Step 1 Lesson Plan - My Portfolio



Author (s): Nick Johnson

Team Members:

|Title of Lesson: Factor and Solve Polynomial Equations

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|Lesson # 2 |Lesson Source (kit, lesson): none |

|Date lesson will be taught: 11/15 and 11/18 | |

|Grade level: high school | |

|Concepts/Main Idea – in paragraph form give a broad, global statement about the concepts and vocabulary you want students to understand as a result of doing this activity: |

|After these lesson’s the students should be able to factor polynomials completely into a product of unfactorable polynomials with integer coefficients. The student’s should be able to recognize special |

|factoring patters like: sum of two cubes and difference of two cubes. The students will also be able to factor polynomials in quadratic form using tools such as: finding a common monomial factor or |

|recognizing a difference of two squares, etc. We will also explore a few real world applications to show how factoring polynomials can help in real life situations. |

|Objective/s- Be specific; prioritize; include higher-order objectives; be sure they are measurable.|Evaluation |

|Write objectives in SWBAT form… |In the space below, explain the type(s) of evaluation that will provide evidence that students have learned |

|The Students Will Be Able To: |the objectives of the lesson (formative and summative). You will provide student copies at the end of the |

| |lesson. |

| |1. Factor 16x^4 – 625 completely. Circle the correct answer. |

|Factor polynomials |a. (4x^2 + 25)(2x + 5)(2x – 5) |

|Find common monomial factor |b. (4x^2 – 25)(2x + 5) |

|Use special factoring patterns |c. (4x^2 – 5)(4x^2 + 5) |

|Solve real life polynomial questions | |

| |2. Factor the polynomial 2y^5 – 18y^3 completely. Circle the correct answer. |

| |a. 2y^3(y^2 – 9) |

| |b. 2y^3(y + 3) |

| |c. 2y^3(y + 3)(y – 3) |

| | |

| |3. Which one of these problems is completely factored? Circle the correct answer. |

| |a. (x^2 – 16)(x – 3) |

| |b. x(x + 5)(x – 3) |

| |c. 2x^2(x – 4) + 10(x – 4) |

| | |

| |4. Find the real number solutions for the equations g^3 + 3g^2 – g – 3 = 0. Circle the correct answer. |

| |a. g = 1,2,3 |

| |b. g = -1,1,-3 |

| |c. g = 0,-1,3 |

| | |

NGSS and Common Core Standards

Math Lessons:

1. Perform arithmetic operations on polynomials. CCSS.Math.Content.HSA-APR.A.1

2. Understand the relationship between zeros and factors of polynomials. CCSS.Math.Content.HSA-APR.B.2, CCSS.Math.Content.HSA-APR.B.3

3. Understand polynomial identities to solve problems. CCSS.Math.Content.HSA-APR.C.4, CCSS.Math.Content.HSA-APR.C.5

4. Create equations that describe numbers or relationships. CCSS.Math.Content.HSA-CED.A.2

5. Interpret the structure of expressions. CCSS.Math.Content.HSA-SSE.A.1, CCSS.Math.Content.HSA-SSE.A.2

|Materials list (BE SPECIFIC about quantities) |Accommodations: Include a general statement and any specific student needs |

|for Whole Class: have evaluation for everyone |There are no specific accommodations for the lesson. For ELL students, everything|

| |that is asked of the students will be written on the board. We will be using |

|per Group: |visuals, numbers and mathematical expressions that should help ELL students. |

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|per Student: calculator, pencil, and paper | |

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|Advance preparation: | |

|Print all activities for the lesson. | |

|Include handouts at the end of this lesson plan document (blank page provided) | |

| |Safety: Include a general statement and any specific safety concerns |

| |There are no physical safety concerns. Student should use basic school safety |

| |rules. Students will be able to feel safe answering all questions as being wrong |

| |in math is often how you can learn. All student should be safe asking any and all|

| |questions, there are truly no stupid questions. |

|Engagement: Estimated Time: 10-15 min |

|What the teacher does AND how will the teacher direct students: |Probing Questions: Critical questions that will connect prior |Expected Student Responses AND Misconceptions – think like a student to |

|(Directions) |knowledge and create a “Need to know” |consider student responses INCLUDING misconceptions: |

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|For the engagement of this lesson I will give all of the students a |1. “Who is going to share the answer they got?” |1. “I got the volume equals 4x^3 – 30x^2 + 54x.” |

|piece of paper they can fold into a rectangle and I’m simply going to |2. “Did anyone get anything different than these two |1. “But I got the volume equals x(9-2x)(6-2x) |

|give the students the dimension’s of the box and ask them to tell me |answers?” |2. “Yeah I got the 1st one, the person with the 2nd answer didn’t |

|the volume of the box. The rectangle they’re given will be 6 inches |3. “Oh okay, they didn’t multiply it all the way through?” |multiply the equation all the way threw. |

|by 9 inches, with a height of x. |3. “Is that answer wrong though? They’re still technically |3. “Yeah I guess they’re technically the same.” |

|After giving them roughly 5-10 minutes to solve the problem, I’ll ask |the same right? I never said if it had to be multiplied all |4. “Yeah it is.” |

|someone to share there answer. |the way through or not.” |5. “It is -1520.” |

|The height is 7.5 |4. “Wouldn’t the 2nd answer given be the first answer |6. “No it can’t work because you can’t have a negative volume.” |

|All though this equation technically doesn’t work because 7.5 is too |factored?” |7. “Then x = 7.5 inches.” |

|big of a height and when plugged into the width or height equation |5. “If I said x = -5 what is the volume?” |8. “Oh okay, after plugging in x the volume does end up being 405.” |

|separately gives a negative value for each. The students still get |6. “So when x = -5 the volume is -1520? Can that answer |9. “X has to be less than 3 for the equation to be real.” |

|the concept. |work? Is there anything wrong with the answer?” | |

| |7. “Exactly, you can’t have a negative volume. Now, I want | |

| |you all to tell me what the length, width, and height of this | |

| |box is when I tell you the volume is 405. (they should get x | |

| |= 7.5) | |

| |8. “Did any one get another answer? What’s an easy way to | |

| |check if your right? Since we already know the width and | |

| |height, just plug 7.5 in for x in the equation.” | |

| |9. “This equation actually can’t be possible, but you all can| |

| |still see the concept. 7.5 is too big of a height because the| |

| |width and length would be negative values. So how big can x | |

| |really be?” | |

|Teacher Decision Point Assessment: | |

|Exploration: Estimated Time: 35 min |

|What the teacher does AND what the teacher will direct students to |Probing Questions: Critical questions that will guide students|Expected Student Responses AND Misconceptions - think like a student to |

|do: (Directions) |to a “Common set of Experiences” |consider student responses INCLUDING misconceptions: |

|My exploration and explanation are going to be interchangeable and I| | |

|will be going back and forth between the two. |1. Let’s first talk about what it means to be completely |1. “The first equations is factored completely and the 2nd one is not.” |

|I’ll start by talking about what it means for a function to be |factored. “Which equation is completely factored? 2(x + 1)(x|2. “I got 6b(5b – 9), z(z – 12)(z + 6), and c(c + 3)(c + 6).” |

|completely factored. |– 4) or 3x(x^2 – 4)?” | |

|You know a function is completely factored when it’s written as a |That’s right, 3x(x^2 – 4) can be factored even more into: 3x(x| |

|product of unfactorable polynomials with integer coefficients. |– 2)(x + 2) | |

|We’ll then work on some problems and go through how to solve them. |2. Now, I’m going to put a couple of problems on the board | |

|The problems we just completed all have a common monomial factor we |and I want everyone to factor these completely: 30b^3 – 54b^2,| |

|recognize which make the factoring a lot easier. |z^3 – 6z^2 – 72z, and c^3 + 9c^2 + 18c. (answers are: 6b(5b – | |

|Remember the special product pattern you learned in earlier sections|9), z(z – 12)(z + 6), and c(c + 3)(c + 6) | |

|about the difference of two squares? Remember it was a^2 – b^2 = (a|“Did anyone get anything different? Why doesn’t someone come | |

|+ b)(a – b). Now were going to learn about the patterns of the sum |up to the board and show us how to do these.” | |

|and difference of two cubes. I’ll show them each formula and go |3. Knowing what a^2-b^2 equals, what do you think a^3+b^3 | |

|through a couple of examples of each and then write a few problems |equals? The sum of two cubes is: (a + b)(a^2 – ab + b^2). | |

|on the board for them to solve and then go over them as a class. |What do you all think a^3 – b^3 equals? The difference of two| |

| |cubes is: (a – b)(a^2 + ab + b^2). Let’s go over a couple of | |

| |examples together. | |

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|Teacher Decision Point Assessment: | |

|Explanation: Estimated Time: 35 min |

|What the teacher does AND what the teacher will direct students to |Clarifying Questions: Critical questions that will help |Expected Student Responses AND Misconceptions - think like a student to |

|do: (Directions) |students “Clarify their Understanding” and introduce |consider student responses INCLUDING misconceptions: |

| |information related to the lesson concepts & vocabulary – | |

| |check for understanding (formative assessment) | |

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|Review a little of what we learned on Friday about the sum and |1. “Can anyone factor the function x^3 + 7x^2 – 9x – 63?” |1. “No I can’t figure it out, none of our special patterns work for this |

|difference of two cubes. And then start into some new material. |If anyone can, I’ll have them come up to the board and show me|one.” |

|Now were going to look at factor by grouping. |how they were able to do it. Then as a class I’ll go over how| |

|After talking about grouping, we’ll move onto factoring polynomials |to solve the problem. We’ll go over a couple more problems as| |

|in quadratic form. |a class and then I’ll have them do a couple by themselves. | |

|Talk about using quadratics and polynomials to find volumes and |The formula for grouping is: ra + rb + sa + sb = r(a + b) + | |

|areas from a rectangle or square. |s(a + b) = (r + s)(a + b) | |

|Last, I will introduce u-substitution and go over a few examples |2. “Can anyone factor the function 4t^6 – 20t^4 + 24t^2?” | |

|with them. |I’ll give them a few minutes to see if they can factor this | |

| |function. After a couple minutes we’ll go over this problem | |

| |as a class. We’ll go over a couple more as a class and then | |

| |I’ll give them a couple of problems to solve by themselves and| |

| |then we’ll talk about them as a class. | |

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|Teacher Decision Point Assessment: | |

|Elaboration: Estimated Time: 15 min |

|What the teacher does AND what the teacher will direct students to |Probing Questions: Critical questions that will help students |Expected Student Responses AND Misconceptions - think like a student to |

|do: (Directions) |“Extend or Apply” their newly acquired concepts/skills in new |consider student responses INCLUDING misconceptions: |

| |situations | |

|I’m pretty sure I won’t have enough time for an elaboration. But if|1. “ If I draw a rectangle on the board and tell you the |1. The answer is 3x^2 + 14x + 8. |

|there was enough time we’d talk about: |dimensions are: (x + 4) by (3x + 2) what is the area?” |2. X equals 2 and -6.67, but x can’t be a negative value, so x equals 2. |

|Finding the volume and area of real world problems. |2. “So if I tell you area is 48, what does x equal?” |3. |

|These would be problems similar to the engagement. These would be |3. Let’s look at a 3d figure, “What is the volume of a | |

|more draw figures on the board and give student the dimensions, then|rectangle with length x, width 2x + 5, and height of 3x?” | |

|as what is x or ask what the volume or area equation would be. |4. “Now tell me, if that’s the volume. What is the length, | |

| |width and height of the rectangle, if the volume is 100?” | |

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|Teacher Decision Point Assessment: | |

|Evaluation: Estimated Time: 15 min |

|Critical questions that ask students to demonstrate their understanding of the lesson’s performance objectives. |

|Formative Assessment(s): In addition to the final assessment (bell ringer or exit slips), how will you determine students’ learning within this lesson: (observations, student responses/elaborations, white |

|boards, student questions, etc.)? |

|I will have numerous times during class, where they will work on problems by themselves or with a partner. From all of these calculations by themselves I’ll be able to tell who is struggling and who isn’t. |

|We will also be doing an activity for the engage and I will be able to see who does well with the activity and who doesn’t. After the students work on the problems together, we’ll go over them as a class |

|and I’ll make sure to call on everyone at least once, so I can get a sense of where everyone’s at. |

|Summative Assessment: Provide a student copy of the final assessment/exit slips or other summative assessments you use in the lesson |

|1. Factor 16x^4 – 625 completely. Circle the correct answer. |

|a. (4x^2 + 25)(2x + 5)(2x – 5) |

|b. (4x^2 – 25)(2x + 5) |

|c. (4x^2 – 5)(4x^2 + 5) |

| |

|2. Factor the polynomial 2y^5 – 18y^3 completely. Circle the correct answer. |

|a. 2y^3(y^2 – 9) |

|b. 2y^3(y + 3) |

|c. 2y^3(y + 3)(y – 3) |

| |

|3. Which one of these problems is completely factored? Circle the correct answer. |

|a. (x^2 – 16)(x – 3) |

|b. x(x + 5)(x – 3) |

|c. 2x^2(x – 4) + 10(x – 4) |

| |

|4. Find the real number solutions for the equations g^3 + 3g^2 – g – 3 = 0. Circle the correct answer. |

|a. g = 1,2,3 |

|b. g = -1,1,-3 |

|c. g = 0,-1,3 |

| |

|Answer is highlight in yellow |

Knowledge Package

Factor and Solve Polynomial Equations

Foundation Concepts

• Factoring polynomials

o General trinomial

o Perfect square trinomial

o Difference of two squares

o Common monomial factor

• Special factoring patterns

o Sum of two cubes

o Difference of two cubes

Target Concepts

• Factor the sum or difference of two cubes

• Factor by grouping

• Factor polynomials in quadratic form

Possible Misconceptions

• When factoring many students may think if a number doesn’t factor to a integer than it doesn’t factor at all. For example asking the students to factor x^2-5, since the square root of 5 isn’t an integer, students won’t think it has an answer.

• The only problems the students usually see all factor perfectly and end up with nice easy answers. The students may think any problem that doesn’t factor nicely doesn’t have an answer.

Future Topics in Curriculum

• Apply the remainder and factor theorems

• Find rational zeros

• Apply the fundamental theorem of algebra

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