Ben Muller
Math 410
Unit Lesson Plan
Radical Functions
This is a Unit Lesson Plan on radical functions. I will be using the textbook Algebra 2:
Explorations and Applications from McDougal Littell for homework and in-class problems, and
also help with the individual lessons. This Unit Lesson Plan will be geared towards an Algebra II
honors class. Graphing calculators will be used during the Domain/Range and Graphing lessons,
but only after the students have learned to graph the functions by hand. I believe that students
need a strong foundation in written math before they take the step to using calculators. It is
important for the students to understand the concepts, not just memorize which buttons to press.
Motivation is one of the most important aspects of a lesson plan. Without it, students will
not care and will not take the initiative to learn. Learning is a two-way street; both the teacher
and student must care. My motivation as a math student my entire life has just been the numbers
and how they relate in amazing and almost magical ways with each other. The simple beauty of
math, not its application to the real world, has been all the motivation I’ve needed. This,
however, is not what it takes to motivate most students. Students need to believe that what they
are learning can and will be useful at some future date. In this lesson plan, I will try to make
math fun and enjoyable, try to make it like an adventure where the students feel satisfied when
they finally comprehend a topic. But this of course will not be enough, so in this lesson plan, real
world examples are used so that students see just how much math is used in everyday life, and
how grasping the concepts will be a great tool in whatever job they decide to pursue.
The teaching style will be based around a lecture, but will heavily incorporate inquiry and
scaffolding. Probing questions will be asked to try to get the students thinking in the right
direction, and problems will be left for the students to try to figure out on their own. This allows
the teacher to lead the discussion, but also lets the students discover truths they may not have
found before. This comes from the NCTM Principles and Standards, where they believe that
students should be taught “to think and reason mathematically, not just to perform routine
operations” and to “promote experimentation and conjecturing.” The main concepts that the
students will be covering in this unit are exponential rules, rational exponents, domain and range
as they apply to radical functions, solving radical functions, and graphing radical functions
The standards from NCTM that this Unit Lesson Plan will hit upon include:
•
•
•
•
•
•
•
Computing powers and roots on the magnitudes of quantities
Understand and perform transformations
Understand and write the meaning of equivalent forms
Judge the meaning, utility, and reasonableness of the results of symbol manipulations
Draw reasonable conclusions about a situation being modeled
Make and investigate all mathematical conjectures
Organize and consolidate their mathematical thinking through communication
The North Carolina Department of Public Instruction’s standards for radical expressions
states that students should be able to:
“Use equations with radical expressions to model and solve problems; justify results.
•
•
Solve using tables, graphs, and algebraic properties.
Interpret the degree, constants, and coefficients in the context of the problem.”
There will be no official writings for the students to turn in, but there will be a couple of
times during the week where the teacher will ask the students to write out what they think is
going on in a problem. Hopefully there will be lots of discourse, both between the teacher and
students and between students in groups. Many of the topics will be covered in a way that allows
for back and forth questioning in the classroom. There are also a couple of instances where the
students will be faced with a problem that they do not know how to solve. This includes finding
which exponential number is equivalent to a square root and how domain and ranges are affected
when negatives are added to the radical functions. The students will hopefully discover for
themselves how to solve these problems.
The examples used in class will not exclude any students and will not give an advantage
to any particular group. The class will also progress slowly and students will have the chance to
work in groups, which will help those who may find learning math difficult. The teacher will
also be available to help those that are having trouble with the lesson.
There will be some form of homework every night, whether it is a worksheet graded on
completion only or problems from the book that will be graded on accuracy. On the final day
there will be a quiz that will test all of the main concepts that will be covered in this unit. This
quiz will include four problems that ask the student to simplify a function, state the domain and
range, graph, and solve for y=0. This will hopefully create incentive for the students to learn and
will allow the teacher to see if they have or have not.
Lesson 1: Rules of Radicals/Rational Exponents
This lesson will focus on individual understanding of basic concepts about exponentials,
so there will not be much group work. It will start as a lecture review, but will turn into a student
driven search for a new representation of the square root function. Some real world uses will be
hit upon in this lesson, but no real world functions will be solved. That will be saved for the
following lessons, but hopefully the search for √ will peak some students’ interests. This search
will consist of synthesis questions that allow the students to think deeply and creatively about
this topic. The assessment for this lesson will consists only of a worksheet where they will be
graded on completion and not performance. An activity the following day will reward the group
if they did well on the questions.
1. Review Rules of Radicals
Write/explain product property on the board:
·
Show an example:
2 ·2
2
2
32
Explain that this is because you can also write it as 2 · 2
five “2”s being multiplied.
2 · 2 2 · 2 · 2 which is a total of
Write quotient property on the board:
Show an example:
3
3
3
3
9
Explain that this is because this is the same as 3 · 3 · 3⁄3 which cancels out one of the top “3”s,
leaving only two “3”s.
Write power property rule:
·
Show an example:
2
2
·
2
Explain that this is the same as writing 2 · 2 · 2
512
2·2·2 2·2·2 2·2·2
2
Write negative exponent rule:
1/
Show an example:
1
4
4
1
16
Tell the students that if they want extra credit they can write out a little (very informal) proof
using the previous rules.
Example:
·
1. Therefore,
·
1 and
1⁄
.
Write the identity rule and show an example:
7
7
Write the zero exponent rule and show an example:
1
55
1
Time: This should take ~10 minutes, since it is only a review, and there should not be too much
discussion or questions.
2. Introduce the Idea of Rational Exponents.
Pose the students a question: What power must x be taken to, to equal √ ?
Write down: √
solve for a.
Ask the students to think about the problem for a couple of minutes. Explain that the students
will need to use one or more of the exponent rules to solve for a. Also explain that what the
problem is asking for is just what exponent is equivalent to the square root symbol.
The general idea of this exercise is to get the students thinking through how powers relate to
roots and in what way, hopefully some will discover it for themselves. So yes, if the students
never answer, the teacher can eventually explain, but hopefully the students will be the ones
leading the discussion towards to answer.
After the students reflect for a little bit, ask if anyone has suggestions. Call on students if no one
has any ideas. If progress isn’t made, ask the students what the definition of √ is. They should
be able to explain it (the square root of a number times another square root of a number, equals
that number) and hopefully write the definition out (√ · √
). They may also say that
√
. The problem can be solved this way as well.
Ask the students what they think should be done next. (Namely, inserting
for √ and
for
). After this, ask the students to manipulate the equation using the aforementioned rules and try
to solve for a again.
They should have
·
and manipulate it to look like
or
.
Hopefully the students can solve for a here, so 2a=1 and a=1/2.
/
. When we plug this back in we see that it gives us / ·
This means that √
Which makes sense, since using the product rule we see that 1/2+1/2 in fact equals 1.
/
.
The next step is to explain that this works for all roots, and that all of the definitions can be used
to simplify statements that have rational and whole number roots.
In general:
√
/
Example:
√ √ √
√
/
This is the same as saying 1/3+1/3+1/3=1. The same proof as above also works for all roots.
Time: Since there will probably be small discussions that occur during this part and it won’t be a
straight forward, step-by-step explanation, it should take ~20 minutes.
3. Everyday Use
Here the teacher will take a couple of minutes to just list off a few ways that radical equations
and function are used in everyday life. In future lessons, the students will actually solve real
world problems. Explain that radical functions are used all the time in physics, including the
equations
2
and
, which is another form of Einstein’s famous equation.
Time: ~3 minutes
4. Worksheet/Homework
This worksheet (see file Worksheet1) will then be passed out to each student in the class. This
should mostly be a time for the individuals to see how much they understand. The teacher will be
walking around the room to assist those who need help. It will be announced that any problems
not finished in class will be assigned for homework. No other homework will be assigned.
The students will not be told that the grade for this homework assignment will only be how many
of the ten problems the students complete, not that they completed them correctly. An extra point
will be available for each of the first two problems if the student obviously put forth an effort in
being creative in finding different equivalent statements. So a perfect score would be 12/10.
Time: ~15 minutes
Lesson 2: Domain and Range
This lesson will reintroduce the idea of domain and range, but will put it into the context
of radical functions. It will introduce some important ideas, such as the fact that the term inside
an even radical must be non-zero and that to make square roots into functions, we have to choose
only certain y’s, otherwise it will not be a function. This lesson will not have the students
working together on any main topics, but the class will be learning together, talking about the
problems every step of the way, and leading the discussion to an answer. One main
misconception that is likely to crop up is that the range of the square root function is not infinity.
Students may also forget that when writing domain and ranges, the smaller number goes first,
and that ∞ ∞ require open brackets, such as these ( ).
1. Quick/Fun Activity
A magnetic Indiana Jones character (made out of paper with a magnet glued to the back) will be
placed on the white board on a horizontal axis, with a magnetic boulder sitting two spaces behind
him. (This little game may be used multiple times throughout the year, so maybe on the first day
of class take five to ten minutes to watch the entire scene in the cave from Temple of Doom.)
The teacher then goes through the worksheet, question by question, calling on random students
to answer the given question. If the student gets it right, Indy moves forward a space. If it’s
wrong, the boulder moves forward a space. If Indy doesn’t get run over by the boulder the
students get two extra percentage points on their quiz at the end of the week. If the boulder
doesn’t move at all (the students get all the questions right) they get two extra percentage points
on their quiz AND they get to watch a fun clip from a classic movie on the review day.
Time: ~7 minutes
2. Introduce Radical Functions
Radical functions are those that have a variable inside a radical. Functions that only have
numerals inside radicals do not count.
Show two examples: 3
17
√
2
Show two examples that are not radical functions:
√2
3
√5
Time: ~3 minutes
3. Review of Domain and Range
This should be a review, so ask the students what they remember about domain and range. Ask a
few different students to give their opinion.
Answers:
The domain of a function f(x) is the set of all values of x for which f(x) is
defined.
The range of a function f(x) is the set of all values of f(x), where x is in the
domain of f.
Show an example on the board and ask the students what they believe the domain and range are:
f(x)
x
1
Æ
2
2
Æ
2
3
-
4
Æ
7
5
Æ
5
6
Æ
11
Explain the answer, that the domain is the set (2, 5, 7, 11), since those are all the x-values for
which f(x) is defined. The range is the set (1, 2, 4, 5, 6) since those are the values for f(x) for
each x in the domain. 3 is not in the range because it does not have a value of x associated with
it.
Time: ~8 minutes
4. Inverse of
Draw the graph of and write the function
on the board.
Ask the students if they remember how to find an inverse of an equation.
First, solve for x to get
then substitute x for y and y for x to get
Ask the students to graph √ and
come up and draw what they have on the board.
√ .
√ on their calculators. Then ask a student to
Explain quickly why the graph looks like this. Include the fact that it touches the point (0, 0) and
that it increases or decreases by less and less each time.
Also state that x can never be negative because negative numbers don’t have (real) square roots.
Ask the students what this must mean about the domain, if x can never be negative.
Ask the students if they see anything wrong with an inverse function that looks like the one of
the board. The problem is that for certain x-values there are multiple y-values. This goes against
the definition of a function.
Since the values of x and y switch from the function to the inverse, if we restrict the domain of
the original function to either
0 or
0 we get a function for the inverse. Draw these on
the board and show how restricting the domain of the original function will restrict the range of
the inverse.
Ask students to write down what they believe the domain and range of the inverse functions to
be. Remind them that these values can be ranges (not to be confused with “the range”) on the
graph, and do not have to just be individual points.
Ask for students’ guesses and explanations. Correct any wrong answers and affirm right ones.
Explain that the domain and range are both 0, ∞ , or all positive real numbers. Also say that all
square root functions build upon this answer for their domains and ranges.
A main misconception here will be the fact that the range goes to positive infinity. Many
students looking at the graph will think that y has a limit somewhere, since the line is not as steep
as it was in the beginning. This would be a good time to explain to the students (even if they
don’t bring up this objection) that the range really does go to positive infinity.
First ask the students to think about why this would be so, even if the graph does not look that
way. Take some suggestions from students.
If needed, explain by asking them for a positive y-value that they think is too big to be in the
range, then square it, and show that that x-value is in the domain. So if an x-value exists in the
domain, then the corresponding y-value is in the range.
Time: ~17 minutes
5. More Complicated Domains and Ranges
Explain that odd radical functions, like cubed and fifth roots, will have domain and ranges from
negative to positive infinity. But even numbered radical functions will not, since the term inside
these radicals cannot be negative, and x-values can only have one corresponding y-value.
Write the equation
3 4. Remind the students that the domain is whatever value
√
of x makes f(x) a workable function. Ask the students what numbers x is allowed to be.
Answer: Domain is 3 to positive infinity since for these values (x-3) is non-negative.
Remind the students that the range is all the possible values y can take with the domain of x
being from 3 to positive infinity. Ask the students to make a table of a few possible values of x
and y, and then to guess what the range is. Call on a student to write their table on the board and
state the domain.
Answer: Range is from 4 to positive infinity. Explain that this is because the lowest value
3 can take is zero (since square roots cannot be less than zero), but it can also have a value
√
up to infinity. So in a table we would see the first value of y being 4, and all the subsequent ones
getting larger without a limit.
Explain that there is an easy way to find the domain and range for square root functions in the
form
. The domain will be all numbers greater than or equal to h, and the
√
range will be all numbers greater than or equal to k.
Ask the class to graph
3 4 on their calculators, and see if they see how the
√
domain and range seems to fit with the graph. Point the discussion in the direction of the starting
point of the graph is at (3,4). Also ask the students to think about what would happen to the
domain and range if there was a negative in front of the x and if there was a negative in front of
the radical.
Time: ~15 minutes
6. Homework
Find the domain and range of the following problems:
5 11
4 1
44 77
13
√
√
√
√
1)
2)
3)
4)
Graph on your calculator and state what the domain and range are for the following problems.
√
5)
6)
7)
√
√
7. Homework Rubric
The homework will be graded based solely on whether or not the students correctly stated the
domain and range. Since there are in reality two questions per question, the homework
assignment will be out of 14 points. This is nice easy assessment, because it sees whether the
students have learned how to shift the domain and range for square root functions, and if they
understand how flipping functions messes with the domain and range.
Answers:
1)
2)
3)
4)
Range: 11, ∞ Domain: 5, ∞
Range: 1, ∞ Domain: 4, ∞
Range: 77, ∞ Domain: 44, ∞
Range: 0, ∞ Domain: 13, ∞
5) Range: 0, ∞ Domain:
∞, 0
6) Range:
7) Range:
∞, 0 Domain: 0, ∞
∞, 0 Domain: ∞, 0
Lesson 3: Graphing Radical Equations
This lesson will introduce the two important features of graphing square roots,
translations and reflections. The students will also work in groups to figure out how to graph
radicals with larger radicands. Some of the motivation for this lesson will be derived from the
problem that involves physics and a roller coaster. Hopefully students will be interested in how
simple the physics of rollercoasters are and maybe remember that most of the cool things in the
world took math to create. A misconception that may pop up involves the negatives in the radical
functions. Some students may try to pull them out of the radical and some may group the
expression (-x-3) into –(x+3). This will cause the student to think the line on the graph starts at
the x coordinate 3 and not -3. Explaining why the negative reflect the graphs and asking for
questions is important.
1. General Form/Translations
√
Write the general form for the square root function on the board.
State that the graph will start at the point (h,k) and be the same shape as the regular square root
function.
Ask the students to take out their calculators and graph
start of the line seems to be.
4
√
Ask the students where the start should be for the function
√
4
1 should be.
1. Ask them where the
State that these translate the line to a new spot on the graph and are called “translations.”
Time: ~5 minutes
2. Reflections
Ask the students to get together in groups of three and try to graph these functions without their
calculators:
√
√ √
Call on three different groups to come up and draw on the board and explain their answer for one
of the functions.
Write these rules on the board: If there are no negative in front of the x or radical, the line will
travel up and to the right. If there is a negative before the x, then the line will travel in the
negative x-direction, or to the left. If there is a negative before the radical, the line will travel in
the negative y-direction, or down. If there are negatives in both of these places, the line will
travel down and to the left.
Explain that any graph that has the form of the ones above will take the same shape, but will only
differ in the place that they start. This will come from the translation, as was just shown before.
Tell the students that using these simple rules of translations and reflections they can easily draw
any square root graph.
Write down two functions on the board and ask the students to graph them.
5
√
5
2
√4
Then draw the graphs of these two functions on the board and ask the students to write the
functions.
√
√
1
2
3
3
Then ask the students to plug these four functions into their calculators one-by-one and see if
they line up with the graphs.
Time: ~15 minutes
3. Exploration of Higher Roots
Break the students off into groups of four and ask them to use tables and graphs (use calculators
only for finding large roots, not for graphing) to find out as much as they can about the functions
of √ , √ , √ , and √ .
Ask them if they see similarities between certain groups of functions, and also what patterns
emerge. Call on the groups for what they’ve found.
Things they should find is that all the roots have the same basic shape, but as the radicand gets
larger, the line gets shallower. They should also notice that odd roots have possible values for x,
since a negative numbered multiplied by itself an odd amount of times will be negative. The
negative part of the domain of these graphs will lie in the negative part of the y-axis. All even
radicands will only have positive domains.
Have each group turn in a sheet of paper with their graphs and thoughts about the graphs. These
will only be graded for completion.
Time: ~15 minutes.
4. Problem from the Book
Ask the students to open their books (or share with a friend) to page 342 and to look at problems
17-19.
The teacher will ask a student to read the introduction out loud and will write the equation
on the board.
Ask a student to read question 17 out loud: “Solve the equation for v. Does the Demon Drop car
drop faster when people are in it or when it is empty? Explain.”
Tell the students to solve for v. Ask for an answer and an explanation. Answer:
no.
2
and
Ask a student to read question 18 out loud: “The Demon Drop ride involves a drop of 60 ft.
Using g=32 ft/s2, find the velocity of the car after it falls 60 ft.”
Tell the students to plug in the numbers and solve. Remind them to watch out for their units. Ask
for an answer and explanation. Answer: v=61.97 ft/s.
Ask a student to read question 19 out loud: “By what factor would you have to change the height
of the Demon Drop ride in order to double the car’s velocity at the bottom?”
Give the students a couple of minutes to think, and then ask for an answer and explanation.
Answer: Since the height is inside a square root, to double the velocity, you must quadruple the
height.
Time: ~15 minutes
5. Homework/Rubric
Any remaining time will be given to the students to start their homework assignment.
Do problems 20-27 on page 343. (All are simple functions in the vein of the examples shown in
Section 2.)
Students will receive 1 point for correct domain and range, 1 point for correct starting point, and
1 point for correct orientation of the line. Homework will be out of 24 points.
Lesson 4: Solving Radical Functions
In this lesson, the students will learn how to solve simple radical equations. There will be
some group work involved, when the students try to figure out how to solve before being told the
steps. Then the class will work through a book problem together, discussing it as they go along.
This problem will also be a form of motivation, since it is based on hurricanes, and will show the
students another instance of math popping up in useful ways. The questions in this lesson will
push the students to understand solving better by letting them try things out on their own first,
and then teaching them how to do it. Some may try to solve for x before isolating the radical,
since this is what they are used to. So make sure they do not bring an x over to the same side of
the equation as the radical and then try to group the “x”s together.
1. Trial Solve
Have students take the equation 4
board.
√3
2 and try to solve for x. Write the equation on the
Have the students compare with a neighbor until a consensus is reached. Once most groups seem
done, have each group compare with another group (so four students) and discuss how they came
to their answer.
Have one representative from each group write their answer on the board and explain how they
got it. Circle all the correct answers on the board. Hopefully this problem will be simple enough
that most groups will be able to figure out how to solve it.
Time: ~10 minutes
2. Step-by-Step Solution
Explain that there are four easy steps to follow for these types of equations:
i.
Isolate the radical
ii.
Raise both sides of the equation to the power that will eliminate the radical
iii.
Solve for the independent variable
iv.
Check for accurate or extraneous solutions
Perform these steps on the board for the previous equation and show the answer of x=4/3.
Time: ~4 minutes
3. Extraneous Solutions
Write √ 3
6
2 on the board. Ask students to solve for x. Remind them that they may
have to factor or use the quadratic equation and that isolating and getting rid of the radical is
always their first step. Go through the first three steps of the solution on the board giving
answers of x = 2 and -1.
Ask students to plug these two values back into the original equation to see if either of them
solves the problem. Ask the class which (if any) of these two possible solutions work.
Explain that when both sides of the equation are raised to an even power, these extraneous
solutions may crop up. Say that only the answers that solve the original equation are solutions.
So the only solution to this problem is x=2. The extraneous solution x = -1 when plugged in give
the equation 3=-3. This happens because both sides were squared, and when this happens
numbers that were previously negative (-1-2=-3, so the right side would be negative) become
positive.
Time: ~10 minutes
4. In-Class Practice
Put two problems on the board for students to solve: 4 √
11
5
7 and 2
Remind the students to use the four steps.
Ask for two volunteers to come up and explain how they got their solution.
Answers: x=16 and p = -3 and -1.
Time: ~8 minutes
5. Solving Using Calculator
6
3.
Take the same two problems and ask the class if they know how to solve them on their
calculators. Give the students a few minutes to try for themselves. They should remember how to
find the intersection of two lines on their calculator from previous lessons.
After they have a chance to try it out on their own, explain to the students how it is done.
First they have to get all the terms with variables on one side of the equation (or all terms on one
side if they wish). They will graph this line and the other side of the equation on the Y1 and Y2
spots.
They will then hit 2nd TRACE (or CALC) and hit 5 (or intersect). The students will then press
ENTER twice to pick the two curves, and then one more time to make a guess. The x-values at
the places where these lines intersect will be the answers to the equations.
Time: ~8 minutes
6. Problem from the Book
Ask the students to open their books (or share with a friend) to page 354 and to look at problems
19-21.
The teacher will ask a student to read the introduction out loud and will write the equation on the
board. “In a hurricane, the mean sustained wind velocity v, measured in meters per second, is
given by
6.3 1013
where p is the air pressure, measured in millibars (mb), at the
center of the hurricane. The map shows the air pressure and wind velocity at various points along
the path of Hurricane Hugo, which passed through the Caribbean in 1989.”
Ask a student to read question 19 out loud: “Estimate the mean sustained wind velocities at 0 h
(midnight) on September 13th, 14th, 15th, and 16th.”
Have the students plug in the given p’s from the map, and write down their answers.
Ask a student to read question 20 out loud: “Estimate the air pressures at 12 h (noon) on
September 13th, 14th, 15th, and 16th.”
Tell the students to plug in the given velocities and solve for p.
Ask a student to read question 21 out loud: “What happens to wind velocity in a hurricane when
air pressure decreases?”
Give the students a couple of minutes to think, and then ask for an answer and explanation. Then
discuss with the class why this happens. Have the students turn in their papers with the answers
on them and grade only for completion.
Time: ~10 minutes
7. Homework/Rubric
Assign problems 23-31 on page 355. (All of them are simple square root problems like the
second example from the in-class practice.)
Give one point if the students isolated the radical correctly. One point if they correctly squared
both sides. One point if they correctly solved for x. Take away one point if they left an
extraneous solution. So each problem is worth a possible 3 points, for a total of 27 points.
Lesson 5: Review and Quiz
1. Homework
Ask the students to turn in their homework. After the homework is collected, ask if they have
any questions on the problems. Go through the problems on the board.
Time: ~5-10 minutes
2. Review
Call on six different students to come up and write out the product, quotient, power, negative,
identity, and zero exponent rules. If any of the students have trouble, remind them what the rule
·
?)
is meant to define and slowly help them remember. (eg.
Ask the students if they have any questions about how to use these rules.
Ask the students if they have any questions about domain and range with regard to square root
functions. Remind them that the domain and ranges either go to positive or negative infinity and
begin (or end, depending on the negative) at the point h for domain and k for range.
Draw the graph for these equations and write the equations on the other side of the board. Ask
the students to write in their notebook which equation lines up with which graph.
√ √
√ √
Ask the students if they have any questions about graphing.
Go over the four steps to solving a radical equation.
v.
Isolate the radical
vi.
Raise both sides of the equation to the power that will eliminate the radical
vii.
Solve for the independent variable
viii.
Check for accurate or extraneous solutions
Solve the problem 0
√
7
3 on the board.
Answer: x=16
Ask if students have any question about solving radical equations.
Time: ~15-20 minutes
3. Quiz/Rubric
“For problems #1-3, first simplify the function using rules of exponents. Put it into standard
square root form, then graph the function and solve for f(x)=0 MATHEMATICALLY. (NO
CALCULATORS). Also state the domain and range. For problem #4, just simplify and solve for
f(x)=0, do not worry about graphing or finding the domain and range.” The students would be
tested on graphing and solving radical equations during the Chapter Test. This quiz was meant to
be graphing calculator free.
1)
25
/
5
√
Answer:
/
x=-25
/
2)
4
2
√
Answer:
x=14
/
3)
5
√
Answer:
1
x=6
4)
4
4
1
/
6
9
/
√2
Answer:
1
3
√
For problems 1-3, students can receive a possible two points for simplification. Zero points if
they get more than half of the process wrong, one point if they get most of it right, and two
points if they get it exactly right.
Students can receive two points for the graphs, one point for correct starting position, and one
point for the correct orientation of the line.
Students can also receive two points for the solving portion. One point if the student sets up the
problem correctly, but makes a calculation error. Two points if the student sets up the problem
correctly and gets the correct answer.
Students will receive one point if they correctly state the domain and one point if they correctly
state the range.
So each of the first three problems are out of a possible 8 points, for a total of 24 points.
For problem four, students again can receive two points for simplification and two points for
solving the function correctly. There will be a bonus point if the student recognizes that there is
an extraneous answer.
The quiz is out of 28 points, but it is possible for a student to receive a 29/28 because of the
bonus point.
Credit will be given for each individual part of a problem done correctly. So an incorrectly done
simplification will not take away points from a graph that correctly represents that wrong
simplification. The graph will still get credit.
Time: Students should have around 25 minutes to complete the quiz.
Possible Fun Movie Clip: The Ghostbusters meet and get owned by a God. Then blow up a giant
Marshmallow Man. “Ray, if someone asks if you’re a God, you say “YES”!)