ufteachkarissabursey.weebly.com · Web viewThis lesson focuses on graphing translations and shifts of the square root function. This relies heavily on the student’s knowledge

Project Based Instruction

Title of Lesson: Totally Radical!UFTeach Students’ Names: Taylor WhitleyTeaching Date and Time: March 13-14, 2013 8:30am-9:30amLength of Lesson: 2-50 minute lessonsGrade / Topic: 9th Grade/Radical Functions, Simplifying Radical ExpressionsSource of the Lesson: N/AConcepts: This lesson focuses on graphing translations and shifts of the square root function. This relies heavily on the student’s knowledge of translations and ability to derive the rule used for graphing these translations. In order to do this, the student must possess a thorough knowledge of how to predict what a graph will look like, how to determine correct points to use to graph the function, and how to actually graph these functions. Students will then use the similarities and differences that they notice to create a general rule that can be used in future graphing problems. The student must be able to develop relationships and observe relationships between these functions. The student must also know how to develop relationships to determine how to simplify radical expressions. They must know perfect squares and how to factor numbers to find the perfect squares to factor them out of the equation. They must possess the knowledge of relationships to determine the Product and Quotient rule. They also must have the previous knowledge of binomials to develop conjugates and rationalize denominators. All of this background knowledge and concepts contributes to the overall knowledge of how to solve the problems.

Florida State Standards (NGSSS) with Cognitive Complexity: Benchmark Number Benchmark Description Cognitive ComplexityMA.912.A.2.13 Solve real-world problems involving relations and

functions.Level 3

MA.912.A.6.1 Simplify radical expressions. Level 2

Performance Objectives: Students will be able to: Graph and analyze dilations of radical functions. Graph and analyze reflections and translations of radical functions. Simplify radical expressions by using the Product Property of Square Roots. Simplify radical expressions by using the Quotient Property of Square Roots.

Materials List and Student HandoutsDay 1:

One copy of Whale Engage Worksheet* Class Set of “Graphing Radical Equations” Explore Worksheet Class Set of “Golden Gate Bridge” Worksheet One “Real World Application” Worksheet* Class set of whiteboards, markers, and erasers

Day 2: One copy of “Bridge” Worksheet Engage* Class Set of “Totally Radical” Worksheet Explore One copy of “Game Directions” Elaborate* One deck of playing cards per two students


Class Set of “Tom’s Thoughts” Worksheets Evaluate Class Set of Calculators

Advance Preparations*Mr. Fayiga does not have a computer connected to his SMART Board, just a document camera. So, instead of a PowerPoint, I will be using the documents that I have indicated with an asterisk. I will also be writing and drawing all of the important ideas on the board.Day 1:

Prepare worksheets and plan the groupings for the students for the activities. Get the whiteboards, markers, and erasers ready to efficiently hand out.

Day 2: Prepare the worksheets. Get the materials ready to distribute easily.

Safety Whiteboards are not to be used to hit each other, and do not smell the markers. Cards are for playing the game only, not for gambling or throwing.


Day 1 5E Lesson:Engagement Time: 5 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsThe teacher will introduce themselves and a very broad topic for the lesson.

Hello! I am Ms. Whitley, and today we are going to be working with radicals!

The teacher will display on the document camera a picture of a whale, and the equation for finding the speed of sound in water.

The teacher will ask students probing questions about the equation, and how to graph it. This will help students begin thinking about what to consider when graphing square root functions.

Let us start today by thinking about whales. Do whales move to different locations around the world? Why? “No, because the water

temperature is always the same.”“Yes, because they have to move to find food.”Other answers may vary.

Actually, whales are known to travel around the world, and scientists work to study this phenomenon.

What do you know about scientists tracking the way whales move around the world? What methods do they use? “Tagging, observations, etc.”

Answers may vary.Good! Another interesting fact is that scientists use the sounds of whales to track their movements. The distance to a whale can be found by relating the speed of sound in the water, using this equation for the speed of sound:

c=√ Ed , where E=elasticity of water, and

d=density of the water.

How could we graph this equation to see the speed of sound in the water if we know our density of the water is 1 g/cm3?

“We could use a graphing calculator.”“We could make a chart of values, etc.”

The teacher will lead a short discussion about different methods of graphing a function.

Those are all great ideas for graphing this function! Other than making a chart of the values and plugging them in, how else could we graph this function without using


a calculator? How would that method work? Answers may vary.

The teacher will introduce the goal for the day.

Those are very interesting ideas. Today we will learn about different ways in which we can graph this function by hand.

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

MisconceptionsThe teacher will begin by reviewing important vocabulary important for the exploration, using the formula from the Engage.

The teacher writes “square root function,” on the main white board.

We will be working on an activity today to help discover another way to graph this kind of function, but first we need to clarify some important vocabulary.

Using the equation for the speed of sound, what is our square root function? Why do you think so?

“Ed , because it is inside the

square root.”“The actual function.”

The actual function is correct. The square root function is the function defined by y=√x. Square root functions contain the square root of a variable.In our example, do we have an equation defined by y=√x? Where? How do you know? “The equation for the speed

of sound in water is a square root function because it has a square root.”

The teacher writes “radical function” and “radicand” on the main white board.

Knowing about the square root function, what is a radical function?

No, we just used that definition for a square root function. What is the different between a radical function and square root function is?

Exactly. You will learn in future math courses other types of radical functions, but for today we are going to stay with the square root functions.

Finally, who can remember and then tell

“A function that contains the square root.”

“A square root function is a type of radical function.”


me what the radicand is?

Good, so in the speed of sound in water, what is the radicand?

What questions do you have about the vocabulary?

“Radicand is the expression under the radical sign.”

“Ed ”

Answers may vary.The teacher will place the “Graphing Radical Functions!” worksheet on the document camera and explain the activity.

The teacher will then pass out the worksheet after asking for the student’s full participation.The teacher will make sure the students are using the correct terminology.

The teacher will walk around and help students find similarities and differences between the different graphs.

Shortly, I will hand out this worksheet. The purpose of this worksheet will be to explore different ways to graph the functions.

We will be doing this worksheet as a class, and will be discussing ideas, so I will need everyone to participate and everyone to listen to all of my directions.

Let’s do the first row together as a class.Student Name, will you read to me the equation from the first row?

So, in the column titled “Prediction Graph,” you will be drawing what you think the graph will look like, without plotting points. There is no wrong answer for this column, just graph what you think the graph would look like. Everyone take a few seconds to graph what you think this function will look like. Look up and put your pencil down when you are finished.

What questions do you have?

You may begin.

Now compare this graph with two people around you. Look at how they graphed theirs, and find similarities and differences. Remember, there are no right answers to this part of the activity.

“y=√x.”

The teacher will now allow students to graph the function using a table of values.

Moving to the third column titled, “Table of Values,” you will choose points to graph, and then in the fourth column, you will graph these points.


Student name, what graph is the class working on?

Student name, what on the table is everyone filling in?

What questions do you have about this part of the activity?

Go ahead and begin the third and fourth columns for the first row now. I will be walking around observing your methods and asking questions.

“y=√x.”

“The third and fourth columns with the table of values and graph.”

Teacher walks around asking probing questions to make the students think about their processes.

As the teacher walks around, a student is selected to go to the board and graph the equation.

What points did you choose?

Why did you choose these points?

Why are there no negative x values on the graph?

Answers may vary.

Answers may vary.

“You can’t take the square root of a negative number.”*Be sure with this response to clarify that you cannot take the square root of a negative number and get a real answer, not that you cannot take the square root of a negative number.

Teacher goes over the example and prepares the students to answer the rest of the examples.

Okay, now we will look at Student Name’s representation of the equation.

I saw this graph on a lot of your papers, what do you think happened if a student didn’t get this graph?

We are going to use this graph as our reference for the rest of this activity.

Now, go ahead and work on the second equation. I will give you 30 seconds to predict what the graph will look like. Then,

“Found the incorrect points.”“Made a mistake in the arithmetic.”


I will tell you to compare with two people around you, like we did with the first graph.

The teacher walks around the class, repeating the process for the rest of the worksheet. The teacher gives 30 seconds for prediction, 30 seconds for review, and then a few minutes for graphing the actual function.

As the teacher is doing this, guiding questions are asked.

As the teacher walks around, students are picked to graph what they have on the board.

Begin your prediction now for the next equation. Based upon what you know from our first graph, what do you think this graph will look like? Write this on your paper.

Now compare with two people around you. Be sure to note any similarities or differences between your graphs.

How did you choose those points to graph?

How do you think you need to choose your points?

That’s right! Because we don’t know how to find the exact square roots of numbers that are not perfect squares without a calculator.

What are some square roots you know that equal whole, real numbers? How can you use these to graph?

What is different from this graph than the first graph we did as a class?

How do you think this difference affects the graph?

Is the radicand different than the first graph we did as a class? How do you know?

“They make a perfect square.”

“So that they make a perfect square.”

“4,9,16,etc.”“I can use these to graph points.”

“It is multiplied by a number.”Answers may vary.

“It changes the points.”

“No, because they both have x inside the square root.”“Yes, what is inside the


Why did you think the graph would look like that?

Is your actual graph and predicted graph the same? Why or why not?

How did you predict that the graph would look like your actual graph?

What previous knowledge of graphs helps you with these?

What if this number was changed? What do you think your graph would look like then?

What if this number was negative? How do you think your graph would be changed? Why?

square root is different.”

Answers may vary.

Answers may vary.

Answers may vary.

“Graphing transformations of quadratic equations.”

Answers may vary.

“The graph would be opposite.”

The teacher wraps up the exploration activity and asks questions to review the graphs that the students drew on the board.

Is this the correct graph? Why?

Why might this graph look different than your graph?

If you have a different graph, what do you think you did differently when graphing?

How do these graphs differ from the first graph we did as a class? What effect does this have when you graph this equation? I will give you a few minutes to write this down in the last column for the rows. Be ready to share your answers with the class.

Answers may vary.

“Different points used.”

“Algebra messed up.”

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

MisconceptionsThe teacher will facilitate a discussion about the differences that the

Based upon your answers, who would like to share a difference that they saw in their graph, compared to their prediction? Why


students saw in the equations and graphs, and how this might affect the graphs.

do you think this difference occurred?

When you compared your predicted graphs to the graphs of your partners, why do you think they were different? Similar?

How do you think these differences affect the way that the equation is graphed?

What similarities were noticed between these graphs and the graph that we did at the beginning of class?

What differences were noticed between these graphs and the graphs that we did at the beginning of class?

How do you think these similarities affect the graph?

Answers may vary.

“Different points used.”“Algebra mistakes.”

“They move the equation, stretch/compress the equation, etc.”

“They are moved around, different points, etc.”

Answers may vary.

Answers may vary.The teacher will guide the students by questioning to help them develop rules for graphing these types of equations.

We can actually develop certain rules based upon these differences to help us graph the equations without using a table of values in the future.

For the second and third rows, what is the variation between that equation and the first equation we did as a class?

How did this affect your points? Hint: Compare them to the points from our first equation.

Now, on the last column of that row, where it says Rule, I want you to write a rule that you could follow if 2 was any number.

When writing your rule, think about how you would describe it to someone who did

“Multiplied by a number.”

“Makes the x values same but y values multiplied by that number.”


not know about the differences between the equations.

The teacher will walk around and look at student’s rules, asking clarification questions, and making sure that the rules are broad enough for any number multiplied by the radical.

The teacher will also ask 3 students to write their rule on the board.

If the multiple was 3, would this work?

What could we use to represent any number in our rule?

How do you know this rule will work?

Will you come write your rule on the board?

Answers may vary depending on the accuracy of the rule.

“c.” *Be careful that students do not reply with x, because x is our main variable in this case. Explain that in properties and rules, mathematicians use letters at the beginning of the alphabet to represent numbers.”

“Test it.”

The teacher will check the students’ rules with the class.

How could we check to see if these rules work?

Let’s do that, by plugging in 5. If we plug in the answer 5, does it give us what the x and y values would be for the actual graph?

How could we modify this rule, so that it is correct?

Why might different students have different rules?

“Test it.”

Answers may vary, depending on the accuracy of the function.


“Different ways of speaking, expressing the rule.”

The teacher will then give the class the general rule for the equation.

We have now modified these rules to be correct.

A more condensed version of the rule is, if c represents any number,


y=c√ x , (x , y∗c), which means that if the square root function is multiplied by some number, the points will have the same x value as the square root function, but the y value will be multiplied by that number.

What questions do you have about this rule?

The teacher will guide the students by questioning to help them develop rules for graphing these types of equations.

For the fourth row, what is the variation between that equation and the first equation we did as a class?



“A number is added to the square root.”

“Adds 1 to the y-value, etc.”

The teacher will walk around and look at student’s rules, asking clarification questions, and making sure that the rules are broad enough for any number multiplied by the radical.


If you were adding 2, would this work?




“Test it.”




How could we modify this rule, so that it is

“Test them.”



correct? Answers may vary, depending on the accuracy of the function.



A more condensed version of the rule is: y=√x+c ,(x , y+c )

which means that if the square root function is added to some number, the points will have the same x value as the square root function, but the y value will be increased by that number.

If we look at the graph, we can even see that graph is simply moved up or down along the y-axis according to the number.


The teacher will guide the students by questioning to help them develop rules for graphing these types of equations.

For the fifth and sixth row, what is the variation between that equation and the first equation we did as a class?

When you were plotting points, did you have to change your x-values? How?



“A number is added/subtracted from the radicand.”

“Yes, changed them so that we could find perfect squares.”

“Shifted our x values.”

The teacher will walk around and look at student’s rules, asking clarification questions, and making sure that the

If you were adding 3, would this work?



Answers may vary,


rules are broad enough for any number multiplied by the radical.



depending on the accuracy of the function.




How could we modify this rule, so that it is correct?

“Test them.”





A more condensed version of the rule is: y=√x+c ,(x+c , y )

which means that if the square root function is added to some number, the points will have the same y value as the square root function, but the x value will be increased by that number.

If we look at the graph, we can even see that graph is simply moved left or right along the x-axis according to the number.


The teacher will review the rules to make sure the students know each of the different questions.

Make sure you write the rules on the worksheet I gave you, so that you can reference them at any time.

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


MisconceptionsThe teacher will hand out the class set of whiteboards, markers, and erasers, and show the “Real-World Application” Worksheet to the class on the document camera and work through the problems as a class.

The teacher will have students explain their reasoning to the class about the ways they used their rules learned earlier to graph the functions.

Now, we will apply the rules that we have learned to some real-world examples.

You have each received a white board and a marker. Please let me know if your marker does not work.

We will be graphing these equations as a class. Let’s start with the first one.

Who will read the first equation?

This one is from the example at the beginning of class, correct? So, how should we graph this if the density of our water is 1g/cm3 and we know that the speed of sound is doubled and our E is our variable?

Graph this on your whiteboard, using your rule.

Show me your whiteboards.

Which rule did you use? Why?

How did you graph this equation based upon that rule?

What misconceptions might other students have if they saw this rule?

Student reads question.

“Multiply by 2, so use our multiplication rule.”

“Multiplication rule.”

“Kept x values and changed y values.”

Answers may vary.Repeat this procedure for the rest of the problems on the document.Evaluation Time: 5 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsThis Evaluation will be given after the Exploration and before the Elaboration Activity.

Now, let’s see how we could apply these rules to another equation.

The teacher will hand out the “Golden Gate Bridge”

Who will volunteer to read the question? Student reads question.


worksheet, and give directions to the students for completing it.

I will now give you 3 minutes to discuss with the person next to you how to solve this problem. The purpose of this is so that you can work with your partner to throw around ideas about how, using your rules, you would graph this problem and find the velocity when it hits the water.

What are your questions?

You may begin.The teacher will observe the student’s answers and ask guiding questions to the groups who are struggling, and ask questions for the students who know the answer to get them to think beyond what they know.

How would you solve it?

What do you know from the question?

How does what you know help you solve the question?

What do you need to know?

How will you find what you need to know?

Answers may vary.

“Gravity.”

“Can plug in and find velocity, it is the constant.”

“Height.”

“Read the last part of the equation.”

The teacher will give students time to write down what they have discussed. The teacher will collect these worksheets and then lead a discussion about their answers.

Now before we discuss our answers, I want you to write down on your worksheet the process you would use to graph this equation—do not actually graph it! And then I want you to actually find the velocity of the object when it hits the water. Be sure to show all of your work and be thorough with your explanation.

When you are finished, please raise your hand and I will collect your papers.

What questions do you have?The teacher will ask students to have a discussion about how they found their answers.

How did you find your answer?

How did you know how to begin?

What information did they give you?

What information did you need?

Answers may vary.

Answers may vary.

“Gravity.”

“Height.”


How did you know this?

Why did you solve the problem this way?

“Read the last part of the equation.”

Answers may vary.

Assessments and Handouts for Day 1:


d=√ Ed


Name: _____________________________________________________________________ Date:_________

Graphing Radical Functions!Equation Prediction Graph Table of Values Actual Graph Observation/Ruley=√x

x y

y=2√ xx y

Rule:

y=−3√xx y

Rule:


y=√x+1x y

Rule:

y=√x+1x y

Rule:

y=√x−2x y

Rule:


Name: _______________________________________________________________________________

Golden Gate Bridge

“The Golden Gate Bridge is about 67 meters above the water. The velocity v of a freely falling object

that has fallen h meters is given byv=√2gh), where g is the constant

9.8 meters per second squared. If an object is dropped from the

bridge, what is its velocity when it hits the water?”

How would we graph this function? Find the velocity.


Real World Applications

Speed of Sound in Water:c=√ Ed

Equation to find the diameter a steel cable should have to support a given weight:

d=√w8


Day 2 5E Lesson:Engagement Time: 5 minutesWhat the Teacher Will Do Teacher Directions and Probing Questions Student Responses/Possible

MisconceptionsThe teacher will introduce themselves and a very broad topic for the lesson.

Hello! I am Ms. Whitley, and today we are going to be continuing our work with radicals!

The teacher will display on the document camera a picture of the Sunshine Skyway Bridge, and the equation for finding the diameter of a steel cable should have to support a given weight.

The teacher will ask students probing questions about the equation, and how to graph it. This will help students begin thinking about what to consider when graphing square root functions.

Let us start today by looking at the Sunshine Skyway Bridge.

Who can tell me where this bridge is located?

Has anyone ever been there?

How strong do you think this bridge needs to be to hold all of that traffic?

Bridges are supported by cables. Do you think that engineers just guess how big the steel cables need to be to hold the weight of the bridge?

No? Then how to they determine it?

“Tampa, Florida.”

Answers may vary.

Answers may vary.

“No.”

Answers may vary.Who remembers the equation about finding the diameter of a steel cable should have to support a given weight?

Who can tell me what they remember about that equation?

d=√w8Looking at this equation, let’s think about if the weight was 258. What would our equation be?

Students may raise their hands.

Answers may vary. Hopefully, a student will remember the equation.

“d=√32”So, if we were asked to simplify this, how do you think we would go about this?

How do you know?

“Use a calculator.”“Estimate with nearest perfect square.”

Answers may vary.Those are all great ways to simplify this radical. Today, we will be creating quick rules, like yesterday, to use when


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

MisconceptionsThe teacher will start the lesson by reviewing what was learned yesterday.

What is one thing that someone remembers learning about yesterday? “Graphing, rules for

transformations, etc.”How did we discover those rules? “We did examples and then

made up the rules from there.”

So, today we will be doing a similar activity, except it will be different than graphing. Today’s topic, simplifying radicals, is very important for use in Ms. Bursey’s lessons on Friday and Monday.

The teacher will put the Totally Radical! Worksheet on the document camera for viewing.

The teacher will pass out the worksheets to the class.

First, when I say “simplifying radicals,” what do you think I mean? “Solving radicals, taking out

perfect squares.”(If students said something like “Taking out perfect squares,” jump to the asterisk.)

Those are all good thoughts, but the simplifying radicals that we will be discussing today means “taking out all of the perfect squares.”

Does everyone understand this? Answers may vary. If students do not understand, you may go over the material a second time, making sure the students remember what “perfect squares,” and other vocabulary means.

*So, keeping that in mind, you will be working on this activity, using the worksheet I just gave you.

The teacher will do an example with the class to help guide them.

To do this activity, we will be working together as individuals and groups within class, and if someone thinks that they have a successful method for simplification, there will be time to discuss your method.Let’s look at the first two rows. Who will


read to me the first radical? Student reads the radical.In the first column, you will estimate what the process will be to simplify the radical.

Who wants to share what they think the process is?

Be careful, do not simply state that the square root of 4 is 2.

“Write square root of 2 times 2 then take out one 2.”

How do you know that the square root of 4 is 2?

So how could we write this process on paper? Go ahead and write that on your column of estimated process on your paper now.

“4 is 2 multiplied by itself.”

“You could write the square root of 2 times 2 is equal to 4.”

As the teacher walks around the classroom, two students will be chosen to write their estimated process on the board up front. While the teacher is walking around, probing questions are being asked.

Why did you write this process?

Would someone who did not understand radicals understand your process?

Why or why not?

Answers may vary.

“Yes, or no.”

Answers may vary.

Now, we will look at the two examples of estimated processes on the board.

Are these correct processes?

How do you know?

Now that you get a general idea of what the processes will look like, go ahead and complete simplifying the rest of the radicals. Note that some radicals may have different processes than the one we just described.

Who will volunteer to describe what you will be doing?

“We will be filling out the


What questions do you have?

estimated process column for all of the radicals.”

The teacher will circle the class as they work on the column for all of the radicals, asking guiding questions to the students. The teacher will also pick one student to come to the board and write their process down per radical.

How did you come up with this process?

Why did you write the process the way that you did?

Could you be more specific in the operations you use for your processes? How?

Can you think of a rule/property we can make from this process?

Answers may vary.

Answers may vary.

Answers may vary.

Answers may vary.The teacher will then lead a discussion based upon the processes that were written on the board, one at a time.

The teacher will continue this questioning for the rest of the radicals.

As we decide which are acceptable processes, you can write these in the process column of your worksheet, and write the answers in the simplified column.

Would you consider this a thorough, accurate process? Why or why not?

How could we make this more detailed?

Would someone who did not know about radicals be able to see how you carried out this process? Why or why not?

Now you can write the simplified Now, we will be using these processes to develop rules/properties, like we did yesterday.

Depends on the process.



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

MisconceptionsThe teacher will lead a discussion based upon similarities that students see about the radical functions to develop rules and properties.

What are some similarities that we see between the processes in the first four radicals?

What are some differences?

How could we develop a rule to describe

“They include perfect squares.”

“Some include more than perfect squares.”


this? I will give you a minute to write this rule down on your paper and then 2-3 minutes to discuss it with the people in your row. Be sure to not differences and similarities. Remember, like yesterday, to not include specific numbers in your rule.

The teacher walks around to ask guiding questions, and select certain students to write their rule on the board.

How did you come up with this rule?

Will this rule work for all numbers?

Is this specific enough?

Will everyone understand this rule?

Answers may vary.

Answers may vary.

Answers may vary.

Answers may vary.The teacher will review the rule for the first four rows.

Let’s look at the rules on the board.

Are these accurate?

Who can tell me how we can find out if these are accurate?

How can we test them?

Good! This rule will work for all numbers. This rule is called the Product Property of Square Roots. It works for all nonnegative real numbers, and the official form is √ab=√a ∙√b. Write this rule down for the space in row 4.

How could we use this rule to solve the fifth row radical with variables?

Answers may vary, depending on the rule.

“Test them.”

“Plug in other numbers, like 32”

“Write out all of the numbers separately under square root symbols and simplify.”

The teacher will now carry out the same process for the 6th and 7th rows.

What are some similarities that we see between the processes in the 6th and 7th radicals?

What are some differences?

How could we develop a rule to describe this? I will give you a minute to write this rule down on your paper and then 2-3

“They include perfect squares.”

“Some include more than perfect squares.”


minutes to discuss it with the people in your row. Be sure to not differences and similarities.






Answers may vary.

Answers may vary.

Answers may vary.

Answers may vary.The teacher will review the rule for the 7th and 8th rows.


Are these accurate?



Good! This rule will work for all numbers. This rule is called the Quotient Property of Square Roots. It works for all real numbers, and the official form is

√ ab=√a√b

. Write this rule down for the

space in row 8.

One thing to remember is that when solving a problem, and you get a radical expression in the denominator, this answer is not classified as “fully simplified.” How could we get rid of the radical in the denominator?

Why would this be “mathematically legal?”


“Test them.”

“Plug in other numbers.”

“Multiply the fraction by the radical in the denominator divided by the same rational.

“Because it is the same as multiplying by 1.”

The teacher will look at the last row’s radical expression to be

How could we develop a rule to describe this? I will give you a minute to write this rule down on your paper and then 2-3


simplified. minutes to discuss it with the people in your row. Be sure to not differences and similarities.






Answers may vary.

Answers may vary.

Answers may vary.

Answers may vary.The teacher will review the rule for the last row.


Are these accurate?



Good! When we multiply by the opposite of the bottom fraction and use our FOILING method, we call this “Multiplying by the Conjugate.” Binomials of the forma√b+c√d and a√b−c √d are called conjugates. The product of two conjugates is a rational number and can be found using the pattern for the difference of square. Write this down for the space in the last row.


“Test them.”

“Plug in other numbers.”

The teacher makes sure that the students understand all rules and properties.

What questions do you have about the properties?

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

MisconceptionsThe teacher will give directions for the activity and leave a copy of the activity’s directions on the document projector for

We will now be playing a card game to discover how much you have retained from our activity today.

Please take out a piece of paper, grab a


the students to reference. calculator from the back of the room and a pencil.

You will be paired up with a partner and will each receive half of a shuffled deck of cards. The goal of this game is to collect a larger number of cards than your partner.

Each card will be assigned a value: Aces=1, 2-10=face value, Jack=11, Queen=12, King=13.

You will each flip over one card each at the same time to form the radicand of the radical that you are meant to simplify.

For example, if a 3 and 2 are flipped over, you will race each other to simplify both the square root of 32 and 23.

I want you to show your work on paper including: the radical to simplify, writing out the perfect squares according to the property, and then the final answer. The first student to write down all of this on the paper accurately acquires both cards.

If, for example, the square root of 23 cannot be simplified, write cannot simplify on your paper.

If there is a disagreement about who “won” the match, raise your hand and I will work it out.

You may use the calculator to find the factors of larger perfect squares, but remember that this may take some time.

What questions do you have about this activity?

Everyone do one problem as practice with your partner and then we will begin.

The teacher will walk


around for the five minutes. If there is a difference in opinion of a win, then the cards are left on the table, two more are drawn, and the winner acquires four cards.Then the teacher pairs up pairs of students to make groups of four, and explains the next portion.

Now, you will be in groups of four and get five more minutes. Each of you will place down a card in a formation that makes a square root fraction.The same rules apply as far as what needs to be written on the paper to consider a “win.”

What questions do you have?The teacher wraps up the activity.

What properties did you use for this activity?

How did you know to use these properties?

“Quotient and Product.”

“Numbers were multiplied, numbers were divided.”

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

MisconceptionsThe teacher will pass out the “Tom’s Thoughts” worksheet to be used as a formative assessment before the elaboration.

Who will volunteer to read the paragraph at the top of the page?

In this activity, you will work by yourself to determine what Tom does wrong in each step. Some might be calculation errors, and some might be rule errors.

I will also need you to show the correct answer in the box provided, and a description of Tom’s Mistakes in the box provided.

What are your questions?

Raise your hand when you are finished and I will come by and collect the paper.

Student reads the paragraph.


Assessments and Handouts for Day 2:


d=√w8SUNSHINE SKYWAY BRIDGE


Name: ____________________________________________________________ Date: ______________Totally Radical!

Radical Estimated Process Process Simplified Valuey=√4

y=√8

y=√16

y=√32

Rule:

y=√90 x5 y3

y=√ 164

y=√ 1220

Rule:

CHALLENGE:

y= 35+√2

Rule:


Two Player:- Each player lays down one card.- Players will calculate all combinations of simplified

radical.- Write:

oRadical(s) to simplify.oProcess used (write out all square roots)o Final Answer

- First person done gets both cards.

- In a tie, play again and winner gains all 4 cards.

Four Player:- Same rules as above.- Card Placement:

o P1 P2

P3 P4

Card ValueAce 12 23 34 45 56 67 78 89 910 10Jack 11Queen 12King 13

1. 2√18+2√32+√72=¿2 (√9∗√2 )+2 (√4∗√2 )+√36∗√2

2. 2 (√9∗√2 )+2 (√4∗√2 )+√36∗√2=

2 (9√2 )+2 (2√2 )+6√23. 2 (9√2 )+2 (2√2 )+6√2=

11√2+4 √2+6√24. 11√2+4 √2+6√2=¿

44 √2+6√25. 44 √2+7 √2=51√2


Name: __________________________________Tom was given the following problem: 2√18+2√32+√72 and asked to simplify.

His thought process is shown below. Does he have the correct answer? Justify your answer by solving the problem correctly and stating where he went wrong in each step.

Correct Answer: (Show your work)

Tom’s Mistakes:1.

2.

3.

4.

5.

Documents

ufteachkarissabursey.weebly.com · Web viewThis lesson focuses on graphing translations and shifts of the square root function. This relies heavily on the student’s knowledge