microsoft word - s1_syllabus.doc€¦ · web viewoperations with radical expressions and radical...

Project Based Instruction

Title of Lesson: Operations with Radical Expressions and Radical EquationsUFTeach Students’ Names: Karissa BurseyTeaching Date and Time: March 15 and 18Length of Lesson: Two 50-minute lessonsGrade / Topic: 9th grade AlgebraSource of the Lesson: Carter, J. A. (2011). Algebra 1 (Florida ed.). Columbus, OH: Glencoe/McGraw-Hill.Concepts: Within these four lessons the students are being introduced to the more complicated properties of radicals. At this point the students already know the basics of square roots and how to simplify radical expressions but not much more. Radicals are defined as “an expression that uses a root, such as a square root, cube root, etc.” The students will be learning the different parts that make up a radical, the degree, the radical symbol, and the radicand. Within these lessons the students will also be learning how to add, subtract, and multiply radicals with both like and different radicands. The students will learn that in order to add or subtract radicals they must have like radicands. In order to help solidify this idea the teacher could make a connection to like terms with variables. If the radicals being added or subtracted do not have like radicands then the students will learn to simplify the radicals to get like radicands. If like radicands are still not present then the students will learn they cannot combine the two radicals together under addition or subtraction. The students will also learn to hold the radicand constant when adding the two radicals and the coefficients are what is added or subtracted. Students will also learn that when multiplying radicals both the coefficients and the radicands and multiplied with each other respectively.

Radical - math word definition - Math Open Reference. (n.d.). Table of Contents - Math Open Reference. Retrieved February 28, 2013, from http://www.mathopenref.com/radical.html

Florida State Standards (NGSSS) with Cognitive Complexity: Benchmark Number Benchmark Description Cognitive ComplexityMA.912.A.6.1 Simplify Radical Expressions Level 2MA.912.A.6.2 Add, Subtract, Multiply, and Divide Radical Expressions

(Square Roots and Higher)Level 2

Performance Objectives: Students will be able to: Day 1

o Add and Subtract Radical Expressionso Multiply Radical Expressions

Day 2o Solve Radical Equations where the only variable is in the radicando Solve Radical Equations where there is a variable on each side

Materials List and Student Handouts Day 1 and 2

o A form of projection system to project the engagement, directions, and worksheetso A copy of Appendix B,C,E, and F for each studento A teacher set of all of the worksheets to project to the students using the elmo. o A timer or clock to keep track of the amount of time spent on the activities


Advance Preparations The teacher will need to review and then alter the worksheets if necessary The teacher will need to print out the appropriate number of worksheets The teacher will need to make sure the projection system in the classroom is running appropriately

Safety The students will be throwing paper balls during the last day’s engagement and the balls must be

thrown carefully and students should be directed to not throw at another student’s head.


5E Lesson:Engagement Time: 5 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsDay One: The teacher will introduce themselves and the day’s lesson.

Good Morning my name is Ms. Bursey and today we will be learning about using operations with radical expressions.Review the operations to be used.

What mathematical operations do you think we will be using in class today?

[Addition, Subtraction, and Multiplication]Some students may also say division but that is not in this lesson.

Present the Engagement Problem in Appendix A (Slide 2 of the PowerPoint).

What methods can we use to solve this problem?

What are some mathematical operations?

When looking at a math expression or equation what are some of the symbols that are can be seen.

Now what math operations do those symbols represent?

Ok let us keep operations in mind while we do our next activity!We will come back to this problem at the end of class.

[Mathematical operations]

[Addition, Subtraction, Multiplication, and Division.](If the students have difficulties the teacher will help them realize what the operations are.)

[+,=,-, × ,÷]

[Addition, equality, subtraction, multiplication, and division.]

Day Two: The teacher will again introduce themselves and


the topic of the day’s lesson.

But before we can start the activity we are going to need to review some directions and rules first.

As the student provides the rule the teacher will write it down on the board.

Good morning! Today we will be learning about Radical Equations, so in order to do that we will be participating in a fun activity!

For this activity you will be given a question. On a piece of paper you will answer this question.

You will need to write each step of the problem and you will need to write out your reasoning for each step in complete sentences.

After you finish answering the question you will crumple up your paper and wait for my say so.

When I say “Go” you will throw the paper balls around the room for one minute.

After the minute is up everyone will pick up a random paper ball and I will choose two random people to read the answers on the papers.

But there are rules.

What is the major rule?

If anyone violates this rule or things get too out of hand then we will end the game and get straight to the lesson.

[Do not hit other people in the face.]


The teacher will project Appendix D.

After two minutes.The teacher will time one minute.

The teacher will then choose two random students to read out the answers on their papers.

After each reading the teacher will ask first the student and then the class if there was anything missing on the paper.

Ok! Here is the question.

Go ahead and take a couple of minutes to answer the question and state your reasons.

Ok now I would like everyone to stand up. Once you are standing up get into your throwing positions. And Go!

Ok the one minute is up and I need everyone to please pick up a paper ball.

Was that a complete answer and justification with every step?

What else could the original writer added?

Great! Thank you guys for participating in that game. Now please take your seats and we will begin working on the activity for today.

[Answer could differ depending on the response being read.]

[Again answer could differ depending on the response being read.]

Exploration Time: 20 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsDay One:Introduce the Exploration Activity.

Exploration Part 1:For this next activity we are going to determine how we can use mathematical operations with radical expressions.

I am now passing out worksheets


(Appendix B). Make sure you write your name on it when you get it. As stated before we are going to be looking at operations with radical expressions. Let us work through the examples together as a class.

Look at Example 1.

Do you think we can add these radicals? Why?

How should we add them?

Let us look at it another way.

What does the 4 in front of 4 √5 mean?

And what does 2√5 mean?

So 2 groups of √5 and 4 more groups of √5 would give us ___ groups of ____?

Correct! So what should our final answer be?

Now let us look at Example 2.What do you think we will do to subtract these two radicals?

Yes! What is the final answer?

Great! Today you will be working in the group you are now seated in to do similar problems. By the end of part one you should be able to develop a general algorithm for adding and subtracting

[Yes.]

[Add the four and the two but keep the square root of five constant, 6√5] (The answer is 6√10.)

[Four groups of √5¿

[Two groups of √5]

[Six groups of √5]

[6√5]

[Subtract the two from the nine but keep the square root of three constant.](0) [7√3]


The teacher will circulate throughout the class and make sure that the students are on track and understanding the material. If a student seems to be having trouble with the information the teacher will ask probing questions to try and get the student back on track.

Move on to Explanation Part 1.

radicals with like radicands.

You have five minutes to finish part one. Begin!

What math operation are you using on this problem?

When we did the examples together as a class, what did both radicals have in common?

What parts of the radicals were we adding (or subtracting) together?

Now what part of the radical remained the same?

What is the part within the radical symbol called?

Now both radicals had the square root of 5 (or 3) and we only added their coefficients. What else does that remind you of?

[The answer could be addition or subtraction.]

[The square root of 5 or 3.]

[The coefficient in front of the radical.]

[The part within the radical symbol.]

[The radicand]

[Adding like terms.]


Explore Part 2: The teacher will now bring the class’ attention back to the front and move on to Part two.

Let us now look at example 3.

Can these two radicals be added together?

If students reply the two radicals cannot be added together the teacher will ask the following questions.

What is needed to be able to add two radicals together?

Are these two radicands equal?

Is there a way we can make them equal?

What is the first step to simplifying the radicands?

Once we do that what do we do?

Then is that radical completely simplified?

After we do that to our example are the two radicands equal?

Can we add them together now?

What should our final answer be?

[Yes.](Some students may reply no as they cannot see how.)

[The radicands must be equal.]

[No.]

[Simplify them (students should know this as this is what they will be learning the two days prior to this lesson.]

[Completely factor the radicand.]

[Move the factors to the appropriate powers outside of the radicand and in front of the radical symbol.]

[Yes.]

[Yes.]

[Yes.]

[12√2]


Correct! Now work with your group to complete part two.

You have five minutes. Begin!

The teacher will again circulate the room and make sure that students are remaining on task.

If the students seem to become confused the teacher will ask questions.

Move on to Explain Part 2.

What is needed for us to be able to add or subtract two radicals?

What similar factor do you think will be the radicand that we are looking for?

How can we get the radicand to equal that number?

After we do that can we then add the two radicals?

[The radicands must be equal.]

[Answer varies based off of question on the worksheet.]

[Factor out the perfect squares.]

[Yes.]

Explore Part 3: The teacher will then bring back attention to the front of the classroom and will continue onto part three.

Perfect! Again you will be

We have so far looked at adding and subtracting radicals. Now we will look at multiplying radicals.

Let us look at example 4.

What do we do to multiply these two radicals together?

Yes! So what is our final answer?

[Multiply the coefficients together and multiply the radicands together.] (Multiply the coefficients together but not the radicands.)

[24√2]


working with your group to work through part three. By the end of part three you should have a general algorithm for multiplying radicals.

You have five minutes. Begin!

Move to Explain Part 3. Day Two: Introduce the Exploration Activity.

Exploration Part 1:

The teacher will work through the first example with the students as a class. The students should write the steps on the worksheet as the class works through the problem, i.e. get the radical by itself, square both sides, etc.

For this next activity we are going to use what we learned last week to now solve radical equations.

I am now passing out worksheets (Appendix E). Make sure you write your name on it when you get it.

As just said we will be solving radical equations and we will be first starting with radical equations with a variable only in the radicand.

Let us look at example one on the worksheet.

What are we solving for?

Correct. What is the first step we must take?

Yes and what would be the next step?

What operation undoes a square root?

Can we only square the one side?

So what do we do?

[a]

[Subtract seven from both sides.]

[Square both sides.]

[Squaring]

[No]

[Square both sides]


Great! Today you will be working in our groups of four to do similar problems.

You have five minutes and you may begin!

Move to Explain Part 1.

And what is done after that?

Perfect, and what do we find to be the value of a?

Ok now if we input 20 into the original equation do we get a balanced equation? Perfect!

[Subtract five from both sides.]

[a=20]

[Yes.]

Exploration Part 2:

We will now be moving onto solving radical equations with variables on both sides.

The teacher will work through the second example with the students as a class.

Let us look at example two on the worksheet.

What are we solving for?

Correct. What is the first step we must take?

Why?

Yes and what would be the next step?

And what is done after that?

After that how do we find the value of k?

[k]

[We square both sides.]

[To eliminate the square root]

[To subtract k from both sides.]

[Subtract 1 from both sides.]

[Factor out the quadratic equation.]


Please get back to your groups to work through part two of our activity!

You have five minutes, begin!

Move to Explain Part 2.

What does the factored equation look like?

So what does k equal?

But what do we have to do before we can definitely say that is the answer?

Correct. Does the original equation balance for both values?

What value does not work and is therefore extraneous?

So what is the value of k?

Correct!

[0=k(k-3)]

[k=0,3]

[Check the values and make sure the original equation balances.]

[No.]

[k=0]

[k=3]

Explanation Time: 15 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsDay One: The teacher will go over the worksheets from the Exploration, Appendix B.

Explain Part 1: To get answers the teacher will ask four of the groups to choose a representative to come up and present their answer. As the teacher moves through each of the problems they will stop to

How did you and your group get the answer to the first problem?

What is the answer?

Correct!

How did you and your group get the answer to number two?

[We added the coefficients three and six together.]

[9√5¿

[We subtracted six from one.]


ask if the class got the same answer.

After all four of the groups have presented their answer the teacher will then work with the class in developing the general algorithms for adding and subtracting two radicals with like radicands.

Now that we have those algorithms we are going to focus on adding and subtracting radicals with unlike radicands.

Go back to Explore Part 2

What is the answer to that question?

Yes and what are those numbers before the radicals called in our expressions?

Continue this line of questioning for numbers three and four.

What is the general algorithm for adding radicals with like radicands?

Yes! Can we use this will all radicals?

What radicals can we use this with?

What is the general algorithm for subtracting radicals with like radicands?

Yes! Can we use this will all radicands?

What radicands can we use this with?

[−5√7]

[Coefficients.]

[a√b+c√b=(a+c)√b, b ≥ 0]

[No]

[They must have like radicands.]

[a√b−c √b=(a−c ) √b ,b ≥ 0]

[No]

[They must be equal.]

Explain Part 2:

To get answers the teacher will ask four different groups to choose a representative to come up and present their answer. As the teacher moves through each of the problems they will stop to ask if the class got the same answer.

How did you and your group get the answer to number one?

What is the answer to the first problem in part two?

Correct!

Continue this line of questioning for numbers two, three and four.

[We simplified the second radical to 4 √5. The two radicals could then be added.] [8√5]


Go back to Explore part 3

What is needed for two radicals to be added or subtracted together?

Yes, now if they do not have like radicands what can we do?

Ok but what if some of the radicands are still not equal? Can you add or subtract them together then?

Why not?

Good what number from part two of the activity shows us this?

Correct! Now we will be moving on to our third and last part of today’s activity.

[The must have like radicands.]

[Simplify s the radicands if not already.]

[No.]

[In order to add or subtract two radicals they must have like radicands.]

[Number three.]

Explain Part 3: To get answers the teacher will ask four different groups to choose a representative to come up and present their answer. As the teacher moves through each of the problems they will stop to ask if the class got the same answer.

If the teacher has time they will move onto the Elaboration.

What is the answer to the first problem in part two?

Correct! Now how did you and your group get this answer?

Correct again!


What is the general algorithm for multiplying two radicals?

Yes! Can we use this with all radicands?

[96¿

[We multiplied the coefficients together and the radicals together.]

[a√b ( f √g )=af √bg ,b≥ 0∧g≥ 0]

[Yes.]

Day Two:

The teacher will go over the worksheets form the Exploration, Appendix E.


Explain Part 1: To get answers the teacher will ask four of the groups to choose a representative to come up and present their answer. As the teacher moves through each of the problems they will stop to ask if the class got the same answer.

Return to Exploration Part 2.

How did you and your group get the answer to number 1?

What is the answer to the first problem in part one?

Correct!


Great! Now how could you check to see if your answer was correct?

[The student will step by step explain the process because each response will be different for each group.]

[h=40]

[Check the value to make sure the equation balances.]

Explain Part 2:

To get answers the teacher will ask four of the groups to choose a representative to come up and present their answer. As the teacher moves through each of the problems they will stop to ask if the class got the same answer.

How did you and your group get the answer to number one?

What is the answer to the first problem in part one?

Correct!


Great! Now what must you always do after you find a value for the variable you are solving for?

Correct!

[The student will step by step explain the process because each response will be different for each group.]

[y=3]

[Check the value to make sure the equation balances.]

Elaboration Time: 10 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsDay One: The teacher will pass out


the Elaboration worksheets, Appendix C, and remind the students to write their names on them once they receive their worksheet.

The teacher will inform the students to work in their same groups to solve the problem.

The teacher will circulate around the room and ask guiding questions if a group seems to become stuck.

What operation will we be using initially in this problem?

Yes so what would you be multiplying together?

Correct and what would we also need to multiply those by to get the area of a rhombus?

Exactly!

[Multiplication.]

[The lengths of the two diagonals.]

[1/2]

The teacher will bring the student’s attention back to the front of the classroom.

The teacher will then ask for volunteers to answer the question.

What is the area of the rhombus?

Correct!

[12(240−16√15−18√30+18√2)

]

Evaluation Time: 5 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsEnd of Day 2The teacher will pass out the Evaluation, Appendix F, and make sure that the students work individually on them.


Attach any assessments and handouts as additional pages in this document. Do not submit them as separate files!

Appendix A:


Jill is going to run in the park to get her ready for a triathlon. She plans to run the course that she has designed twice each day.

x √5

How far does Jill have to run to complete the course that she has designed?

How far does she run every day?

Appendix B: Exploration Day One

x

2x


Name:

Part One:Example1: 4 √5+2√5 Example2: 9√3−2√3

Directions: Work in a group of three to simplify each expression. Show all of your work.

1. 3√5+6√5 3. 7√5+4 √5

2. √7−6√7 4. 10√2−6√2


Part Two: Example 3: 2√18+√72


1. 4 √5+2√20 3. √8+√12+√18

2. √12−√3 4. 3√50−3√32+√8


Part Three:Example 4: 3√2∗4√6


1. 4 √3 (8√3) 3. 5√3(6√10−6 √3)

2. √3(√7+3√2) 4. (5√2+3√5 )(2√10+5)

Appendix C: Elaboration


Name:

Directions: Work in your groups of three to find the area of the given rhombus. The area A of

a rhombus can be found using the equation A=12

d1d2, where d1 and d2 are the lengths of the

diagonals. What is the area of the rhombus below using the given diagonal lengths? Make sure to show all of your work.

8√5−3√6

6√5−2√3

Area=


Appendix D: Day Two Engage

Directions: Working alone solve the below simplify the problem below, make sure you show all of your work and explain each step.

While finishing his homework Bob came across the following problem.

Simplify: 2√3 (√5 x+√24 )−3 √2

If Bob simplified the problem correctly what should his answer be?

Make sure to show your work and explain why.

a. 2√8x+2√24−3√2

b. 2√15 x+9√2

c. 2√15 x+4 √6

Appendix E: Exploration Day Two


Name:

Part One: Example 1: √a+5+7=12

Directions: Solve each equation. Make sure to show all of your work and check your solution.

1. √10 h+1=21 3. √h−5=2√3

2. √7 r+2+3=7 4. √2k+8=3√2

Part Two:


Example 2: √k+1=k−1

Directions: Solve each equation. Make sure to show all of your work and check your solution.

1. y=√12− y 3. √r+3=r−3

2. √1−2 t=1+t 4. √3 x−5=x−5

Appendix F: Evaluation

Name:


Directions: Simplify each expression. Show all of your work. Please circle your final answers.

1. 3√2+5√2−11√2 2. (3√5+√7 )(√20−2√3)

Directions: Solve each equation. Show all of your work. Please circle your final answers.

3 √5 x+6−1=3 4. √3 x2−32=x

Exploration Day One Answer Key:

Part 1: 1.9√52.−5√7 3.11√5 4. 4√2


Part 2: 1.8√52.√33.5√2+2√34. 5√2Part 3: 1.96 2.√21+3√6 3.30√30−90 4.35√5+55√2

Elaboration Day One Answer Key:

120−8√15−9√30+9√2

Exploration Day Two Answer Key:

Part One: 1. H=402. R=23. H=174. K=5

Part Two:1. Y=32. T=03. R=6

4. 13± 72

Evaluation Answer Key:1. −3√22. 30−6√15+2√35−2√213. X=24. X=4

microsoft word - s1_syllabus.doc€¦ · web viewoperations with radical expressions and radical...

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