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Strategy Lesson 1

Sarah DouglasEDRS 610

University of Maryland, University College

Strategy Lesson 1 Application of Polynomials Sarah Douglas

Name: Sarah Douglas Grade: Ninth

Unit: Polynomials Content Area: Algebra

Lesson Topic: Application of Polynomials Time Allotted: 90 minutes

Type of Lesson: Developmental

Context of Learning:

The students have been working on a unit on polynomials. They have worked on adding and subtracting polynomials, and the day before this lesson they learned about multiplying polynomials. The next lesson will involve reviewing all of the operations before they take their pre-test and test. The students have shown that they can add and subtract complicated polynomials, like (3x3-2x2+5)+(4x2+5x3+6x)-(4x3+5x-9). The students have been introduced to the FOIL method of multiplying polynomials and multiplying a binomial by a polynomial. They have made a connection between multiplying polynomials and the distributive property. This lesson is to review their knowledge of multiplying polynomials by adding to their flip charts and to apply their knowledge of polynomials to real-world situations. The lesson uses the available technology present in classroom where I observed. The students have access to graphing calculators, which they can use for multiplying coefficients. The Interactive Whiteboard will be used for displaying the worksheets and directions and demonstrating directions and problems. The Interactive Whiteboard allows for the use of different colors; the students tend to understand a problem better when it is color coded. The students have access to colored pencils whenever they need them. The students sit at tables of two, so they have a partner to ask questions to and get assistance from.

Curriculum Standard Addressed: (from Maryland Department of Education)

Goal 1: Functions and Algebra

- The student will demonstrate the ability to investigate, interpret, and communicate solutions to mathematical and real-world problems using patterns, functions, and algebra.

- Indicator 1.1.3: The student will apply addition, subtraction, multiplication, and/or division of algebraic expressions to mathematical and real-world problems.

Math Standards – Algebra

- Arithmetic with Polynomials and Rational Expressionso Perform arithmetic operations on polynomials.

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Objectives:

Students will update their flip charts to include multiplication of polynomials.

Students will practice adding, subtracting, and multiplying polynomials.

Materials:

- A special education teacher- Interactive Whiteboard- Polynomial Flip Charts- Markers/colored pencils- Calculators- Constructed Response worksheet- Multiplying Polynomials worksheet- Individual Whiteboards and markers- Jackie Robinson coloring polynomials worksheet

Proactive Behavior Management:

The lesson is fairly structured with time restrains because this group of students can get loud and distracted if there is any drop in instruction. The suggested time limits are there so the students know that they need to get their work done and avoid off-topic discussions. If the students are working and need extra time, the extra time will be given. The students will be working in pairs. The students need a partner to assist them and ask questions, but the more students in a group the greater the chances the students will get distracted. Students, who get easily distracted, will be separated from the students, who antagonize them. Students, who get easily distracted, may be placed with students, who tend to concentrate on the work, or at the front of the room. This way the distractible students are more likely to pay attention. I will stick to the classroom rules and know when to ask the students, who are being disruptive, to go into the hallway to calm down. If the student, who is distracting other students, is taken out of the situation, the rest of the class can get back to the classwork. I will need to be prepared to notice the point when the class is going to get overly distracted.

Provisions for Student Grouping:

The students are arranged in tables of two students. The students’ tablemates will be their partner throughout the lesson. Groups are limited to two students, because of the distracted nature of the class. If more students were in the groups, the chance of the students getting off task will be higher. Half the students are qualified as special needs students. There is one student who needs a lot of help but is not qualified as a special needs student. These

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students will be placed with a partner who does not have learning disabilities or qualify as a special needs student. Two students with special needs will not be placed together unless one of the students shows excellent performance with applying operations to polynomials. The groups will be rearranged to accommodate for students who were absent the day before. Students returning to the class will be placed with a partner who was present the day before. Present students can help fill in the gaps for the absentee student. This lesson is at the end of a unit on polynomials and by this point groups will be formed so that the students complement each other. Students, who are normally disruptive, will not be place together or near each other. If a group is not working well together and a compromised cannot be reached, I will switch the student groups.

Procedures

Warm-Up/Opening:

The objective and warm-up will be written on the board, so the students can begin the lesson when they enter the classroom. The normal routine is for students to enter the classroom, pick up a calculator, find their seats, and begin working on the warm-up. If the students meander off course, I will remind them that they should be seated and working on their warm-up when the bell rings. The warm-up is two questions involving multiplying polynomials, which is a review of previous day’s lesson. I will ask for volunteers to present their work on the board. If no one volunteers, I will select two students to present on the board and then two more students to explain the process. I will lead the discussion on corrections needed, the process used, and where to find information if the students did not know the process.

After the warm-up, we will discuss the students’ problems with the homework. The students were to complete the ‘Multiplying Polynomials’ worksheet for homework. I will instruct the students to pick three problems that they struggled with to go over together. The worksheet will be displayed on the Interactive Whiteboard. I will write the steps to the solution on the board as we discuss them.

Motivator/Bridge:

The students have been creating Flip charts for the operations applied to polynomials. After the previous lesson on multiplying polynomials, the students are ready to add multiplying polynomials to their flip charts. I will instruct the students to get out their flip charts, and choose one student to pass out the markers/colored pencils. The flip charts are created with copy paper or construction paper, and were constructed at the beginning to the lesson on polynomials. An example of a polynomials flip chart is included below. Before the students

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begin, I will display the directions for the flip charts on the Interactive Whiteboard. I will remind them of the required parts of the flip chart and encourage them to work with a partner. The flip chart will help them with the next activity and the unit review at the beginning of the following week. Each section of the flip chart features one operation. The students are required to list two examples of each operation and the process needed to complete the problems, and to write one sentence that will help them remember the procedure. If the students are having trouble with the assignment, I will direct them toward the previous day’s worksheet and their partner. The examples can be homework problems. The students will be encouraged to complete their examples on a separate sheet of paper and verify they are correct before writing them on the flip chart. Once the students have completed their flip charts, they have to show the special education teacher or me their examples for correctness. Students, who finish early, can work on the homework sheet if it has not been completed or a coloring polynomials worksheet (included below). While the students are working on their flip charts, the special education teacher and I will walk around the room and write down a grade for the students’ homework. The class will be given about fifteen minutes to work on their flip charts. If they begin to get off-task, I will remind them of the time they need to be finished by. If the students finish before fifteen minutes, we will move onto the constructed response activity.

Procedural Activities:

First, I will have one student pass out the Constructed Response worksheet and rubric and another student pass out the individual whiteboards with markers. Each pair will need one whiteboard. I will display the rubric on the Interactive Whiteboard and direct the students to look at their rubric attached to the worksheet. Depending on how distracted the students are, either I will read the rubric or I will have the students read the rubric aloud. The students often focus better if their attention is directed toward the whiteboard instead of individual students. This rubric will be the same for all constructed response prompts throughout the year. As this is the last one of the year, the students should be familiar with the rubric. Next, I will display the constructed response directions on the Interactive Whiteboard. I will ask for volunteers to explain each step in their own words. I will pause for a moment to ask the students if they have any questions about the rubric or directions.

After all of the directions have been read, I will display the first constructed response prompt and read it aloud. As I read the prompt a second time, I will have the students call out the important information that needs to be underlined. I will instruct them to pay attention to mathematical terms and equations present in the prompt. I will use different colors while underlining the important information and offer the students colored pencils to do the same. As I read, I will think aloud and ask myself questions, like “this is a math expression, it is probably needed to solve the problem” and “area is a math vocabulary word so it will probably

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be important.” Next, I will instruct the students that they need to make a plan for solving the prompt and remind them that diagrams and drawings are normally helpful. As they make a plan, I will ask them what the formula for area is and write the answer on the Interactive Whiteboard. If the students are generally having a tough time drawing a picture to represent the problem, I will direct their attention back to the Interactive Whiteboard. I will think aloud how I would draw the information presented in the prompt. Then, I will ask them what they think the next step may be and ask guiding questions that will lead them to calculate an equation for the area of the rectangular flower beds. I will give them time to complete their solutions. I will walk around the room and answer any questions the students have. The students will write their plan, drawing, solution and short explanation on the individual whiteboards and display them around the room. If the individual whiteboards are too small, the pairs and pick a space on the whiteboards at the front and back of the room. They will have eight minutes or so to walk around the room and look at other groups’ plans for solving the prompt, and write two sentences about what they saw and how it relates to their plan. I will guide the students in a discussion about what they noticed about the other plans and what their next step is going to be in solving the problem. Then the students will have about fifteen minutes to edit their solution and write their final response before we move onto the next constructed response activity. I will remind them that they will be allowed to take the constructed response home to finish, but if they finish in class they don’t have to worry about it.

For the multiplying polynomials constructed response, I will take a step back. A student will read the prompt, and the students will work in their pairs to underline the important information. When they finish underlining, I will ask them what they underlined and underline the same information on the Interactive Whiteboard using different colors. The students will be in charge of creating their own plan for solving the prompt and creating an equation. The special education teacher and I will be available to answer questions and direct them to the part of the activity they should be on. After the students create a plan and find a solution, they will display their plan and solution on an individual whiteboard and display it around the room. The students will walk around the room and observe other pairs’ solutions. After the students write down what they observed, I will lead them in a discussion about the solutions they saw. I will prompt them to kindly point out any problems they saw with their classmates’ solutions. Then the students will edit their solutions and write their final response. If time runs out the students will take home their constructed responses to finish over the weekend.

Adaptations:

A special education teacher is present in this class. The special education teacher will be walking around during the lesson to answer individual student questions, guide students to

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correct answers, and keep students on task. The students in this class need individual attention and the special education teacher is there to help give the students the attention they need and deserve. I will be walking around to help individual students as well in between giving directions and explaining the lesson. Two teachers will help us get to more students. The constructed response has two parts so that the students can practice the directions and activity before completing the second part of the task. The students are allowed to use notes on quizzes and pre-tests. The flip charts are being created to help organize their notes and provide a reference point when they are completing activities. I will have a sample flip chart for the students to look at if they get really stuck on what to write on their flip charts.

Assessment:

The homework grade is based off of completion and effort. As long as the student attempted most of the problems he/she will receive a perfect score. While the students are working their flip charts, the special education teacher and I will point out problems that need corrections. The students will most-likely be using the problems from this worksheet in their flip charts; we will take this opportunity to offer advice for corrections. The flip charts are for the students to use for studying and practice, and will be reviewed for a participation grade at the end of the unit. After each section, the flip charts are reviewed for correctness so the students are not studying incorrect examples. The constructed responses will be graded using a rubric (included below). The class will be allowed to take the constructed responses worksheet home over the weekend to finalize their final responses to the problems. The coloring worksheet will be graded for effort.

Summary/Closure:

In the last five minutes, I will ask the students if they have any questions or concerns about their constructed responses before they take them home for the weekend. I will remind them that I am available through email if they have additional questions or major problems.

Extension Activity:

The extension activity is a coloring worksheet (included below). On one side the students solve multiplying polynomial problems, and on the other the answers correspond to a section of the picture. Each problem is assigned a color, which will be used to color the correct answer section of the picture. The students will be working on this activity after they finish their flip chart activity and after the constructed response activity. It will be graded for effort and not for correctness. If the students research and list a fact about Jackie Robinson, they can earn an extra point on their test.

Review/Reinforcement (Homework):

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For homework, the students will be finishing their final answers to the constructed response prompts. Even if they have already written their final response, they can take the weekend to edit and finalize their response. In addition, they will be taking home the extension activity or coloring polynomials worksheet. This worksheet will be graded for an effort grade similar to the homework grade.

Reflection:

This lesson is based off of the lesson I observed at Huntingtown High School on June 3rd, 2011. Mrs. Schmidt, the content teacher, and Mr. Green, the special education teacher, were conducting a lesson and review of multiplying polynomials. For the most part the students were given a problem and told to find the solution throughout the lesson. A series of worksheets and a PowerPoint presentation listed problems to be completed. My lesson is a different version of the lesson I observed; it covers the same material and would be conducted in the same classroom. The difference is that I added some literacy strategies and a constructed response section to get a better understanding of how the students were thinking about polynomial problems.

Janet Allen (2008) discusses how she uses flip charts “to help students keep track of the language rules they were creating.” In the class I observed, the students are allowed to use their notes on quizzes but their notes are small sections on multiple pieces of paper. I adapted the flip charts, so the students could keep track of the polynomial problems they were learning. Instead of looking through multiple pieces of paper to get help for a problem, the students would be able to simply flip open their flip charts to the appropriate section. The flip charts include examples for each operation and an explanation of the process. While a couple of problems are presented on the board in Mrs. Schmidt’s class, the homework sheets are not graded for accuracy and nothing guarantees that the students copy the answers correctly. I added into my lesson a moment to check that the problems and solutions being written into the flip charts were correct. The students need to be studying and using correct problems.

Since this lesson is based on Mrs. Schmidt’s classroom, I decided to only include the technology available in her classroom. This technology includes TI-83 calculators and an Interactive Whiteboard. The calculators do not have the capability to solve polynomial expressions for the students, and these students will not learn how to graph polynomials until next year. Part of the daily procedures requires the students to grab a calculator when they enter the classroom. The calculators are used as an assistance device when they are adding, subtracting or multiplying numbers. While these students know how to add and multiply numbers, they often get frustrated and stuck on simple operations. To help the students concentrate on the more advanced problems, they can use the calculators to reduce the frustration. If the students ask what the answer is to a simple multiplication problem, they will

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be directed to use the calculator. The hope is that the students will depend on their available recourses instead of counting on the teacher, because outside the classroom they will not have a teacher to guide them.

The Interactive Whiteboard offers a place to display directions, worksheets and notes. The homework sheet can be displayed on the board and the students can follow along. The documents displayed on the board look exactly like the worksheets the students are using, so the students can spend less time trying to find where the teacher is. Another feature that tends to help the students in this class is the use of multiple colors. The distributive arrow is displayed in the same color as the answer to the distribution step. The next distributive arrow and answer is written in a different color. The students can see and identify each step easily. The students often use different colors when they are doing problems on their worksheets. During the discussions of the constructed response plans and solutions, I will write the plans the students came up with and the steps the students created to solving the problems. The students can then use the Interactive Whiteboard as a reference point for their own corrections and final response.

After observing in Mrs. Schmidt’s classroom several times and talking to her about the students, I have realized that her classroom frame work of endless worksheets with problems does not always help the students. The students take two tests a couple days a part with the same content, and after the first test is corrected and discussed the students are still getting the second test wrong. For my lesson, I wanted to move away from worksheets with lists of problems and examine the students’ thinking. Angela Zehner (2010, pp. 46) states, “If we want students to do this type of thinking – to justify their answers and prove their logic is sound – we must give them time to talk, discuss, and even argue about the content.” Zehner’s constructed response model gives this time to the students. Throughout the constructed responses the students are writing about, discussing, and examining their thinking. When the students display their work, they can use their classmates work to evaluate and edit their own work (Zehner, 2010). The constructed responses grow in complexity so the students are not overwhelmed. The guided practice part of the lesson features a constructed response involving adding and subtracting polynomials. Once they have gotten comfortable and familiar with the constructed response prompts, the students will be completing a constructed response on multiplying polynomials. The students will be offered guidance in completing the last constructed response, but the hope is that the class will be able to guide the independent practice on their own.

Grades are based on the effort the students put into their assignments to help their grades. The homework assignment and coloring activity will be discussed and corrections will be made, but the actual grade will be on effort only. The students will hand in the assignments, and I will mark where the students need to make corrections. This way they know which

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problems are correct and which need corrections. An option on the coloring activity is to research a fact about the person the class is coloring. If the students write down a fact about Jackie Robinson, they will receive an extra point on their unit test. The constructed responses will be graded based on a rubric. I want to give the students credit for examining their thinking and completing a challenging task.

The extension activity is a coloring page of Jackie Robinson. If I knew what the history class was currently working on, I would choose a figure or coloring scene that relates to the history class. For this lesson, I chose Jackie Robinson because these students live in a prominently white community. Jackie Robinson is a prominent figure in African American history. The coloring page gives a very brief history of Robinson that the students can read as they complete the activity. I chose to let the students work on the extension activity throughout the class to fill time as some students wait for others to finish. The class can learn more about Robinson by researching. The students will receive credit for their research in the form of an extra point on their test.

References:

Allen, J. (2008). More tools for teaching content literacy. Portland, ME: Stenhouse Publishers.

Crayola. Jackie Robinson baseball player coloring page. Retrieved from http://www.crayola.com/free-coloring-pages/print/jackie-robinson-baseball-player-coloring-page/

KUTA Software, LLC (2011). Kuta Software – Infinite Algebra 1: Multiplying Polynomials. Retrieved from http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Multiplying%20Polynomials.pdf

Maryland State Department of Education (2001, March). High school core learning goals. Retrieved from http://mdk12.org/share/clg/source/mathematics_goals2001.pdf

Maryland State Department of Education. Common core state standards for mathematics. Retrieved from http://mdk12.org/instruction/commoncore/CCSSI_Math_Standards.pdf

Oswego City School District Regents Exam Prep Center (2011). Applied Practice with Adding Polynomials. Retrieved from http://www.regentsprep.org/Regents/math/ALGEBRA/AV2/sapplied.htm

Oswego City School District Regents Exam Prep Center (2011). Applied Practice with Multiplying Polynomials. Retrieved from http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/Papply.htm

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http://www.regentsprep.org/Regents/math/ALGEBRA/AV3/Papply.htm

http://www.regentsprep.org/Regents/math/ALGEBRA/AV2/sapplied.htm

http://mdk12.org/instruction/commoncore/CCSSI_Math_Standards.pdf

http://mdk12.org/share/clg/source/mathematics_goals2001.pdf

http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Multiplying%20Polynomials.pdf

http://www.kutasoftware.com/FreeWorksheets/Alg1Worksheets/Multiplying%20Polynomials.pdf

http://www.crayola.com/free-coloring-pages/print/jackie-robinson-baseball-player-coloring-page/

http://www.crayola.com/free-coloring-pages/print/jackie-robinson-baseball-player-coloring-page/


Nevada Department of Education. Math constructed-response items rubric guide. Retrieved from http://www.doe.nv.gov/Assessment/CRT/Math_Rubric.pdf

Zehner, A. (2010). Not my enemy, but my friend: How literacy serves content-area goals. In S. Plaut, The right to literacy in secondary schools: Creating a culture of thinking (pp. 36-48). New York, NY: Teachers College Press.

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http://www.doe.nv.gov/Assessment/CRT/Math_Rubric.pdf


Warm-up

Problems: (x2+4)(x-2) (x+2)(3x2+6x+2)

Solutions: x3-2x2+4x-8 3x3+6x2+2x+6x2+12x+4

3x3+12x2+14x+4

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Example of Flip Chart

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Polynomials Constructed Response

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Name:____________________________________________________ Date:_____________________

Directions:

1. Read.2. Reread and code the text.3. Make a plan of action.4. Analyze and share thinking.5. Edit and evaluate answer.

A circular courtyard has an area of 10-2x2. There are two rectangular flower beds in the courtyard. The first one has width 2x and length x. The second box has width 3x and length x. Write an expression for the area of the lawn using polynomials. Simplify.

Plan:

Observations of other groups:

Final Response:

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A rectangular swimming pool is twice as long as it is wide. A small concrete walkway surrounds the pool. The walkway is a constant 2 feet wide and has an area of 196 square feet. Find the dimensions of the pool.

Plan:

Observations of other groups:

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Final Response:

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Math Constructed-Response ItemsRubric Guide

The constructed-response questions are worth up to 3 points. Use the rubric below to guide your responses.

Score ExpectionsFull Credit Your response addresses all parts of the

question clearly and correctly. You use and label the proper math terms

in your answer. Your response shows all the steps you took

to solve the problem.Partial Credit Your response addresses most parts of the

question correctly. Your response does not show all of your

work or does not completely explain the steps you took to solve the problem.

Minimal Credit Your response addresses only one part of the question correctly and explains the steps you took to solve that one part. In answering the remaining parts of the question, your response is incomplete or incorrect.

Your response does not show all of your work or does not explain all of the steps you took to solve the problem.

No Credit Your response is incomplete.

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Coloring Polynomials

Name:______________________________________________Date:___________________

Find each product.

1. 6v(2v + 3) Yellow 2. -4(v + 1) Red

3. (4n + 1)(2n + 6) Brown 4. (6p + 8)(5p – 8) Green

5. (5n + 6)(5n – 5) Blue 6. (4p – 1)2 Purple

7. (3x – 4)(4x + 3) Black 8. (7k – 3)(k2 – 2k + 7) Red

9. (4a + 2)(6a2 – a + 2) Yellow 10. (n2 + 6n – 4)(2n – 4) Blue

Write down a fact or two about Jackie Robinson. (1 point will be added to your test score)

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Coloring Polynomials (Solutions)

Name:______________________________________________Date:___________________

Find each product.

1. 6v(2v + 3) Yellow 2. -4(v + 1) Red

12v2 + 18v -4v - 4

3. (4n + 1)(2n + 6) Brown 4. (6p + 8)(5p – 8) Green

8n2 + 26n + 6 30p2 – 8p - 64

5. (5n + 6)(5n – 5) Blue 6. (4p – 1)2 Purple

25n2 + 5n – 30 16p2 – 8p + 1

7. (3x – 4)(4x + 3) Black 8. (7k – 3)(k2 – 2k + 7) Red

12x2 – 7x – 12 7k3 – 17k2 + 55k – 21

9. (4a + 2)(6a2 – a + 2) Yellow 10. (n2 + 6n – 4)(2n – 4) Blue

24a3 + 8a2 + 6a + 4 2n3 + 8n2 – 32n + 16

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