes/base-10 blocks do indeed provide a helpful visual representation for the students with LDs and low academic achievement. How might the teacher modify the number choice in the problems? In what ways could the Array Activity be modified to meet these students’ level of ability?

Comment [LAL48]: Excellent approach.

Comment [LAL49]: See notes above about the use of the word “model.”

Comment [LAL50]: How is group/partner work an effective modification in terms of classroom communication for students with moderate autism and disability?

Lesson Plan Format Date: November 4, 2013 Subject: Mathematics Grade: 3rd Grade Lesson Topic: Single Digit Division Class/Group Size: 25 Common Core State Standards for Content: CCSS.Math.Content.3.OA.A.2 Interpret whole-number quotients of whole numbers, e.g., interpret 56 ÷ 8 as the number of objects in each share when 56 objects are partitioned equally into 8 shares, or as a number of shares when 56 objects are partitioned into equal shares of 8 objects each. For example, describe a context in which a number of shares or a number of groups can be expressed as 56 ÷ 8. CCSS.Math.Content.3.OA.A.4 Determine the unknown whole number in a multiplication or division equation relating three whole numbers. For example, determine the unknown number that makes the equation true in each of the equations 8 × ? = 48, 5 = _ ÷ 3, 6 × 6 = ? CCSS.Math.Content.3.OA.B.6 Understand division as an unknown-factor problem. For example, find 32 ÷ 8 by finding the number that makes 32 when multiplied by 8. Learning Objective(s): Students will explore the concept of single-digit division through the use of manipulatives, array modeling, and other mathematical tools and representations. This introductory lesson to single-digit division will focus on division by grouping (Measurement Division) and sharing (Partitive Division). Students’ prior knowledge of single-digit multiplication as creating equal groups will be used to develop their understanding of single-digit division problems with the number of items in each equal-sized group (Measurement) and number of groups (Partitive) as the unknown. What assessment will you use to assess whether students have learned this objective? Formative assessments will be collected throughout the lesson by observation of students at work. While the students are working in pairs the teacher will walk around to individual pairs, asking students to explain the strategies they are using to solve the problems. More advanced students will be asked to recall their multiplication facts to quickly assess ‘how many’ groups can you make of the ‘total amount.’ Students will frequently be asked to think about the largest amount they can put into each group at one time for assessment of the sharing strategy. The closing activity will be a graded classroom activity that will serve as a formal way (summative assessment) to assess the students overall understanding of the lesson.


I Need: 1. Manipulatives


2. Practice Worksheets with arrays 3. Chalk board and chalk 4. Overhead Transparencies

x Hundreds chart x Multiplication Table

Students Need: 1. Manipulatives

x Unifix cubes or counters x Base-Ten Blocks x Paper plates x Counting Beans

2. Writing Utensil 3. Hundreds Chart 4. Multiplication Table


a. Organization of Students: - Students will work individually during the introduction where they will work on the multiplication problem. - Then they will work in partners for the first division problem. Since they are learning a new concept, they may all

have some sort of difficulty learning this new concept. Students will be paired strategically with another partner based on ability. This will allow both partners to work together to collaborate and come to a solution.

Comment [LAL1]: Excellent work in selecting the appropriate Mathematical Content CCSS here!

Comment [LAL2]: Nicely-stated learning objectives here.

Comment [LAL3]: Please see my notes below about the Closure activity. It might be appropriate to conclude with the array modeling activity for timing purposes. Also, the array model is a nice way to offer a familiar visualization from single-digit division that they can use to make meaningful sense of the division process.

Comment [LAL4]: What about student organization for the array modeling and guided practice activities in the lesson?

- The teacher will walk around during the problem solving stage and ask students to demonstrate their thinking and show how each pair came to find the answer by thinking aloud and writing down their strategies.

b. Prerequisite Knowledge: - Students will be proficient working with whole unknown (multiplication) types of problems. - Students know how to both make sense and create an array.

o 10 rows with 3 columns will yield a total array with 30 parts. - Students are able to solve multiplication problems using ‘grouping’ language. - Students are familiar with solving multiplication problems. We will go over a single digit multiplication problem

in the beginning of class using size unknown (partitive division) in order to scaffold them into the division lesson using prerequisite knowledge.

- Students should have prior place-value knowledge for flexibly modeling two-digit dividends with base-10 blocks from subtraction (i.e. trading one 10-rod for 10 individual cubes).

c. Key vocabulary and terms: - Dividends - Divisor - Quotient

d. Describe potential difficulties students may experience with the content: - Students will solve the problems using manipulatives. They will be encouraged to use them because it will allow

them to understand the process of division and what it means to divide. Also, when learning the two strategies of sharing and grouping, it should be easy for them to see the difference between the two strategies.

- Some students may be more advanced, and may not need to use manipulatives. Some students may take advantage of the hundreds chart. The difficulty may come in terms of counting, when students must decide whether to count in ones, tens or use skip counting.

- The most advanced students will remember their multiplication facts, and will not need to make use of either manipulatives or the hundreds chart. They will use derived facts to come to the answer.

o For example, a student may use derived facts to answer 54 ÷ 9. The student may recall that 9 x 5 = 45. Using this fact, the student will add 9 more, giving them the answer of 6.

- For all students, we will challenge them to think of at least two ways to find the solution to the problem. This will help both advanced learners and students who still need to use manipulatives come up with alternative ways to solve one problem. This way, students who are finished quickly will have another task to do which won’t leave them bored, while struggling students will have another chance to discover the correct answer.

- Student’s single digit multiplication facts are strong. To support students who cannot remember the few missing facts, we can use other derived facts to help them reach the unknown answer. For example, if a student struggles to solve 42 ÷ 6 = 7, we can ask them to recall their multiplication facts of 6. Most will remember that 6x6=36 and knowing that it must be one more group of 6, they should be able to solve the problem. This could be seen during a partitive division problem.

- The teacher will support students in recording their thinking and applying strategies with the single-digit division problems. For example, students will record the beginning ‘do now’ as written on the board. 3 X _ = 12

- To answer the problem, they find the number in each group. They will then be instructed to construct a sentence explaining the entire multiplication problems, correctly using groups and the number in each group.

o IE. Johnny has 3 baskets of apples. If he has 12 total apples, he can put 4 apples into each basket. - They will be recording their strategies during the second half of the class on their worksheet. Since they are

using manipulatives to solve problems, they will be required to write down the problems, the answer, and a sentence to demonstrate understanding of total/# of groups/# in each group

- In order to accommodate students who lack prerequisite skills we will supply students with support if they need to check their multiplication facts, or if they need to use the hundreds chart if they are applying a skip counting method.

o Hundreds chart o Multiplication table o Manipulatives

Comment [LAL5]: Excellent work in identifying these perquisite knowledge and skill sets.

Comment [LAL6]: Very detailed list here of students’ potential difficulties as well as how students at different levels of understanding would approach the given tasks.



- The ‘Do Now’ will activate students’ prior knowledge by using single-digit multiplication story problems from the previous lessons and changing their unknowns to generate Measurement and Partitive division problems. After each problem is solved, I will debrief the students on the strategy that they used to solve each problem.

- We will first display (and read aloud as a whole class) a multiplication story problem for 3 x 4 = 12 o Kate has 3 boxes of lollipops. If each box contains 4 lollipops, how many lollipops does Kate have in

total? o We will be looking for students to write out the problem equation (3x4=___) and then to use their

manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to explain and demonstrate their thinking.

o Students will work independently on this problem. o Debrief the students by showing them that in this problem the total was missing. Have students come

up and demonstrate how they solved this problem and what strategy they used (manipulatives, drawing, recall, etc.)

- The ‘Do Now’ continues with another story problem to practice (measurement division) multiplication where the number of groups in the equation is unknown.

o We will first display (and read aloud as a whole class) the story problem. Bill has 21 cookies. If he can place 7 cookies in each box, how many boxes will he need?

o We will be looking for students to write out the problem equation (___x7=21) and then to use their manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to model their thinking.

o Students will be allowed to work with a partner to solve this problem. o Debrief the students by showing them that in this problem; the number of groups was missing. Have

students come up and demonstrate how they solved this problem and what strategy they used (manipulatives, drawing, recall, etc.)

- The ‘Do Now’ wraps up with a story problem to practice size unknown (partitive division) multiplication where the number in each group is the unknown part of the equation.

o We will first display (and read aloud) the story problem. You have 15 cupcakes and 3 boxes. You want to put the same number of cupcakes in each box. How many cupcakes can you put in each box?

o We will be looking for students to write out the problem equation (3x___=15) and then to use their manipulatives of choice to solve the equation. Teachers will be monitoring students, asking students to model their thinking.

o Students will be allowed to work with a partner to solve this problem. o Debrief the students by showing them that in this problem; the number of how many were in each

group was missing. Have students come up and demonstrate how they solved this problem and what strategy they used (manipulatives, drawing, recall, etc.)

- Array Activity - I will pose one multiplication problem to represent and solve with an array/area to make sure the students are

using it correctly, or else they will be unable to proceed to visually representing division with arrays. Ex: make 1 array for 6, (ie, 1x6, 2x3, 3x2, 6x1)

- Now that we remember how to correctly use an array, we can move on to some different types of problems that can also be solved with an array. Students will work in pairs to create different arrays with the number provided to them, which is 12. Then, students will make up their own problems using the arrays they created. Tell students an example would be that they have 12 cookies to work with, and they have to give them out to a certain number of friends. The problems that they make will follow this model, because they will have to examine the different number of cookies depending on how many people there are, based on the columns and rows of the array. Students will be using division without even realizing they’re doing so! This problem will help the teacher gauge whether students are using the grouping or sharing strategy.

o Grouping -- Students will make the array of 12 using (1x12, 2x6, 3x4, 4x3, 6x2, and 12x1). If students are

Comment [LAL7]: Your problem-posing here does a nice job in connecting the lesson to the students’ previous mathematical encounter with single-digit multiplication. However, how will you connect the single-digit division lesson to the students’ everyday lives? It might be useful to preface the Do Now problems by asking students how they share groups of items with their friends and family members. That way, these Do Now problems uphold a sense of day-to-day familiarity to the students.

Comment [LAL8]: The last two Do Now problems’ strategies need to be modeled before moving onto the Guided/Independent Practice activity.

Comment [LAL9]: Nice consistency across the series of Do Now problems listed here.

Comment [LAL10]: As discussed during the class presentation, it is very important to highlight the distinctions across these problems in terms of the varying unknown value.

Comment [LAL11]: Excellent way to draw connection between the array models and a story context.

grouping they will look at the columns. o Sharing -- students will make the array of 12 using (1x12, 2x6, 3x4, 4x3, 6x2, and 12x1). If students are

sharing they will look at the rows. - If they have 12 cookies, and they are to give the cookies out to friends, we are looking for the students to know

that the number of friends (row) x number of cookies (column) = total array. - Students will have instructions on the board, and grid paper to work out the array on with their partners. - We will give the students the numbers to work with. We are using single digit division without remainders, so

we want to make sure the numbers divide evenly with no remainders, so that the array will work. Students will make all the arrays for the number 12. This way there can be multiple arrays using (1x12, 2x6, 3x4, 4x3, 6x2, and 12x1), so this is a great number to work with to show how many cookies (rows) each of the students’ friends (columns) will get if the student has 12 cookies to start with.

- To check for assessment, we will have students construct number sentences to model how many cookies each friend will get if they have 12 total cookies. We will make it clear to students that they will NOT be including themselves in the count of friends. They will do this for more than one array model.

- Students will work in pairs to create division story problems from the arrays that they just made. This will be the same problem posed to all students as they work with their pairs, to examine the different division story problems that can be configured by using the same numbers.

o If we see students struggling with the problems, we can point them in the direction of the ‘Do Now’ story problems where all three problem-types were modeled.

- Students will make the arrays with their partners in a pair/share situation, so that students can individually attempt the problem, and then work with a partner to share what they have made. Then, those pairs will share with another pair if they finish quickly. If there is only one pair finished much earlier, students will be instructed to come up with more story problems with additional numbers.

o 24 – 1x24, 2x12, 3x8, 4x6, 6x4, 12x2, 1x24 o 16 – 1x16, 2x8, 4x4, 8x2, 16x1

Supporting Students

- If a student is struggling with moving on to an advanced strategy (ie, skip-counting on fingers, deriving a multiplication fact, etc.) we can first determine what types of problems are giving them difficulty.

- With measurement division, a more advanced student will understand what he or she needs to skip count by, and they will count until they reach the total.

- However, with partitive division, the student will understand how many times they need to skip count, but they do not know what they are counting by, so they will start to make guesses as to how many would be in each group.

o Student guessing demonstrates that a student may not be ready to work without manipulatives for partitive division problems. Support students with another partitive division problem.

- We will proceed with the strategy that we have noticed the majority of the students tended to use, with the least difficulty. This is only an introductory activity, so it’s ok for students to be unsure, because we are just beginning to explore what it means to divide. To avoid any major conflicts, students will be strategically paired so that there will be minimized distractions and more focus on the problem.

- The nature of this problem is not to use the proper terms of “division” yet, so that they can see that they can divide by using concepts that they are already comfortable with. With these problems that the students are forming, they will not have to name the operation used, and can form equal groups from single digit multiplication to devise the different single digit division problems.

- If there is too much confusion, or if everyone seems to be finishing quickly, this is a good time to redirect attention to explaining 1 specific strategy to use for these types of problems (division problems). This way, we can use what was just discovered in their activity to explain how to continue with division.

- To move past difficulties during the activity, students can look at other students’ examples for support, or look at another number array if that helps to put the concept into perspective. Questioning and Student Sharing

- During the opening activity, all students should have come to the same answer, even though their strategies may have been different. If their strategies were different, students are to make note of the other strategies in

Comment [LAL12]: Good thinking on number choice here.

Comment [LAL13]: These number sentences should be consistent with multiplication number sentences having varying unknowns.

Comment [LAL14]: Nice form of differentiated instruction planned here.

Comment [LAL15]: What forms of teacher questioning will support students in identifying division from the story’s context?

Comment [LAL16]: Nice way to build on students’ prior knowledge to support them.

their notebooks. o What did you do to solve this problem? o Did anybody approach the problem in a different way? o What was my number of groups in this problem? Did we already know how many items each group

received? o Can you show me how you know the number of groups is missing from this problem? o How did you represent the number of groups in this problem? (drawing, using a single unifix cube, etc.)

- Students are asked to present one of their array models to the class. Then, they are instructed to tell the class how they constructed their volunteered array modeling; and which parts of the array they looked at to make a different division story problem (row vs. column).

o Following the number sentence guidance, students are expected to fill in how many groups there are, followed by how many are in each group, and that will equal the total amount they have in their array.

- Students will be asked to think about the question, and if the numbers worked based on the arrays that they made (since they should have made the same arrays).

- Students sharing their solutions to problems will be asked how they made their array, and to share their number sentence with the class.

- We will check for understanding by assessing whether or not students use of rows vs. columns match the correct number of friends vs. cookies.

- If a student gives an incorrect answer we will address it immediately. We will ask if another student received a different answer to the problem. When a student arrives at the correct answer, we will ask how the student came to the answer, and if they can find the mistake in the previous student’s response. If no one can address the mistake, we will address the mistake.

o A sample mistake may be if a student creates their array with the total being used as one of the rows or columns and the other dividend as the other row or column.

o If this happens, we will reread the problem to the student, asking he or she to tell us how many are in total of the given problem. This questioning will guide the student to realize that the total is how many pieces are in the total array, and not used as a part of the array.

- Students will be chosen to share by having raised their hands first. At each part of the lesson, based on hands raised, we will chose students who have not participated to involve the most amounts of students. If we notice that the same 2-3 students are the only students to participate, we will begin to call on students based on teacher observations. Also, if there is a different type of technique used, we will call on that student that we noticed during observations to highlight the transferrable nature of these math problems.

- During the lesson, as we have walked around to students and pairs, we are observing student work. We will make note of different and correct modeling techniques and chose students to share.

o By having students show different techniques, this will help students realize that there can be more than one correct way to solve the problems.

- Students may choose to use the base-10 blocks during the guided practice/independent practice portion of the lesson.

o We will ask students if we are able to evenly distribute 2 base-10 rods and 4 single cubes among 4 friends.

o The students will be asked what they are able to do if they want to distribute the blocks evenly. This questioning will allow the students to realize that they need to trade in the base-10 blocks for single cubes in order to be evenly distributed.

Guided Practice/Independent Practice

- Now we have demonstrated the grouping vs. sharing methods of division to the students. o Grouping demonstrates measurement division. o Sharing demonstrates partitive division

- Students will work on another activity using manipulatives and other mathematical tools including the hundreds chart and multiplication table, and will demonstrate knowledge of one of the methods.

- We will present another activity to further practice these methods. Students will work independently.

- Problem1:

Comment [LAL17]: These are exactly the questions to address my notes from above regarding making the measurement/partitive distinction as well as the multiplication/division distinct in general.

Comment [LAL18]: Another nice prediction of a student strategy here. I am still curious to hear about some incorrect strategies especially during the Do Now activity segment. There seems to be plenty of discussion here surrounding the array model exercise.

Comment [LAL19]: This guided/independent practice activity should be included between the Do Now/lesson opener and the array modeling activity. However, I do not see them included in your lesson plan’s Time Table.

Comment [LAL20]: It appears as though you are using these two problems as guided practice to be debrief during the class. If you would like to assess students’ independent approaches with these different single-digit division story problems, it is probably helpful to include at least 1-2 more problem as independent practice.

- Mike shares his 24 balloons among 4 friends. How many balloons will each friend get? We want the students to demonstrate this division problem using paper plates to represent the four friends, and the beans to represent the balloons. We can see students’ method use by watching how they divide out the beans, and whether or not they have grouped or shared. By using the word share, we expect most students to give out 1 bean to each plate until they have run out of beans, resulting in 6 beans per plate. However, students who know their multiplication facts more fluently may want to group and just immediately put 6 beans per plate, which is also acceptable.

- Problem2: - There are 16 brownies. If you are to give 2 brownies to each friend, how many friends will get brownies? - We want the students to demonstrate this division problem by realizing that they can count out groups of 2

“brownies” until they reach 16 “brownies.” When they have finished counting, the total number of groups will represent the number of friends.

- Time Table: Clock reading

during the lesson

“Title of the


10-10:20 (20 min long)

Do Now

1. Solving 3x4=___, writing/drawing how they got the answer, and writing the number sentence

2. ___x7=21, writing/drawing how they got the answer, and writing the number sentence

3. 3x___=15, writing/drawing how they got the answer, and writing number sentence

Walking around the room to see which strategy is used the most, and making sure students are remembering to categorize each part of the problem into a number sentence.

10:20-10:45 (25 min long)

Array Activity

Creating array for 12 cookies and number sentence (# Friends vs. # cookies)

Monitoring progress, explaining Friends=groups & cookies=rows

Comment [LAL21]: Nice prediction of a correct student strategy here. What are some potential incorrect student strategies? How would you address them in the classroom?

Comment [LAL22]: Some additional detail on debriefing the guided practice activity in terms of establishing connections across student strategies would be appropriate here. The section on students’ potential difficulties lists some of these correct approaches. However, how will you has the teacher demonstrate the Unifix cube/base-10 block, Hundreds Chart, and number fact approaches are valid? For example, suppose students are completing the first problem on dividing 24 balloons among 4 friends. What is teacher dialogue that demonstrates how the 6 cubes “dealt out” to each friend/group is equivalent to counting back 6 spaces from 24 four times on the Hundreds Chart? It is important to continue using the grouping language while highlighting these mathematical connections across the student strategies.

Comment [LAL23]: Please see my notes above about incorporating the Guided/Independent Practice segment into the lesson’s Time Table. You may need to allocate your time allotments here by shortening the length of the array modeling activity. The Guided/Independent Practice should take more time as they array modeling will serve as an additional visualization of single-digit division for the lesson.

10:45-10:53 (8 min long)

Additional Problem Solving

1. 4x___=24

2. ___x2=16


10:53-11:00 (7 min long)

Closure Array

Creating array for 18 to model the different ways to construct the quotient.

Waiting as students work independently and then ask students to share their responses.

IV: Closure

How will you end the lesson? - To end the lesson we will ask the students to find as many different ways to divide 18 into equal groups. They

will work individually for the first 5 minutes or so then they will have the chance to talk to their partners to see if they found any different ways. We will them ask them to share their ways and will write their responses on the board. Counters will be available if needed.

o Students will be encouraged to use the array model to visually capture the number sentences. o 1x18, 2x9, 3x6, 6x3, 9x2, 18x1

- The purpose of this activity is to check the students’ understanding of the process of division and to make sure they can work out the problems on their own.

- Since the students have already worked with the quotients that have more than one array, this will encourage students to see that there is more than one way to construct an array of numbers, such as 18.


How will you differentiate instruction for: 1. English language learners? - These students will be paired with students who are bilingual, if possible, in order to assist the students during

the parts of the lesson where they are instructed to write sentences. - For the use of sentence frames in examples such as partitive division, visual aid will be provided. While the word

problem should still be given to the student, the number sentence will also already be written out for the student as well so that he/she can relate the number sentence to the word problem. For example, for the first Do Now problem, the word sentence will be given but also the number sentence (3x4=___).

- If the student is struggling during independent work, the student will also be paired with a bilingual student so that both students can feed off of each other and achieve a higher level of thinking.

- If there are multiple English Language Learners in the classroom, translations of the ‘Do Now’ will be done ahead of time, and presented along with the English versions.

2. Students with LD or low achievement? - If they are having difficulty understanding the meaning of division, we will enforce division as sharing among

groups of equal size. We will give a number of problems one at a time and ask them to figure it out with unifix cubes and plastic bags. For example we will ask them to share 14 blocks between themselves and a partner. Then ask them to share them between themselves and three or four classmates (depending on the group size). Then we will review what we did by asking them “How many cubes did we start with? How many were in you group? How many counters did each of you end up with?”

- Using an array model, we will have the students remember the multiplication parts of the array. o # Of groups (rows) times # in each group (column) = the total amount of the array

- For partitive division o They will be able to easily see the total amount of the array. o We will demonstrate that in order to find the missing piece; we can look for the number of columns. o In order to modify an array problem for low achievement or LD students, we will modify the number

choices in the problems to base-10 numbers instead of numbers, which may be difficult for them such as

Comment [LAL24]: With the inclusion of the guided/independent practice segment into the lesson, I am not sure how much time you will have to conduct this closure activity. You may want to reserve this focus on the number 18 for students who complete the array modeling earlier than the others. If you end up having time, I would recommend using the number 24 instead as it begins to use multidigit factors (e.g. 2 x 12).

Comment [LAL25]: Well-thought modifications for diverse learners in the mathematics classroom. Great considerations specific to the mathematical demands of the lesson at hand.

18. - For measurement division

o They will be able to easily see the total amount of the array. o We will demonstrate that in order to find the missing piece; we can look for the number of rows.

3. Student with autism or moderate disability? - We will enforce the use of unifix cubes in order to help the students visualize the mathematics-taking place. We

will use our sharing strategy in order to divide the counters equally. We will give them a number such as 10 and ask them to split the cubes equally and gradually moving up to more difficult numbers (ex: 15, 21, etc.). One of the teachers will give the students a problem and ask them to explain their process of thinking and how they were able to solve the problem. The students will be allowed to work in partners or groups depending on their preference. If working with a partner is not a suitable option, the student will work closely with an aide or teacher if possible.

Instructor Feedback (on Sakai):

Hi Group 3 Lesson Plan Members!

Thank you for the final draft submission of your group lesson plan. Below is your completed grading rubric and comments regarding your final submission and in-class presentation:

Content and Standards: 2.5/2.5

Context for Learning: 2/2.5

Instruction: 3/5

Assessment: 2.5/2.5

Modifications/Accommodations: 2.5/2.5

TOTAL: 12.5/15 (A-)

Comments: Your group did a fabulous job in setting up the mathematical framework for the single-digit division lesson in terms of identifying appropriate CCSS for mathematical content, activating students' prior knowledge of single-digit multiplication and array modeling, and outlining well-phrased learning objectives for conceptual understanding. The expectations of students' potential difficulties and intended instructional modifications for diverse learning offers the mathematics teacher with informed perspective on how to effectively tailor the lesson plan's activities.

For future implementation of this single-digit division lesson plan, I would advise more explicit discussion of modeling the two last Do Now problems as well as debriefing the Guided/Independent Practice activities. It is particularly important to highlight how students' varying strategies with different mathematical tools and number fact knowledge are mathematically equivalent. In the attached feedback document, I included an example of one possible way in talking across students' volunteered strategies for the Guided/Independent Practice problem involving balloons. Before or during the presentation of the Do Now problems on single-digit division, I would ask students about their past experiences with equally sharing a group of items among people such as family members and friends. This fair-sharing prelude offers a familiar context to the Measurement and Partitive Division exercises' problem-solving strategies that connects to the students' everyday

lives. Your lesson plan does an excellent job in predicting students' correct or advanced strategies; however, I am still left wondering about specific instances of incorrect student reasoning especially for the Do Now and Guided/Independent Practice activities. What might these look like and how would you address them in terms of whole-class questioning?

Overall, a very strong lesson plan that thoughtfully considers the use of prior mathematical knowledge and meaningful visualizations in building up the concept of single-digit division. Nice work!

If you have any questions or concerns regarding your group lesson plan feedback, please feel free to e-mail me ([email protected]) or discuss in person with me. Hope this helps!

- Luis

Names: Grade: 2nd Topic: Two digit subtraction Title of Lesson: Two Digit Subtraction with Subtraction Houses NJ Core CurriculumCommon Core State Content Standards: (Can be found at: http://www.state.nj.us/education/cccs/) CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. CCSS.Math.Content.2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.1 Materials/Resources:

1. Teacher: House model, whiteboard, white board markers, worksheet

1. Student: pencil/eraser, scrap paper, practice houses (teacher will provide) Prior Knowledge: Students should be able to subtract single digit numbers. Students should know the tens and ones places in two digit numbers up to 20. Important Ideas/Real Life Connections/Rationale: Students will need to know how to subtract using two-digit numbers because this is a daily life skill. The teacher will mention to the class, for example, that “if you need to buy 12 oranges to bake a cake, but the store only sells oranges in bags of 20, how many oranges will you have left over if you buy the whole bag?” The teacher will use examples that the students understand and can relate to. If students can picture what is being subtracted, they will have a better idea of how to apply it to real life. It is also important that the teacher makes sure that every student in the class is able to relate, otherwise they will fall behind and become confused, not because of the math content, but because of bad examples. Objectives: Students will be able to complete 2 digit subtraction. Students will be able to borrow from the tens place when needed. They will also be able to accurately and

Comment [LAL1]: Please make sure to use the lesson plan template found under Sakai Resources when writing your groups’ drafts. This template will ensure that you are addressing all necessary lesson plan content. A series of guiding questions are included in the lesson plan template document to support you in addressing all of these main ideas.

Comment [LAL2]: All of the CCSS listed for mathematical content are appropriate for the second grade level and subject matter of multidigit subtraction. Excellent job!

Comment [LAL3]: Since the lesson is focusing on multidigit numerical values, it will be important for students to use base-10 blocks and the hundreds chart in their problem-solving processes. Unifix cubes should also be included so students build on their prior experience in solving single-digit subtraction tasks with them in the lesson. We will further explore how to incorporate these mathematical tools during our class on multidigit arithmetic.

Comment [LAL4]: Please see my note below about including what students already know about 2-digit subtraction here. This will be important if your group prepares a lesson plan focused on problems with regrouping. In addition, this section should make mention of how the teacher will activate students’ prerequisite mathematical knowledge. You begin to talk about this when students are solving some single-digit and “simple” single-digit subtraction problems at the start of the mathematics lesson below. It is important to purposefully select these introductory problems so you can segue into the lesson’s subject matter of 2-digit subtraction with regrouping strategies.

Comment [LAL5]: What are key mathematical vocabulary terms that will be explored in the lesson?

Comment [LAL6]: This is indeed a good example of posing a 2-digit subtraction problem with regrouping that is contextualized in a real-life situation (namely, food shopping). Please see my notes below about providing specific information on the word problems (with real-life contexts) presented to students.

Comment [LAL7]: Authentic problem-solving contexts relatable to students’ lives is also an effective way of capturing students’ attention in the 2-digit subtraction lesson. What are some different interesting contexts and connections that will be raised in the lesson?

efficiently subtract two-digit numbers using the house method. Once students master subtraction they will be able to apply it to real life situations. Assessments: Worksheet will be done in class to see how the student understands and applies the house model. Teacher will also walk around to see who needs help and scaffold when necessary. Description of Lesson/Agenda: - The lesson will start with the teacher at the board discussing some single-digit subtraction problems with students participating in the discussion. - Students will reflect on what they already know about subtraction. - Teacher will then introduce two-digit subtraction for the second time. (Assume that students have seen simple two-digit subtraction before) - Teacher will then put some problems on the board and re-teach the concept of two-digit subtraction. - Teacher will then introduce the house model (shown below) to the students. Teacher will explain what each component of the house is for and how to use it properly to organize subtraction problems. - Teacher will give some examples of two-digit subtraction problems using the house method. (teacher will have mini house models to give to the students so they can visualize) - Students will work in pairs/independently to solve three two-digit subtraction problems on a worksheet using the house model. - Answers to the problems will be addressed to the whole class. Students will volunteer to go to the board and show how they solved the problem to the class and the students sitting down can agree or disagree if the answer is incorrect. This will serve as an informal assessment. - Teacher will then show some problems that require regrouping and have students respond as a class. - Students will then work again in small groups to solve the next three two-digit subtraction problems on the worksheet with regrouping using the house model. - Answers to the problems will be addressed to the whole class. Students will volunteer to go to the board and show how they solved the problem to the class and the students sitting down can agree or disagree if the answer is incorrect. This will again serve as an informal assessment.

Comment [LAL8]: This is a great start in writing the learning objectives for your lesson on 2-digit subtraction. How will your lesson address building students’ problem-solving flexibility with the 2-digit subtraction content? In what ways will the students apply place-value understanding of multidigit numbers to inform their problem-solving approaches? How does this knowledge support students in understanding the mathematical meaning behind regrouping and the standard algorithm? We will discuss these mathematical concepts in additional detail during our next class session.

Comment [LAL9]: What tasks will be included in this worksheet? How are the items included in the worksheet aligned with the CCSS listed? Will the worksheet be completed as independent practice or in ...Comment [LAL10]: The “Lesson Description/Agenda” section of your draft provides a great overview of what your 2-digit subtraction lesson will look like. However, ...Comment [LAL11]: What are these problems exactly? Why did you choose these problems in particular? How will segue into ...Comment [LAL12]: What do you mean by this? What did the previous lesson on 2-digit subtraction cover? If students have been previously exposed to multidigit subtraction, ...Comment [LAL13]: Your group does a fantastic job in posing problems as the starting points for the mathematics lesson. (We will ...Comment [LAL14]: This is a very interesting visual for multidigit subtraction. I agree with you that the model offers a helpful mode of organization when students venture ...Comment [LAL15]: Please include what specifics problems will be used and why (e.g. number choice, problem type). Keep in mind that all problems should be presented using ...Comment [LAL16]: Great job in having students present their respective solutions to the class. What are different solution strategies that you expect to consider in this debrief? In ...Comment [LAL17]: Your group exemplifies wonderful mathematics teacher moves in purposefully separating problems that involve and do not involve regrouping. Since you ...Comment [LAL18]: Please see my note above about focusing only on problems with regrouping as a follow-up lesson on 2-digit ...Comment [LAL19]: Additional detail on how the teacher will highlight the mathematics associated regrouping in 2-digit subtraction (e.g. trading 10 ones for 1 ten). In what ways ...

Potential Difficulties: Students may not understand borrowing and regrouping. Students may not understand how to apply house model. Students may have trouble checking their work. Modifications: For students with special needs, depending on their disability, modifications will need to be made. For ELL students, the house model should not need to be modified, because it is a visual aid requiring no knowledge of the Eenglish language. For students who are incapable of working independently, it will be required for them to work with a partner, as opposed to working alone on the in-class problems. Also, the house can be modified to a simple box, if the extremities of the house itself can be a distraction to certain students. Homework: For homework, the students will be asked to take their math dice home and roll numbers to fill in subtraction houses. They will roll 4 times to come up with two 2-digit numbers. (The directions will explicitly say to make sure that the higher number is on the top level of the house.) This is so that the numbers are random and the students will have a fun time coming up with and solving the problems, rather than just completing the same worksheet in a mundane fashion.

Comment [LAL20]: It is great that your group is already actively making predictions about students’ potential difficulties with the 2-digit subtraction lesson. Some specifics about these potential difficulties and mathematical misconceptions will need to be outlined here. In what ways will students struggle with the mathematical ideas of “borrowing and regrouping”? How will these struggles appear in the students’ work? In what ways will the teacher intervene to address these difficulties? Please the list of guiding questions provided in the lesson plan template document.

Comment [LAL21]: Agreed. However, the nature of these modifications needs to be specifically indicated here. It looks like your group is encouraging partner-work and groupwork throughout the lesson. How will this be used to support students with learning disabilities? In what ways will these partner and group selections be purposefully made? What are mathematical tools and instructional accommodations that will be used in effectively communicating the mathematics to these students with learning disabilities?

Comment [LAL22]: The “house model” can indeed to be a supportive organizational visual for ELL students. How will the teacher support ELL students in making meaning of the mathematics vocabulary discussed in the lesson? What are some forms of scaffolding and problem modifications that can be made to help them in their problem solving?

Comment [LAL23]: How will your lesson plan make accommodations for students with low achievement in mathematics or overall?

Comment [LAL24]: A lesson closure that highlights or extends the class session’s mathematics on 2-digit subtraction should be included. How does your lesson re-cap the lesson content or preview the next lesson through its concluding segment?

Comment [LAL25]: This is a very interesting homework assignment approach. It is indeed open-ended in nature and encourages students to take ownership in creating the problems. How are students recording their mathematical strategies in solving the homework tasks’ multidigt subtraction problems? Will the homework assignment be reviewed during the following class? Will it be graded?

Instructor Feedback (via E-mail): Hello, Group 1 Lesson Plan Members! Hope you are all having a wonderful weekend and had a great week. I am writing this e-mail message to provide you with your first round of feedback on your multidigit subtraction lesson plan. Attached to this message you will find a MSWord document containing my feedback via TrackChange comments. Please do not be discouraged by the large number of TrackChange comments provided in the document as every group's first draft typically receives the same amount of feedback. You will notice the feedback volume will gradually lessen in your subsequent drafts after you see the lesson modeled during tomorrow's class and receive additional suggestions after your lesson plan presentation on March 3rd. (Remember that you are our class's lesson plan pioneers as the first group to submit a preliminary draft!) Your group did an excellent job in identifying the appropriate CCSS for mathematical content coverage, purposefully separating multidigit subtraction problems with and without regrouping, and building off of students' original solution strategies in the lesson development. The next lesson plan draft should use the lesson plan template document (with guiding questions) found under Sakai Resources to ensure that you are addressing all specifics that were missing in the preliminary draft. Since your group is approaching the lesson as a continuation of multidigit subtraction, I would recommend focusing solely on two-digit subtraction tasks that involve regrouping strategies even though it seems to me that this is what your group was initially attempting to do in the first draft. Please see my notes in the MSWord feedback document about being more specific about the selected problems, the teacher's modeling approaches and discussion facilitation, and support strategies to address student difficulties. Overall, this is a very strong first approach to your lesson plan draft especially considering that you are yet to review what a problem-posing lesson looks like and observe the mathematical content of multidigit arithmetic being modeled for you. My feedback, therefore, will make much more sense to you after tomorrow morning's class session. I am providing you the feedback in advance to tomorrow's class so you can feel free to review it and ask me any questions you have tomorrow before or after class. If your group would like to meet this week, the following is my availability for an in-building GSE meeting: Monday, 2/24 - After class Tuesday, 2/25 - Before 4:00pm Thursday, 2/27 - Before 2:30pm Friday, 2/28 - After 11:30am Please feel free to e-mail me back with any questions or concerns that you have in the meantime. If your group members would like to meet with me and discuss feedback in person, please let me know what date(s) and time(s) works best for you. My desk area is in the office suite located in Room 229 of the Rutgers GSE building. Hope this helps! See you tomorrow morning for class. - Luis

Lesson Plan Format Date: 3/3/14 Subject: Math Grade: 2nd Lesson Topic: Two Digit Subtraction Class/Group Size: 20 New Jersey Common Core State Standards for Content: CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. CCSS.Math.Content.2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

Learning Objective(s): Students will be able to complete 2 digit subtraction. Students will be able to borrow from the tens place when needed. They will also be able to accurately and efficiently subtract two-digit numbers using the house method. Once students master subtraction they will be able to apply it to real life situations. What assessment will you use to assess whether students have learned this objective? Students will be able to complete 2 digit subtraction. Students will be able to borrow from the tens place when needed. They will also be able to accurately and efficiently subtract two-digit numbers using the house method. Once students master subtraction they will be able to apply it to real life situations.

Core and Supplemental Materials I Need: House model, whiteboard, whiteboard markers, worksheet

Students Need: Pencil, eraser, scrap paper, practice houses (teacher will provide) base 10 blocks, hundred chart, unifix cubes (teacher will provide)

I. Context for Learning

Comment [LAL1]: The lesson objectives look great overall. Please see my notes below for minor revisions. One thing to incorporate in the learning objectives for your next draft is HOW they will be addressed in the lesson (e.g. student strategies, use of mathematical tools).

Comment [LAL2]: The learning objectives should include the mathematical language of “regrouping.” You can use the term “borrowing” in the lesson, but it is important for the students to know what they are mathematically doing when “borrowing” from the tens place (namely, exchanging 1 ten for 10 ones).

Comment [LAL3]: I would not make a distinction between the “house method” and “not the house method.” The students should just be employing different strategies in subtracting 2-digit numbers. Even though the “house” visual illustration is indeed student-friendly for organizational purposes of subtracting numbers via the standard algorithm, students actually will not learn about the standard algorithm until 4th grade. Do not worry about including the “house” for this lesson plan.

Comment [LAL4]: The word problems presented in this lesson plan should be realistic situations that the students would find relatable to their age group. Think about the contexts used in the problems that we observe during our class meetings.

Comment [LAL5]: I am confused. These are your mathematical learning objectives. What assessments (informal and formal) will be used to gauge student understanding of the subject matter?

Comment [LAL6]: Nice work. Additional insight on how both the teacher and student are using these mathematical tools in the lesson.

a. Organization of Students: Students will start out working independently at their desks working on the “do now” problem. Teacher will conduct whole class instruction on multi-digit subtraction with borrowing. Next students will work with a partner or in small groups to solve several problems while teacher offers help to those who need. Students will work individually to create their own word problem that incorporates learned content.

b. Prerequisite Knowledge: Students should be able to subtract single digit numbers. Students should know the tens and ones places in two digit numbers up to 20. Students should be able to subtract two digit numbers without borrowing (for example 18 – 12 = 6). Teacher will activate student’s prior knowledge with the “do now” that will focus on simple subtraction without borrowing and regrouping.

Do Now: 9 – 5 = ? 19 – 15 = ?

c. Key vocabulary and terms: borrowing, regrouping, subtraction, tens place, ones place, difference d. Describe potential difficulties students may experience with the content: Students may not understand how to

apply the house model. Students may have trouble checking their work. Students may have trouble borrowing and regrouping. Students may have trouble with the concept that a number in the tens place represents the number of groups of ten. These struggles may appear in students’ work when they have to show their work to demonstrate the process of crossing out a number in the tens column to add value to the ones column. We predict that when students use the borrowing method they may have trouble identifying that the tens column represents tens and the ones represents ones. If these problems do occur, the teacher will model how to accurately solve the problem in front of the class while explaining each step. The teacher will use base 10 blocks and ones block to visually represent the numbers in the problem. The teacher will provide each student with the opportunity to work with these manipulatives if necessary.


Comment [LAL7]: Please keep in mind our class conversations about how student strategies should be driving the teacher instruction.

Comment [LAL8]: This section on prerequisite knowledge looks great. What mathematical strategies do you expect to be familiar for students?

Comment [LAL9]: These problems should be presented with relatable word problem contexts.

Comment [LAL10]: You will not actually say this term during class.

Comment [LAL11]: Excellent! How will you address this student struggle with place-value understanding in class?

Comment [LAL12]: You are referring to what students will do when they are applying the standard algorithm. The students will not be presented to this technique until a later grade.

Comment [LAL13]: Great. I see that you are beginning to address these student struggles here. Some additional detail on exchanging the tens and ones should be explicitly discussed.

Description of the lesson: Introduction We will activate prior knowledge by beginning the lesson with a “do now” activity. We will write two problems on the board; 9-5, which activates single digit subtraction knowledge and 19-15, which activates double-digit subtraction without borrowing. The students will work on these problems at their desk individually and then we will go over the answers as a class. We will then introduce two-digit subtraction with borrowing. We will show the problem 20 – 17 and ask students what strategy they think should be used to solve this problem. Then we will ask the students to come up to the board and ask them to share their methods of solving the problem. We will discuss and evaluate the accuracy of each strategy and solution. At this point, after the students have had their first exposure to double-digit subtraction with borrowing we will introduce the house model and how it can be used to organize similar problems. We will again work with the problem 20-17 to show that the “roof” of the house model can be used to record the borrowed ten and the new amount of ones. This will be done orally in front of the class as they sit at their desks. We will repeat this model with the problem 44 – 39. At this point teacher will provide three examples of two-digit subtraction problems on the board for students to work out in pairs or in small groups. Students will be encouraged to use the provided manipulatives and their own house model to solve each problem. The problems will be: 36 – 19 54 – 27 83 – 76 If students finish these problems early, they will be encouraged to come up with different strategies to solve each problem.

Supporting Students Teacher will walk around the room going from group to group assessing their progress and scaffolding. If the students seem to be struggling, teacher will show again how this can be modeled by using manipulatives. Rather than teacher explicitly stating what to do, she/he should probe students by asking questions to activate knowledge. For example: teacher will ask students “if you cannot take 7 away from 4, what could you do next?” ***We are unsure about the question “How will you redirect student attention back to the problem?” Questioning and Student Sharing While other students are sharing their strategies to solve the problem, other students should be taking notes and writing down these strategies as well. This will be helpful to solving other similar problems and for students who have difficulties. While students are sharing, other students who are listening will be encouraged to ask questions if they are unsure, to learn from one another. To engage students who are not participating in the discussion teacher will directly ask if they used this method to solve or another method and if another to explicitly state/demonstrate their strategy. To capture student attention, teacher can introduce the example problems in word problem form using familiar names and items that may interest them. For example: Kelly had 47 doughnuts and she gave 19 to Brian. How many doughnuts does Kelly have now? In order to make sharing explicit to the class, students will be asked to tell AND show the exact methodology used to solve the problem. For example: teacher will ask questions like “why did you decide to cross that number out?” “How did you know to do that?” and “How many tens are left now?” etc. To respond to students who share a mistake or misconception, teacher will ask the other students if they agree or disagree. If and when another student disagrees, ask them to explain why. To share, students will be encouraged to raise their hands.

Guided Practice/Independent Practice *** confused about the critical mathematics section. Can you please explain more *** Time Table:

Comment [LAL14]: Nice job! Purposeful selection of problem task and number choice.

Comment [LAL15]: Additional detail needs to be provided on how you will “go over the answers as a class.” Please remember that this should be focused on students’ original strategies and not just their final answers. What are some mathematical strategies (e.g. direct modeling, counting, derived facts) that you expect to encounter for single-digit subtraction? This would be a great spot to highlight these familiar student strategies before delving into the lesson’s main content.

Comment [LAL16]: The “20-17” problem is an example of guided practice for your lesson plan. Please see my note about including assessments.

Comment [LAL17]: The segue from the Do Now to this multidigit subtraction problem should be made more explicit. How can the single-digit subtraction strategies from the Do Now be used to solve this? See my note about providing problems with context.)

Comment [LAL18]: Similar to my previous note, the details of how the classroom teacher is modeling student ...Comment [LAL19]: Please see my note above about delaying the use of the “house model.” It would be more effect to ...Comment [LAL20]: This would be an example of independent practice in your lesson plan. Please see note above ...Comment [LAL21]: One of the main elements missing in your problem-posing is how the teacher will discuss how place ...Comment [LAL22]: Please see notes above about providing contexts for all problems presented to students.

Comment [LAL23]: How are you supporting students to advance their problem-solving approaches from concrete ...Comment [LAL24]: Your use of teacher questioning to provide student support is great here. However, the sample question ...Comment [LAL25]: Please see my note above about how students are yet to be ...Comment [LAL26]: What are some examples of common errors and misconceptions associated with this ...Comment [LAL27]: In what ways are you going to use informal observations of students’ independent/group problem ...Comment [LAL28]: Please see my notes above where independent and guided ...

Clock reading

during the lesson

“Title of the activity”

Students doing Me doing

0 – 5 min (5 min long)

DO now

Students will be working on the Do Now problems independently at their desks. Once finished, students will share their answers

Teacher will be walking around asking questions. After finished, teacher will go over the problems on the board and evaluate.

15-20 Introduction

Students will be listening to example problems and providing possible solutions

Teacher will be explain example problems and evaluating students answers

10-20 Small Group Partner Practice

Working with a partner or a small group to strategize and solve three example problems

Walking around and observing. She will scaffold and answer questions when necessary

5 minutes Wrap-Up and Closure

Listening and providing feedback to the conversation about what was learned

Leading the discussion

IV: Closure How will you end the lesson?


How will you differentiate instruction for: For students with special needs, depending on their disability, modifications will need to be made. For ELL students, the house model should not need to be modified, because it is a visual aid requiring no knowledge of the english language. For students who are incapable of working independently, it will be required for them to work with a partner, as opposed to working alone on the in-class problems. Also, the house can be modified to a simple box, if the extremities of the house itself can be a distraction to certain students.

Comment [LAL29]: It looks like your group did not get to this section yet. Maybe we can talk more about this tomorrow in our meeting?

Comment [LAL30]: Like the closure, it looks like you still did not get to make revisions here. How do the modifications that we discussed in our last class when comparing the procedural/conceptual lesson plan play a role here?

Lesson Plan Format

Date: 3/3/14 Subject: Math Grade: 2nd

Lesson Topic: Two Digit Subtraction Class/Group Size: 20 Common Core State Standards for Content: CCSS.Math.Content.2.NBT.B.5 Fluently add and subtract within 100 using strategies based on place value, properties of operations, and/or the relationship between addition and subtraction CCSS.Math.Content.2.OA.B.2 Fluently add and subtract within 20 using mental strategies.2 By end of Grade 2, know from memory all sums of two one-digit numbers. CCSS.Math.Content.2.NBT.B.9 Explain why addition and subtraction strategies work, using place value and the properties of operations.

Learning Objective(s): Students will be able to complete 2-digit subtraction. Students will be able to regroup from the tens place when needed using mathematical tools. They will also be able to use several different strategies to accurately and efficiently subtract two-digit numbers. Once students master subtraction they will be able to apply it to real life situations. What assessment will you use to assess whether students have learned this objective? We will use formal assessment. The formal assessment will be independent practice. Students will work independently to solve problems using a strategy of their choice and explaining why they used that strategy. We will use an informal assessment through guided practice where they question and observe modeling by the teacher.

Core and Supplemental Materials

I Need: Base 10 blocks, counting blocks, hundreds chart, whiteboard, whiteboard markers, worksheet

Students Need: Pencil, eraser, scrap paper, base 10 blocks, hundred chart, unifix cubes (teacher will provide)

I. Context for Learning

Comment [LAL1]: Learning Objectives: 2/3 CCSS: 1/1 Texts/Resources: 1/1 All of the CCSS and texts/resources identified are appropriate for the planned lesson in multidigit subtraction. Your mathematical learning objectives discuss what the lesson’s content focus will be and how you expect different strategies to be adopted. However, the reader is left wondering how you will build this concept of regrouping using students’ place-value understanding, basic addition/subtraction knowledge, etc. as outlined in the CCSS.

Comment [LAL2]: How are you building on the concepts of place value and basic addition/subtraction facts to accomplish these learning objectives? Keep in mind that the learning objectives should be directly aligned with the cited CCSS.

Comment [LAL3]: Like what?

Comment [LAL4]: All of the problems presented throughout the lesson should be presented in real-life contexts.

Comment [LAL5]: Appropriate Assessments: 1.5/2 Independent/Guided Practice: 1.5/3 All of the assessments (including the independent and guided practice) included in the lesson plan are open-ended, meaningful, and aligned with strategy development. Please see my note below about what the focus of your informal class observations will be. How is the aligned with the learning benchmarks and strategies outlined in the CCSS? Also, the text following Questioning and Student Sharing got cut off. If you have any additional text after presenting the “common misconception,” I was not able to read it in the document.

Comment [LAL6]: What specifically are you looking for in these observations? What are some learning benchmarks and strategies of which you will take particular note?

Comment [LAL7]: Prerequisite Knowledge: 2/2 Potential Student Difficulties: 1.25/2 Real-Life Connections: 1/ 1 Your lesson plan does an excellent job in identifying the prerequisite knowledge from single-digit arithmetic and non-regrouping arithmetic that will support them with this new lesson. All of the subtraction tasks in the lesson are presented in meaningful, real-life contexts that will help students make sense of the problem in their solution approaches. The lesson plan outlines various student potential difficulties with two-digit subtraction stemming from poor place-value knowledge. Please see my note below about how this can be addressed via questioning/scaffolding and explicitly showing students what to do. Also, how are you supporting students with recording their strategies in solving the problems?

a. Organization of Students: Students will start out working independently at their desks working on the “do now” problem. Teacher will conduct whole class instruction on multi-digit subtraction with borrowing. This instruction will be led by the students. Teacher will ask questions such as: “What is the first step?” “What should we do next?” “Do we all agree?” This way, the students will be actively participating in the lesson. Next students will work with a partner or in small groups to solve several problems while teacher offers help to those who need. Students will work individually to create their own word problem that incorporates learned content.

b. Prerequisite Knowledge: Students should be able to subtract single digit numbers. Students should know the tens and ones places in two digit numbers up to 20. Students should be able to subtract two digit numbers without borrowing (for example 18 – 12 = 6). Teacher will activate student’s prior knowledge with the “do now” that will focus on simple subtraction without borrowing and regrouping. Do Now: Kelly has 9 jelly beans. Michael came over and he took 5 of her jelly beans. How many jelly beans does Kelly have now? Marissa was given 19 crayons. Ben came over and took 15 of her crayons. How many does Marissa have left? 9 – 5 = ? 19 – 15 = ? After the Do Now the teacher will go over different strategies as to how the students came up with their answer

c. Key vocabulary and terms: subtraction, tens place, ones place, difference, regroup d. Describe potential difficulties students may experience with the content: Students may not understand how to

apply the house model. Students may have trouble checking their work. Students may have trouble borrowing and regrouping. Students may have trouble with the concept that a number in the tens place represents the number of groups of ten. We predict that when students use the borrowing methodregroup, they may have trouble identifying that the tens column represents tens and the ones represents ones. If these problems do occur, the teacher will model how to accurately solve the problem in front of the class while explaining each step. The teacher will use base 10 blocks and ones block to visually represent the numbers in the problem. The teacher will provide each student with the opportunity to work with these manipulatives if necessary The teacher will explicitly show that there are ten one blocks and one ten blocks.


Comment [LAL8]: Please see my note below in ensuring that both concrete and abstract strategies are being acknowledged in the lesson. As you mentioned, students will have prior knowledge of basic addition/subtraction and place-value knowledge for 2-digit numbers. How will students apply this knowledge in the lesson so they are not dependent on direct modeling with the Unifix cubes or base-10 blocks?

Comment [LAL9]: This takes away the problem-solving ownership from the student though. How might you use questioning about the digits’ corresponding place values to support them in representing the number using the base-10 blocks? If students are still using the Unifix cubes to represent the numbers, how might you support them so they can segue into using the base-10 blocks?

Comment [LAL10]: What do you mean here?

Comment [LAL11]: Mathematical Ideas: 1.5/2 Student Strategies: 1/2 Questioning: 1.75/2 Teacher Modeling: 1/2 Lesson Closure: 0.75/1 Lesson Agenda: 1/1 Your lesson plan does a very nice job in outlining specific questions that will be posed in supporting students throughout the problem-solving process. At the same time, the lesson also acknowledges that the teacher will build on students’ original strategies to meaningful present the process of regrouping in 2-digit subtraction. There seems to be a strong focus on direct modeling, however, throughout the entire lesson. Additional discussion of how the teacher will connect these concrete approaches to more abstract ones (e.g. counting, derived facts) needs to be included in the lesson. Please see some TrackChange comments where questioning/probing (as opposed to showing) should come into play. In addition, your use of place-value knowledge to develop the regrouping concept is great. Some of the problems can be solved without having to engage in direct modeling of the regrouping process. With that said, how are students thinking flexibly about the numbers using place value understanding? For example, 83 = 76 is quite easy to solve and may not entail the entire regrouping process. How are student being supported in observing that? On a related note, how you supporting students in advancing to more abstract ways of solving the problem – moving away from direct modeling with cubes and blocks? Please see my note below about including more insight on the mathematics highlighted in the lesson closure with the problem involving the number 100.


We will activate prior knowledge by beginning the lesson with a “do now” activity. We will write two problems on the board; 9-5, which activates single digit subtraction knowledge and 19-15, which activates double-digit subtraction without borrowing. The students will work on these problems at their desk individually and then we will go over the answers as a class. The strategies we will go over include direct modeling using base 10 blocks and counting blocks and a hundreds chart. We will show that there is one ten and nine ones and from there you can take away one ten and five ones in order to get your answer. The teacher will discuss place value by stating that 19 has one ten and nine ones. We will then introduce two-digit subtraction with regrouping. The teacher will then explain that we are now going to try something more difficult and will instruct the students to think about what we can do if we need to take 19 away from 36. The teacher will first show 36 counting blocks and then take away 19 counting blocks to show that it is possible to take 19 away from 36. After, the teacher will ask the class if there is another way we can do this that would be easier knowing what we already know about 10s and 1s and how to group them. After the class provides possible answers, the teacher will model the problem using base 10 blocks and group the ones blocks into groups of 10. The teacher will use the same method and let the students come up with the strategies for future problems. We will show the problem 20 – 17 and ask students what strategy they think should be used to solve this problem. Then we will ask the students to come up to the board and ask them to share their methods of solving the problem. We will discuss and evaluate the accuracy of each strategy and solution. At this point, after the students have had their first exposure to double-digit subtraction with borrowing we will introduce the house model and how it can be used to organize similar problems. We will again work with the problem 20-17 to show that base 10 blocks, counting blocks, and the hundreds chart can be used to understand the regrouping of tens and ones in multi-digit subtraction. This will be done orally in front of the class as they sit at their desks. The students will participate as well as answer questions and offer strategies to solve the problem. We will repeat this model with the problem 44 – 39. At this point teacher will provide three examples of two-digit subtraction problems on the board for students to work out in pairs or in small groups. Students will be encouraged to use the provided manipulatives and their own house model to solve each problem. The problems will be: Ashley has 36 donuts and wants to give away 19 to the class. How many donuts will she have left? Monica started with 54 stickers. She put 27 stickers on the board. How many stickers are left? Holli bought 83 gumballs. She sold 76 of them to her classmates. How many gumballs does Holli have left? If students finish these problems early, they will be encouraged to come up with different strategies to solve each problem. Supporting Students Teacher will walk around the room going from group to group assessing their progress and scaffolding. If the students seem to be struggling, teacher will show again how this can be modeled by using manipulatives. Rather than teacher explicitly stating what to do, she/he should probe students by asking questions to activate knowledge. For example: teacher will ask students “if you cannot take 7 away from 4, what could you do next?” The teacher will redirect student attention to the problem by asking students to show that 47 is equivalent to 4 tens and 7 ones in whatever strategy they come up with. For example, they can show it by using the base 10 blocks, counting blocks, or the hundreds chart. Questioning and Student Sharing While other students are sharing their strategies to solve the problem, other students should be taking notes and writing down these strategies as well. This will be helpful to solving other similar problems and for students who have difficulties. While students are sharing, other students who are listening will be encouraged to ask questions if they are unsure, to learn from one another. To engage students who are not participating in the discussion teacher will directly ask if they used this method to solve or another method and if another to explicitly state/demonstrate their strategy. To capture student attention, teacher can introduce the example problems in word problem form using familiar names and items that may interest them. For example: Kelly had 47 doughnuts and she gave 19 to Brian. How many doughnuts does Kelly have now? In order to make sharing explicit to the class, students will be asked to tell AND show the exact methodology used to solve the problem. For example: teacher will ask questions like “What was the first thing you did?”, “How did you know to do that?”, and “How many tens are left now?” etc. A common misconception we anticipate to see would be the student automatically subtracting the larger number in the ones column from the smaller number in the ones column, even though it should be reversed. To respond to students who share a mistake or misconception, teacher will ask the other students if they agree or disagree and why. If and when another student disagrees, ask them to explain why. To share, students will be encouraged to raise their hands. We will get students to participate in the lesson by working in

Comment [LAL12]: I would not even mention that the next problem will be more difficult. Just pose the problem and see how students attempt to solve it. You might turn students away from the problem by presenting it as more challenging.

Comment [LAL13]: Students should be granted the opportunity to tackle this more challenging problem that involves regrouping prior to having the teacher highlight the mathematics involved.

Comment [LAL14]: What are the key mathematical ideas at play here when students regroup? For example, the students are trading in 1 ten for 10 ones to be able to subtract. Why is this true? In addition, the strategies highlighted here focus only direct modeling using the base-10 blocks. It is mentioned earlier in your lesson plan that you would highlight additional student strategies such as counting down, derived facts, etc. How are these being addressed in the lesson?

Comment [LAL15]: How?

Comment [LAL16]: Your number choice throughout the lesson plan is excellent.

Comment [LAL17]: One major recommendation for further improvement is providing additional detail on how the teacher will connect students’ different mathematical strategies. For example, how is the mathematics the same in using the base-10 blocks and applying a derived fact to solve a specific subtraction task?

Comment [LAL18]: The students should be doing this for the teacher – not the other way around.

Comment [LAL19]: How can this be acknowledged through questioning? Keep in mind, as you previously stated, the students will have strong place-value understanding when they come into this mathematics lesson.

Comment [LAL20]: Please see my note above inquiring on how you support students with their recording strategies and writing the subtraction number sentences.

Comment [LAL21]: How can you use your informal class observations to ensure that a wide variety of strategies are presented?

Comment [LAL22]: This got cut off in your lesson plan draft. I do see that you are bringing up the common student error in reversing the subtraction order. However, any text beyond that is not viewable.

Clock reading

during the lesson

“Title of the activity” Students doing Me doing

0 – 5 min (5 min long)

DO now

Students will be working on the Do Now problems independently at their desks. Once finished, students will share their answers

Teacher will be walking around asking questions. After finished, teacher will go over the problems on the board and evaluate.

15-20 Introduction

Students will be listening to example problems and providing possible solutions

Teacher will be explain example problems and evaluating students answers

10-20 Small Group Partner Practice

Working with a partner or a small group to strategize and solve three example problems

Walking around and observing. She will scaffold and answer questions when necessary

5 minutes Wrap-Up and Closure

Listening and providing feedback to the conversation about what was learned. The class will be provide possible answers for the 100 problem

Leading the discussion as well as posing an extension problem using three digits.

IV: Closure

How will you end the lesson? The closure will be an extension of what we are learning. The teacher will go into the 100 place value. For example, the teacher might pose a problem such as, “Jen has 100 pencils but gave away 15. How many pencils does Jen have left?”


Comment [LAL23]: Interesting! Some additional insight should be included here on how the teacher will highlight and support student’s extension of 2-digit subtraction to execute 3-digit subtraction. What are the place-value ideas that need to be acknowledged here? In what ways is the regrouping similar to 2-digit subtraction?

Comment [LAL24]: ELL Students: 0.5/2 Students with Disabilities: 0.25/2 Low Achievement: 0/1 This section on instructional modifications needs to be more fully developed. Please refer to the sample modifications presented in the ELL text’s chapters as well as the Behrend/Witzel articles on students with LDs. Think about how these modifications can be tailored to your lesson’s content on multidigit subtraction. Also, there is no mention here on modifying instruction for students with low mathematical achievement who may struggle with fundamental idea like basic addition/subtraction, place value, etc. How is the instruction and activities being tailored to support these students?

How will you differentiate instruction for: For students with special needs, depending on their disability, modifications will need to be made. We will differentiate instruction by allowing the students to choose which strategy to use for multi-digit subtraction problems. If they are visual learners they can use the base 10 blocks or counting blocks. We will also introduce the ones and tens blocks as well as the ones and tens column in Spanish for ELL students. For students who are incapable of working independently, it will be required for them to work with a partner, as opposed to working alone on the in-class problems.

Instructor Feedback (via E-mail):

Hello again, Group 1 Lesson Plan Members! I am writing this e-mail message to provide you with graded feedback on your group lesson plan assignment for two-digit subtraction with regrouping. Attached to this e-mail message is a MSWord document containing your feedback on your final submission. In the document, you will find a TrackChange comment that corresponds to each segment of the grading rubric -- Mathematical Content & Standards, Context for Learning, etc. These specific comment bubbles contain the grading breakdown aligned with the 30 possible points that can be earned for the final lesson plan draft as well as comments for further improvement. Your group did a fantastic job in outlining the appropriate mathematical content standards and describing the students' context for mathematical learning (e.g. prerequisite knowledge/skills, potential difficulties/misconceptions). Please refer to my comments outlined under the Mathematical Instruction and Modifications for Diverse Learners sections that provide insight on room for improvement. It sounded like your group still wanted to make some additional changes to your lesson plan draft prior to final submission which explains why some sections needed some further development including the learning modifications part. Please feel free to e-mail me back with any questions or concerns. As I mentioned in class, your group is welcome to revise and re-submit the lesson plan for a higher score that will replace the original one. Your group will receive the class presentation grade and feedback in a separate message. I am more than happy to continue supporting you through the revision process as well. Hope this helps! GRADE: 19/30

Comment [LAL25]: What are some modifications that we observed in the Behrend and Witzel pieces that can be incorporated in your lesson? Think about how students with LDs may struggle with making sense of the problems, communicating their strategies, recalling prior knowledge, etc.

Comment [LAL26]: There are no columns in this lesson since students are not learning the standard algorithm in second grade. How are the manipulatives particularly helpful for ELLs? In what ways is communication in the classroom being supported among the ELL students? How are these students making meaning of the mathematical vocabulary in the lesson?

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