Download Developmental Math – An Open Curriculum Instructor Guide

Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Table of Contents and
Learning Objectives
Unit 16: Radical Expressions and Quadratic Equations
Unit Table of Contents
Lesson 1: Introduction to Roots and Rational Exponents
Topic 1: Roots
Learning Objectives
• Find principal square roots and their opposites.
• Approximate square roots and find exact roots with a calculator.
Topic 2: Squares, Cubes and Beyond
Learning Objectives
• Simplify square roots.
• Find cube roots.
• Simplify expressions with odd and even roots.
Topic 3: Rational Exponents
Learning Objectives
• Convert radicals to expressions with rational exponents.
• Convert expressions with rational exponents to their radical equivalent.
• Use the laws of exponents to simplify expressions with rational exponents.
• Use rational exponents to simplify radical expressions.
Lesson 2: Operations with Radicals
Topic 1: Multiplying and Dividing Radical Expressions
Learning Objectives
• Multiply and simplify radical expressions that contain a single term.
• Divide and simplify radical expressions that contain a single term.
Topic 2: Adding and Subtracting Radicals
Learning Objectives
• Add radicals and simplify.
• Subtract radicals and simplify.
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Monterey Institute for Technology and Education (MITE) 2012
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1.1#
Developmental Math – An Open Curriculum
Instructor Guide
Topic 3: Multiplication of Multiple Term Radicals
Learning Objectives
• Multiply and simplify radical expressions that contain more than one term.
Topic 4: Rationalizing Denominators
Learning Objectives
• Rationalize a denominator with a monomial containing a square root.
• Rationalize a denominator that contains two terms.
Lesson 3: Radical Equations
Topic 1: Solving Radical Equations
Learning Objectives
• Solve equations containing radicals.
• Recognize extraneous solutions.
• Solve application problems that involve radical equations as part of the solution.
Lesson 4: Complex Numbers
Topic 1: Complex Numbers
Learning Objectives
• Express roots of negative numbers in terms of i.
• Express imaginary numbers as bi and complex numbers as a + bi.
Topic 2: Operations with Complex Numbers
Learning Objectives
• Add complex numbers.
• Subtract complex numbers.
• Multiply complex numbers.
• Find conjugates of complex numbers.
• Divide complex numbers.
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Developmental Math – An Open Curriculum
Instructor Guide
Lesson 5: Solving Quadratic Equations
Topic 1: Square Roots and Completing the Square
Learning Objectives
• Solve quadratic equations by using the Square Root Property.
• Identify and complete perfect square trinomials.
• Solve quadratic equations by completing the square.
Topic 2: The Quadratic Formula
Learning Objectives
• Write a quadratic equation in standard form and identify the values of a, b, and c in a
standard form quadratic equation.
• Use the Quadratic Formula to find all real solutions.
• Use the Quadratic Formula to find all complex solutions.
• Compute the discriminant and state the number and type of solutions.
• Solve application problems requiring the use of the Quadratic Formula.
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Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Instructor Notes
Unit 16: Radical Expressions and Quadratic Equations
Instructor Notes
The Mathematics of Radical Expressions and Quadratic Equations
To a large degree this unit is transitional–focusing more on extending known concepts than on
introducing new ones. Most of the unit covers more complex notations and procedures for
working with exponents and roots and further work with simplifying radical expressions and
solving radical equations.
There are two new topics, complex and imaginary numbers. Students are also introduced to the
techniques of completing the square and using the quadratic formula to solve quadratic
equations.
Teaching Tips: Challenges and Approaches
One of the underlying themes of teaching developmental math is that each concept and
technique builds upon earlier, often easier, ideas. This unit exemplifies that. Although many
students will be intimidated by their first glimpse of radical expressions and equations, you can
help them conquer their apprehensions by making sure they understand the fundamentals and
building confidence by starting simply with skills they already know. Insist that students write
down every step at first, and question them about the purpose of each step. Give them lots and
lots of practice.
Root Basics
Simplifying radical expressions is a common stumbling block in this unit but it's a skill students
have to master to succeed in developmental mathematics. The combination of radicals and
exponents looks daunting and can require multiple skills to solve. In order to simplify radicals
effectively, students need to be comfortable factoring numbers. While this skill was likely
covered in earlier courses, you may find that students need to brush up before they can begin
the real work of this unit.
Go over the terminology of roots and exponents—for example, raising something to the second
power is the same as squaring, and a quantity to the third power is the same thing as that
quantity cubed.
Students have been told for a long time that every positive number has two real-number square
roots and that the number 0 has just one square root (0). Up until now, negative numbers did
not have real-number square roots. Now is the time to inform students that there is a way to
consider the square root of a negative number. Put a number like
up on the board and
have students guess what they think the answer might be. Most will guess -5! Be sure to
stress the differences among
,
, and
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Developmental Math – An Open Curriculum
Instructor Guide
Use problems like the one solved below to point out that some roots represent rational numbers
and others irrational numbers:
[From Lesson 1, Topic 1, Topic Text]
Make sure your students know how to find the square root of a number on their calculator
because every calculator works differently. Students will want to say that
or
. They may not be familiar with the symbol ” ”. You will also need to
remind your students to make certain that their radicals are simplified. To them
better than
looks far
.
Simplifying Radicals
Once the fundamentals are in place, introduce radicals containing variables with simple
expressions such as
. It will make sense to students that since x3 is equivalent to
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Developmental Math – An Open Curriculum
Instructor Guide
is the same as finding
. This is easy to simplify to
and finally to
.
Once students understand that their first goal is to find and isolate all the square terms under
the radicand, you can introduce more difficult problems. The key is to have students practice
this technique with problem sets that gradually increase in complexity. We recommend that you
follow the examples used in the topic texts, presentations, and worked examples—show every
step and explain how each advances the goal of pulling out and simplifying the squares.
It is important that when students work on their own they also write out all the steps, at least
initially. While the course materials do clearly explain that the square root of a product is equal
to the product of the square roots, this may be easily misunderstood at first. As shown in the
example below, when factoring a radicand like 16x2y4, take the time to write out that the square
root of 16 is 4, the square root of x2 is x, and so on:
[From Lesson 1, Topic 2, Presentation]
Informally question students as they work problems like these, until they can explain each of the
steps.
Once students are comfortable pulling out the square terms, you'll need to introduce the notion
of variable roots and absolute value:
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Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 1, Topic 2, Topic Text]
Show that if
,
does not equal x. The addition of absolute value notation is difficult
for students to understand but it is technically necessary if an even root is being taken.
In the beginning, restrict discussion to square roots. Move on to cube roots and the
generalization to odd and even roots only after students are proficient at simplifying squares.
Remind students that it is okay to have a negative sign in the radicand if the root is odd.
A good review of the properties of exponents will help students be able to simplify radical
expressions beyond square roots. When
is converted to
, it’s easy to see that both
2
are therefore equivalent to a . Try to work through a number of these problems so students will
see that
.
Students will need to be reminded that
and
are not the same. This can be
illustrated very easily by converting the exponential representations to radicals:
and
Multiplying and Dividing Radicals
The properties of exponents lead to rules for the multiplication and division of radicals. These
rules are straightforward but students need to be reminded that the rules apply only when the
roots of the radicals are the same. Sometimes it may be easier for the radicals to be simplified
before using the rules. Be sure to work out a problem both ways to illustrate this, for example:
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Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 2, Topic 1, Topic Text]
This solution has two larger numbers (18 and 16) being multiplied together. If the individual
radicals are simplified first, the numbers being used are much smaller.
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Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 2, Topic 1, Topic Text]
Adding and Subtracting Radicals
Adding or subtracting radicals is only possible when the index and the radicand of two or more
radicals are the same. Help students remember this by comparing it to adding like terms
together in beginning algebra. Point out that like radicals can be added and subtracted the
same way that like variables can be added or subtracted.
There are two common problems that students have when trying to add and subtract radicals.
First, since they have already seen that
, they may think that
. Second, they may assume that a problem can’t be done at all when terms
don’t have the same radicand. Encourage your students to simplify all radicals before giving up,
because the radicals might be like after all, as seen here:
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Developmental Math – An Open Curriculum
Instructor Guide
[From Lesson 2, Topic 2, Topic Text]
Solving Radical Equations
Students tend to do strange things when solving radical equations. Remind your students to
isolate the radical as the first step in solving. For example, if the equation is
,
the correct start is to subtract 5 from both sides of the equation. However, creative students will
square both sides and get 2x ─ 7 + 25 = 64. Other students will decide that the 5 belongs as
part of the radicand, and then square that side of the equation as a unit. Someone else might
decide to add 7 to both sides to try to get the x term by itself.
As always, lots of practice combined with instructor feedback is vital. As your students make
their mistakes, be sure to explain why what they are doing isn’t correct. Also, because
extraneous solutions are introduced when squaring both sides, make sure your students check
all answers. Group and classroom problem solving is an efficient way of alerting students to
these types of errors.
Imaginary and Complex Numbers
Once your students understand how to simplify and carry out operations on radicals, it is time to
introduce the concept of imaginary and complex numbers. Explain that they need to step
outside the real number system in order to define the square root of a negative number.
After defining i and the terms imaginary and complex numbers, stress that complex numbers
can be added, subtracted, multiplied, and divided using the same ideas that are used for
radicals and variables. There really is only one more step and that is replacing any i2 with −1 in
order to simplify completely.
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Developmental Math – An Open Curriculum
Instructor Guide
Quadratic Equations
Earlier in this course, students were introduced to quadratic equations and learned how to use
factoring to solve them. Now they'll see that radicals and complex numbers are useful for
solving quadratic equations that can’t be factored. Two methods are presented, completing the
square and using the quadratic formula.
It is a good idea to compare methods of finding solutions to quadratic equations. Begin with an
equation like x2 + 2x = 15 and solve by factoring, completing the square (which also uses the
square root property), and the quadratic formula. Then move on to examples that can’t be
solved by factoring. Include equations whose solutions contain radicals as well as complex
numbers. Also, make sure that students can use both methods even if they develop a favorite
technique.
Students will undoubtedly have trouble remembering the quadratic formula at first. One way to
help them is to put it to music. There are many videos on YouTube that do this already, such as
http://www.youtube.com/watch?v=2lbABbfU6Zc&feature=related.
Practice is needed to ensure that students use the quadratic formula correctly. A common error
is neglecting to put the quadratic equation into standard form first before identifying a, b, and c.
Most developmental math students will be unfamiliar with the discriminant, so stress its value in
predicting what the solutions to a quadratic equation will look like without going to the trouble of
actually solving it. Introduce this topic with many examples, such as the following:
[From Lesson 5, Topic 2, Worked Example 4]
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Developmental Math – An Open Curriculum
Instructor Guide
Keep in Mind
This unit is a challenging one. Although it mainly involves skills that students learned in
beginning and intermediate algebra, the inclusion of radicals make problems look much harder.
Whenever possible, remind your students that they actually have done most of this before—for
example, multiplying binomials with radicals is really the same thing as multiplying two binomials
together. Remind them as well that adding two complex numbers together is really the same
thing as adding like terms. Offer support and a lot of practice to ensure that the concepts are
understood.
Most of the material in this unit has been geared toward intermediate algebra students.
However, it is appropriate to use this material for enrichment in the beginning algebra class.
Additional Resources
In all mathematics, the best way to really learn new skills and ideas is repetition. Problem
solving is woven into every aspect of this course—each topic includes warm-up, practice, and
review problems for students to solve on their own. The presentations, worked examples, and
topic texts demonstrate how to tackle even more problems. But practice makes perfect, and
some students will benefit from additional work.
Practice finding roots and converting between radical expressions and exponential equivalents
can be found at
http://www.zweigmedia.com/RealWorld/tut_alg_review/framesA_2B.html
A really good site to practice taking square and cube roots is
http://www.district87.org/bhs/math/practice/radicals/radicalpractice.htm
Get help solving radical equations at
http://www.sosmath.com/algebra/solve/solve0/solve0.html#radical
For practice multiplying complex numbers, use
http://www.ltcconline.net/greenl/java/BasicAlgebra/ComplexMultiplication/ComplexMultiplication.
html
For help finding conjugates of complex numbers and dividing complex numbers, see
http://www.ltcconline.net/greenl/java/BasicAlgebra/ComplexDivision/ComplexDivision.html
To add, subtract, multiply and divide complex numbers, go to
http://www.ltcconline.net/greenl/java/BasicAlgebra/ComplexArithmetic/ComplexArithmetic.html
Get practice solving quadratic equations using the principle of square roots at
http://www.mathsnet.net/algebra/e31.html
(You can get additional problems on this site by clicking on “change” or “more on this topic.”)
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Developmental Math – An Open Curriculum
Instructor Guide
Find help solving quadratic equations at
http://www.sosmath.com/algebra/solve/solve0/solve0.html#quadratic
Quadratic formula help can be found at
http://www.mathsnet.net/algebra/e53.html
(Get additional problems on this site by clicking on “change” or “more on this topic.”)
Summary
This unit expands previously taught ideas and techniques for simplifying radical expressions
and expressions with exponents. Students also learn new procedures for solving quadratic
equations and are introduced to complex and imaginary numbers for the first time. All these
concepts are critical to the successful completion of this course and subsequent courses in
upper levels of mathematics. Students can master this material if they are reminded of their
existing knowledge of factoring, radicals, and exponents, and if they are taken through the new
procedures step-by-step, with a gradual increase in complexity.
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Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Tutor Simulation
Unit 16: Radical Expressions and Quadratic Equations
Instructor Overview
Tutor Simulation: Storing Sand and Salt
Purpose
This simulation allows students to demonstrate their ability to work with radical expressions and
quadratic equations in a real world problem. Students will be asked to apply what they have
learned to solve a problem involving:
•
•
•
Solving Radical Equations
Understanding Word Problems
Rewriting Formulas with Radicals
Problem
Students are presented with the following problem:
You will be helping out the Northern Sand, Salt and Gravel Company. They provide sand to
various highway departments to spread on icy roads for winter road safety.
Your tasks will include helping them determine how much room they'll need to store a winter's
supply of sand, and how large a tarp would be required to protect the sand from the elements.
Recommendations
Tutor simulations are designed to give students a chance to assess their understanding of unit
material in a personal, risk-free situation. Before directing students to the simulation,
•
•
•
Make sure they have completed all other unit material.
Explain the mechanics of tutor simulations:
o Students will be given a problem and then guided through its solution by a video
tutor;
o After each answer is chosen, students should wait for tutor feedback before
continuing;
o After the simulation is completed, students will be given an assessment of their
efforts. If areas of concern are found, the students should review unit materials or
seek help from their instructor.
Emphasize that this is an exploration, not an exam.
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Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Puzzle
Unit 16: Radical Expressions and Quadratic Equations
Instructor Overview
Puzzle: Shape Shifter
Objectives
Shape Shifter is a manipulative puzzle that tests a student's understanding of the graphs of
quadratic functions. Players are given the vertex form of the equation for a parabola, y = a(x –
h)2, which describes the shape, direction, and position of the parabola on a graph. They're then
asked to tweak the coefficients of the equation until its parabola achieves a desired shape.
This game reinforces how the values in an equation control the appearance and placement of
its graph. Because each parabola is manipulated to match the curve of a real-world object—
bridge cables, the support legs of the Eiffel Tower, the path of water in a fountain, a satellite
dish, and even the familiar double curve of McDonald's golden arches—players also gain an
appreciation for the prevalence of parabolas and the importance of quadratic functions in their
everyday lives.
Figure 1. Shape Shifter asks players to change the coefficients in a quadratic function so that its parabola
matches a given shape.
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Developmental Math – An Open Curriculum
Instructor Guide
Description
Shape Shifter takes players through a sequence of ten puzzles. Each puzzle shows a quadratic
equation in vertex form and two parabolas. One parabola, shown in red, is the graph of the
given equation. Players are asked to use buttons to change the coefficients of the equation until
its graph matches the second parabola, shown in green. When they've done so, the parabolas
are replaced by a picture of the real-world object they model. Players can then move on to the
next puzzle.
Shape Shifter is designed as a single player game, but could be used in a classroom by asking
the group to shout out or take turns suggesting which variables need to be adjusted in which
direction for the curve to fall into place.
1.16#
Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Project
Unit 16: Radical Expressions and Quadratic Equations
Instructor Overview
Project: Variables Involved in Sunscreen
Student Instructions
Introduction
You are working in a research laboratory for a producer of sunscreen and want to make a new
type of sunscreen aimed at men and women between the ages of 18 and 25. You will use your
ability to analyze data and set-up and solve applications involving variation to make
recommendations for the new product.
Task
In this project you will play the part of a researcher for a sunscreen manufacturer. When
considering different brands of sunscreen products with the same SPF (sun protection factor),
research has shown that consumers are interested in three main factors: 1) fragrance, 2)
coverage, and 3) price. Working together with your group, you will analyze data and make
calculations to determine models for two of the three factors. You will not be analyzing pricing
for this project.
Instructions
Solve each problem in order and save your work along the way, as you will create a
professional report at the conclusion of the project.
•
First problem: Traditional Sunscreen Fragrance
• Since there is a fragrance added to the sunscreen, you need to understand how
scent spreads, so the people (target ages 18-25) will smell nice under any
conditions. Your lab has collected experimental data on the traditional sunscreen
fragrance to determine how long it takes the fragrance of the sunscreen to reach
varying distances. Smell travels very differently than light or sound. It is not a wave
in the air, but it comes from molecules that evaporate from the source of the smell.
Each molecule bounces around at random with the greatest concentration near the
source of the smell. Draw a graph of the distance in meters versus time to reach that
distance. Explain any other variables you think will affect how quickly a smell can
travel or why the speed at which the fragrance reaches certain distances seems to
change. You may want to conduct an internet search as a reference.
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Developmental Math – An Open Curriculum
Instructor Guide
•
Time (in seconds)
Distance in meters
15
0.5
56
1
135
1.5
252
2
Second Problem: Fragrance Model
•
You need to develop a mathematical model to show how the scent of the fragrance
spreads. By looking at the curvature at the graph, you suspect that it may be directly
proportional and of the form:
, where t is time in seconds and D is distance
in meters. For each distance, solve for the exact k with simplified square roots and
rationalized denominators. Next, use your calculator to approximate k for each
distance. Finally, determine your mathematical model.
Distance in meters
Time (in seconds)
0.5
15
1
56
1.5
135
2
252
1.18#
Exact k
Approximate k
Developmental Math – An Open Curriculum
Instructor Guide
•
•
Finally, graph your model with the data. Since the sunscreen is aimed at men and
women between the ages of 18 and 25, do you think that the fragrance is the right
strength, should it be a stronger fragrance, or a lighter fragrance? Along with your
given recommendations, base some of your suggestions on the data.
Third Problem: Coverage
•
Experts recommend that an adult should use about 1 ounce of sunscreen for
adequate coverage. All body parts that are exposed to the sun need to be protected.
Interestingly, the amount of skin depends on a person’s height and weight. If you
assume that when they refer to an adult, they are speaking of a person weighing 144
pounds and is 5 feet 6 inches tall, how much sunscreen should other adults of
different body sizes use? To figure out adults’ body surface areas (BSA), you need to
find the height in inches, h, and weight in pounds, w and then use the Mosteller
formula:
•
Fill in the chart by converting the heights to inches and then calculate the BSA for
each body size. Use your calculator to get an approximate BSA. Then, using
proportions, figure out the amount of sunscreen that each adult should use based on
the BSA.
Body Surface Area
(BSA)
Amount of Sunscreen
Height
Height
(inches)
Weight
(pounds)
5'2"
118
5'6"
144
5'10"
170
6'2"
187
•
(square meters)
(ounces)
1
Based on your calculations, do you think the recommendation that adults should use
1 ounce of sunscreen is accurate? Explain your reasoning.
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Developmental Math – An Open Curriculum
Instructor Guide
Collaboration
Get together with another group to compare your answers to each of the three problems.
Discuss how you might combine your answers to make a more complete and convincing
analysis of the situation.
Conclusions
Present your solution in a way that makes it easy for the sunscreen manufacturer to understand
your results. Be sure to clearly explain your reasoning at each stage and conclude with
recommendations about fragrance and coverage. You should create an advertising campaign
relating this information to the target audience of 18-25 year-olds. Be sure to explain why the
new sunscreen is more beneficial for consumers.
Instructor Notes
Assignment Procedures
This project contains several different types of problems that give students practice in
simplifying square roots, approximating square roots, and using the quadratic formula. The first
two problems are connected. However, the third problem does not depend on the first two
problems. Therefore, this project can be easily tailored by assigning only those problems
corresponding to those skills you would like to reinforce.
Problem 1
The graphs may differ a little depending on how smooth they made the curve. The explanation
may vary slightly based on research conducted.
1.20#
Developmental Math – An Open Curriculum
Instructor Guide
It appears that at close distances the aroma travels rapidly, but then it does not keep up the
same rate. The variables that affect how quickly a smell can travel are: 1) the amount of
chemicals, 2) the temperature, and 3) the mass of the chemicals (lighter molecules travel
faster).
Problem 2
All of the models should simplify to the given answers.
Distance in meters
Time (in seconds)
0.5
15
0.129
1
56
0.134
1.5
135
0.129
2
252
0.126
1.21#
Exact k
Approximate k
Developmental Math – An Open Curriculum
Instructor Guide
The exact k values were found by:
Distance in
meters
Equation
Simplify and Rationalize the
Denominator
Exact k
0.5
1
1.5
2
Note to instructor: Students should find their model using mathematical reasoning. Two different
models are given below with a rationale.
Remember, these are experimental data, so not all of the k values will match. The average of
the approximate k values is 0.1295 found by (.129+.134+.129+.126)/4. If an approximate
answer is desired, I would use 0.1295 as the k value. The model then becomes
.
Since the average is really close to the 0.129 k value, which occurs twice in the data if an exact
value is required the model becomes
.
1.22#
Developmental Math – An Open Curriculum
Instructor Guide
The graphs may differ a little depending on how smooth they made the curve. The explanation
may vary slightly based on their beliefs and how they examined the data.
Since the sunscreen is aimed at men and women between the ages of 18 and 25, I would think
that the sunscreen producers might want a stronger fragrance meaning that it gives off a more
powerful smell right away. Based on the data, it takes the fragrance over 4 minutes (252/60) to
reach 2 meters, which is approximately 6 feet.
Problem 3
All of the calculations should be the same. The amount of sunscreen should be calculated
based on the BSA of a 5 foot 6 inch tall person.
Body Surface Area
(BSA)
(square
meters)
Height
Height
(inches)
Weight
(pounds)
5'2"
62
118
Amount of Sunscreen
(ounces)
=
1.529
1.23#
1.529/1.742 = 0.88
ounces
Developmental Math – An Open Curriculum
Instructor Guide
5'6"
66
=
144
1 ounce- given
1.742
5'10"
70
170
=
1.950/1.742 = 1.12
ounces
=
2.102/1.742 = 1.207
ounces
1.950
6'2"
74
187
2.102
Students’ answers may vary. Below is one type of response.
The recommendation of 1 ounce seems reasonable because the range is 0.88 ounces to 1.207
ounces, which are all very close to 1 ounce.
Technology Integration
This project provides abundant opportunities for technology integration, and gives students the
chance to research and collaborate using online technology. The students’ instructions list
several websites that provide information on numbering systems, game design, and graphics.
The following are other examples of free Internet resources that can be used to support this
project:
http://www.moodle.org
An Open Source Course Management System (CMS), also known as a Learning Management
System (LMS) or a Virtual Learning Environment (VLE). Moodle has become very popular
among educators around the world as a tool for creating online dynamic websites for their
students.
http://www.wikispaces.com/site/for/teachers or http://pbworks.com/content/edu+overview
Allows you to create a secure online Wiki workspace in about 60 seconds. Encourage
classroom participation with interactive Wiki pages that students can view and edit from any
computer. Share class resources and completed student work with parents.
http://www.docs.google.com
Allows students to collaborate in real-time from any computer. Google Docs provides free
access and storage for word processing, spreadsheets, presentations, and surveys. This is
ideal for group projects.
http://why.openoffice.org/
1.24#
Developmental Math – An Open Curriculum
Instructor Guide
The leading open-source office software suite for word processing, spreadsheets,
presentations, graphics, databases and more. It can read and write files from other common
office software packages like Microsoft Word or Excel and MacWorks. It can be downloaded
and used completely free of charge for any purpose.
Rubric
Score
Content
•
•
4
•
•
•
•
3
•
•
•
•
2
•
•
Presentation/Communication
The solution shows a deep understanding of
the problem including the ability to identify
the appropriate mathematical concepts and
the information necessary for its solution.
The solution completely addresses all
mathematical components presented in the
task.
The solution puts to use the underlying
mathematical concepts upon which the task
is designed and applies procedures
accurately to correctly solve the problem
and verify the results.
Mathematically relevant observations and/or
connections are made.
•
The solution shows that the student has a
broad understanding of the problem and the
major concepts necessary for its solution.
The solution addresses all of the
mathematical components presented in the
task.
The student uses a strategy that includes
mathematical procedures and some
mathematical reasoning that leads to a
solution of the problem.
Most parts of the project are correct with
only minor mathematical errors.
The solution is not complete indicating that
parts of the problem are not understood.
The solution addresses some, but not all of
the mathematical components presented in
the task.
The student uses a strategy that is partially
useful, and demonstrates some evidence of
mathematical reasoning.
Some parts of the project may be correct,
but major errors are noted and the student
could not completely carry out mathematical
procedures.
•
•
1.25#
•
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•
•
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There is a clear, effective explanation
detailing how the problem is solved.
All of the steps are included so that
the reader does not need to infer
how and why decisions were made.
Mathematical representation is
actively used as a means of
communicating ideas related to the
solution of the problem.
There is precise and appropriate use
of mathematical terminology and
notation.
Your project is professional looking
with graphics and effective use of
color.
There is a clear explanation.
There is appropriate use of accurate
mathematical representation.
There is effective use of
mathematical terminology and
notation.
Your project is neat with graphics
and effective use of color.
Your project is hard to follow
because the material is presented in
a manner that jumps around between
unconnected topics.
There is some use of appropriate
mathematical representation.
There is some use of mathematical
terminology and notation appropriate
to the problem.
Your project contains low quality
graphics and colors that do not add
interest to the project.
Developmental Math – An Open Curriculum
Instructor Guide
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There is no solution, or the solution has no
relationship to the task.
No evidence of a strategy, procedure, or
mathematical reasoning and/or uses a
strategy that does not help solve the
problem.
The solution addresses none of the
mathematical components presented in the
task.
There were so many errors in mathematical
procedures that the problem could not be
solved.
1.26#
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There is no explanation of the
solution, the explanation cannot be
understood or it is unrelated to the
problem.
There is no use or inappropriate use
of mathematical representations (e.g.
figures, diagrams, graphs, tables,
etc.).
There is no use, or mostly
inappropriate use, of mathematical
terminology and notation.
Your project is missing graphics and
uses little to no color.
Developmental Math – An Open Curriculum
Instructor Guide
Unit 16 – Correlation to Common
Core Standards
Learning Objectives
Unit 16: Radical Expressions and Quadratic Equations
Common Core Standards
Unit#16,#Lesson#1,#Topic#1:#Roots#
Grade:#8#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.NS.'
The#Number#System#
CATEGORY'/'CLUSTER'
''
STANDARD'
8.NS.2.'
STRAND'/'DOMAIN'
CC.8.EE.'
Know#that#there#are#numbers#that#are#not#rational,#and#
approximate#them#by#rational#numbers.#
Use#rational#approximations#of#irrational#numbers#to#
compare#the#size#of#irrational#numbers,#locate#them#
approximately#on#a#number#line#diagram,#and#estimate#the#
value#of#expressions#(e.g.,#pi^2).#For#example,#by#truncating#
the#decimal#expansion#of#square#root#of#2,#show#that#square#
root#of#2#is#between#1#and#2,#then#between#1.4#and#1.5,#and#
explain#how#to#continue#on#to#get#better#approximations.#
Expressions#and#Equations#
CATEGORY'/'CLUSTER'
''
Work#with#radicals#and#integer#exponents.#
STANDARD'
8.EE.2.'
Use#square#root#and#cube#root#symbols#to#represent#
solutions#to#equations#of#the#form#x^2#=#p#and#x^3#=#p,#
where#p#is#a#positive#rational#number.#Evaluate#square#roots#
of#small#perfect#squares#and#cube#roots#of#small#perfect#
cubes.#Know#that#square#root#of#2#is#irrational.#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.2.'
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Grade:#9?12#?#Adopted#2010#
Unit#16,#Lesson#1,#Topic#2:##Squares,#Cubes,#and#Beyond#
Grade:#8#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.NS.'
The#Number#System#
CATEGORY'/'CLUSTER'
''
Know#that#there#are#numbers#that#are#not#rational,#and#
approximate#them#by#rational#numbers.#
1.27#
Developmental Math – An Open Curriculum
Instructor Guide
STANDARD'
8.NS.2.'
STRAND'/'DOMAIN'
CC.8.EE.'
Use#rational#approximations#of#irrational#numbers#to#
compare#the#size#of#irrational#numbers,#locate#them#
approximately#on#a#number#line#diagram,#and#estimate#the#
value#of#expressions#(e.g.,#pi^2).#For#example,#by#truncating#
the#decimal#expansion#of#square#root#of#2,#show#that#square#
root#of#2#is#between#1#and#2,#then#between#1.4#and#1.5,#and#
explain#how#to#continue#on#to#get#better#approximations.#
Expressions#and#Equations#
CATEGORY'/'CLUSTER'
''
Work#with#radicals#and#integer#exponents.#
STANDARD'
8.EE.2.'
Use#square#root#and#cube#root#symbols#to#represent#
solutions#to#equations#of#the#form#x^2#=#p#and#x^3#=#p,#
where#p#is#a#positive#rational#number.#Evaluate#square#roots#
of#small#perfect#squares#and#cube#roots#of#small#perfect#
cubes.#Know#that#square#root#of#2#is#irrational.#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.2.'
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Unit#16,#Lesson#1,#Topic#3:##Rational#Exponents#
Grade:#8#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.EE.'
Expressions#and#Equations#
CATEGORY'/'CLUSTER'
''
Work#with#radicals#and#integer#exponents.#
STANDARD'
8.EE.1.'
Know#and#apply#the#properties#of#integer#exponents#to#
generate#equivalent#numerical#expressions.#For#example,#
3^2#x#3^?5#=#3^?3#=#1/3^3#=#1/27.#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.1.'
EXPECTATION'
N5RN.2.'
Explain#how#the#definition#of#the#meaning#of#rational#
exponents#follows#from#extending#the#properties#of#integer#
exponents#to#those#values,#allowing#for#a#notation#for#
radicals#in#terms#of#rational#exponents.#For#example,#we#
define#5^1/3#to#be#the#cube#root#of#5#because#we#want#
(5^1/3)^3#=#5^(1/3)^3#to#hold,#so#(5^1/3)^3#must#equal#5.#
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Unit#16,#Lesson#2,#Topic#1:##Multiplying#and#Dividing#Radical#Expressions#
1.28#
Developmental Math – An Open Curriculum
Instructor Guide
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.2.'
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Unit#16,#Lesson#2,#Topic#2:##Adding#and#Subtracting#Radicals#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.2.'
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Unit#16,#Lesson#2,#Topic#3:##Multiplication#of#Multiple#Term#Radicals#
No#Correlations#
Unit#16,#Lesson#2,#Topic#4:##Rationalizing#Denominators#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5RN.'
The#Real#Number#System#
STANDARD'
''
Extend#the#properties#of#exponents#to#rational#exponents.#
EXPECTATION'
N5RN.2.'
Rewrite#expressions#involving#radicals#and#rational#
exponents#using#the#properties#of#exponents.#
Unit#16,#Lesson#3,#Topic#1:##Solving#Radical#Equations#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.A.'
Algebra#
CATEGORY'/'CLUSTER'
A5REI.'
Reasoning#with#Equations#and#Inequalities#
STANDARD'
''
Understand#solving#equations#as#a#process#of#reasoning#and#explain#
the#reasoning.#
EXPECTATION'
A5REI.2.'
STRAND'/'DOMAIN'
CC.F.'
Solve#simple#rational#and#radical#equations#in#one#variable,#
and#give#examples#showing#how#extraneous#solutions#may#
arise.#
Functions#
CATEGORY'/'CLUSTER'
F5IF.'
Interpreting#Functions#
STANDARD'
''
Analyze#functions#using#different#representations.#
1.29#
Developmental Math – An Open Curriculum
Instructor Guide
EXPECTATION'
F5IF.7.'
GRADE'EXPECTATION'
F5IF.7(b)'
Graph#functions#expressed#symbolically#and#show#key#
features#of#the#graph,#by#hand#in#simple#cases#and#using#
technology#for#more#complicated#cases.#
Graph#square#root,#cube#root,#and#piecewise?defined#
functions,#including#step#functions#and#absolute#value#
functions.#
Unit#16,#Lesson#4,#Topic#1:##Complex#Numbers#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5CN.'
The#Complex#Number#System#
STANDARD'
''
Perform#arithmetic#operations#with#complex#numbers.#
EXPECTATION'
N5CN.1.'
Know#there#is#a#complex#number#i#such#that#i^2#=#?1,#and#
every#complex#number#has#the#form#a#+#bi#with#a#and#b#real.#
EXPECTATION'
N5CN.2.'
STRAND'/'DOMAIN'
CC.N.'
Use#the#relation#i^2#=#?1#and#the#commutative,#associative,#
and#distributive#properties#to#add,#subtract,#and#multiply#
complex#numbers.#
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5CN.'
The#Complex#Number#System#
STANDARD'
''
Represent#complex#numbers#and#their#operations#on#the#complex#
plane.#
EXPECTATION'
N5CN.5.'
(+)#Represent#addition,#subtraction,#multiplication,#and#
conjugation#of#complex#numbers#geometrically#on#the#
complex#plane;#use#properties#of#this#representation#for#
computation.#For#example,#(1#?#square#root#of#3i)^3#=#8#
because#(1#?#square#root#of#3i)#has#modulus#2#and#argument#
120#degrees.#
Unit#16,#Lesson#4,#Topic#2:##Operations#with#Complex#Numbers#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5CN.'
The#Complex#Number#System#
STANDARD'
''
Perform#arithmetic#operations#with#complex#numbers.#
EXPECTATION'
N5CN.1.'
Know#there#is#a#complex#number#i#such#that#i^2#=#?1,#and#
every#complex#number#has#the#form#a#+#bi#with#a#and#b#real.#
EXPECTATION'
N5CN.2.'
EXPECTATION'
N5CN.3.'
Use#the#relation#i^2#=#?1#and#the#commutative,#associative,#
and#distributive#properties#to#add,#subtract,#and#multiply#
complex#numbers.#
(+)#Find#the#conjugate#of#a#complex#number;#use#conjugates#
to#find#moduli#and#quotients#of#complex#numbers.#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
1.30#
Developmental Math – An Open Curriculum
Instructor Guide
CATEGORY'/'CLUSTER'
N5CN.'
The#Complex#Number#System#
STANDARD'
''
Represent#complex#numbers#and#their#operations#on#the#complex#
plane.#
EXPECTATION'
N5CN.5.'
(+)#Represent#addition,#subtraction,#multiplication,#and#
conjugation#of#complex#numbers#geometrically#on#the#
complex#plane;#use#properties#of#this#representation#for#
computation.#For#example,#(1#?#square#root#of#3i)^3#=#8#
because#(1#?#square#root#of#3i)#has#modulus#2#and#argument#
120#degrees.#
Unit#16,#Lesson#5,#Topic#1:##Square#Roots#and#Completing#the#Square#
Grade:#8#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.8.EE.'
Expressions#and#Equations#
CATEGORY'/'CLUSTER'
''
Work#with#radicals#and#integer#exponents.#
STANDARD'
8.EE.2.'
Use#square#root#and#cube#root#symbols#to#represent#
solutions#to#equations#of#the#form#x^2#=#p#and#x^3#=#p,#
where#p#is#a#positive#rational#number.#Evaluate#square#roots#
of#small#perfect#squares#and#cube#roots#of#small#perfect#
cubes.#Know#that#square#root#of#2#is#irrational.#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.A.'
Algebra#
CATEGORY'/'CLUSTER'
A5REI.'
Reasoning#with#Equations#and#Inequalities#
STANDARD'
''
Solve#equations#and#inequalities#in#one#variable.#
EXPECTATION'
A5REI.4.'
Solve#quadratic#equations#in#one#variable.#
GRADE'EXPECTATION'
A5REI.4(a)'
GRADE'EXPECTATION'
A5REI.4(b)'
STRAND'/'DOMAIN'
CC.F.'
Use#the#method#of#completing#the#square#to#transform#any#
quadratic#equation#in#x#into#an#equation#of#the#form#(x#?#p)^2#
=#q#that#has#the#same#solutions.#Derive#the#quadratic#formula#
from#this#form.#
Solve#quadratic#equations#by#inspection#(e.g.,#for#x^2#=#49),#
taking#square#roots,#completing#the#square,#the#quadratic#
formula#and#factoring,#as#appropriate#to#the#initial#form#of#
the#equation.#Recognize#when#the#quadratic#formula#gives#
complex#solutions#and#write#them#as#a#plus?minus#bi#for#real#
numbers#a#and#b.#
Functions#
CATEGORY'/'CLUSTER'
F5IF.'
Interpreting#Functions#
STANDARD'
''
Analyze#functions#using#different#representations.#
EXPECTATION'
F5IF.8.'
GRADE'EXPECTATION'
F5IF.8(a)'
Write#a#function#defined#by#an#expression#in#different#but#
equivalent#forms#to#reveal#and#explain#different#properties#of#
the#function.#
Use#the#process#of#factoring#and#completing#the#square#in#a#
quadratic#function#to#show#zeros,#extreme#values,#and#
symmetry#of#the#graph,#and#interpret#these#in#terms#of#a#
1.31#
Developmental Math – An Open Curriculum
Instructor Guide
context.#
##
Unit#16,#Lesson#5,#Topic#2:##The#Quadratic#Formula#
Grade:#9?12#?#Adopted#2010#
STRAND'/'DOMAIN'
CC.N.'
Number#and#Quantity#
CATEGORY'/'CLUSTER'
N5CN.'
The#Complex#Number#System#
STANDARD'
''
Use#complex#numbers#in#polynomial#identities#and#equations.#
EXPECTATION'
N5CN.7.'
STRAND'/'DOMAIN'
CC.A.'
Solve#quadratic#equations#with#real#coefficients#that#have#
complex#solutions.#
Algebra#
CATEGORY'/'CLUSTER'
A5REI.'
Reasoning#with#Equations#and#Inequalities#
STANDARD'
''
Solve#equations#and#inequalities#in#one#variable.#
EXPECTATION'
A5REI.4.'
Solve#quadratic#equations#in#one#variable.#
GRADE'EXPECTATION'
A5REI.4(a)'
GRADE'EXPECTATION'
A5REI.4(b)'
Use#the#method#of#completing#the#square#to#transform#any#
quadratic#equation#in#x#into#an#equation#of#the#form#(x#?#p)^2#
=#q#that#has#the#same#solutions.#Derive#the#quadratic#formula#
from#this#form.#
Solve#quadratic#equations#by#inspection#(e.g.,#for#x^2#=#49),#
taking#square#roots,#completing#the#square,#the#quadratic#
formula#and#factoring,#as#appropriate#to#the#initial#form#of#
the#equation.#Recognize#when#the#quadratic#formula#gives#
complex#solutions#and#write#them#as#a#plus?minus#bi#for#real#
numbers#a#and#b.#
1.32#