Kaitlynn GilbertTE 801, Section 002

Scott School Intern, 3rd Grade

Project Two: Unit Plan

Part 1: Big Ideas and Standards

Big Ideas: 1.) Multiplication is used to find the total (product) in a number of equal groups. 2.) Multiplication and division are related because both involve groups of equal size. 3.) Multiplication is commutative and distributive.

Common Core Standards: 3.OA.1: Interpret products of whole numbers 3.OA.2: Interpret whole-number quotients of whole numbers. 3.OA.5: Apply properties of operations as strategies to multiply and divide. 3.OA.6: Understand division as an unknown-factor problem.

These two standards are integrated across all the lessons, but are not the “focal point” of the unit.

o 3.OA.3: Use multiplication and division within 100 to solve word problems in situations involving equal groups, arrays, and measurement quantities.

o 3.OA.4: Determine the unknown whole number in a multiplication or division equation relating three whole numbers.

Standards for Mathematical Practice: MP.1: Make sense of problems and persevere in solving them.

o I chose this mathematical practice to focus on, as I believe that my students give up very easily if a problem is too difficult for them. I would like to build a sense of community throughout the unit that it is okay to have productive struggle while doing math. The students will need to model multiplication and division problems using an array model, and they will be solving word problems. In order to do these, they will need to understand and persevere with the task, even if it is a problem they cannot solve right away. I plan to help students develop this practice through challenging them with high-level tasks and the ‘5 practices’ model. By having students work through a problem and share their strategies/thinking with their peers, the students will learn from one another and be able to connect their strategies with the mathematical concepts of the unit. Sharing with others also will help to build students’ perseverance through this unit.

MP.3: Construct viable arguments and critique the reasoning of others.

o One of my goals for this unit is for students to clearly explain their thinking. As the students are comparing methods they use to solve multiplication and division problems (arrays, equations, pictures, manipulatives, etc.), I would like them to explain these strategies and why a certain one works best for them. The problem-solving tasks and real-life applicable tasks will need to be solved and reasoned with their classmates. I have seen this as a struggle for my students thus far, so I plan to support this by constantly asking the students to explain their thinking. I hope to make it a natural reaction that students defend their answers without being reminded. I will also stress the importance of explaining thinking so that other classmates may learn from them by using the “thumbs up/thumbs down” method. Students will put their thumb up if they agree with their classmate’s thinking, but they also must be able to defend why they agree.

Learning Targets/Objectives: Lesson 1: Students will connect repeated addition to multiplication through identifying

equal groups of objects and finding totals. Lesson 2: Students will compare the equal groups of objects in scattered configurations

to the array model, exploring the correspondence between 1 equal group and 1 equal row.

Lesson 3: Students will analyze equal groups given in scattered configurations and organized into arrays to determine whether factors represent the number of groups or the size of a group.

Lesson 4/5: Students will find the meaning of the unknown in division as the size of groups or the number of groups through word problems that help give meaning through context.

Lesson 6: Students will use arrays to write multiplication sentences and find unknown factors, then write division sentences where the quotient represents the same as the unknown factor.

Lesson 7/8: Students will demonstrate the commutativity of multiplication through array models by writing two different multiplication sentences to interpret the same array.

Lesson 9: Students will use arrays to decompose unknown facts as the sum or difference of two known facts.

Lesson 10: Students will write decomposition as (ac) + (bd) = acbd and explain each step of the solving process.

I used the Eureka Curriculum, Module 1 to help format/organize my learning targets. (http://commoncore.org/maps/math/grade-3/module-1).

Section 2A: Pre-Assessment Design

The pre-assessment for this unit is a paper and pencil assessment. Students will be given three questions to complete. The questions are examples of high-level tasks because there are multiple entry points and multiple correct answers for each question. There is no set procedure on how to solve these questions, and students are asked to connect the problems to the overlying mathematical concepts of my unit. The questions allow students to show their math strengths in multiple ways, including drawing pictures, writing sentences, and filling in equations. This pre-assessment will allow me to see the different strengths of my students, and I will also be able to identify how students are succeeding with the mathematical practices I hope to achieve in my unit. I will see how well they persevere in solving these high-level questions and also if they are able to explain their thinking well. The questions are also applicable to the students’ lives, as they are about the farmer’s market in DeWitt and the Scott School garden. The pre-assessment demonstrates to students that math is applicable outside of school, which is another aspect that will be highlighted during my unit.

Question #1: Part A: This question connects to students drawing an array to determine which factors

(or the size of groups and number of groups) will make the correct product, which is the learning target in Lesson 3. (3.OA.1)

Part B: This correlates to the learning target in Lesson 2, as students are explaining the correspondence between 1 equal group and 1 equal row. (3.OA.1)

Part C: This shows whether students see the connection between multiplication and division. Students are asked to find the meaning of the unknown in both division and multiplication problems, which is the learning target in Lesson 4/5. Students are also using the array they draw to write a multiplication sentence and a division sentence from the same fact family, which is the learning target in Lesson 6. (3.OA.6 and 3.OA.2)

Question #2: This question connects to students’ ability to demonstrate what they know about the

commutative property through an array model. This is the learning target in Lesson 7/8. (3.OA.5)

Question #3: This question connects with the learning targets in Lessons 9 and 10. Students are asked

to demonstrate the distributive property using an array model and then write as many number sentences as they can showing the total number of objects in the array. (3.OA.5)

Properties of Multiplication and Division Pre-Assessment

Name: ______________________________ Date: ________________

1.) a.) The DeWitt Farmer’s Market has 36 booths selling fruits, vegetables, and flowers. The owner of the farmer’s market wants the booths set up in equal rows. Draw an array with the booths in equal rows. (You may use an X to show each booth)

b.) Does your array show equal groups? Why or why not? If it does show equal groups, circle one equal group in your picture.

c.) Write one multiplication and one division sentence for the array you drew.

__________ x ____________ = _________________

__________ ÷ ____________ = ________________

What does each number in your multiplication and division sentence represent?(Example: 36 is the total number of booths)

2.) x x x x x xx x x x x xMs. Gilbert’s Array x x

x x Ms. Zarotney’s Array

Tell me what you notice about these arrays using math vocabulary.

3.) Below is a picture of the garden that the multi-age classroom planted outside of Scott School. The plot size of the garden is 4x6. They planted tomatoes and carrots.

On the picture, draw one slice or cut through the garden to divide the plot so that the tomatoes and carrots are separated.

After you draw the slice, write as many number sentences as you can think of to show the total number of vegetables:

__________ ___________ = ______________

__________ ___________ = ______________

__________ ___________ = ______________

__________ ___________ = ______________

Section 2B: Pre-Assessment Results

Question #1A: Students are drawing an array to determine which factors (or the size of groups and number of groups) will make the correct product. (Lesson 3)

Notes about Student Performance: 10/25 students were able to draw an array that had equal rows, equal columns, and made

the correct product of 36.o Examples: 4x9, 12x3, 6x6, and 18x2

5/25 students were able to draw an array that had equal rows and equal columns, but it made a product other than 36.

o Examples: 13x3, 7x5, 7x6, 12x4, 9x3 10/25 students were unable to draw an array with equal rows and equal columns.

o Examples: rows with different amounts, no rows or columns present (scattered)

Question #1B: Students are explaining the meaning of equal groups & the correspondence between 1 equal group and 1 equal row. (Lesson 2)

Notes about Student Performance: 1/25 students were able to clearly explain that an equal group but must have the same

number of objects in each group, and that 1 equal group=1 equal row in their array. o Example: “It does show equal groups because each row has nine booths.”

14/25 students were able to explain that an equal group has the same number of objects in each group, but they could not relate this back to their array.

o Example: “They have 18 x’s in them. 18+18-36.” o “It does because there are 7 in each group.”o “It does because I put it in groups of 8.”

10/25 students were unable to explain why their array showed equal groups.o Example: “Yes, it is equal because the number 6 is equal. And 6 plus 30 equals

36, an equal number.” (confusing equal and even)o “My array shows equal groups because they’re set up in order in rows nice and

neat.”

Question #1C: Students are asked to use the array they drew to write a multiplication and division sentence from the same fact family. This also tested their connection between multiplication and division. They were asked to write what each number in their number sentences represented in their array—which tests if they are able to find the meaning of the unknowns in division and multiplication problems. (Lessons 4, 5, 6)

Notes on Student Performance (Writing the Number Sentences) 2/25 students were able to write an accurate multiplication and division number

sentence that described the array they drew.

o Example: 3x12=36 and 36/12=3; 6x6=36 and 36/6=6 4/25 students were able to write an accurate multiplication sentence for their array but

NOT a division sentence.o Example: 2x18=36 and 2/36=18o 2x18=36 and 2/36=23o 6x6=36 and 36/6=36

6/25 students were able to write an accurate multiplication sentence but it did not describe the array they drew.

o Example: 5x6=30 for a 6x6 arrayo 3x12=36 for a 7x5 arrayo 36x1=36 for an array that has unequal rows and columns

13/25 students were unable to write an accurate multiplication or division number sentence that represented the array they drew.

o Example: 12x12=36 and 36/12=12o 8x5=37 and 10/2=8o 20x36=46 and 36/20=46o 1x5=5 and 1/5=3o 7x6=36 and 7/6=36

Notes on Student Performance (Explaining what each number represents in the array) 1/25 student was able to state that the factors represented how many rows there are

and how many booths in each row, and the 36 was the total number of booths.o Example “The four is how many rows there are, the 9 is how many are in each

row, and the 36 is how many booths there are in all.” 8/25 students were able to state the meaning of at least one factor in their array (either

the number of rows or the number of booths in each row).o Example: “18 is how many are in the groups”o 1”8 is how many are in the row.”o “13 is how many I have in each row.” o “The 8 is what I grouped it into. I had 5 rows of 8, and I got 37.”

16/25 students were not able to state what each factor or product in their number sentence represented.

o Example: “The 12 means you have to do it by 12”o “1 farmer. 36 booths. 30 people want to buy something and they are all in one of

the 36 booths.” o “Well because 36 times 1 equals 36 and 30 divided by 1 equals 36.” o “The 3 represents the 6 booths and the 9 is to represent the 9 booths.”

Question #2: Students were to demonstrate what they know about the commutative property through an array model. (Lesson 7-8)

Notes on Student Performance:

6/25 students were able to describe the directional differences of the arrays and also that the arrays have the same product.

o Example: “Ms. Gilbert’s is side to side and Mrs. Zarotney’s is up and down, but they both has much as each other. It has 8 on both.”

o “They both have eight X’s, but they are in different directions.” o “They both have the same amount as one another. One is up and down and the

other is side to side.”o “They both have 8 X’s and they’re going different. But they have the same

number but different lines and that both are the same array.” 8/25 students were able to tell that the two arrays had the same product, but they did

not mention any directional differences (such as 4 rows or 2 rows).o Example: “Well Ms. Zarotney’s has 8. You have 8. So that is equal.”o “They have the same number.” o “Ms. Gilbert’s array is the same as Mrs. Zarotney’s array.”o “They both have 8 of them. They are equal.”

10/25 students were able to describe directional differences with the arrays but did not state that the product of the arrays were equal.

o Example: “Ms. Gilbert’s array is sideways and Mrs. Zarotney’s array is up and down.”

o “Ms. Gilbert’s is vertical and Ms. Zarotney’s is upright.” 1/25 student was not able to explain anything he noticed about the two arrays.

Question #3: Students were asked to demonstrate slicing an array into two parts and writing number sentences to describe the array in two pieces. This was supposed to demonstrate their knowledge of the distributive property. (Lesson 9-10)

Notes on Student Performance: 9/25 students were able to draw one slice through the array and write at least one

number sentence that accurately described the new array.o All students sliced the array in half (2x6 and 2x6)o Example number sentences: 12+12=24

7/25 students were able to draw one slice through the array, write a true number sentence, but the number sentence did NOT represent the new array.

o All students sliced the array in half (2x6 and 2x6)o Example number sentences: 1x8=8, 1x20=20, 3x3=9, 38-10=28, 6+6=12, 12-6=6,

4x6=24, 6x4=24, 36-1=35, 2+36=38, 5/25 students were able to draw one slice through the array but were NOT able to write

a true number sentence or a number sentence that represented their array. o All students sliced the array in half (2x6 and 2x6)o Example number sentences: 2x12=25, 4x8=24o Many of these students did not include an operation sign (+, -, x, or /)

4/25 students were not able to draw one slice through the array or write a number sentence that represented their array.

o Example: split array into 4 pieces, drew no cut through the array, shaded a few boxes of the array

0/25 students were able to create a decomposition number sentence that described the product of 24 (i.e. (2x6) + (2x6) = 24).

Explanation of what I learned from the results & how to use this to plan my unit:

Looking at the results of the pre-assessment, every question had more students who missed or are working towards the objective than students who have mastered the objective. This is very much what I expected for the pre-assessment. These results tell me that I will need to spend ample time on each objective in my lesson to be sure that students are learning and mastering the content of my unit. As seen above in the notes about the students’ performance on the pre-assessment, I was able to categorize students’ answers based on the common misconceptions of the class.

Lesson 1 & 2: A majority of the class was able to explain that an equal group has the same number of objects in each group. They were not able to relate this back to an array model, however. I believe this is because the array model is something new that many students are not familiar with. I will be sure that once I introduce an array model in lesson 2, I will relate this back to their previous knowledge of equal groups.

Lesson 3: A majority of the class was able to draw an array that had equal rows and columns. This made me confident that many students have seen an array before, but they may not be comfortable working with them. One misconception with the arrays was students were not able to determine the dimensions of the array to get the correct product of 36. I will be sure to address how the number of rows and columns in an array is important. A little less than half of the class was able to correctly draw an array. I will be sure to explicitly explain what an array is, and if these students are still struggling, I may have to work with them in a small group.

Lesson 4-6: This is the area where students struggled the most. More than half of the class was unable to write a correct multiplication and division sentence that represented their array. Only 2 students in the entire class were able to write a multiplication and division sentence that represented their array. These 3 lessons will be critical while teaching my unit. I will need to be explicit in my teaching and provide many examples of ways students can write multiplication and division sentences. I also know that none of the students in my class have experience with division. Division is going to be a whole new concept for them, which means I will need to be clearly explaining the foundational concepts of division (equal groups, relationship to multiplication, etc.). Over half of the class was also unable to clearly explain what their numbers represented in their number sentences. It will be important to model for the students what a clear explanation sounds like. I will also be explaining how to determine what the meanings of the unknowns are in both multiplication and division sentences through exposing the

students to different word problems. We will also work on creating our own word problems.

Lesson 7 & 8: A majority of the class was able to notice the directional differences in an array that is horizontal and one that is vertical. I will need to expand on these students’ knowledge and show them that even though the directions are different, the two arrays are the same. By teaching the vocabulary of rows, columns, and product, this should help to expand their explanations. I will also be able to use this foundational knowledge about directional differences for modeling the commutative properties with arrays. I will be able to use what they know about the directional differences in the arrays to represent that two multiplication number sentences can be created for one array.

Lesson 9 & 10: Over half of the students were able to separate an array into two parts. All of the students, however, sliced the array in half. I will need to show them that the array could be sliced in other ways, also. Less than half of the students were able to create accurate number sentences that described their sliced array. None of the students in the class created a decomposition number sentence. This tells me that the distributive property does not come naturally to the students. I will need to be sure to focus heavily on these two lessons. I was able to see that a majority of the students were able to create number sentences that were accurate, even if they did not describe their array. I will be able to use this skill that the students have when teaching the steps of the decomposition process.

Section 2C: Formative and Summative Assessments

Formative Assessments:

1.) Observations & Monitoring:Throughout my unit I will constantly be observing and monitoring students’ progress. I hope to integrate the ‘5 practices’ into my unit, so after I introduce a task to the students, I will monitor the students’ strategies, solutions, and thinking on a clipboard. I will then use the notes I have taken to select and sequence the students to share with the class. I will also use these observation notes as a form of assessment, as I can determine which students are meeting the learning targets of the lesson and which are not. These observations will also help me to determine which students may need to work in a small group with myself and which students may need to be challenged in the next lesson.

2.) Student Whiteboards:Each of the students in my class has a whiteboard they keep in their desk. I will use these whiteboards throughout my lesson as a quick check to see which students understand the concepts. I may ask students to answer a quick math problem or have them show a strategy that they used to solve a problem. When the students are finished answering, I will count to three and they will hold their boards up. I can then view the answers and/or strategies on the whiteboard to see where my students stand throughout the lesson. The whiteboards are a great way to assess at the beginning of a lesson (during the “launch”) to see what students remember from the previous lesson. I may also use the whiteboards in the middle of a lesson after a new concept is introduced to see if more scaffolding needs to be done on this concept before I allow students to work on it independently.

3.) Thumbs Up/Thumbs Down:My mentor teacher already uses the thumbs up/thumbs down method very frequently throughout the day. I will continue to use this type of formative assessment to check for many things: when students are ready to begin a lesson (ending a transition), if students understand directions, or if students agree/disagree with a classmate’s answer/strategy. The thumbs up/thumbs down method is a quick way to check for student readiness and understanding during a lesson.

4.) Exit Tickets:The Eureka curriculum that my class will be using for this unit has pre-made exit tickets for each lesson. I may use these (or adapt them to a higher-level) as a formative assessment for the end of the lesson. Each student will be given an exit ticket and asked to answer a question or two based on what we have covered that day. The questions will vary each day, sometimes it may just be a simple definition or other times it may ask to show a strategy they learned.

5.) Math Journals/Homework Assignments:The Eureka curriculum also has homework assignments that align with each lesson. While I do not plan on giving much homework throughout the unit, I may assign some homework weekly for students to practice the skills that we are learning. I will take some of the questions from the Eureka homework. The students also have math journals where they can take notes and complete tasks. I will collect this homework, along with their math journals, to grade for completion and see if students are meeting the learning goals of the lessons.

Formative Assessment System:Since I do have a wide variety of assessment strategies, I plan to keep an Excel spreadsheet with each student’s name. I will keep different columns with the date and the standard of the lesson. Under each column, I will combine any of the formative assessment notes I have on this student to record a summary of the students’ progress. For example, on October 28th, I will look over the observation notes I have for each student (monitoring, whiteboards, thumbs up/thumbs down), exit tickets, math journal, and homework assignments and write a short bulleted list of whether the student is meeting the goals of the unit. I will not do this every day for every student, but I will try to do at least 5-10 students a day. I will also use my classroom’s scoring of 1-4 to assign students a score for any formative assessments (4=above grade level, 1=very below grade level). I will be able to look at these notes/scores on the spreadsheet and determine the next best steps or the changes I need to make for the next day. My observation notes will be kept on a clipboard with each student’s name and date. When I monitor each group, I will make notes of these students on the clipboard.

Summative Assessment:Below is a copy of the summative assessment I created for my unit. The questions are designed to be high-level tasks that have students directly applying what they learned in the unit. The questions are high-level tasks because they have multiple entry points and most of the questions do not have a set procedure of how to find the answer. Most of the questions also have more than one correct answer, like when the students have to draw an array or explain their thinking. The test will also test student’s perseverance, as it is long and asks them to defend their thinking in almost every question.

Question #1: 3.OA.1 & 3.OA.3 Part A/B: Correlates directly to the learning target for lesson #1, where students are

identifying multiplication as equal groups. They are also practicing the vocabulary of groups and size of groups.

Part C: This question relates to learning target for lesson #2, as students are comparing equal groups of scattered configurations to the array model. This question will also show me how well students understand the definition of array (1 equal group= 1 equal row).

Part D: This question will test whether students can find the total of a group of objects using both repeated addition and multiplication, which is a skill tested in Lesson 1 and 2.

Question #2: 3.OA.1 Part A and B: These two questions align with the learning target for Lesson 3, in which

students are analyzing equal groups in an array to determine whether factors represent the number of groups or the size of groups. I will also be testing the students’ proficiency with the word factors, groups, size of groups, or number of groups.

Part C: This question asks students to use an array to solve a multiplication problem. This question will also test whether students can clearly explain their thinking to defend a student’s answer (a mathematical practice I am focusing on).

Question #3: 3.OA.2, 3.OA.3, 3.OA.4, 3.OA.6 Part A, B, C: These two questions correlate with the learning targets for Lesson 4 and 5.

These questions will test whether students understand that the unknown in a division problem can represent the number of groups or the size of a group.

Part D: This question will test students’ knowledge about the connection between multiplication and division. They will show how multiplication and division number sentences are connected, as the quotient represents the same as the unknown factor. This is aligned to the learning target for lesson #6.

Question #4: 3.OA.1, 3.OA.5, 3.OA.3, 3.OA.4 This tests students’ abilities to demonstrate and identify the commutative property.

They will use the arrays given to model that two different multiplication sentences can be written for the same array. This correlates to the learning targets in lessons #7-8.

Question #5: 3.OA.1, 3.OA.5, 3.OA.3, 3.OA.4 Part A: Students will create slice an array into two parts to help decompose a unknown

fact. This will help to demonstrate their knowledge of the distributive property, which aligns with the learning target in lesson #9.

Part B: Students will be asked to explain the steps of the decomposition process using an array for guidance. This correlates to the learning target in lesson #10.

Eureka, Module 1, Lessons 1-10

Properties of Multiplication and Division Post-Assessment

Name: ______________________________________ #: ___________

1.) Ms. Gilbert went to Uncle John’s Apple Orchard. She bought a pile of apples, and when she got home she put them all on her counter. Below is a picture of the apples on her counter.

a.) Circle the equal groups in the picture. How do you know they are equal groups?

b.) How many groups do you have? ____________

What is the size of the group? ____________

c.) Ms. Gilbert wants her apples arranged into an array. Draw the apples in an array:

d.) How many total apples are there?

Show how to solve using repeated addition:

Show how to solve using multiplication:

2.) The DeWitt quarterback passed the football 8 times each quarter. If there are 4 quarters in a football game, how many times did he pass the ball total during the game?

Bobby’s Answer:

4 x 8= 32. The DeWitt quarterback passed the ball 32 times during the game.

a.) What does the 4 represent in Bobby’s array?

b.) What does the 8 represent in his array?

c.) Is Bobby’s answer correct? Explain using our math vocabulary why his answer is right or wrong.

3.) The multi-age classroom plants 15 sunflower seeds in the Scott School garden. The seeds are planted in rows.

a.) Draw an array of the garden:

b.) Write a division sentence in the spaces below to find how many seeds are in each row.

_____________ ÷ ____________ = ________________

c.) What does your quotient represent in part B? (Hint: Use words like factor, rows, number, size)

d.) How could I use what I know about division to help me solve 3 x ______ = 15?

4.) Ms. Gilbert thinks 6x3=3x6.

Do you agree or disagree? Draw 2 arrays to help explain your thinking.

5.) Mrs. Webb is making a photo album with pictures from this school year. Her photo album has 7 rows with 4 pictures in each row.

a.) Slice the 7x4 array into two parts and fill in the multiplication sentences to find how many total pictures Mrs. Webb has in her photo album.

7x4

b.) Explain using number sentences how you would use the distributive property to find the product of 7x4 (use your arrays in part A for help).

7x4

Section 3: Differentiation Strategies

Choose three or four different strategies that you will use throughout the unit to differentiate instruction and/or learning:

1.) Working with Small Groups Many of my lessons that I create for this unit will involve whole group, small group,

partner, and individual work. This is to accommodate those students who learn best from different types of interaction. When students are working individually, I will be pulling small groups of students who are struggling with the concepts of the lesson. I will determine which students to pull based on the observational notes I take, the whiteboard activities we complete, and their work on the daily exit tickets. Already in my math class, either my mentor teacher or I will work with a small group of students who are failing to meet the objectives. I will continue this to make sure that all students are able to learn the concepts of the unit.

2.) Scribing/Reading Instructions for Students 3 of the students who have IEPs in my classroom have accommodations for teachers to

scribe for them or read instructions to them. I will continue these accommodations throughout my unit. If a task, exit ticket, or homework assignment requires reading, I will be sure that my mentor teacher or myself reads these 3 students the directions. If my mentor teacher and I find that we cannot read their explanations, we will ask the students to verbally state their thinking while we scribe for them.

3.) Use of easier or more challenging numbers Throughout this unit, I am focusing on the mathematical practice of perseverance. In

order for this to apply to all students, I will need to adapt some of the tasks that the students complete. For example, I may have students work with partners of similar math abilities based on the results from my formative assessments. I will have those who are excelling work with larger numbers, and those who are struggling work with smaller numbers. For example, when creating arrays, I will have students who are excelling work with larger factors, such as those above 20. For those students who are struggling, I will keep them working with factors under 10. During individual work, if students finish quickly, I will have them do a problem again with more challenging numbers or ask them to find as many strategies as possible. This will push the students to persevere and clearly explain their thinking.

4.) Clear modeling for word problems and open-ended tasks Based on the results from my pre-assessment, I found that a majority of my class is

unable to clearly explain their thinking using math vocabulary. In order to help those

students who struggle to complete large word problems, I will provide scaffolding. I will model a different example before they work on the problem, give them some time to think about it, and then show them how to translate their own spoken ideas into a math sentence. We have not practiced this in their math class yet, and I feel that all of the students would really benefit from learning to dissect word problems and large tasks. This may be a differentiation I do in the small groups I pull during independent work time.

Explain how you will use the other adults in your room to maximize student learning.

My MT will be a great resource for me as I teach this unit. If the task requires it, she will help me to model how students complete an activity or work with their partners. We have found this to be a great system if students are struggling to begin a task. My MT will also be able to work with a small group of students who are struggling and would benefit from individualized instruction. I will use the formative assessments as a way to determine the small group that will work with her. By having a small group work with her at least once a week, they will be receiving additional support that does not interfere with the whole group lesson.

The resource room teacher typically does not visit our room during math time; however, we are able to send her students with IEP if they are really struggling to stay on task or complete their work. I feel that working with a small group during independent time will alleviate some of this; however, if the students need one-on-one support, they will be able to go to her room for this. The resource room teacher will also be able to help me decide on appropriate homework and/assessments to give the students with IEPs. For example, she may be able to help me decide how many problems or which problems are appropriate for these students when working on the formative and/or summative assessments.

Explain how you will provide the scaffolding and support needed for students with IEPs and other special needs.

We do have 4 students in my class with IEPs, so much of the scaffolding and support that I have mentioned above are for these students. As I said above, we will be reading directions for these students and scribing for them if need be. According to her IEP, one of the students is allowed to use a calculator, so she will be given this option if need be. I will help scaffold these students by providing clear directions and modeling for these students what I would like them to do during the task. I will also provide many visuals for these students, such as bringing in real-life examples of arrays or graphic organizers of repeated addition vs. equal groups vs. array. I would like to keep a consistent vocabulary for these students, also. For example, I will need to decide if I would like to use the word “product” or “total” and “multiply” or “times”. By keeping this vocabulary consistent throughout the unit, students are less likely to be confused by the

terminology of the tasks. These students with an IEP will also be working in a small group with my MT or myself if I see that this instruction would benefit them.

Some of the students who do not have IEPs in my class also struggle with attention problems. I will help to support these students by providing many transitions for them, such as transitioning from whole group to partner to individual. I also hope to include some movement in my lessons or having students model some of the concepts we are discussing, such as repeated addition and equal groups. By creating tasks that are engaging and applicable to students’ lives, I believe this will help students stay focused.

Explain how you are using what you learned from Project One to plan your unit and differentiate instruction.

After filling out the chart in Project One Part I, I learned that many students do not believe they are good at math because they have trouble passing Rocket Math (timed tests). Many students did not seem aware of the other types of math smarts that we have discussed in class. I want to help make students mindful of the different types of smarts that are used in math tasks, such as organization, drawing diagrams, explaining thinking on paper and out loud, calculating quickly in your head, or building models. In order to do this, when I assign a group or partner task, I will let the students know of the different types of math smarts that are needed in order to complete the task. I will encourage groups to work cohesively together so that everyone has a role. I want everyone in the group to be able to tell me what is going on and answer my questions when I talk to them. I might even try to use the group roles we discussed in class for one of my lessons. This differentiates instruction because the group mates have to work together to formulate a strategy so that all group mates are involved and understand what is going on. I have witnessed many times where two of the students are doing all the work, and the others are left out on the sidelines. In order to have everyone be responsible for the material, I will highlight the different math smarts that are needed and make sure everyone has a role in the group.

In Project One Part II, I learned about many of the students’ interests and places they like to visit in the DeWitt community. I plan to use this information to create tasks that relate directly to their interests. For example, many of my students are interested in sports, so I will be creating a few tasks that involve football, gymnastics, and swimming. My pre-assessment had word problems that related to the DeWitt farmer’s market and a garden that one of the classrooms planted at our school. I will continue this trend with the lessons I teach in order to keep the students engaged. I am hoping that this will show the students more examples of how math is applicable in their life outside of school.

(See Updated Chart on the Wiki Page)

Section 4: Projected Sequence of Lessons

Date: 10/28/13

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will connect repeated addition to multiplication through identifying equal groups of objects and finding totals.

Rationale: The idea of repeated addition is familiar to students from second grade. They will use this background knowledge they have of repeated addition to find a total number of objects. This lesson is important as it introduces the necessary foundational skills of multiplication, such as “group” and the multiplication symbol, that are integral in future grades. This lesson helps students move from addition, which is a mastered skill for most students, to using multiplication as a more efficient way to find totals (or products). In project 1, many students said they enjoyed playing or watching sports, so students will be working with a task that involves football.

Brief description/overview of lesson:

Launch of task: When a field goal in football is “good”, the referee holds up two arms. I have will have 10 students come to the front and pretend to be referees and hold up their arms. As a class, we will practice skip counting their arms (skip counting by two’s) to find how many arms we have altogether As students are skip counting, I will write 2+2+2+2+2+2+2+2+2+2=20. I will think aloud, “I wish there were an easier way figure out the number of arms besides having to write all of these 2’s.” Students will turn and talk with their group about an easier way to write the repeated addition sentence, looking for students to think of equal groups. We will then discuss how many groups of two arms there are, what is one group, and how many are in each group. Underneath the addition sentence, I will write “10 2’s”, “10 groups of 2”, “10x2”.

Partner work task: Students will work with their eyeball partners to find the relationship between repeated addition, counting groups, and multiplication. I will give each pair 12 counters (unfix cubes). The partners will practice making groups of two with their counters. For those students who finish early, I will challenge them to find as many different equal groups as they can create.

Discussion of task: Partners will share the equal groups they created. Students will write on their whiteboards the repeated addition, equal groups, and multiplication sentences for each pair that share. I will use this whiteboard share as a formative assessment. We will discuss if these three number sentences are all saying the same thing. After a few pairs share, I will draw

equal groups of footballs and non-equal groups of footballs and their related equations on the overhead. Students will put their thumbs up or down if they agree that my work is correct. They must be able to defend why or why not. I will wrap up by reminding the students that you need equal groups to multiply.

Independent Work: Students will complete the exit ticket for this lesson. I will read aloud the questions to the students who an IEP. I will use this exit ticket to determine the students who will need to pulled for a small group for tomorrow’s lesson.

Homework: I will ask students to find an example of equal groups at their house and draw a picture of it.

Materials: 12 unifix cubes for every student, exit ticket for every student, projector, student whiteboards

Plans for Formative Assessment: I will use the whiteboard responses and the thumbs up/down that students were giving to determine who I need to monitor while students are working on the exit ticket. I will use the exit ticket to structure tomorrow’s lesson and decide if we need to review equal groups again before moving to the array model.

Daily Reflection: Today’s lesson went very well! I realized very quickly that the students had a lot of prior

knowledge about multiplication. In the opening activity, I had them talk with their groups about what they know about multiplication. We were able to compile a list of many things that were going to be covered in that day’s lesson. Because so many students were familiar with multiplication (from second grade), this lesson was a review for most of them. The launch of the problem, which involved having students stand as football referees in the front of the class, went very well. The students were engaged, and they were able to quickly find ways to count the referee’s arms faster, such as skip counting and multiplication. The students then worked in partners to make equal groups with unifix cubes. Many of the pairs did this very quickly, so I challenged these students to find as many equal groups as they could. This was a great extension, as this kept them busy while I worked with the students who needed extra guidance. I recorded brief notes as I went around to talk with some of the partners. I recorded that all of the pairs were able to easily create equal groups, but they struggled to explain why they had equal groups and how they created equal groups. I pushed the students to be able to explain the strategy they used to create equal groups. During the discussion portion of my lesson, I had 2 pairs share the different equal groups they had created. The first pair had 2 groups of 6, while the second pair had 3 groups of 4. Each pair was able to show their equal groups to the class, but I had to prompt them with many questions about how they knew they had equal groups and the strategy they used to find equal groups. None of the pairs used the “fair-share” method to find equal groups. A majority of the students used skip counting (grouping the unifix cubes by 2, 4, 6, 8, etc.). While these pairs were sharing, I had the rest of the class record a repeated addition and multiplication sentence for the pairs’ equal groups. I used this whiteboard activity as a quick check to see which students understood the connections between multiplication, equal groups, and repeated addition, and which students did not. I then targeted these students who were struggling while they worked on the exit ticket. The students who were struggling with the whiteboard activity seemed to have problems writing the multiplication sentence. I stressed to them that the multiplication sign means “groups of”, and that we first write the number of groups and

then write the size of each group. I had the students complete the exit tickets independently. I made sure to check on those

students who were having problems during the whiteboard activity. After I collected and graded these tickets, only 5 of the students received a “2” and 1 student received a “1”. The students who received 2’s were able to correctly answer one question out of two. The student who received a 1 was not able to answer either question correctly. The majority of these students were not able to write a multiplication sentence to represent a picture of 3+3+3=9. To be sure that these students do not fall behind, my mentor teacher will be working with them in a small group during tomorrow’s lesson. She will review equal groups, repeated addition, and the meaning of multiplication. Tomorrow’s lesson is an extension of today’s lesson, so all students will be reviewing what we learned today; however, these 6 students will be getting more individualized support.

I did not have to change too much during the lesson. I had to delete some of the conclusion of my lesson because we ran out of time. I was not able to get to the example where I draw groups that are unequal and students explain why multiplication does not work. Tomorrow’s lesson will go as planned because the class overall did very well with this first lesson. They were engaged and participating. They were also working hard. We will be pulling those 6 students to work in a small group, but otherwise, tomorrow’s lesson will be introducing the array model and relating it back to what we learned today.

Date: 10/29/13

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will compare the equal groups of objects in scattered configurations to the array model, exploring the correspondence between 1 equal group and 1 equal row.

Rationale: This lesson continues from yesterday’s lesson as students relate the equal groups in scattered configurations to equal groups in an array. They will begin to distinguish the number of groups and the size of groups when they count rows and how many in 1 row. The arrays will continue to be used as a model throughout the entire unit, so this is a foundational lesson that students really need to understand and master before moving any further into the unit. Students will recognize the real-world applications of arrays, and the efficiency of these arrays as they skip-count to find totals.

Brief description/overview of lesson:

Launch of task: I will show students pictures of many arrays they can find in real life: ice cube tray, muffin pan, Legos, egg carton, eye shadow palette, and a paint tray. I will ask students to turn and talk with their neighbors about arrays they may have seen before. I will then pose this question to students: “Jordan uses 2 ice cubes in 1 glass of lemonade. He makes 4 glasses. How many ice cubes does he use altogether?” I will have students work individually in their math journals. We will then discuss ways that students solved this and draw on what we learned

yesterday about equal groups. I will show students how I used my ice cube tray (an array) as a faster way to organize my groups and count the number of total ice cubes. I will show students the ice cube tray and discuss the features of an array (straight lines=rows, equal amount in each row). I will also talk about how to write the multiplication fact for the array: first write the number of groups and then the size of each group, then skip-count to find the total number of objects in the array. Students will write this example in their math journals for future reference throughout the unit.

Partner Work Task: Students will create their own array posters. I will model the required information they need to have on this poster. They are able to draw any array that they want, but they must be able to explain how many rows are in their array, how many objects are in each row, and the corresponding repeated addition and multiplication sentence for the array. We will hang these posters up for future reference throughout the unit.

Discussion of Task: Students will share their array poster. I will be taking notes as to which pairs are able to successfully draw an array, explain how many groups, how many objects in each group, and write a multiplication sentence for that array.

Independent Work: Students will complete the exit ticket independently. I will monitor students’ thinking as they do this. I will also pull students for a small group at this time if they need more instruction from yesterday. I will use some of the problem set problems found in the curriculum to help these students work on skip counting with equal groups.

Materials: ice tray, pictures of real-life arrays, chart paper & markers (one for each pair), math journals, 1 exit ticket for every student

Plans for Formative Assessment: I will look over the students’ array posters and their exit tickets to determine which students will need to work in a small group for tomorrow’s lesson. I will also monitor students’ work as they write in their math notebooks and create the array posters. Since we will be working with arrays throughout the entire unit, students will be receiving a lot of practice with these.

Daily Reflection:After yesterday’s great lesson, I was expecting much of the same; however, this was not entirely

true. The students were engaged and participating during my lesson, but I had a lot more students with difficulties than the previous lesson. The lesson began with myself introducing real-life examples of arrays. I asked the students what they noticed about these arrays. The students were able to say that there was the same amount of objects going side to side and up and down. I was able to use this explanation to introduce the words row and column. From there, we moved to our high-level word problem. Students solved this independently in their math journals, and I stressed that they could use any strategy they would like (from yesterday or today), but they had to be able to explain their thinking with pictures and a multiplication sentence. During this time, my MT pulled the small group that I had identified yesterday and worked on this task with them at the carpet area. As I walked around to monitor, most of the students were using the equal groups strategy, drawing 5 ice cubes in each glass of lemonade. There were 2 students who used the array model that had just been introduced. A handful of

the students finished this task quite quickly, so I had them do the same problem using larger numbers (11 x 6) since this is not a fact that many students have memorized. This kept the students busy while the others finished the original task. My teacher was able to monitor those in her small group and would send them back to work in their seats when she saw that they understood yesterday and today’s lesson. By the end of the task, she only had 1 student left in her small group. I made note of this to watch him closely during the remainder of the lesson.

I then had 2 students share their work with the class. The first student shared how she used equal groups to solve the problem. The second student shared how she used an array to solve the problem. I again had to do most the prompting to these students, such as the number of groups, the size of each group, what each number represents in their multiplication sentence. I then had the rest of the students give thumbs up or down if they agreed or disagreed with the work that was shared. Almost all students had their thumbs up. In tomorrow’s lesson, I plan on asking these students to defend why they agree or disagree with their classmate’s work. This is a new way of discussing for them, so I am trying to ease them into it. I then connected the equal groups to the array and why they both work to solve the same problem.

Students then worked with their shoulder partners to create an array poster. I was very explicit in giving directions and modeling exactly what needed to be on the poster. The students were free to choose any array they wanted to create, both in numbers and design. This task was very eye opening for me, and I was glad I supplemented it into the curriculum. I was able to move from group to group and examine their misconceptions about an array. The main issues that I noticed for most groups were fine-motor problems. We have a few students that struggle with fine motor, so it was hard for them to line up their drawings into rows and columns. Their rows would often be uneven. We had to stress the importance of why the arrays need to be organized, and we had students erase and start over, using their notebook to help them line up each row they drew. I was very happy with the way the students pushed themselves, however. Almost all pairs tried to create an array that was harder than the one we had worked on before. For example, pairs tried a 6x7 array, a 8x4 array, and a 9x9 array! Another misconception that students had was writing the multiplication sentence to represent their array. A few pairs were not writing the factors in the correct order (number of groups x size of groups). This is going to be something that I discuss in great detail during tomorrow’s lesson because it was a big error in almost every poster. There was one pair that was way off base with their multiplication sentence, using numbers that were not present in their array. This pair also had the one student that was left in my MT’s small group. All but 4 pairs were able to finish their posters. I will be giving those pairs that did not finish 10 minutes during the beginning of math to complete the poster. I have also noted on my spreadsheet which pairs did not finish and the main struggles they were having (i.e., incorrect order of factors). These students will be working with my MT tomorrow during the task to review drawing an array, and how we can write a multiplication sentence to represent the array. We will be working with arrays throughout the entire unit, so I believe that some of the fine motor problems that a few of the students are having will dwindle as we practice drawing arrays every day.

I did not change too much throughout the lesson. Again, I ran out of time at the end of the lesson, so students were not able to complete the exit ticket. They will be completing this tomorrow during their math warm-up time instead. If I were to do this lesson again, I would like to stress more of the connection between equal groups and arrays. I would like to show students exactly how they are related (1 equal group=1 row). I will now be doing this during a review problem for tomorrow’s lesson. I will be sure to emphasize how we can skip count each row in an array just like we skip count each group to find the total. Tomorrow’s lesson is a continuation of today. Students will be reviewing many of the main ideas that have been introduced. I will also be introducing the definition of factors as the number of groups or the size of groups. I’m hoping that tomorrow’s lesson will also clear up some of the

confusion about first writing multiplication sentences as _______ groups of _________ = ________ and then _______ x _______ = ___________. I will be using tomorrow’s problem set as a type of review from the first 3 lessons. This will help me determine who may need extra practice problems or needs to be in a small group every day.Date: 10/30/13

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will analyze equal groups given in scattered configurations and organized into arrays to determine whether factors represent the number of groups or the size of a group.

Rationale: This lesson is a continuation from yesterday’s lesson, as it encourages students to think about what the meaning of the factors are as they begin to multiply. It asks them to apply what they learned about the array model and the multiplication sentences they created for the array model when thinking about factors. This lesson will benefit students because they will begin to see the meaning and purpose behind multiplication aside from what they know when memorizing facts for a timed test. This will also benefit students when creating their own multiplication word problems, as they must realize that each factor in a multiplication number sentence has a purpose. They will transition from knowing 8x4=32 to knowing why 8x4=32. This lesson will relate to the students because they will be a part of an equal group, which will require them to move around the room. This will benefit those students who have trouble focusing at their seats and need movement throughout a lesson.

Brief description/overview of lesson:

Launch of task: I will ask students to divide themselves (plus the two teachers) into 4 equal groups and each group will stand in a corner of the room. The students have to do this with a level 0 voice, however. I will give them a minute to think about their plan for dividing the class. Once the minute is up, they will divide themselves without talking, and their group will give me a thumbs up when they’re ready. We will then discuss how many equal groups we made and how many students are in each group. I will have students come back to their seats and work with their group to determine what the multiplication sentence would look like for our activity. When each group has figured out the sentence, I will introduce the new words factors—the number of groups and the number in each group.

Partner Task: I will pose the following task to students: “Pretend our class divided themselves into 4 equal groups and there were 9 students in each group. How many people are in our pretend class?” Students will work with their partner to draw an array, write a multiplication sentence, and write underneath each factor what it represents in the array. Students will do this in their math journals. I will monitor students as they do this and take observation notes. I will use my notes to select and sequence at least 2 pairs to share.

Discussion of Task: The pairs I choose will share their arrays and multiplication sentences with

the class. Students will give thumbs up and thumbs down if they agree or disagree with their work. These students will have to defend why they agree or disagree. I will again reiterate the meaning of the word factor and how we can figure out if the factor is the number of groups or the size of each group in an array. Independent Work: Students will complete an individualized graphic organizer in which there is a multiplication problem in the middle (more challenging numbers for those students who are excelling). Surrounding the middle, there are 4 squares. Each square is labeled with what the student must write: skip counting, repeated addition, equal groups, and array. I will not be helping students as much on this because I would like to use this as a more “formal” assessment for how the students are doing for the first 3 lessons of the unit.

Materials: math journals, graphic organizer for every student

Plans for Formative Assessment: I will be taking observational notes as the students work with their partners. I will look for who is able to apply what we have been learning about arrays to the new work with factors. I will use also the graphic organizers as an assessment to see if students have grasped the objectives from the first 3 lessons. I will grade these and determine which students need small group and individual attention. This graphic organizer will also help me determine if we can safely move onto division in our next few lessons. I do not want to move to division before the foundational pieces of multiplication are comfortable for all students.

Daily Reflection:The first activity that asked students to divide themselves into 4 equal groups went

smoother than expected. I could tell that it was very challenging for the students to divide themselves across the room without talking. I did find, however, that a couple of students were dominating this activity and pointing at people to tell them where they needed to go. There were definitely a lot of “followers” during this activity. I do like this activity, however, because it gets them moving out of their seats and does require some teamwork. During the partner task, those students who I noted on the spreadsheet that struggled with yesterday’s lesson worked in a small group with my mentor teacher. The rest of the students worked in partners to complete the task listed above. They quickly went through this task and were able to identify what each of the factors in the problem represented, so I then gave these students the same task but with more challenging numbers (12 students in each group, 4 equal groups). This was much harder for these students because they were unable to draw from the facts they have memorized. They had to use some of the strategies we have been discussing, such as skip counting and arrays. I held the discussion for this lesson with only those students were not working in the small group. Those students that I asked to share were able to explain their thinking clearly to the class, however, we do need to practice defending why we agree or disagree with our peers. We will continue to practice repeating and revoicing. I decided against the graphic organizer for their independent work because we had run out of time, and a small group of about 11 students was not able to take place in the discussion. I instead gave a homework assignment (Eureka problem set) with review problems from the first three lessons. I also compiled all of my observational notes from the first 3 lessons and found that 6 of my

students were excelling and need to be challenged in the next few lessons. Many of the struggles that my students had from these first 3 lessons were writing the factors in the wrong order when writing a multiplication sentence (# of rows x size of each row). While they get the same product, their sentences do not represent their arrays. A few of the students are still struggling to identify rows and size of rows in an array to even write a multiplication sentence at all. Both of these misconceptions will be addressed as we continue to practice with arrays. Tomorrow, because it is Halloween, I will be pushing my unit back one day as we review all of the concepts learned thus far in the unit.

Date: 11/1/13

CCSS(s): 3.OA.2: Interpret whole-number quotients of whole numbers.3.OA.6: Understand division as an unknown-factor problem.

Learning Target(s)/Objective(s): Students will find the meaning of the unknown in division as the size of groups through word problems that help give meaning through context.

Rationale: Students are now moving from division to multiplication. As they do this, however, we will be linking the study of factors that was discussed in the previous lesson to the use of factors in division. Students will learn to understand division as an unknown factor problem. We will first introduce the idea of the unknown factor as the size of group in division during this lesson. This lesson is very important for future grades, as they are learning division conceptually and how to interpret problems by writing division expressions. If students are not learning the foundational skills of division, then they will suffer in future grades when they begin division with larger numbers. This lesson will relate to students’ lives as we will discuss a place in the community that many students are familiar with—the DeWitt Farmer’s Market. The farmer’s market example was used in their pre-assessment, so my goal is to make connections between what the students knew on the pre-assessment to what they are learning about division now.

Brief description/overview of lesson:

Because many of the students have never worked with division before, I will be modeling a task for the students as they move from representing division as “fair-share” to pictures to abstract. I will not be holding a typical “task discussion” as I have in previous lessons.

Concrete to abstract: “Yesterday, my neighbor bought a bag of 18 apples from the farmer’s market. He wanted to share them with me, so he divided them into 2 equal groups. Do you know how many he gave me?” Students will each be given 18 counters to represent the apples. They will divide the counters into 2 equal groups by giving one to my neighbor and one to me, one to my neighbor, one to me (model this on the whiteboard). We will then discuss the answer and write a number sentence, making sure to discuss what the unknown in the

sentence represents. Students will try the same task with 15/3.

Pictorial to abstract: I will draw a picture with 3 groups of 4 apples. We will discuss what 4, 3, and 12 represent in the picture and write a division sentence showing this. I will also discuss the difference between 12/3=4 and 12/4=3. I will talk about how we want the division sentence that represents the size of the group, so we want 12/3=4.

Abstract to pictorial: I will write 8/4= _______ on the board. I will say, “If 8 is the total and 4 is the number of groups, then what does the unknown factor represent?”. Students will discuss this with a partner and draw a picture on their whiteboard to go with the division sentence.

Independent work: I will have students complete the exit ticket for this lesson. On the bottom of the exit ticket, I will have students will create one own word problem that describes the unknown as the size of the group. During this time, I will also work with a small group of students based on the independent work from yesterday (graphic organizer) to focus on the different multiplication strategies we learned: array, repeated addition, skip counting, and equal groups.

Materials: 18 counters for each student, student whiteboards, 1 exit ticket per student

Plans for Formative Assessment: I will be using the exit ticket to determine which students understand that an unknown in division could represent the size of the group. I will be looking to see if students can do this with both pictures and equations. This will help me decide which students will need to be pulled to review the lesson and work on additional problems during small group tomorrow. I will be observing and monitoring students as they work with their counters and whiteboards to create pictures and number sentences that accurately depict the task.

Daily Reflection:Today marked the first day of learning about division! I was a little nervous that students

were not solid enough with multiplication to move on to division, but my MT encouraged me that we needed to keep moving and they will continue to practice multiplication. I could tell that the students were also a little nervous about learning division, as it seemed like this “foreign concept” to them; however, by the end of the lesson, one girl told me that “division is easy!”. I liked the flow of the lesson, moving from manipulatives to pictures to abstract. I do, however, think I should have given the students more movement throughout the lesson. I found that it was a long lesson, and they were at their seats for most of it. I should have moved from their seats to the carpet or let them move around during partner work. Many of the students felt comfortable using the “fair-share” method with the manipulatives; however, they struggled a little bit with creating pictures. I had to remind them to draw their “circles” (or groups) first and then make tally marks in each group using the fair-share method. The biggest misconceptions I saw were when writing a division sentence for their pictures. I found that I should have introduced the vocabulary of quotient, divisor, and dividend to them at this point. If I could change the lesson, I would introduce this at the start. Students were confused about

the order of the numbers, and struggled to see that the “total” comes first when writing a division sentence. My MT was gone during this class period, so I did not have a small group work today. She will be working with the small group on Monday with the 7 students who were either absent or received “below grade level” on their exit tickets.

Date: 11/4/13

CCSS(s): 3.OA.2: Interpret whole-number quotients of whole numbers.3.OA.6: Understand division as an unknown-factor problem.

Learning Target(s)/Objective(s): Students will find the meaning of the unknown in division as the number of groups through word problems that help give meaning through context.

Rationale: This lesson directly extends from yesterday’s introduction to division. Students will continue to learn division as an unknown factor problem. This lesson will focus on the unknown factor as the number of groups in division. This lesson is very important for future grades, as they are learning division conceptually and how to interpret problems by writing division expressions. If students are not learning the foundational skills of division, then they will suffer in future grades when they begin division with larger numbers. This lesson will relate to students’ lives as we do a gymnastics-related task. Many of the students expressed an interest in sports (specifically gymnastics), so this will task will engage them and help them realize the real-life applications to math.

Brief description/overview of lesson:

Launch of task: We will move to the carpet and students will work on the problem together: “Next weekend is a gymnastics competition at the school. 18 judges will be at the competition. They need our help to set up the tables for the judges to sit at. We know that 6 judges can sit at each table, but we’re not sure how many tables we’ll need.” They will turn and talk to their neighbor about what information we already have. Students will model with their counters how to divide the 18 counters in groups of 6. I will watch for a student to use the “fair share” strategy and have an equal number of 6 in each group. I will share this example with the class. When everyone in the class agrees we need 6 tables, I will write the equation 18/6=3 and ask students how this equation relates to our problem.

Partner Task: This question will be used as the high-level task for this lesson: “There will be concessions stands at the gymnastics meet. Norah plans to buy 15 candy bars. 3 candy bars come in a pack. How many packs should she buy?” Students will work on this task in their math journal. They will include the following: what the numbers 15 and 3 represent, what the unknown represents (size or number of groups), and write a division sentence to find the answer. I will be monitoring as students work to see which strategies students used to find the answer: pictures or numbers or words.

Discussion of Task: I will call on a few pairs to share their strategies. Students will put their thumbs up or down if they agree/disagree with the partners. If a pair does not use the strategy of drawing a picture showing equal groups, I will draw 5 groups with 3 in each group and write 3, 6, 9, 12, and 15 under each group. I will have students share what they think I did and what this reminds them of from earlier in the unit (skip-counting). I will have students practice this with 21/3 = ______ and 14/2= _______.Independent work: I will have students complete the exit ticket for this lesson. On the bottom of the exit ticket, I will have students will create one own word problem that describes the unknown as the number of groups. During this time, I will also work with a small group of students based on the exit tickets from yesterday’s lesson. I will pull those students who struggled to use pictures and numbers to complete simple division word problems where the unknown was the size of the groups.

Materials: 18 counters for each student, 1 exit ticket per student, student math journals

Plans for Formative Assessment: I will be using the exit ticket to determine which students understand that an unknown in division could represent the number of groups. I will be looking to see if students can do this with both pictures and equations. This will help me decide which students will need to be pulled to review the lesson and work on additional problems during small group tomorrow. I will be observing and monitoring students as they work with their partners to complete the task. I will record the strategies that students are using and how comfortable they seem working with pictures and equations.

Daily Reflection:Again, the students practiced drawing pictures, arrays, and writing division sentences. The

biggest struggle that students had today was determining what the meaning of their quotient should be. Yesterday they worked with quotients that represented the size of the groups. Today they worked with quotients that represented the number of groups. We continued to work with writing the total number of objects first in the number sentence. I also found while grading their exit tickets today that that was the biggest misconception for most students: determining whether their unknown was the number or size of groups. Those students who I identified that needed a challenge, I had them write division word problems on their exit ticket where the unknown was the number or the size of the groups. We had 7 students again who were either absent or received “below grade level” on their exit tickets. Tomorrow, instead of moving on to Lesson 6, we will be reviewing division again with those students who struggled. The other students will be completing one of our required journal tasks, practicing a high-level asking them to identify factors and create as many arrays as they can for the number 20.

I also find that I am struggling with the lack of technology that our classroom/school has. In our room, we only have an overhead projector. I find that it is hard for students to share their work (and also takes up a lot of time) because they have to re-write whatever was in their math journal to share with the class. I also find that when I am modeling something or want to introduce a new problem, I have to re-write it on the overhead. While it is doable, it takes up a lot of time. I really wish that we had a document camera so students could put their authentic work underneath and we could quickly start a discussion. I also find that it would be easier for

all students to see, as most students are not great at writing legibly and large enough for everyone to see on the whiteboard.

Date: 11/6/13

CCSS(s): 3.OA.2: Interpret whole-number quotients of whole numbers.3.OA.6: Understand division as an unknown-factor problem.

Learning Target(s)/Objective(s): Students will use arrays to write multiplication sentences and find unknown factors, then write division sentences where the quotient represents the same as the unknown factor.

Rationale: This lesson has students applying what they’ve learned about division so far (unknown factor problems) to the array model that was introduced during multiplication. Students will interpret these arrays to write division sentences. This lesson is important because it really shows the connection between multiplication and division. Students can use the skills that they have from multiplication already (including their timed facts) to help them solve division problems. They will do this through relating the unknown factor in multiplication to the quotient in division. This lesson really helps move students toward solidifying their understanding of both multiplication and division. I will again relate today’s lesson to the interests of the students, including the students’ love of sports. The topic today will be basketball!

Brief description/overview of lesson:

Launch of task: I will pose the following question to students: “20 college basketball players are playing in the NCAA tournament. There are 5 players on each team. How many teams are playing in the tournament?” Students will work independently on their whiteboards to create a division sentence for this problem. While students are doing this, I will be drawing an array that represents this problem on the overhead. We will discuss if the division sentence they created and the array I drew are similar. We will talk about the array and what the total number of dots, number of dots in each row, and number of rows represents in our problem.

Partner Task: Students will work in partners to complete the same task as above but using different numbers. I will have students work with a person of similar ability. The students who have been succeeding during the unit will be given more challenging numbers to work with. Those students who are struggling with the unit will be given easier numbers to work with. The students will work in their math notebooks to draw an array from the division sentence I give them. Those students who struggle with this task may underline each row to literally show division and circle each row to show the size of each group. I will help them explain each step they take.

Discussion of Task: Students will share the arrays they drew with a “math talk”. Similar to the other lessons, students will be asked to give thumbs up or down if they agree or disagree with the arrays. I will pick one array to focus on (i.e. 15/3=5 where the quotient represents the size of the group) during the discussion. I will ask students to write both a division and multiplication sentence for the array. I will ask, “Where do you find the quotient in our multiplication sentence?” (second number, size of the groups, factor). I will solidify that the unknown factor represents the same as the quotient.

Independent work: Students will complete two assignments during independent time. They will first create a word problem solved with an array where 21/3=7. Students will then complete the exit ticket for the lesson, which has them draw an array and complete number a multiplication and division number sentence based on a word problem. The students also have to explain what the unknown factor and quotient represent. I will also work with a small group of students to give additional guidance based on yesterday’s exit tickets. We will work on solving with equations and pictures a division problem where the unknown is the number of groups.

Materials: student whiteboards, 1 exit ticket per student, student math journals

Plans for Formative Assessment: I will be using the whiteboard activity to monitor what students remember from yesterday’s lesson in creating division number sentences where the unknown is the number of groups. I will use the exit ticket and independent work time to determine my plans for tomorrow’s lesson. I will look to see that students understand the connection between division and multiplication, specifically the relationship between the unknown factor and quotient. If students are really struggling to grasp this, we may spend an extra day on this before moving to the commutative property of multiplication.

Daily Reflection: Lesson 6 has been the most challenging lesson/concept taught so far in this unit. Students

had to apply what they’ve learned about both multiplication and division and connect the two properties together. Before I began this lesson, I changed my original ideas and had the students complete a vocabulary page in their math journals. We defined the words factor, product, and quotient. While I had mentioned these words, having the students write them down solidified this for them. They could also refer back to it throughout their tasks. The lesson was similar to those in the past. I should have incorporated more movement throughout the lesson, as I find that our curriculum could all be taught while students are sitting in their seats and I’m talking. While students did work in partners, they were not given the opportunity to move very much with their partner throughout the lesson. I held a discussion again with those 21 students who were not working in a small group with my mentor teacher. I’m finding it hard to include a discussion where everyone is involved because the students that are not in the small group typically finish quicker and are ready to share and discuss their answers. I want to have everyone involved in the discussion! I am finding that the students are becoming more comfortable with what I call “math talks”. When I’m monitoring and observing partners, many

ask me if they can be the group to share with the class. Many of the students seem to have no fear in sharing their work with the class. The students have also been very respectful if they disagree with someone’s work. I tend to choose at least one group that had a misconception, so we can address this in front of the class. The class is very polite in stating why they disagree, and I have not had any hurt feelings about this. The biggest change I would like to make to tomorrow’s lesson and future lessons is really using the vocabulary of factors, products, and quotients. The more students hear it, the more they will become comfortable using it. I also am going to make vocabulary signs to hang up in front of the classroom as a reference to students.

Date: 11/7/13

CCSS(s): 3.OA.2: Interpret products of whole numbers3.OA.5: Apply properties of operations as strategies to multiply and divide.

Learning Target(s)/Objective(s): Students will demonstrate the commutativity of multiplication through array models by writing two different multiplication sentences to interpret the same array.

Rationale: Students will draw on their previous experience with arrays in this unit by being introduced to manipulating arrays to study the commutative property. In my pre-assessment, I noticed that many students are able to recognize the directional differences when an array is rotated 90 degrees. I will use this knowledge to teach students that the meaning of the factors changes depending on the orientation of the array. This lesson is very important as students move into higher grades, as the commutative property is an efficient way to solve multiplication with larger numbers. If they know that they can switch the order of the numbers they multiply, they will see that they can use this strategy to solve twice as many facts. Manipulating the arrays is a great visual and concrete way for students to understand why the commutative property works. I learned from Project 1 that many students do not believe highly of themselves in math and think they are smart at math only if they are succeeding at Rocket Math timed tests. I will use today’s small group work to point out all the different smarts that will be needed to complete the assignment.

Brief description/overview of lesson:

Launch of task: Students will move to the carpet with their whiteboards. I will have them rotate their whiteboards so it’s horizontal. I will have them draw a line down the middle to make 2 sides. On the left side they will skip count by two 4 times and write each number. On the right side, they will skip-count by four 2 times. I will have them turn and talk to discuss how these two sides are related. I will then have them draw an array under each side to match their skip counting. I will also be doing this on chart paper to model it for the students. Students will then describe what they notice about the arrays. We will then write and solve multiplication sentences to show the total objects in each array. (2x4=8 and 4x2=8). I will guide them to notice that the factors switch places when the array rotates by asking the following questions: Is there

the same total for both arrays? Where do you see the number of groups? Is 2x4 the same as 4x2?

Small Group Task: Students will go back to their seats and work with their team to figure out why 2x4=4x2. They will work together, and I will emphasize the different smarts that may be needed to complete the task. I will be monitoring as groups work together, making sure that students are using the vocabulary: rows, columns, factors, skip counting, and total. Discussion of Task: Each group will present to the class their findings about why this statement is true. I will finish the discussion by explaining that when multiplying (just like addition), changing the order of the factors does not change the total. This is called commutative.

Independent work: Students will complete the exit ticket for this lesson by explaining why they agree or disagree with 2x5=5x2 using both an array and skip counting. I will be working with a small group during this time based on the exit tickets from yesterday. I will be working with students who struggled with connecting multiplication to division using the array model and unknown factors.

Materials: student whiteboards, 1 exit ticket per student

Plans for Formative Assessment: I will be using the group presentations to show me that they are able to explain their thinking clearly to the class. I will be looking to see students using the math vocabulary of rows, columns, total, array, factors, and skip counting. I will record notes to be sure that students understand why the order of the factors in a multiplication problem can be switched and how this relates to the array rotating. I will use the exit tickets from today to determine how to proceed for tomorrow’s lesson. Tomorrow’s lesson is a continuation of the commutative property, so I will use this to see which students may need more explicit instruction than just the group presentations.

Daily Reflection: The students were familiar with the commutative property of addition, as we had learned

about this earlier in the year, so the commutative property of multiplication was an easy transition for them to think about. The major difficulty that students had was explaining why the commutative property works. One of my math practices for this unit was defending their thinking and explaining their work clearly. Students really worked hard at that during this lesson. I did change this lesson a little from what I had originally planned. Instead of the small group task, students worked in partners to create a “community of the commutative property” poster. The students were able to choose their own factors (less than 12), create 2 arrays with these factors, and explain why the commutative property is true for their arrays. I emphasized to students that they use the vocabulary: row, columns, factors, and product. We did not have time for each pair to share their poster to the class, but I was able to collect these posters so I was able to look over their work. Students did very well choosing factors and drawing arrays with these factors. The main aspect that all groups struggled with was explaining why the commutative property worked for their arrays. Most could explain that their factors were switched and equaled the same product; however, they could not connect that to their arrays.

The commutative property is taught again in tomorrow’s lesson, so I will be sure to reinforce how the arrays represent the commutative property. Students did do very well with the exit ticket, however, because they were not asked to explain their thinking. Only 3 students received “below grade level” on the exit ticket.

Date: 11/8/13

CCSS(s): 3.OA.2: Interpret products of whole numbers3.OA.5: Apply properties of operations as strategies to multiply and divide.

Learning Target(s)/Objective(s): Students will demonstrate the commutativity of multiplication through array models by writing two different multiplication sentences to interpret the same array.

Rationale: Students will again practice using arrays to model the commutative property. We will be digging into the idea of columns becoming rows and rows becoming columns when an array is rotated. This lesson is very important as students move into higher grades, as the commutative property is an efficient way to solve multiplication with larger numbers. If they know that they can switch the order of the numbers they multiply, they will see that they can use this strategy to solve twice as many facts. Manipulating the arrays is a great visual and concrete way for students to understand why the commutative property works. This lesson will relate to the students, as they will be able to model with their bodies how the commutative property works. By getting the students moving and active in the problem, it should help to make the lesson more concrete.

Brief description/overview of lesson:

Review of yesterday: I will have students create arrays with their body as a review from yesterday. I will choose 18 students to come model the problem: “Children sit in 2 rows of 9 on the carpet for math time. Bryson says, “We make 2 equal groups.” Don says, “We make 9 equal groups.” Who is correct? Explain how you know.” Students will be given the opportunity to share with the class who they think is right, but they need to defend their argument with math vocabulary that was introduced yesterday.

Launch of Task: I will have students come to the carpet with their whiteboards. They will turn whiteboard so it’s vertical. We will practice skip counting by threes 4 times and drawing an array to match this. We will discuss how many columns and rows there are. Then, they will turn their board horizontal. We will discuss again how many columns and rows, and review the differences between the horizontal and vertical arrays. I will be sure to point out that the total number of dots did not change, and review the term commutative.

Partner Task: Students will practice with their eyeball partner to skip count, draw arrays, and write multiplication sentences. I will have those students at a higher-level work with more

challenging numbers, and those students at a lower level work with easier numbers. They will do this in their math journals. During this time, I will work with those partners in a small group who struggled yesterday on the exit ticket.

Discussion of Task: I will have a few pairs share their arrays and multiplication sentences on the overhead. Students will agree or disagree with their thumbs up/down. At the end of sharing, I will pose this question: “For an array that has 3 rows and 9 columns, is 3 groups of 9 and 9 groups of 3 equal?” I will have students talk with their partner to discuss this. As a wrap-up, I will be summarize the lesson by discussing that skip counting helps us draw arrays because skip counting is repeated addition, which is multiplication. I will also state how factors switch places in arrays when they rotate, and this is called the commutative property.

Materials: student whiteboards, math journals

Plans for Formative Assessment: I will be using the student whiteboards as a way to review from yesterday’s lesson. I will see if students remember the basics of the commutative property from arrays, and this will help me determine if students need more modeling of this before they work with partners. As partners are working in their journals, I will be monitoring and observing how they are skip counting, creating an array, and an accurate multiplication sentence. I will use the homework as a more “formal” assessment for the past 2 lessons, as I grade the homework to determine if students understand the foundation of the commutative property and are able to explain why it’s true for all numbers using arrays.

Daily Reflection: Instead of doing a review by having the students create arrays with their bodies, I had

them share their commutative property posters with one another. We did a gallery walk where each pair was able to look at their peers’ posters and determine if they agree or disagree with their explanation of why the commutative property works. I really enjoyed this activity because it allowed the students to move, and it also exposed them to different explanations that their peers had. By the end of this activity, most students were able to solidify that the commutative property works because when you switch the order of the factors, the product stays the same. What I wanted to work on after that activity was explaining how the rows and columns of an array also change, but the array still equals the same total. Students came to the carpet and we worked with our whiteboards on creating arrays and turning our whiteboards to notice how the rows and columns change places. We wrote multiplication sentences for the array when it is horizontal and when it is vertical. From observing their whiteboards, most students seemed to be able to quickly draw an array and write a multiplication sentence (in the correct factor order) for each array. I still had some students who were confusing rows and columns. If I could go back and change this lesson, I would have the students use arm motions when explaining rows and columns. I would have them put their arms straight up for columns and their arms straight out for rows. I think this movement would help to settle this confusion. Other than the confusion with the rows and columns, all students (except 7) were able to identify and show the commutative property using arrays. We will continue working on clear explanations as the unit wraps up.

Date: 11/11/13

CCSS(s): 3.OA.2: Interpret products of whole numbers3.OA.5: Apply properties of operations as strategies to multiply and divide.

Learning Target(s)/Objective(s): Students will use arrays to decompose unknown facts as the sum or difference of two known facts.

Rationale: As our works with properties continue, this lesson will serve as a basis to introducing the distributive property to students. This lesson is important because it teaches another strategy to solving multiplication facts. The distributive property also comes up again many times throughout algebra. Students will use arrays to help them visualize how a product can be decomposed into different groups of rows added together. This lesson is reflective of Project 1, as many students said they liked using manipulatives and they need more modeling. This lesson will be very structured and modeled, as opposed to many of our other lessons.

Brief description/overview of lesson: Modeling (“I do”)/Guided Practice (“We do”): Students will be given the “Three Arrays No Fill Template” and their whiteboards. They will come to the carpet. I will have them cover part of the array to show 5 rows of 3. They will draw a box around the uncovered array and write a multiplication sentence to describe it (5x3=15). Students will move the paper to show 7x3. They will shade the rows they added and write a multiplication sentence to describe the shaded part of the array (2x3=6). We will discuss how many threes are in 5x3 and how many threes we added to make the array show 7x3. I will write 7 threes=5 threes + 2 threes on chart paper, and 7x3= 5x3 + 2x3. Students will give me a thumbs up or down if they agree/disagree. We will then complete the problem by writing the totals for the two parts of the array: 21= 15 + 6. We will then move to subtracting 2 known smaller facts to solve an unknown large fact, with the example 10x3 to solve 9x3.

Independent Work (“They do”): The task today will be done individually because I want to see if students understand the lesson that was taught and if more modeling and scaffolding needs to be done because this is a tricky concept. Students will be given a distributive property mat. They will create an array on the left side of the mat with cubes. They will then draw the array and write their multiplication sentence. On the left side of the mat, they will break their array into two parts with their cubes, draw these two arrays, and write the multiplication sentences for the new arrays (i.e. (4x3) + (4x3) = 24).

Materials: student whiteboards, “Three Arrays No Fill Template” 1 per student, chart paper, student distribute property mat, cubes

Plans for Formative Assessment: I will be using the student whiteboards as a way to practice the guided release of responsibility model. I will first model how to decompose arrays using

sums, and students will practice this on their whiteboards. I will then model how to decompose using differences, and students will practice this. I will be observing and taking mental notes of how students are doing throughout this part of the lesson for who I need to target during independent time. I will take the distributive property mat that students complete to determine who may need to be pulled for a small group during tomorrow’s lesson.

Daily Reflection:This lesson changed quite a bit from my original plan. I began using the “Three Arrays No Fill

Template” that the Eureka curriculum provided; however, I found that this was not concrete enough for the students, and they were struggling to see how you can split an array into two parts to solve a multiplication problem. In the middle of the lesson, I decided to try something different and use the distributive property mat that I was going to use later in the lesson. I had the students move back to their seats, and we got out unifix cubes and “ones” cubes to work with. I had the students create different arrays for more challenging multiplication problems, such as 7x6 and 8x7. I had them “slice” their arrays (like surgeons, I told them) into two smaller arrays that were “friendly numbers”. I reinforced to students that they are able to choose how they slice the array because they know what numbers they are comfortable working with. Once the students seemed to see the process of actually slicing an array into two parts, I then had them move to just drawing the arrays. Throughout this, we were practicing writing multiplication sentences for the two sliced arrays and adding them together to find the total. I found that my higher-level students did not see the purpose of the distributive property because they could already multiply 7x6 and 8x7 in their heads. I reminded them that once they reach older grades, they will be working with much larger numbers that they will not have memorized. In order to solve these larger multiplication problems, splitting an array and using the distributive property is a strategy they can use. After this explanation, these students began to see the purpose of the lesson. At the end of the lesson, I had the students begin working on a worksheet that had 8 problems that were similar to what they had been doing in the lesson: drawing an array and slicing it into two parts and adding the multiplication sentences together to get the product. I like this worksheet because it was good practice for drawing arrays and writing correct multiplication sentences for these arrays.

Date: 11/12/13

CCSS(s): 3.OA.2: Interpret products of whole numbers3.OA.5: Apply properties of operations as strategies to multiply and divide.

Learning Target(s)/Objective(s): Students will write decomposition as (ac) + (bd) = acbd and explain each step of the solving process.

Rationale: Students will continue to explore the distributive property that was introduced yesterday. Today, however, it will be taken one step further. Students will actually learn to write the decomposition as (5x3) + (2x3) =21. They will also have to explain each step of the solving process, which is a much need skill to have as they get older. This lesson is important

because it teaches another strategy to solving multiplication facts. The distributive property also comes up again many times throughout algebra. Students will use arrays to help them visualize how a product can be decomposed into different groups of rows added together. Students will be solving a task that was on the pre-assessment that relates to a real life situation at Scott School.

Brief description/overview of lesson:

Launch of task: I will draw an array on the board that is 4x6 and explain that this is the plot of the Scott School garden. I will discuss how this summer they planted tomatoes and carrots, but they needed to keep them separated in the garden. I need students to draw a slice through the array to separate the vegetables and then write corresponding multiplication sentences to represent the total number of vegetables.

Partner Task: Students will work with partners to complete the task in their math journals. I will be looking to see if students are successfully drawing a 4x6 with a slice through it and writing the corresponding number sentences next to the array. I will be monitoring and taking notes as students complete the task to select and sequence a few pairs to share their work.

Discussion of Task: The pairs that I select will share their arrays with students on the overhead. We will discuss how they sliced their array and how they used the two multiplication facts to find the total number of vegetables. I will ask students how they know that the 2 number sentences are equal. For each array that is presented, we will practice writing the steps of the decomposition process, and explain why each step works. I will also model using the language of the number of groups and the size of groups to reflect on previous lessons. Students will turn and talk with their partner about the steps using the 3 equations of the decomposition process.

Independent Work: Students will complete the exit ticket individually, which has them completing number sentences of arrays split into two parts and completing the equations of the decomposition process.

Materials: student math journals, 1 exit ticket per student, overhead

Plans for Formative Assessment: I will be monitoring and taking notes as students work with their partners to complete the task. I will be taking notes on whether students are able to draw the correct array, split the array into two parts, and write correct number sentences for each step of solving the problem. I will also look over the exit tickets to determine who has met the objectives of the past two lessons, and who I need to review with before they take their summative assessment.

Daily Reflection: I had the students complete the distributive property worksheet that I gave yesterday as a

source of review from yesterday. Those students who received “below grade level” and were

absent yesterday worked with my mentor teacher during this time to review the splitting of the arrays into two arrays. After this, students worked in pairs to complete the task that is listed above. The students remembered this task from their pre-assessment, and I told them that they are going to be very surprised at how much they have learned since that assessment! The students did very well this task, and I asked many groups to split their array into a second or third way because they finished rather quickly. Once all pairs had at least one way to split their array and find the total number of vegetables, we had a discussion. This was the last discussion of my unit, and we had not had one in a few lessons; however, the students did well. They were again all eager to be picked to share in front of the class. I chose one pair that split their array in half and used skip counting to find their answer. I chose another pair that split their array into 1x6 and 3x6 and used the distributive property to find their answer. The rest of the class put their thumbs up or down if they agreed or disagreed with their work. Since both pairs were able to correctly find the total number of vegetables, all students agreed. I pushed the pairs who shared to explain their thinking clearly to the class, and I also pushed the class to defend why they agree. Once both pairs had shared, I then connected the group’s work that had used the distributive property to explain the steps of the decomposition process. I had students turn and talk about how I was combining steps and why at the end it was equal to my original problem of 4x6. I found that this involved a lot of pre-algebra, including parentheses and looking at both sides of the equal signs. When I gave the exit ticket, most of the class struggled with explaining the steps of the distributive property and combining the equations into a simpler form. I find that I am more focused on them adding the two multiplication sentences together to get the total product, then how they can simplify equations. As long as they can clearly explain their steps, I am okay with whatever way they choose to solve it. However, because many students did struggle to explain their steps, we will be reviewing this process again during our review day tomorrow before the summative assessment.

Section 5: Detailed Lesson Plans for the First Three Lessons

LESSON #1

Date: 10/28/13

Overall lesson topic/title: Equal Groups as Multiplication

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will connect repeated addition to multiplication through identifying equal groups of objects and finding totals.

Rationale: The idea of repeated addition is familiar to students from second grade. They will use this background knowledge they have of repeated addition to find a total number of objects. This

lesson is important as it introduces the necessary foundational skills of multiplication, such as “group” and the multiplication symbol, that are integral in future grades. This lesson helps students move from addition, which is a mastered skill for most students, to using multiplication as a more efficient way to find totals (or products). In project 1, many students said they enjoyed playing or watching sports, so students will be working with a task that involves football.

Materials: 12 unifix cubes for every student, exit ticket for every student, projector, student whiteboards

Procedures and approximate time allocated for each event

LAUNCH: Introduction to the Lesson (15 minutes)

How will I establish clear expectations for how they will be working and what products I expect? Today, we will be working with the whole group and in partners. We are going to review some behaviors that

I will be looking for as we work. I will have students share some of the behaviors they believe are appropriate when working with partners:

o Being respectful and using a level 0 voice when others are talking and sharing with the classo Participating by sharing my thinkingo If I disagree with someone’s answer, I need to do so respectfully.

How will I help them make connections to prior lessons or experiences in and out of school? I will ask students to quickly turn and talk with their team what they know about multiplication. I will walk

around to listen to the teams as they discuss. I will have some of the students share what their team discussed. I will write some of the common statements on the whiteboard.

This will be a quick activity, as I was would like to spend more time discussing what they learned through the task.

What will I say to help children understand the purpose of the lesson? “Today will be our first day as we begin to learn multiplication! I am going to read the “I Can” statement

that I have posted on the board. It says: I can connect repeated addition to multiplication through identifying equal groups of objects.”

“Today we will be learning the meaning of multiplication, which is having equal groups of something. We will work with a few strategies to learn multiplication today, and a few might be ones that you have heard of! Give me a thumbs up if you’ve heard of skip counting, repeated addition, or equal groups.”

How will I motivate them to become engaged in the lesson and understand its real world purpose? “How many of you think that multiplication is important in real life?” I will have students share a couple

examples of why they think it is important. I will be looking for students to share:o Trying to determine how many items to buy if items are in groups (i.e. tennis balls)o Finding the area of a football field or a backyardo Baking brownies or ordering a pizza and trying to decide how many pieces each person should get

“You’re going to see one example right now of a way that multiplication could be used in real life!”

How will I launch the problems? I will draw Popsicle sticks and have 10 people come to the front of the room. I will state the following: “When

a field goal in football is “good”, the referee holds up two arms. I’m going to have our volunteers pretend to be referees and hold up their arms. Let’s figure out a way to count their arms to figure out how many total arms there are. I’m going to count each arm individually first.” (students count with me 1, 2, 3, ….”)

“Is there an easier way to do this?” Students will turn and talk about a way that could be easier or faster to

count the total amount of arms. I will monitor students’ discussion, looking for students who are giving suggestions of skip counting.

When and how will I distribute materials and supplies? After we have finished the “launch” and discussed skip counting as a more efficient way to count, I will pass

out 12 unifix cubes to each partner pair. 12 unifix cubes will be given to each student because 12 has many factors, which means there are multiple answers and entry points for these to students to create equal groups.

Whole-Group Discussion of Launch Activity: (5 minutes)I will have students share who concluded that skip counting would be a more efficient way to count the total number of arms. I will have the students skip-count aloud (2, 4, 6, 8…) and write this addition sentence on the board: 2+2+2+2+2+2+2+2+2+2+2=20. Thumbs up if you see the correct amount of 2’s! I will think aloud, “Skip counting was faster than counting each arm individually, but I still wish there was an easier way than having to write all these 2’s.” Students will turn and talk with their group to about an easier way to write the repeated addition, looking for students to think of equal groups. I will call on a couple students to share that were able to think of equal groups (or multiplication).

“How many groups of two arms are there?” (10) “How many arms are in each group?” (2) Underneath the addition sentence, I will write “10 twos” and “10 groups of 2 is 20”. “We will be exploring the relationship between repeated addition, skip counting, and equal groups in your

partner task today!”

EXPLORE: Outline of Key Events During the lesson (20 minutes)

Partner Task:Students will work with their eyeball partners to find the relationship between repeated addition, counting groups, and multiplication. At this time, I will pass out 12 unifix cubes to each pair.

“You will have 12 cubes for each of you. I would like you to use your counters to make equal groups.” Differentiation: I will challenge those students who finish early to find as many different equal groups as they

can create with their 12 cubes.

As students are working, I will be monitoring and looking for students’ different strategies to create equal groups. I anticipate that students will use the fair-share method with their partner (one for you, one for me, etc.), or will use skip counting by 2s, 3s, or 4s to create equal groups. I will also be monitoring by having students check their equal groups:

How do you know that you have equal groups? How many are in each group? How many groups do you have? What strategy did you use to be sure you have equal groups?

SUMMARIZE: Closing Summary for the Lesson (20-25 minutes)

Whole-Group Discussion of Task: (15 minutes)I will select 3 pairs to share that used different strategies to find equal groups with their 12 counters: fair-share method and skip counting. I will sequence students’ responses by choosing the student who used the skip counting method last, as this will transition to our discussion of skip counting (repeated addition), equal groups, and multiplication. To connect to the objective of the lesson, as the groups are sharing, I will have the other students record the repeated addition and equal group statements on their whiteboards (i.e. 4+4+4=12 and 3 fours and 3 groups of 4).

If none of the students brought up the idea of multiplication while working with their cubes, I will lead the discussion toward introducing the multiplication sign. For example, if a group shares 4+4+4=12 and 3 groups of 4, I will say, “3x4 is another way to write 4+4+4”.

Connecting: “Are all of these number sentences saying the same thing?” Students will do a think pair share to discuss why the number sentences are all equal. Anticipated responses:

o They all have 4s in them and the answer is 12.o I think the 3 shows how many 4s there are.o You have to count four 3 times because there are 3 groups of them. That’s how you get 3 times 4.o 3x4 might be an easier way to write a long addition sentence.

“Ways that are easier and faster are efficient. When we have equal groups, like you made with your counters, multiplication is a more efficient way of showing the total than repeated addition.”

Independent Exit Ticket Activity: (5-10 minutes)Students will complete the exit ticket for this lesson.

Differentiation: I will read aloud the questions to those students with an IEP.

How will I structure the closing of the lesson? (5 minutes)I will first collect the exit tickets that students have completed.

I will the draw equal groups of footballs on the projector: 2 groups of 4 footballs. Students will put their thumbs up or down if they agree or disagree that there are equal groups. “Why are there equal groups?” (same number of footballs in each group). I will call on students who need to be able to defend their answer. Students will then write the multiplication and repeated addition sentences on their whiteboard. I will have them hold up their answers so I can observe which students understand how repeated addition and multiplication sentences are related. I will then draw unequal groups of footballs with the number sentence 3x4=12. Students will put their thumbs up or down if they agree or disagree with my work. I will call on students to defend their reasoning. Anticipated responses:

I disagree because my addition sentence equals 11, not 12 It’s because the last group doesn’t have 4 footballs.

You can do multiplication when the groups are equal.“I hear most students disagreeing because my groups are not equal. To multiply, you must have equal groups.”

How will I facilitate the sharing of student thinking?As stated above, during the closing of the lesson, I will call on students to explain whether they agree or disagree with my work. I will also have students record on their whiteboards so I am able to see quickly which students understand the basics of the lesson. During the whole-group discussion that I described above, I will facilitate the sharing of student thinking by selecting and sequencing 2 pairs to share that used different strategies to find equal groups. I will be sure to connect all three strategies: repeated addition, equal groups, and skip counting to the multiplication equation. I will also be sure to restate what students share throughout the discussion, so that members of the class will hear and understand what their peers’ thinking/strategies are.

How will I help students actively listen to each other?During the discussion and closing summary of the lesson, I will remind students to use a level 0 voice and be respectful of students who are sharing. We will review these expectations at the beginning of the lesson and also again before we hold our discussion. As I wrote above, I will also have students be writing multiplication and repeated addition sentences for the work that pairs are sharing, so this holds them accountable for listening and really understanding what their peers are explaining.

How will I help them make connections to prior lessons or prepare for future experiences?After we wrap-up by having students look at examples and non-examples of equal groups, I will explain the importance of equal groups for future experiences. “We are going to be working on multiplication as equal groups throughout the next two weeks. It is very important that we understand what multiplication is, so we can learn strategies to solve it. Tomorrow, we will be working with the same strategies again—repeated addition, skip counting, and equal groups, but we will also be learning what an array is and how we use it to solve multiplication problems.

How will I summarize the main ideas of the lesson/bring closure to the lesson and help children reflect on their experiences?“I would like you to first think in your head for about 30 seconds about what you learned about multiplication today”. After 30 seconds, I will have students share with their partner what they learned about multiplication.

Did your idea of multiplication grow from our list on the board at the beginning of the lesson? Did your idea change?

I will not have students share their answers to the questions, but rather just reflect in their mind.

What kind of feedback would I like from students at this time?Tonight, you have your first homework assignment! I would like you to find an example of equal groups at your house and draw a picture of it. I will give you time to share these tomorrow before math, and we will discuss how your equal groups shows multiplication! You could even challenge your parents or brothers and sisters to solve your multiplication problem!

Description of Formative Assessment

I will use the whiteboard responses and the thumbs up/down that students were giving to

determine who I need to monitor while students are working on the exit ticket. I will also be monitoring and recording notes while the partners work on the task. This will help me select and sequence who I would like to share. I will use the exit ticket to structure tomorrow’s lesson and decide if we need to review equal groups again before moving to the array model.

LESSON #2

Date: 10/29/13

Overall lesson topic/title: Relating Multiplication to the Array Model

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will compare the equal groups of objects in scattered configurations to the array model, exploring the correspondence between 1 equal group and 1 equal row.

Rationale: This lesson continues from yesterday’s lesson as students relate the equal groups in scattered configurations to equal groups in an array. They will begin to distinguish the number of groups and the size of groups when they count rows and how many in 1 row. The arrays will continue to be used as a model throughout the entire unit, so this is a foundational lesson that students really need to understand and master before moving any further into the unit. Students will recognize the real-world applications of arrays, and the efficiency of these arrays as they skip-count to find totals.

Materials: ice tray, pictures/transparencies of real-life arrays, chart paper & markers (1 for each pair), math journals, 1 exit ticket for every student

Procedures and approximate time allocated for each event

LAUNCH: Introduction to the Lesson (15 minutes)

How will I help them make connections to prior lessons or experiences in and out of school? I will ask students to turn and talk with their team about how they would describe an equal group. I will give

them about one minute to complete this, and then I will pull Popsicle sticks to have people share what their group said. We will take student responses to quickly review that equal groups means there is the same number of each object in the group. Multiplication can only be done with equal groups!

“Today we will be continuing to talk about equal groups multiplication as we learn a new model called the array model.”

What will I say to help children understand the purpose of the lesson? “Today will be our first day working with the array model. I am going to read the “I Can” statement that I

have posted on the board. It says: I can compare the equal groups of objects in scattered configurations to the array model. This means you will be able to compare what you learned yesterday with what we learn today about arrays. Give me a thumbs up if you are ready to do this!”

How will I motivate them to become engaged in the lesson and understand its real world purpose? Before I teach you what the array model of multiplication is, I want to show you a few pictures of real-life

arrays. As you look at these pictures, think about if you have seen an array at home before:o Ice cube trayo Muffin pano Lego’so Egg cartono Eye shadow paletteo Paint tray

“Turn and talk to your shoulder partner about an array that you may have seen at home or at school before.” I will call on 3 students to share where they’ve seen an array. As you can see, arrays are everywhere, which means multiplication is everywhere! This is why it is so important we learn about this today.”

How will I build background knowledge? “As you can see from the examples I showed you and the examples you shared, an array is a special way we

arrange groups of objects into rows and columns that make it easier for us to create multiplication sentences.” I will then draw an example of array labeling the rows and columns for those students who not familiar with this vocabulary.

How will I establish clear expectations for how they will be working and what products I expect? “Just like yesterday, we will be working with the whole group and in partners. We are going to review some

behaviors that I will be looking for as we work.” I will have students share some of the behaviors they believe are appropriate when working with partners:

o Being respectful and using a level 0 voice when others are talking and sharing with the classo Participating by sharing my thinkingo If I disagree with someone’s answer, I need to do so respectfully.

How will I launch the problems? Let’s see how we could solve this problem: “Jordan uses 5 ice cubes in 1 glass of lemonade. He makes 4

glasses of lemonade. How many ice cubes does he use altogether?”

When and how will I distribute materials and supplies? After we have finished the launch of the problem, students will get out their math journals to try to do this

task independently. After we discuss the task, I will then pass out chart paper and markers to each group as they create their array posters.

EXPLORE: Outline of Key Events During the lesson

Independent Task (15 minutes):Students will work independently to solve this task in their math journals. I chose to have them work independently to vary the learning styles of those in our class. Some students like working better by themselves while other like working better in partners. I think it is best to switch the type of interaction often throughout a lesson.

Differentiation: I will challenge those students who finish early to find their answer using as many strategies as possible (equal groups, array, skip counting, repeated addition, multiplication).

Differentiation: Those students that had poor performance on yesterday’s exit ticket will work with my MT on this task as a small group, being sure to review what an equal group is and how it relates to multiplication.

As students are working, I will be monitoring and looking for students’ different strategies to create equal groups. I anticipate that most students will use the equal groups method by drawing a picture of 4 groups of 5 ice cubes. I also anticipate that most students will be able to use the multiplication sentence or skip counting to solve this problem. I

am hoping that some students will use the array model that I briefly introduced. I will also be monitoring and taking notes on the strategies students are using. I will also promote higher-level thinking by asking:

How do you know that you have equal groups? How many are in each group? How many groups do you have? What is another way that you could draw your picture? Is there an easier way to find the total number of ice cubes besides equal groups?

Whole-Group Discussion of Task (10 minutes):I will select 2-3 students to share that used different strategies. I will sequence students’ responses by having the student who used equal groups go first as I anticipate that is how many students solved it. I will then have a student share that solved it using an array (if there are any). I will have the rest of the class give thumbs up or down if they used a similar strategy to the one being shared. I will also ask for students to repeat the strategies that their classmates used.

Connecting the two strategies. Pretend I was to give you this problem: “Pretend Jordan has an ice cube tray with 4 rows of 5 ice cubes. How many total ice cubes can he make in 1 tray? Would you solve this using an array or equal groups? Which one would be easier for the context of the problem?”

o “You could use equal groups or an array to solve this problem and the problem I gave you earlier. However, it’s important to use the story of the problem to see what strategy to use. In the first problem, you can imagine dropping ice cubes into the 4 glasses, so it might be easier to use equal groups. In this new problem, you can picture the 5 rows of an ice cube tray in your mind, so an array would be the natural strategy to use.”

o “It’s important in future word problems to read the problem carefully and decide which strategy makes the most sense for you.”

I will lead the discussion toward comparing the way the ice cubes are organized in equal groups with the way they’re organized in the array.

o “In the equal groups, the ice cubes are touching each other, but the circles have space between them.”

o “Each line in the array shows 5, like each group of ice cubes.”o “The array is organized with everything in straight lines.”

“I see that many students are noticing straight lines on the template. Let’s call a straight line going across a row. Let’s skip count each row at a time and I will write the total next to each row (5, 10, 15, 20…).”

o “What is the total number of circles we counted?” (20)o “Take 10 seconds to find how many rows of 5 you counted.” (4)o “Does 4 rows of 5 circles equal 20 circles?”

I will then write 4x5=20 on the board. Please turn and talk with your partner about why this equation is true using the array this group shared. Anticipated responses:

o “Yesterday we learned that we can multiply equal groups.”o “We skip counted 4 rows of 5 circles each and the total is 20.”o “It just means 4 groups of 5 and when you add 4 fives, you get 20!”o “Writing 4x5 is a lot easier than 5+5+5+5.”

“You have just shown me that an array model and equal groups both get me the same answer in a multiplication sentence. Both models can be useful depending on the story. Sometimes an equal groups model will make more sense to you, and sometimes an array might make more sense for you. In problems where they tell you objects are arranged in rows, it might be easier to use an array model when solving the problem.”

o “When we write a multiplication sentence, we first write the number of groups. How many groups?” I will write 4x______

o “Next we write the size of the group.” 4x5o “Skip-count to find the total number of circles in the array”. 4x5=20

“We just found the answer to the multiplication fact that represents the array.”

Partner Exit Tickets (20 minutes)

Students will make an array poster with their shoulder partner. I will model the required information that needs to go on the poster on the overhead to provide scaffolding for those who struggle with multi-step directions: one array, number of objects in each row, number of rows, total number of objects, multiplication fact to describe the total number of objects. I will allow the pairs to draw any array they want, using any numbers. I will challenge those students who are succeeding in math to choose an array that is larger.

Students will share their array poster with their teammates. I will be taking notes throughout this as to which pairs are able to successfully draw an array, explain how many groups, how many objects in each group, and write a multiplication sentence that describes the array.

We will be hanging up these posters in the classroom for future reference throughout the unit.

If there is time, students will complete the exit ticket independently. If there is not time, I will send this home as homework. I will monitor students’ thinking as they work on this.

SUMMARIZE: Closing Summary for the Lesson (5 minutes)

How will I structure the closing of the lesson?I will first collect the array posters and exit tickets that students have completed.

I will pull sticks and have 6 students create an array with their bodies in the front of the classroom. We will discuss how we know they created an array:

Organized with rows and columns Equal groups: how many? What size?

We will then talk about what the multiplication sentence of this array would be and how we know that (what each number represents).

How will I facilitate the sharing of student thinking?As stated above, during the closing of the lesson, I will call on students to explain how they know the students have created an array. I will also call on other students to repeat and explain what their

classmates said. During the discussion of my lesson, I will have 2-3 students share the strategies they used to solve the task. At this time, students will give thumbs up or down if they used a similar strategy as the person sharing. I will also call on students to repeat or explain their classmates’ strategies. These are similar to many of the ‘talk moves’ that we discussed in class.

How will I help students actively listen to each other?During the discussion and closing summary of the lesson, I will remind students to use a level 0 voice and be respectful of students who are sharing. We will review these expectations at the beginning of the lesson and also again before we hold our discussion. As I wrote above, students will be expected to repeat and explain their classmates strategies, so they need to be listening and comprehending what is going on at all times. I will also reinforce the behavior expectations before the partners work to create their array posters. I will stress that the work needs to be done evenly between the two people, and when I come to check on their work, both partners will be able to answer any questions I have.How will I help them make connections to prior lessons or prepare for future experiences?I will explain the importance of the array model for future experiences. “We are going to be working with arrays throughout most of this unit. If you feel uncomfortable working with arrays right now, please let me know after math and maybe we can do some practice problems. I want everyone to understand what an array is, how to draw one, and how it can be used to solve multiplication. The array is a great tool for you to have when you are solving multiplication problems.”

How will I summarize the main ideas of the lesson/bring closure to the lesson and help children reflect on their experiences?“I would like you to first think in your head for about 30 seconds about what you learned about an array today”. After 30 seconds, I will have students share with their partner what they learned about multiplication.

How does an array help you solve multiplication problems? Do you use any arrays at home to help you?

I will not have students share their answers to the questions, but rather just reflect in their mind.

Description of Formative Assessment

I will look over the students’ array posters and their exit tickets to determine which students will need to work in a small group for tomorrow’s lesson to get more practice drawing arrays. I will also monitor students’ work as they try to solve the task in their math notebooks and create the array posters. The monitoring will help me in selecting and sequencing those individual students who should share. The array posters will be used to determine if students have met the objectives of the lesson. Can they draw an accurate array? Can they determine how many groups and the size of the groups? Can they write an accurate multiplication sentence based on their array?

LESSON #3

Date: 10/30/13

Overall lesson topic/title: Interpreting the Meaning of Factors

CCSS(s): 3.OA.1: Interpret products of whole numbers

Learning Target(s)/Objective(s): Students will analyze equal groups given in scattered configurations and organized into arrays to determine whether factors represent the number of groups or the size of a group.

Rationale: This lesson is a continuation from yesterday’s lesson, as it encourages students to think about what the meaning of the factors are as they begin to multiply. It asks them to apply what they learned about the array model and the multiplication sentences they created for the array model when thinking about factors. This lesson will benefit students because they will begin to see the meaning and purpose behind multiplication aside from what they know when memorizing facts for a timed test. This will also benefit students when creating their own multiplication word problems, as they must realize that each factor in a multiplication number sentence has a purpose. They will transition from knowing 8x4=32 to knowing why 8x4=32. This lesson will relate to the students because they will be a part of an equal group, which will require them to move around the room. This will benefit those students who have trouble focusing at their seats and need movement throughout a lesson.

Materials: math journals, graphic organizer for every student

Procedures and approximate time allocated for each event

LAUNCH: Introduction to the Lesson (15 minutes)

How will I help them make connections to prior lessons or experiences in and out of school? “Every morning since the first day of school, we complete the factor tree on the board. Today, for example,

was the 45th day of school. You all do a great job of persevering and figuring out the factors of 45, but do we know what the word ‘factor’ even means? Do you know why 9 and 5 are factors of 45, or why this important?”

Today, we will solve the answers to all these questions!

What will I say to help children understand the purpose of the lesson? “Today, we will continue what we did yesterday with the array model and learn the meaning of the word

factors. I am going to read the “I Can” statement that I have posted on the board. It says: I can analyze equal groups given in scattered configurations and organized into arrays to determine whether factors represent the number of groups or the size of groups. All this really means is that you will be able to tell me what the factors 9 and 5 mean in the multiplication sentence 9x5=45, and how to find these factors using an array instead of the factor tree.”

How will I motivate them to become engaged in the lesson and understand its real world purpose?

“Finding factors will become even more important when you move into fourth grade. In fourth grade, you will be working with much larger numbers, and prime and composite numbers. In order to be able to work with these numbers, we have to first know what a factor is and why it is important.”

“Are we ready to become fourth grade mathematicians? Give me a thumbs up if you are ready for the challenge!”

How will I establish clear expectations for how they will be working and what products I expect? “Just like yesterday, we will be working with the whole group and in partners. We are going to review some

behaviors that I will be looking for as we work.” I will have students share some of the behaviors they believe are appropriate when working with partners:

o Being respectful and using a level 0 voice when others are talking and sharing with the classo Participating by sharing my thinkingo If I disagree with someone’s answer, I need to do so respectfully.

“We will also be moving around quite a bit today. While you are moving to different places in the room, please be using a level 0 voice and keeping your body to yourselves. Please also move quickly as we have a lot of fun activities to do today!”

How will I launch the problems/provide background knowledge? “Here are the rules for our first activity! I will write these on the board so you will not forget:

o 1. Divide yourselves (plus the 2 teachers) into 4 equal groups.o 2. Each group will stand in a corner of the room.o 3. Divide silently. You can use body movements to gesture, but no words.”

“When you believe that your group is ready and equal with all the other groups in the room, you’re going to show me a thumbs up. Be sure to look around the room to double check that all 4 groups are equal!”

Students will then move around the room silently until there are 4 equal groups (28 people total)= 7 people in each group.

Students will stay standing. “Show me on your fingers how many equal groups we’ve made. (4)” I will write 4x ________ = 28. “Show me on your fingers the size of each group.” (7) I will fill in 7 on the board.

“These numbers—the number of groups and the number of each group—are called factors. “Please head back to your seats. We’re going to work on a similar problem now with our eyeball partners,

but this time we’re not going to move around the room. Pretend Mrs. Waterson’s class divided themselves into 4 equal groups and there were 9 people in each group. How many people are in Mrs. Waterson’s class?”

When and how will I distribute materials and supplies? After we have finished the launch of the problem, students will head back to their seats and get out their

math journals. I will pass out the exit ticket after the discussion of the task is complete.

EXPLORE: Outline of Key Events During the lesson

Partner Task (15 minutes):Students will work with their eyeball partners through this task. I will tell students that they need to defend their answer they need to draw an array, write a multiplication sentence, and write underneath each factor what it represents in the array.

Differentiation: I will challenge those students who finish early to figure out how many people are in a class

that divides themselves into 12 equal groups with 6 people in each group. Differentiation: For those students who struggle with high-level tasks and following a long list of directions, I

will provide a visual on a personal whiteboard of how they should set-up their math journal to complete this problem.

As students are working, I will be monitoring and looking for how well students are explaining their thinking and creating a viable argument for their answer (one of my mathematical practices). I anticipate that most students will draw an array with the number of groups as the rows (4) and the number of people in each row as 9. However, I do anticipate some misconceptions. I think some may confuse the difference between the number of groups and the size of the groups by drawing an array with 9 rows and 4 people in each row. I also anticipate students not labeling each factor correctly, again confusing the difference between the number of groups and the size of the groups. I will also be monitoring and taking notes on the arrays that students are drawing and the corresponding factors in their multiplication sentences. I will also promote higher-level thinking by asking:

How did you know where to start drawing your array? How many rows did you know to draw? Where is this number in your multiplication sentence? How did you know how many to put in each row? Where is this number in your multiplication sentence? How do you use your array to figure out the answer? If you were to create a factor tree of your multiplication sentence, what would it look like?

Whole-Group Discussion of Task (10 minutes):I will select 2-3 pairs to share that had different arrays and multiplication sentences for this problem. I will sequence students’ responses by having the pair who drew an incorrect array and multiplication sentence first. I will then have a pair share that drew a correct array but did not identify the factors correctly. Lastly, I will have a pair share that drew a correct array and was able to identify the meaning of each factor correctly. I like to address the misconceptions first and then work towards what I was expecting students to have done in the task. I will have the rest of the class give thumbs up or down if they agree or disagree with their work. I will ask students to defend why they agree or disagree respectfully.

Connecting the misconceptions to the accurate answer:o “Factors of a multiplication problem can either mean the number of groups or the size of the

group.”o “First let’s figure out how many groups we have. Let’s re-read the problem. It says Mrs. Waterson’s

class divided themselves into 4 equal groups. So how many groups do we have? (4).”o “Then, let’s determine how many are in each group. It said there were 9 students in each group, so

how many students were in each group? (9).”o “In the first group to share, it looks like they drew their array so there was 9 equal groups with 4 in

each group instead of 4 equal groups with 9 in each group. Remember the number of rows in the array represents the size of the group. The number in each row represents the number in each group!”

o “Let’s work on writing the multiplication sentence to this array. First when writing a multiplication sentence, we write the number of groups, so I’m going to write 4x ____ = _____.”

o “Then we multiply by the number in each group, so I’m going to write 4 x 9 = ______.”o “We can use our different strategies to solve this problem, such as skip counting on the array or

repeated addition. Let’s use skip counting! (skip count by 9’s to 36). So, our answer is 36. There were 36 people in Mrs. Waterson’s class.”

o “Our second group to share was able to draw the array we see on the board, but their factors were backwards of what their picture showed.”

o “This last group was able to draw the correct array and represent each factor correctly.”

Independent Exit Tickets (10 minutes)Students will complete an individualized graphic organizer in which there is a multiplication problem in the middle.

Surrounding the middle, there are 4 squares. Each square is labeled with what the student must write: skip counting, repeated addition, equal groups, and array.

Differentiation: Those students who performed poorly on the exit ticket or array poster yesterday will be given a problem with smaller factors and an array that was modeled in yesterday’s class (2x4).

Differentiation: Those students who performed highly on the exit ticket or array poster will be given a problem with larger factors (9x8).

I will be monitoring students’ work; however, I will not be helping students as much on this because I would like to use this as a more “formal” assessment for how the students are doing for the first 3 lessons of the unit.

(see underneath this lesson plan for copy of graphic organizer)

SUMMARIZE: Closing Summary for the Lesson (10 minutes)

How will I structure the closing of the lesson?I will first collect the graphic organizers from the students.

Our closing today will be a debrief of the lesson which is intended to invite reflection and active processing of the lesson. I will pull Popsicle sticks to have students answer the following questions:

Why do you think I started the lesson by asking you to divide yourselves into equal groups in the corners of the room?

We have been introduced to many new vocabulary words: row, array, number of groups, size of groups, and factor. Choose one of those words that you feel confident about and turn to a partner and explain this word. (I will then call on one student that chose each word to review the meaning)

o **If I find that students really struggle with this, we may create a vocabulary page in our math journals before the next lesson.

How will I facilitate the sharing of student thinking?As stated above, during the closing of the lesson, I will pull Popsicle sticks to have students recall and reflect on the lesson. They will also be thinking to themselves and then sharing with a partner. During the discussion of my lesson, I will have 3 students share the arrays and multiplication sentences they created. At this time, students will give thumbs up or down if they agree or disagree with their classmates. Students will practice respectfully disagreeing with those students who had misconceptions about the task and explaining why they disagree.

How will I help students actively listen to each other?During the discussion and closing summary of the lesson, I will remind students to use a level 0 voice and be respectful of students who are sharing. We will review these expectations at the beginning of the lesson and also again before we hold our discussion. As I wrote above, students will be defend why they agree or disagree their classmates strategies, so they need to be listening and comprehending what is going on at all times. I will also reinforce the behavior expectations before the partners work to do the task. I will stress that when I come to check on their work, both partners need to be able to answer any questions I have.

How will I help them make connections to prior lessons or prepare for future experiences?“Tomorrow’s lesson we are going to move to learning about division, which I know was a goal for many of you this year during third grade. You will see that knowing what factors are will help you tremendously in solving division problems! Division and multiplication are very similar and knowing how to find factors and what they mean in story problems means you will be able to solve division and multiplication.”

How will I summarize the main ideas of the lesson/bring closure to the lesson and help children reflect on their experiences?“Remember, factors are either the size of the group or the number of groups. Turn to your elbow partner and state the definition of factors.”

Description of Formative AssessmentI will be taking observational notes as the students work with their partners. I will look for who is able to apply what we have been learning about arrays to the new idea of factors. I will use also the graphic organizers as an assessment to see if students have grasped the objectives from the first 3 lessons. I will grade these and determine which students need small group and individual attention as we move forward. This graphic organizer will also help me determine if we can safely move onto division in our next few lessons. I do not want to move to division before the foundational pieces of multiplication are comfortable for all students.

SKIP COUNTING REPEATED ADDITION

EQUAL GROUPSARRAY

4x9

Section 6: Family Involvement and Communication

Dear Room 20 Parents,

I have loved working with your children and watching them become “responsible, respectful, and safe” third graders here at Scott School! I am very excited about our next math unit that began on Monday, October 28 and will continue through the end of November. Your students have been working very hard this year at persevering through tasks and explaining their thinking using multiple strategies. This will continue as we move into our next unit, which is titled “Properties of Multiplication and Division and Solving Problems with Units of 2-5 and 10”. We will be shifting from the CCGPS (Georgia) curriculum to the Eureka curriculum. The Eureka curriculum was published in 2012 by the Common Core, which is a group of master teachers, scholars, and current and former school, district, and state education leaders. Eureka is a state-of-the-art curriculum that adheres to the Common Core State Standards.

During this next unit, we will be specifically focusing on the foundational skills of both multiplication and division and how these two operations are related. Specifically, the students will be building on their knowledge of repeated addition from second grade to understand multiplication as equal groups. They will also be relating multiplication and division to the array model and interpreting the meaning of factors. As we move towards division, students will learn to connect division to multiplication through finding the meaning of unknown factors either as the size of the group or the number of groups. Lastly, students will again manipulate arrays to learn both the commutative and distributive properties of multiplication. These strategies will help them solve multiplication and division problems with larger numbers.

Your students will be learning to persevere and explain their thinking clearly through this unit with our “math talks”. We will specifically be working on high-level tasks and word problems most days in order to build up their perseverance. Examples of problems that students will be solving are as follows:

Multiplication as equal groups:1.) The picture below shows 4 groups of 2 stars. Write repeated addition and multiplication sentences to represent the picture.

2+ ______ + _______ + _____ = _________

4 x ______ = ___________

Array model of multiplication: 2.) Judy collects seashells. She arranges them in 3 rows of 6. Draw Judy’s arrays to show how many seashells she has all together. Then write a multiplication sentence to describe the array.

Finding the meaning of factors:3.) Mrs. Zarotney’s class last year divided itself into 4 equal groups with 9 students in each group. How many total students did she have last year?

Draw an array, write a multiplication sentence, and write what each factor represents in your sentence.

Understanding the meaning of the unknowns in division:4.) a.) 15 ÷ _______ = 5Write a division word problem where the unknown equals the size of the groups.

b.) 15 ÷ _______ = 5Write a division word problem where the unknown equals the number of groups.

Commutative property:

5.)

Do you agree or disagree with the statement in the box? Draw arrays and use skip counting to explain your thinking.

As you can see from the above sample questions, students will be working on various strategies to solve multiplication and division problems. We will be focusing on visual strategies throughout most lessons, such as arrays and pictures. Students will gradually move from representing multiplication and division with manipulatives and pictures to a more abstract representation (using only numbers). We will also be learning strategies to solve higher-level word problems, like determining what is given in the problem and what we need to find. Students will learn to work with their classmates to clearly explain their thinking on paper and verbally using words, pictures, and numbers. The students have already been doing a great job sharing their strategies with one another during our daily math warm-up, and we will continue to have these math discussions throughout this unit. I believe it is very important for students to learn from each other! These “math talks” will become a part of our math routine a few times a week.

This unit will move your child’s understanding of multiplication and division as a set of facts to memorize towards a more developed understanding of these operations.

2x5 = 5x2

Most of the work for this unit will be done in class, but there may be times that I assign homework based on students’ individual needs. The homework will vary in nature. Some homework will be additional practice of what was covered in that day’s lesson, and other days the homework may be finding real life examples of the math concepts at home or in the community. However, if you do notice only small amounts of work coming home as homework, that does not mean there is nothing that can be done. As always, multiplication facts can be practiced at home, and I encourage students to create their own word problems using these multiplication facts. Problem solving and perseverance can also be practiced, and these are skills that are beneficial not only in math but all subjects! I know it may be difficult to do, but let your child struggle through a problem. Multiplication and division are new concepts for most of the students, so it is okay for them not to understand a problem the first or second time they try it. Encourage them to push through and keep trying! I also urge you to ask your child what is happening in their math class, and try to have your child complete some type of math activity each night. Feel free to have your child bring in any math work they complete at home! Below are a few activities that will help to further students’ math knowledge throughout this unit:

Make arrays out of household items (e.g. pennies, beans, blocks, Legos). Determine how many are in each row.

Hunt for equal groups in your home. Use repeated addition and multiplication to find the totals.

Roll two number cubes (dice). Find the product of the factors. Act out division problems with counters. Order a pizza or bake brownies and figure out how to divide it evenly among all

family members. Roll 2 dice and write the fact families. For example, for rolls of 4 and 6, write

4x6=24 and 24/6=4. Ask your child to find the missing factor. For example, 5 x _____= 35? Think about and write all the ways that you are math smart (ask your child about

this)!

Thank you for your continued support in your child’s learning. For further information about this new curriculum and unit, feel free to visit the Eureka website: http://commoncore.org/maps/math/grade-3/. For upcoming information about this unit, please look at the bi-weekly newsletters and weekly e-mails sent by Mrs. Zarotney. Feel free to e-mail me any frustrations or successes that your child is having throughout this unit so that I can adapt my lessons to meet their needs. If you feel that your child needs to be challenged or needs more individualized instruction, please let me know!

Sincerely,

Kaitlynn GilbertDescribe any other ways you will communicate or engage with families:

We send out bi-weekly newsletters and weekly e-mails to families. I plan to let parents know what concepts we will be learning in the next week. The newsletter and e-mail will also allow me remind parents to be engaging their child in some type of math every night, whether it is practicing multiplication facts, having a “math talk”, or finding real-life examples of math concepts. If parents have any concerns that they e-mail to my MT or myself, I will also address those through e-mail.

I will also keep the students’ summative assessment at the end of the unit for parent-teacher conferences. This will provide the parents with a concrete representation of their students’ mastery of these objectives.

Students take home an assignment folder every night that lists their homework. I will be sure to have students write in any math homework that they have so parents can look at this and keep up to date with my expectations.

Section 7: Final Reflection and Analysis

1.) Write about what your students did and did not learn.

Throughout my unit, I collected daily evidence about my students’ progress in meeting the objectives of each lesson. After (almost) every lesson was complete, the students independently filled out an exit ticket that covered the big ideas from that day’s lesson. I also have records of two homework assignments, one in-class journal task, one in-class worksheet, and 2 posters that students created throughout the unit. I also have results from both the pre-assessment and summative assessments. I also was able to keep very brief anecdotal notes for each student after every couple of lessons. All of this information has been compiled on an Excel spreadsheet. I will be providing evidence from this spreadsheet to explain what my students have learned throughout this unit and what they have not learned.

The first big idea of the unit involved students learning multiplication as equal groups. During my pre-assessment, 15/25 students were able to successfully identify and explain equal groups. Because this was an area that most students were familiar with, the first two lessons of the unit were easy for most students. My results for the formative assessments showed that only 6 students had problems identifying equal groups. However, with small group work differentiation throughout the unit, only 4 students in the class could not explain and identify equal groups on the summative assessment. Overall, the class improved on understanding the meaning of multiplication as equal groups by 6 students.

The second big idea for the unit involved creating arrays, writing multiplication sentences for the array, and identifying the meaning of the factors as either the number of groups or the size of groups. During the pre-assessment, only 10 students were able

to draw an accurate array with equal rows. Out of these 10 students, only 6 were able to write an accurate multiplication sentence for this array, and only 1 was able to state what the factors represented. I believe this area is where students improved the most during my unit. We practiced drawing arrays and writing multiplication sentences for these arrays every day. Throughout my formative assessments of array posters, exit tickets, and homework problems, students gradually improved in this area. During the homework set for this concept, 12 students received a 2 or 1, which is below grade level. On the exit ticket, the number decreased to only 9 students who received a 2 or 1. On my summative assessment, only 7 students were not able to meet grade level expectations for drawing an array, identifying the meaning of factors, and writing an accurate multiplication sentence. The main idea that these students did not learn or remember on the assessment is that the order of the factors matters when writing a multiplication sentence. You must be able to determine which factor is the number of groups and which is the size of the groups: # of groups x size of groups = product.

The third big idea that a majority of students learned was division, including drawing an array, writing a division number sentence to describe the array, and explaining the connection between division and multiplication. On my pre-assessment, only 2 students were able to write an accurate division sentence for an array. On my summative assessment, however, 13 students were able to do this! These students were able to learn the meaning of division and that the quotient of division can either be the size of the groups or the number of groups. Those students who were not able to do this on the summative assessment also struggled during the exit tickets and homework problems. They received a 2 or 1 on the formative assessments for not writing an accurate division number sentence (numbers out of order) or failing to correctly identify the quotient as the size or number of groups. A concept that we will need to reteach to all students is explaining the connection between division and multiplication, which is that the unknown factor in multiplication is the same as the quotient in division. On my summative assessment, only 3 students were able to successfully explain this relationship.

The last big idea for my unit was the use of the commutative and distributive properties to solve multiplication problems. The students had a higher success rate with the commutative property than the distributive. On my pre-assessment, only 6 students were able to describe using the directional differences of two arrays why the commutative property works. After collecting my exit tickets for these two lessons, I found that after the first day of the commutative property lesson, 7 students were not able to meet the objective; however, after the second day, only 3 students were not able to meet the objective. While creating the commutative property posters, students were successful at drawing arrays and writing multiplication sentences to represent the commutative property; however, they still struggled to explain why the property works. In my summative assessment, 18 out of 24 students were able to draw arrays to represent the commutative property; however, only 7 students were able to explain clearly using words like row, columns, factor, and product, why the commutative property is true. As you can see, clear written explanations are an aspect of my unit that a majority of students did not learn. Lastly, during my pre-assessment, only 9 students

were able to draw a slice through an array and write one number sentence that described their new arrays. Through much practice and formative assessments, I saw steady growth in this concept of the distributive property. When I gave a worksheet to practice the distributive property with arrays, only 3 students did not meet grade level expectations. During my summative assessment, only 4 students did not meet grade level expectations for showing how to slice an array into two arrays with “friendly numbers”. However, to continue with the pattern of the unit, students still struggled to explain the steps of the distributive property. During my formative assessment exit tickets, very few students were able to explain these steps, and on my summative assessment, only 4 students were able to do this. While the class is able to do the computation effectively with arrays, they are not able to explain how they used the property to find the product.

Conclusively, I saw growth in all major concepts of my unit from the pre-assessment to the formative assessments to the summative assessment. The only area that I did not see growth (or very little growth) was in written explanations. This may have been because we focused more on the mathematical processes and computations during the unit than the explanations. Students did have to explain their work verbally quite often; however, explanations on paper were done less frequently. Since constructing and defending their explanations was a mathematical principle throughout the unit, I will need to continue working with students on explaining their thinking in writing.

2.) How did you use formative assessment to adjust and differentiate your instruction to maximize student learning? How much did your daily lessons change from your original plans?

I had a formative assessment every day that was used. Almost every day, I gave students an exit ticket to complete that was 1-2 questions about the concepts covered during the lesson. Other formative assessments that I used were 2 homework sets, 1 journal task, and 2 posters created by the students. I also recorded observations of students’ progress in their math journals and whiteboards. I recorded all of these formative assessments on an Excel spreadsheet. At the end of each lesson, I would highlight in yellow the students who were not “at grade level” for these formative assessments. These students would work in a small group with my mentor teacher or I during the next day’s lesson. For the students to not be “at grade level”, they would receive a 2 or 1 on their formative assessments, which meant they were working towards the objective but had not quite reached it. We used the differentiation of small groups almost daily. My mentor teacher or I would review the concepts from the day before and model and practice sample problems with these students. I also highlighted those students in orange who were excelling during the beginning of the unit. I would provide differentiation for these students by giving them a task with more challenging numbers and providing tasks for them that had multiple strategies or answers. I would encourage them to find “as many ways as possible” to solve the task.

My daily lessons did not change tremendously from what I had planned. The concepts and ideas were very similar to what I had planned, but the execution did change. For example, during my distributive property lesson, I found that many students were struggling while completing a whiteboard activity, so I stopped the activity, and we went back to our seats and began the lesson again working with manipulatives. After students felt comfortable making and slicing arrays with manipulatives, we moved to drawing pictures of arrays. I also added a review day after our first 3 lessons because I found through my formative assessments that we needed to review the ideas of multiplication more before we moved on to division. I also spent a day longer on the division lessons, as I found through my anecdotal notes that many students were struggling to understand the difference between problems where the quotient is the size of the group and problems where the quotient is the number of groups. I also added a few extra “activities” that were not in the curriculum. For example, students created a vocabulary page in their math journal, as I found that the curriculum did not establish concrete vocabulary. In addition to the vocabulary pages, I also made a math “word wall” for our classroom. In a different activity, I had students create a commutative property poster with a partner. This was not in my intended plans, but based on my formative assessments, the students understood this property pretty well after the first day, so I wanted to challenge them with something different on the second day. This poster helped them practice writing and defending their work.

(SEE COPY OF MY EXCEL SPREADHSEET ON THE WIKI FOR A COPY OF MY FORMATIVE ASSESSMENT/SUMMATIVE ASSESSMENT DATA)

3.) What did you learn from the results of your summative assessment? Explain any adjustments you made to the original version of your summative assessment.

I gave many of the results from my summative assessment in question #1; however, I will summarize these results. I graded my summative assessment based on my school/classroom’s grading system, which is a score of 4, 3, 2, or 1. 4 means above grade level expectations, 3 means at grade level expectations, 2 means working towards expectations, and 1 means struggling to meet expectations. For my summative assessments, 9 out of 24 students received a “2” overall, and 15 out of 24 students received a “3” overall. The results showed me that from my pre-assessment, my students have grown in the areas of equal groups, arrays, repeated addition, multiplication, division, commutative property, and distributive property. The summative results also were able to show me what Common Core Standards each student is meeting based on their responses. Each question of my summative assessment is correlated with a standard, so I am able to see which students are meeting the CCSS for this unit and which students are not. The results also show me what areas we will need to reteach. The areas that most students seemed to struggle with were drawing an accurate array from a scattered configuration, writing division sentences for an array, and identifying the meaning of the quotient in a particular word

problem. All students struggled with the questions that asked them to explain their thinking, such as defending why they agree or disagree with a students’ work, explaining how division can be used to solve a multiplication problem, and explaining why the commutative property and distributive property are true for a particular multiplication problem. I will discuss these results with my mentor teacher to determine how we will re-teach these concepts and where the continuation of this unit should be focused.

I made a few slight changes to my summative assessment from the original version. My first summative assessment had students creating two different division word problems; however, we did not practice generating our own word problems during the unit, so I decided to restructure the question. Instead, I gave students the word problem and they had to create an array, write a division sentence, and identify the meaning of their quotient. I also changed the question about the distributive property slightly. Before, I had students filling in number sentences for an array that was already split into two smaller arrays. I changed this question to allow the students to split the array however they chose and then write their own multiplication sentences. I believe that my new question was of a higher-level, plus it was similar to the tasks we had completed in class. I also added a question to the summative assessment, asking students to explain the steps of the distributive property instead of filling in number sentences. I knew that this would be a challenge for students, but it did align with my mathematical practice for the unit.

(SEE EXCEL SPREADSHEET ON WIKI FOR A COPY OF MY SUMMATIVE ASSESSMENT RESULTS)

4.) Discuss how you would modify your teaching, participation structures, and/ or tasks if you were to teach this unit again to children of the same age.

If I were to teach this unit again, I would be sure to incorporate more movement into the lessons. The curriculum is very dry, and most of the lessons could be completed with students in their seats. Toward the end of my unit, I began to have students move quite frequently from their seats to the carpet and throughout the room with partners. I would have liked to implement this earlier in the unit to make sure students remained focused and engaged. I also had students working with partners pretty exclusively throughout the unit. I would like to incorporate more small group or “team” work. While doing this, I would like to try the group roles that were mentioned in class, as I was unable to have students work in small groups (aside from those receiving extra support) during my unit. The Eureka curriculum being used has many small tasks throughout one lesson. When writing my unit plan, I tried to take those small tasks and consolidate them to one or two high-level tasks that had multiple entry points and strategies to solve. I found that as my unit progressed, I was struggling to find the creativity to modify the entire curriculum into high-level tasks. Before I was to teach this unit again, I would like to keep working on adapting mandated curriculum to not only be of a high cognitive demand but also meet the needs of my students. I would also like to

modify my teaching by creating more student-led discussion. I did use “math talks” very frequently in my unit; however, I was the discussion leader and seemed to do most of the talking. The students were very receptive and eager to share during the unit, so I would like to harness this enthusiasm and let them try to lead the discussion while I become more of a listener.

5.) Reflect on how you used the information you gathered in Project One to help you plan and teach your unit.

I learned about the students’ community and places they like to visit in the community during project one. I also was able to learn about the students’ interests in project one. I was able to use this information and create tasks that were applicable to their lives. For example, I created tasks that related to football, gymnastics, basketball, the DeWitt Farmer’s Market, and the Scott School garden. I also learned throughout project one that many of my students thought being “smart” in math was excelling at Rocket Math (timed tests). I tried to make it a goal to change this perception. By the end of my unit, many students loved when it was time for math and said that they were enjoying math. I could tell that their confidence in math has increased from the beginning of the year. I think for many students this was the first time they had felt this way this year, as I really tried to stress the different types of smarts that are used in math.

6.) Reflect on what you learned about teaching mathematics and about teaching in general.

While I was teaching my students about perseverance, I learned to have perseverance myself. I sometimes would be frustrated after a lesson because I felt like my students “weren’t getting it”. My mentor teacher reminded me that it is a gradual process, and not all students will understand the concepts right away. I have learned that each day students will improve, and learning new mathematical ideas takes time. I believe the results from my summative assessment prove to me that most students were able to meet many of my objectives with time. I also learned that in teaching you must be flexible. There were many times when I had to adjust my lessons because of holiday parties, meetings, issues with students outside of the classroom, and days to reteach. My mentor teacher helped me realize that teaching is all about flexibility. Things happen and it is important to “go with the flow”. My 10 lessons took me about a week longer than I had anticipated, and this has helped me realize that even with all the planning and preparation, there will be always be things that are unexpected and you must be flexible.

I learned that teaching mathematics is very complex. There is a lot to think about! It is a constant balance between high-level tasks, discussions, and student

understanding. It is most important to look at the needs of your students and determine what they need in order to succeed in a lesson. I also learned that it is important to give students a solid foundation for the concepts you are teaching. For example, it is important to introduce math vocabulary from the beginning so they are able to talk and write in a clear way that all will be able to understand.