Revised Unit Plan

EDCP 342A Unit planning: Rationale and overview for planning a 3 to 4-week unit of work in secondary school mathematics

Your name: Niloofar Razzaghi

School, grade & course: R.E. Mountain Secondary School, Grade 11,

Pre-Calculus

Topic of unit:  Radical Expressions & Equations

(1) Why do we teach this unit to secondary school students? Research and talk about the following: Why is this topic included in the curriculum? Why is it important that students learn it? What learning do you hope they will take with them from this? What is intrinsically interesting, useful, beautiful about this topic? (150 words) The goal is to show students how to solve equations with rational exponents and/or radicals. And to extend students' knowledge of solving to where an equation has either two radicals or two expressions with rational exponents. This lesson is taught to the grade 11 students and this topic is included in the curriculum because: 1) To show students what solving radical equations means. 2)To Give students the expression Isolate-Eliminate-Solve, to help them remember what they need to do first in solving. 3)To Give students practice solving equations with radicals and equations with rational exponents. 4)To show students how to solve equations that have more than one radical or expression with a rational exponent. 5)Students get some practice solving with multiple radicals and/or rational exponents. After learning Radical Expressions and Equations, students would be able to simplify radical expressions and perform simple operations such as adding, subtracting, multiplying and dividing these expressions. With this learning, they would be able to create and solve attractive and beautiful real-life examples:  One of the simplest formulas in electrical engineering is for voltage, V = √PR, where P is the power in watts and R is the resistance in ohms.

(2) A mathematics project connected to this unit: Plan and describe a student mathematics project that will form part of this unit. Describe the topic, aims, process and timing, and what the students will be asked to produce, and how you will assess the project. (250 words) Time: Due on the first week of March 2018 Title: Radical Expressions and Equations  The main goal and the big idea of having this project for the students is to show them that there are large connections between the real-life Experiences and math and particularly with Radical Expressions and equation. Rational expressions are used in many fields such as Financial Planning, Radioactive half-life, /Meteorology/Oceanography Electrical Engineering, medicine, Physics/Engineering and Supply Chain Management/Business. The topic of the project is related in medicine. Since this unit heavily emphasized on computations and calculation, I hope this project will provide the students with an idea of how real-life applications of radical equations look like. Moreover, students will be researching and considering about other field than mathematics in this project; thus, it provides a cross-disciplinary experience.  Students will be asked to choose at least ten people of interest and calculate their body surface areas (in square meters) using the formula: BSA= √(W*H/3600) Where W is the weight of the person in kilo-grams and H is the height in centimeters. Students then are supposed to research about the body surface areas and do a simple analysis, and discuss their findings. The analysis and discussion are open-ended. students are free to draw any kinds of conclusion from their data as long as it is appropriate; for examples, they can draw possible connection between individual’s BSA and their health, or comparing the BSA among different gender/age groups (if students are doing the comparison, they may collaborate with other groups). Students must cite their resources. Students will be assessed on the following criteria:       1.      Accuracy of their calculation, units, rounding.       2.      The written discussion       3.      Quality of the written work: grammar, clarity, citation, etc.

(3) Assessment and evaluation: How will you build a fair and well-rounded assessment and evaluation plan for this unit? Include formative and summative, informal/ observational and more formal assessment modes. (100 words) Formative assessment As formative assessment we can use Kahoot.it, quizzes, worksheets, example, unit test, and IXL.  Students as group or individual will have Kahoot quizzes as formative assessment, and the detailed solution will be given to check their answers. Also, IXL, which is online learning/ homework, will be used to check student’s progress in each section. in class participation, inquiry based questions also are part of formative assessment. Summative assessment As Summative assessment, I will use quizzes, unit test, homework assignments, project works which are used to evaluate students’ unit understanding and to determine whether they have learned the material are being taught in the class.

Lesson Plan (1)

Subject:  Pre-Cal	Grade: 11	Date: ------	Duration: 70 min
Lesson Overview	Multiplying and Dividing Rational Expression
Class Profile	There are 30 students in this class,10 of them are ELLs. As a classroom with different English language skills they all need to learn about the unfamiliar words in the new lesson, so I will prepare a list of the new vocabularies before starting to teach the new lesson. As some students might be visual learners, I might have to draw some words for them to fully understand. For e.g. for the word Reciprocal I might have to show them by writing/drawing a sample, or somehow make it comprehended if is any problem exited.
Prior Knowledge	Simplifying Rational Expression
Objectives  (Pedagogical Goal)	Multiplication and division of Rational expressions is in some ways easier than addition and subtraction of rational expressions because a common denominator does not have to be found. A lot of times in math students must use past concepts to be able to work all the way through the unfamiliar problems.  In this lesson students will have to remember how to factor, simplify rational expressions and multiply polynomials to be able to complete the multiplication or division problems. These would be my pedagogical goals for all my classes including this one: ·         To improve understanding of some of the previous concepts which would be related to the new ones  ·         To improve understanding of the nature of the subjects: what is important, how it is practiced…. ·         To improve understanding of the historical development of selected topics. ·         To develop an unobstructed vision of mathematics.        And …. To improve teaching strategies and have better understanding of students’ strengths and weaknesses.

Materials and Equipment Needed for this Lesson

·         White board ·         Tablet ·         Markers ·         Computers

Lesson Stages	Learning Activities	Time Allotted
Warm-up (Open-ended problem solving in groups)	Most of the time in math, students must use past concepts to be able to work all the way through the unfamiliar problems and new concepts.  In this section students will have to remember how to factor, simplify rational expressions and multiply polynomials to be able to complete the multiplication or division problems.           ·      First, I will have them discuss solving this problem and see if they can remember how to do multiplication in numerical fractions:           3/ 4 * 8 /9        then ask students what they noticed about simplifying rational numbers. Did they have to be multiplying by the common factor to simplify? Could they do if they were adding or subtracting by a common factor?	10 min
Presentation	The rules for multiplying and dividing rational expressions are the same as the rules for multiplying and dividing rational numbers, I will start by reviewing multiplication and division of fractions. When we multiply two fractions we multiply the numerators and denominators separately: a/b* c/d =a*c / b* d When we divide two fractions, we replace the second fraction with its reciprocal (I must make sure they know the meaning) and multiply, since that’s mathematically the same operation: a/b÷c/d =a/b*d/c=ad/bc If students are comfortable with multiplying/dividing the given examples, then I start giving them a new problem adding an unknown, like: x or …. And begin to give the practice and the new lesson.	10 min
Practice and Production	Multiplying Rational Expressions (Q and S CAN NOT BE equal 0) Step 1: Factor both the numerator and the denominator. Step 2: Write as one fraction. Step 3: Simplify the rational expression. Step 4: Multiply any remaining factors in the numerator and/or denominator. I will show the students an example using multiplication and go through step by step to make sure the students understand the process. Multiply Step 1: Factor both the numerator and the denominator. Since our problem’s denominators are already factored, we can move on. Step 2: Write as one fraction.                 Step 3: Simplify the rational expression.                                 (Exclude values of original data making denom. = 0) There are no remaining factors in the numerator or denominator, so we are done. Dividing Rational Expressions (Q, S, and R CAN NOT BE equal 0) Step 1: Write as multiplication of the reciprocal. Step 2: Multiply the rational expressions as shown above. Divide Step 1: Write as multiplication of the reciprocal. AND Step 2: Multiply the rational expressions as shown above.                 =                                   *Multiply by the reciprocal.                           *Simplify by dividing out common factors.                                          *Multiply denom. and num. out.                         *Exclude values making denom. zero. I can come up with some more examples for students to assure that the new lesson is mostly comprehended. As they need to practice on their own, I will also assign some problems from the text book for students to practice more.	40 min
Closing (Arts and Mathematics)	The lesson will end by allowing the students to practice problems together. Ask students to summarize and response to the important parts of the lesson using writing and to what they have learned today in writing, or by showing some examples with a partner, individually, or as a class. I use this as an opportunity to informally assess understanding of the lesson. Ask students to describe the processes for multiplying and dividing rational expressions and simplifying complex rational expressions either verbally or symbolically or using any type or art. I can also give them more time till the next class, if they need to express the lesson using any arts.	10 min

Lesson Plan (2)

Subject: Pre-Cal	Grade: 11	Date: ------	Duration: 70 min
Lesson Overview	Adding and Subtracting Rational Expression              (same denominator)
Class Profile	There are 30 students in this class,10 of them are ELLs. As a classroom with different English language skills they all need to learn about the unfamiliar words in the new lesson, so I will prepare a list of the new vocabularies before starting to teach the new lesson. As some students might be visual learners, I might have to draw some words for them to fully understand. For example, as I will use commutative/associative properties in this lesson, I will remind them the properties as well as giving them the definition and description for some ELL students.
Prior Knowledge	Simplifying Rational Expression, and multiplying …the Previous lesson
Objectives  (Pedagogical Goal)	This lesson reviews addition and subtraction of fractions using the familiar techniques that students have seen in earlier grades, then the lesson will move towards the process for adding and subtracting rational expressions by converting to equivalent rational expressions with a common/different denominator.  The goals are: 1) Add or subtract rational expressions with common(unlike) denominators 2) Identify the least common denominator of two or more rational expressions 3) Add or subtract rational expressions with unlike denominators 4) Realize the connection between adding/subtracting rational numbers and adding/subtracting rational expressions These would be my pedagogical goals for all my classes including this one:  To improve understanding of some of the previous concepts which would be related to the new ones  ·         To improve understanding of the nature of the subjects: what is important, how it is practiced…. ·         To improve understanding of the historical development of selected topics. ·         To develop an unobstructed vision of mathematics.        And …. To improve teaching strategies and have better understanding of students’ strengths and weaknesses.

Materials and Equipment Needed for this Lesson

·               White board ·               Tablet ·               Markers ·               Computers

Lesson Stages	Learning Activities	Time Allotted
Warm-up (Open-ended problem solving in groups)	First, I remind the students how to add fractions with the same denominator. Allow them to work through the following sum individually.  Calculate the following sum, and subtraction: 𝟑 /𝟏𝟎 + 𝟔/ 𝟏𝟎 ,5/13-3/13. The solution should be available to the class either by the teacher or by a student because the process of adding fractions will be extended to the new process of adding rational expressions. Then I will ask them to solve two problems with unlike denominators. 15/26+6/13, and 11/21-6/7 If they are done, allow them to think about how to approach those problems, which involves adding few rational expressions.	10 min
Presentation	Ask students for help in stating the rule for adding and subtracting rational numbers with the same denominator. The result below is valid for real numbers 𝑎, 𝑏, and 𝑐.  𝑎/ 𝑏 + 𝑐/ 𝑏 = 𝑎 + 𝑐/ 𝑏 and 𝑎 /𝑏 – 𝑐/ 𝑏 = 𝑎 – 𝑐/ 𝑏. They could generalize the process for two rational expressions, rearrange terms using the commutative property (should have reminded them as they have learned it in earlier grades and was remined earlier already) to combine the terms with the same denominator, and then add using the above process, or they could group the addends using the associative property (again, should be remined to them earlier) and perform addition twice.	10 min
Practice and Production	I should plan on demonstrating the connections between fractions and simple rational expressions. Thus, we begin with discussing the following examples of fractions to help students recall some basics.  1/8 +5/ 8 and 2/ 3 +5 /7 Then we begin the additional contents as follows: I write two examples with common denominators on the board and discuss solutions with the class, asking questions of the students and getting suggestions for each step.  2/x+3/x and 3x/x+2 - 2x/ x+2 Give the students a similar problem to work on individually or in pairs. Then the students will provide the instructor with the solution. Write several examples on the board and discuss solutions with the class. These examples should contain rational expressions with un-like denominators and should increase in difficulty level with the instructor still prompting students for input in the working of the problem. I will try to ease student anxiety by providing a list of steps, demonstrating the steps on several increasingly difficult problems, and showing the students, that even very complicated looking problems should be worked in the same manner as simple rational expressions. Therefore, after the first example, I should discuss these general steps for solving a problem with un-like denominators, list them on the board, and pass out the handout of general steps for the students to reference. The I will continue discussing the solutions to the remaining examples demonstrating the steps on these more difficult examples. To add/subtract rational expressions with the same denominator Steps to Add and Subtract Rational Expressions: 1. Factor denominators. 2. Find Least Common Denominator (LCD). 3. For each rational expression, compare denominator to LCD and multiply numerator by missing factors from LCD. 4. Combine numerators of rational expressions and put over LCD. 5. Simplify result by factoring numerator and canceling factors common with denominator.	40 min
Closing (Arts and Mathematics)	The lesson will end by allowing the students to practice problems together. Ask students to summarize and response to the important parts of the lesson using writing and to what they have learned today in writing, or by showing some examples with a partner, individually, or as a class. I use this as an opportunity to informally assess understanding of the lesson. Ask students to describe the processes for adding/subtracting rational expressions and simplifying complex rational expressions either verbally or symbolically or using any type or art. I can also give them more time till the next class, if they need to express the lesson using any arts.	10 min

Lesson Plan (3)

Subject:  Pre-Cal	Grade: 11	Date: ------	Duration: 70 min
Lesson Overview	Solve rational equations
Class Profile	There are 30 students in this class,10 of them are ELLs. As a classroom with different English language skills they all need to learn about the unfamiliar words in the new lesson, so I will prepare a list of the new vocabularies before starting to teach the new lesson. As some students might be visual learners, I might have to draw some words for them to fully understand. the words extraneous solutions need to be explained and defied for the whole class as it will be used in the lesson
Prior Knowledge	Simplifying Rational Expression , and multiplying …
Objectives  (Pedagogical Goal)	After this lesson, students will be able to: define and provide examples of rational equations describe the steps to solve a rational equation solve rational equations. In the previous lessons, students learned to add, subtract, multiply, and   divide rational expressions. In this lesson they can solve equations involving rational expressions There is more than one method to solve a rational equation, and in this section. The first method is to multiply both sides by the common denominator to clear fractions. The second method is to find equivalent forms of all expressions with a common denominator, set the numerators equal, and solve the resulting equation. These would be my pedagogical goals for all my classes including this one:  To improve understanding of some of the previous concepts which would be related to the new ones  ·         To improve understanding of the nature of the subjects: what is important, how it is practiced…. ·         To improve understanding of the historical development of selected topics. ·         To develop an unobstructed vision of mathematics.        And …. ·              To improve teaching strategies and have better understanding of students’ strengths and weaknesses.
Materials and Equipment Needed for this Lesson
·              White board ·              Tablet ·              Markers ·             Computers

Lesson Stages	Learning Activities	Time Allotted
Warm-up (Open-ended problem solving in groups)	I will introduce lesson by reminding students or asking students what solving means. Then I will explain that this lesson is on solving rational equations. I will give them examples of quadratic, and radical equations that they have solved in previous chapters. Now that students know how to add, subtract, multiply, and divide rational expressions, it is time to use some of those basic operations to solve equations involving rational expressions. An equation involving rational expressions is called a rational equation. Students will benefit from first solving the equation 𝑥/ 5 – 2/ 5 = 1/ 5. More advanced students may try to solve other harder examples.	10 min
Presentation	·               I will then get students help and allow them to tell me what the think the lesson will be about: ·               To solve some rational equations, you can multiply each side of the equation by the LCD (Least Common Denominator) You can use cross product to solve rational equations.   You need to check for extraneous solutions.  Extraneous solutions are solutions that are derived but not from the original equations. (Definition   has been already explained in the beginning of the lesson to the class to make sure all students even ELL comprehended it) ·                 Check all results to find the solutions for the original equations.	10 min
Practice and Production	Give students a few minutes to discuss extraneous solutions with a partner to try to fully get the definition. When do they occur, and how do you know when you have one? Extraneous solutions occur when one of the solutions found does not make a true number sentence when substituted into the original equation. The only way to know there is one is to note the values of the variable that will cause a part of the equation to be undefined. This lesson is concerned with division by zero; later lessons exclude values of the variable that would cause the square root of a negative number. I will make sure that all students understand extraneous solutions before proceeding. Then, have them work in pairs on the following exercises.  Now I will have students to work with partners to solve the following equation. I will circulate the room and observe student progress; if necessary, offer the following hints and reminders:  Reminder: Ask students to identify excluded values of x. Suggest that they factor the denominator x ^2 − 4. They should discover that x ≠ 2 and x ≠ −2 must be specified. § Hint 1: Ask students to identify a common denominator of the three expressions in the equation. They should respond with (x − 2) (x + 2), or equivalently, x ^2 −4. §  Hint 2: What do we need to do with this common denominator? They should determine that they need to find equivalent rational expressions for each of the terms with denominator (x −2) (x +2). Solve the following equation for x: 𝟏/ x+𝟐 + 𝟏/ x−𝟐 = 𝟒/ x^𝟐−𝟒 . we applied what we have learned in the past two lessons about addition, subtraction, multiplication, and division of rational expressions to solve rational equations. An extraneous solution is a solution to a transformed equation that is not a solution to the original equation. For rational functions, extraneous solutions come from the excluded values of the variable. Rational equations can be solved one of two ways: 1. Write each side of the equation as an equivalent rational expression with the same denominator and equate the numerators. Solve the resulting polynomial equation, and check for extraneous solutions. 2. Multiply both sides of the equation by an expression that is the common denominator of all terms in the equation. Solve the resulting polynomial equation, and check for extraneous solutions.	40 min
Closing (Arts and Mathematics)	The lesson will end by allowing the students to practice problems together. Ask students to summarize and response to the important parts of the lesson using writing and to what they have learned today in writing, or by showing some examples with a partner, individually, or as a class. And to tell me which way they would prefer. I use this as an opportunity to informally assess understanding of the lesson. Ask students to describe the processes for solving rational expressions and simplifying complex rational expressions either verbally or symbolically or using any type or art. I can also give them more time till the next class, if they need to express the lesson using any arts.	10 min