developmental math – an open curriculum instructor guide · lesson 3: formulas and variation...

Developmental Math – An Open Curriculum Instructor Guide

1.1#

Unit 15: Rational Expressions

Unit Table of Contents Lesson 1: Operations with Rational Expressions Topic 1: Introduction to Rational Expressions Learning Objectives

• Find values of a variable that make a rational expression undefined. • Simplify rational expressions. Topic 2: Multiplying and Dividing Rational Expressions Learning Objectives • Multiply rational expressions and simplify. • Divide rational expressions and simplify.

Topic 3: Adding and Subtracting Rational Expressions Learning Objectives

• Add rational expressions and simplify. • Subtract rational expressions and simplify. • Find the Least Common Multiple of several algebraic expressions. • Simplify problems that combine both adding and subtracting. Topic 4: Complex Rational Expressions Learning Objectives • Simplify complex rational expressions.

Lesson 2: Rational Equations

Topic 1: Solving Rational Equations and Applications Learning Objectives

• Solve rational equations. • Check for extraneous solutions. • Solve application problems involving rational equations.

Some rights reserved. See our complete Terms of Use.

Monterey Institute for Technology and Education (MITE) 2012 To see these and all other available Instructor Resources, visit the NROC Network.

Unit 15 – Table of Contents and Learning Objectives

http://creativecommons.org/licenses/by/3.0/

http://www.montereyinstitute.org/license/pd.html

http://www.montereyinstitute.org/

http://nrocnetwork.org/


1.2#

Lesson 3: Formulas and Variation

Topic 1: Rational Formulas and Variation Learning Objectives

• Solve a formula for a specified letter. • Identify direct, inverse, and joint variation. • Find the unknown in a variation problem. • Solve application problems involving direct variation. • Solve application problems involving inverse variation. • Solve application problems involving joint variation.


1.3#


Instructor Notes The Mathematics of Rational Expressions

In this unit, students will learn how to carry out basic mathematical operations on rational expressions — in other words, how to simplify, add, subtract, multiply, and divide fractions that contain polynomials. They'll also see how they can tackle rational equations by combining these new skills with known procedures for solving basic algebraic equations. Finally, they'll explore using rational formulas with direct, inverse and joint variation to describe real-life situations and to find answers to real problems. Teaching Tips: Challenges and Approaches

The keys to learning how to work with rational expressions are preparation and practice. Students will need to apply previously learned techniques for factoring binomials and trinomials and for simplifying and performing operations on fractions in order to succeed. We suggest reviewing these concepts in class on simple, non-polynomial examples first before introducing their use on rational expressions. Once students grasp the basics of rational expressions, have them practice until they're comfortable. Domain The concept of domain is new to students so it's crucial to introduce this topic since they'll need to identify the domain before simplifying any rational expression. Discuss finding the values of a variable that make a rational expression undefined, or equivalently, finding the domain of possible values of the variable. Point out that division by zero is not defined, and will make a rational expression not defined as well. Be sure to work through a problem that shows how simplifying first would miss an excluded value, such as the following:

Unit 15 – Instructor Notes


1.4#

[From Lesson 1, Topic 1, Topic Text]

Operations with Rational Expressions Ease students into operations on rational expressions by running through operations on numeric fractions first. Then show how the very same techniques work on rational expressions. Multiplication and division of rational expressions should be taught first, because addition and subtraction of fractions with unlike denominators is more challenging. When students are simplifying, be alert for common mistakes, such as those that occur when

they try removing factors of 1. For example, many students will simplify a fraction like to


1.5#

5 instead of . As always, encourage students to save themselves by checking

their answers in the original expression. Once they plug x = 5 into and get 2, they'll see

they made a mistake. Warn your students to be careful with signs and parentheses—they're easy to lose track of in these problems. In the following example, parentheses are correctly placed around the expression 2x + 8:

#[From Lesson 1, Topic 3, Topic Text]

Your students will try to do the work for this problem in their heads and come up incorrectly with

. Stress that, especially with subtraction, all work should be shown in order to prevent

careless errors. Although most students will have worked with fractions with different denominators since 5th grade or so, these types of problems will still confuse and intimidate many. With rational expressions, it's best if students find common denominators by using prime factorization instead of by multiplying. It is a good idea to review this technique by working through a few strictly numeric fractions, and then move on to applying it to polynomial fractions.


1.6#

Complex Fractions The term 'complex fractions' tends to amuse students because they think all fractions are complex to begin with. It is important for students to understand that just like unsimplified fractions, complex fractions are not considered to be simplified. Be sure to show the two ways to simplify complex fractions:

• Performing the operations in the numerator and denominator first then rewriting the problem as a division problem with fractions as shown in the following example:

[From Lesson 1, Topic 4, Presentation]

• Finding the common denominator for all the rational expressions and multiplying the

complex fraction by the name for one with this common denominator in its numerator and denominator. In the example above, the common denominator of all the denominators would be (x + 2)(x + 3) and therefore the name for one would be

.

Usually the second way is generally easier for students, but some will prefer the first way. Solving Rational Equations When working with equations, students usually prefer to remove fractions as soon as possible. Be sure they understand that most problems can be solved in two ways, by clearing the fractions first and then proceeding as usual or by rewriting the expressions to have a common denominator and then working with just the numerators, as seen in this example:


1.7#


Work this and similar problems both ways, discussing when each method is better used. When solving rational equations, remind students to figure out what values for the variable need to be excluded because they lead to division by zero. Show a rational equation that leads to extraneous solutions and point out the relationship between excluded and extraneous values, such as in the problem below:

[From Lesson 2, Topic 1, Worked Example 2]


1.8#

If students haven’t been checking their solutions, now is the time they will need to because of the possibility of extraneous values. Work Problems The real conceptual challenge in this unit is the logic behind using rational equations to solve work problems. It's important to very carefully explain the idea that individual rates of work can be added together to get a combined rate. Be sure to work through several of these problems as a group, using real world examples that students can easily understand.

[From Lesson 2, Topic 1, Presentation]

After students are comfortable with this straightforward type of work problem where individual rates are added together, introduce more complex scenarios. These include scenarios where rates are multiples of one another, or when individual rates must be calculated from a given group rate. Once students understand how rates can be combined in a rational expression, they'll have a much easier time writing and solving these kinds of word problems.

Since applying rational equations to work problems can be highly counterintuitive, urge students to use common sense as a quick check on their answers—does a calculated rate make sense in comparison to the given rates? For example, if Amy does a job in 2 hours and Joe does the same job in 3 hours, does it make sense that they would take 5 hours to do the job working together? Shouldn’t they be able to complete the job working together in less than 2 hours?

Rational Formulas and Variation


1.9#

As seen with work problems, rational formulas are useful ways to represent real-life situations. Students are already familiar with some common formulas such as A = lw or d = rt, and most have had practice rewriting them to solve for different variables. But you may wish to spend a little time reviewing the basics of formulas before introducing variation. It's likely that students will never have heard of variation, let alone the different types of variation. Carefully distinguish between different types of variation by showing graphs and tables, stressing what happens to the second quantity if the first quantity changes. Have a chart available for quick reference, at least until students have had some time to familiarize themselves with variation.


It is important for students to be able to distinguish if an application varies directly, inversely, or jointly. Practice with real-life examples until students are easily able to determine which kind of variation is at work.

Keep in Mind

Working with rational expressions requires students to really use their factoring skills. Because they may have only just learned factoring techniques, this unit may prove challenging. Keep stressing that fractions, in general, and specifically rational expressions and equations, do become easier to do with lots of practice. Most of the material in this unit has been geared towards 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.


1.10#

Help for simplifying rational expressions can be found at http://www.hostsrv.com/webmab/app1/MSP/quickmath/02/pageGenerate?site=quickmath&s1=algebra&s2=simplify&s3=basic. Students can click on "random" to generate and expression, work on it offline, then click "simplify" to see the answer.

Practice adding rational expressions can be found at http://www.ltcconline.net/greenl/java/BasicAlgebra/RationalExpressions/RationalExpressions.html.

Help with dividing polynomials can be found at http://www.sosmath.com/algebra/factor/fac01/fac01.html. There is a review as well as a few examples to try.

Examples of solving rational equations are found at http://www.sosmath.com/algebra/solve/solve0/solve0.html#fraction and http://www.mrperezonlinemathtutor.com/A2/8_1_Rational_Equations.html.

Summary

This unit on working with rational expressions and equations tackles some fairly complex procedures and difficult ideas. Just the appearance of complex fractions will intimidate many students. We suggest making sure students are fluent with factoring and working with numerical fractions before you begin. Then as they work with polynomial fractions, and especially rate-related word problems, they will obtain a better understanding of fractions and factoring.


1.11#


Instructor Overview Tutor Simulation: Profits and the Rising Cost of Fuel

Purpose

This simulation allows students to demonstrate their ability to work with rational expressions in a real world problem. Students will be asked to apply what they have learned to solve a problem involving:

• Simplifying Rational Expressions • Adding Rational Expressions • Multiplying and Dividing Rational Expressions • Solving Rational Equations

Problem Students are presented with the following problem: You are a manager of a local manufacturing company. You have asked three market analysts to forecast the average profit of your product for the next fiscal year based on rising fuel costs. Each of the analysts presented you with a different mathematical model. Your job is to simplify these models using algebraic techniques to determine if they are equivalent. 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.

Unit 15 – Tutor Simulation


1.12#

• 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.


1.13#


Instructor Overview Puzzle: We Can Work it Out

Objectives

We Can Work It Out is a puzzle that requires students to write and solve rational expressions in order to find the answers to rate problems. It tests a player's grasp of the reasoning behind using polynomial fractions to calculate combined rates, as well as their ability to add and subtract these types of expressions.

Figure 1. Players need to use the rate = work/time formula to write a rational equation that will lead to the correct combination of workers.

Description

This puzzle presents ten workplace scenarios that each ask players to choose the combination of workers that will complete a given task in a specified amount of time. To do so, players must

Unit 15 – Puzzle


1.14#

write a rational expression to describe the output of each of several workers. They then must calculate how to add the rates to find a set of workers that together will produce the desired group rate.

Players earn points for choosing the right combination of workers. The puzzle does not advance until the correct answer is reached.

We Can Work It Out is most effective as a single player game so that each player has time to work through the problems at his own speed. It could be used in a classroom by asking students to shout out answers or to take turns calculating rates and suggesting combinations.


1.15#


Instructor Overview Project: Patient Care and Dosing Information

Student Instructions

Introduction #

There are many things to consider when administering medications to patients. You are working for a major pharmaceutical company and they want to develop mathematical models for a new trial drug. You will use your ability to analyze rational expressions and set up and solve applications involving variation to make recommendations for the healthcare industry.#

Task

In this project you will play the part of a consultant to a pharmaceutical company, which is working closely with hospitals to insure the best care possible for all patients. Working together with your group, you will analyze data and make calculations to determine how prescription drugs enter the bloodstream and the needs associated with patient care. You will make a chart that will allow medical staff to make recommendations for drug dosing.#

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. If required, round to the hundredths place (two decimal places). #

• First problem: Established Drug

• A consultant for the pharmaceutical company analyzed data and found that when a certain drug is administered to a patient by injection into the muscle, the following mathematical equation closely models the vital concentration (C) in milligrams per Liter (mg/L) of the drug in the bloodstream at time , where t is measured in hours since giving the drug:

Unit 15 – Project


1.16#

You need to analyze this model to see how the drug enters and leaves the bloodstream in order to answer several questions from healthcare professionals and compare it to a new trial drug. Since the domain is all real numbers greater than or equal to 0, you want to record how the drug changes each hour. Using the model from above, fill in the table:

Time (t) in hours since drug was administered

Concentration of drug (mg/L)

0

1

2

3

4

5

6

7

8

9

10

11

12

• Draw a graph of the drug concentration in mg/L over the twelve hour period. Then answer the following healthcare professionals’ questions (Note: You may want to analyze additional data points to answer some of these questions): • What happens to the drug concentration over the 12-hour period? Explain in

layman terms—everyday words. Include significant details such as what is the highest concentration of drug that is reached in the patient’s bloodstream, when is the highest concentration reached, and how long does the medicine last in the patient’s system.

• It is recommended that the patient always have at minimum 2 mg/L of the drug in the bloodstream. At what point should the patient be given a second dose of the medicine?


1.17#

• Second Problem: Trial Drug

• The pharmaceutical company is almost finished with a new trial drug that may replace the established one from Problem 1. This drug will be injected directly into the bloodstream. They have asked their analysts to develop a mathematical model for the vital concentration (C) in milligrams per Liter (mg/L) of the drug in the bloodstream at time , where t is measured in hours since giving the drug. Since it is difficult to model data with rational expressions, your job is to simplify the analysts’ models and compare them to externally collected data to determine which one best fits the data.

Model 1: ,

Model 2: ,

Model 3:

• External data was collected on patients by observing and recording the concentration in mg/L of the trial drug. Which of the three simplified models best fits this data?

Time (t) in hours since drug was

administered

External Data


Model 1


Model 2


Model 3


0 28

1 14

2 10

3 7

4 6

5 5

6 4

7 3.5

8 3

9 2.8


1.18#

10 2.6

11 2.3

12 2.2

• Finally, you need to make an analysis similar to before by drawing a graph of the simplified model that you determined best fits the external data. Then answer the following healthcare professionals’ questions (Note: You may want to analyze additional data points to answer some of these questions): • What happens to the drug concentration over the 12-hour period? Explain in

layman terms—everyday words. Include significant details such as what is the highest concentration of drug that is reached in the patient’s bloodstream, when is the highest concentration reached, and how long does the medicine last in the patient’s system.

• It is recommended that the patient always have at minimum 2 mg/L of the drug in the bloodstream. At what point should the patient be given a second dose of the medicine?

• Why would the trial drug be considered an improvement over the previous medicine? Why would the established drug from Problem 1 be considered better than the trial drug? You need to determine which drug (established or trial) your group wants to use in patient care. You will use this model in Problem 4.

• Third Problem:

• One of the problems with the dosing of medicine is that the amount of blood in a human is proportional to its body size. This means that a larger person has more blood and thus, needs a larger dose of medicine. You find that the volume of blood has a direct variation to body weight. The data for different adults are shown in the chart below:

Body Weight in Pounds

Blood Volume in Liters

200 7.2

154 5.5

100 3.6

The direct variation equation would be of the form , where V is the volume of blood in Liters, k is the constant of proportionality, and W is the body weight in pounds. Solve for the constant of proportionality. Check your results with all of the data values to determine if this is truly a direct variation situation.


1.19#

• Based on your direct variation equation, fill in the following table for each body weight, so that healthcare professionals can easily determine the amount of blood in liters of a patient based on his/her recorded weight.



100

110

120

130

140

150

160

170

180

190

200

• Fourth Problem:

• In order to determine the critical amount of medicine in milligrams that is needed for a patient at a given time, use the table from the Established Drug in Problem 1 or the Trial Drug in Problem 2 (whichever one your group selected in Problem 2) and multiply by the amount of liters of blood for a 150 and 180 pound person. This calculation will determine the number of milligrams of medicine that is needed for a 150 and a 180 pound person to reach the vital concentration. Then calculate the difference in milligrams.



(Established or Trial)

150 pound

Amount of drug

(mg)

180 pound

Amount of drug

(mg)

Difference per 30 pounds

(mg)


1.20#

0

1

2

3

4

5

6

7

8

9

10

11

12

• You need to determine how much additional medicine should be given to patients with different weight requirements. Since the direct variation equation is linear, the calculated difference will be the same for each additional 30 pounds. If you consider the difference in milligrams in the first five hours after the medicine is administered, what recommendation would you make? Your statement may be something such as, “For each additional 30 pounds above 150 pounds, I would increase the medicine by _____________ milligrams”.

Collaboration

Get together with another group to compare your answers to each of the four 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 healthcare professionals, such as doctors and nurses, to understand your results. Be sure to clearly explain your reasoning at each stage and conclude with a recommendation to either pursue the trial drug or stick with the established


1.21#

drug. Regardless which drug your group recommends, be sure to explain why the drug is more beneficial to patients. Finally, explain how patients’ weights will have an impact on the critical concentration of the drug in the bloodstream.

Instructor Notes

If you want to shorten the project, have students solve only the first and second problem. Assignment Procedures Problem 1



0 0

1 4

2 5

3 4.62

4 4

5 3.45

6 3

7 2.64

8 2.35

9 2.12

10 1.92

11 1.76

12 1.62

All of the answers should be similar for each group, but the wording may vary slightly. The graphs may differ a little depending on how smooth they made the curve and if additional data points were analyzed.


1.22#

The drug concentration peaks at 2 hours after being injected into the muscle with the initial dose with the highest concentration being 5 mg/L. The drug rapidly enters the bloodstream and leaves the bloodstream at a slower rate after this peak. The medicine seems to linger in the bloodstream for longer than 12 hours and appears to gradually diminish. Since the recommendation is to always have at minimum 2 mg/L of concentration in the bloodstream, it would be advisable to give a second dose between 9 and 10 hours after the first injection.

Problem 2 All of the models should simplify to the given answers. Simplified Model 1:

It simplifies to .

Simplified Model 2:


1.23#

It simplifies to .

Simplified Model 3:

It simplifies to .

Students should use the simplified models to fill in the chart. The chart should look like the one below.

Time (t) in hours since drug was

administered

External Data


Model 1


Model 2


Model 3


0 28 25 29 6.27

1 14 12.5 14.33 5.17

2 10 8.33 9.5 4.4

3 7 6.25 7.1 3.83

4 6 5 5.67 3.40

5 5 4.17 4.71 3.05

6 4 3.57 4.04 2.77

7 3.5 3.13 3.53 2.53

8 3 2.78 3.13 2.34

9 2.8 2.5 2.82 2.17


1.24#

10 2.6 2.27 2.56 2.02

11 2.3 2.08 2.35 1.89

12 2.2 1.92 2.16 1.78

Model 3 definitely does not fit the external data. Model 1 and Model 2 are very similar. However, Model 2 is a better fit for this data. It hits very closely to each data point.

All of the groups should have selected Model 2 as the best fit for the data. Their answers for the first two bulleted questions should be similar, but the wording may vary slightly. The graphs may differ a little depending on how smooth they made the curve and if additional data points were analyzed. The last bulleted point about the trial drug versus the established drug may vary. You may want to encourage them to take opposing views (some groups argue for the established drug while others argue for the new trial drug).

Graph of Model 2:


1.25#

The drug concentration peaks right at 0 hours immediately after the patient is injected with the initial dose directly into the bloodstream with the highest concentration being 29 mg/L. The drug rapidly leaves the bloodstream for the first three hours and then leaves at a slower rate leveling out for the next 8 hours. The medicine seems to linger in the bloodstream for longer than 12 hours and appears to gradually diminish. Since the recommendation is to always have at minimum 2 mg/L of concentration in the bloodstream, it would be advisable to give a second dose between 13 and 14 hours after the first injection.

Trial Drug an Improvement:

It provides a peak concentration immediately after being injected and a second dose is not needed until between 13 and 14 hours later.

Established Drug Better:

The peak concentration does not come so immediate (it takes 2 hours) and the concentration does not reach such high levels (only 5mg/L compared to 29 mg/L).

Each#group#should#select#either#the#Established#Drug#Model#or#Model#2#from#the#Trial#Drug#Model.#

Problem 3

Students should solve this equation for k, and then solve for the constant of

proportionality for each situation (see chart below).



200 7.2 0.036

154 5.5 0.036

100 3.5 0.036

It is a direct variation situation and the equation is .

All groups should have the same chart shown below:




1.26#

100 3.6

110 3.96

120 4.32

130 4.68

140 5.04

150 5.4

160 5.76

170 6.12

180 6.48

190 6.84

200 7.2

Problem 4 Each group should have one of the following two tables depending on whether they selected to use the established drug or the trial drug. I used the given External Data for the trial drug. However, students could have used the data from Model 2. It is very similar.



(Established Drug)

150 pound

Amount of drug

(mg)

5. 4 Liters of Blood

180 pound

Amount of drug

(mg)

6.48 Liters of Blood


(mg)

0 0 0 0 0

1 4 21.6 25.92 4.32

2 5 27 32.40 5.40


1.27#

3 4.62 24.95 29.94 4.99

4 4 21.6 25.92 4.32

5 3.45 18.63 22.36 3.73

6 3 16.20 19.44 3.24

7 2.64 14.26 17.11 2.85

8 2.35 12.69 15.23 2.54

9 2.12 11.45 13.74 2.29

10 1.92 10.37 12.44 2.07

11 1.76 9.5 11.40 1.90

12 1.50 8.10 9.72 1.62



(Trial Drug)

150 pound

Amount of drug

(mg)

5. 4 Liters of blood

180 pound

Amount of drug

(mg)

6.48 Liters of blood


(mg)

0 28 151.2 181.44 30.24

1 14 75.6 90.72 15.12

2 10 54 64.80 10.80

3 7 37.8 45.36 7.56

4 6 32.4 38.88 6.48

5 5 27 32.40 5.40

6 4 21.6 25.92 4.32

7 3.5 18.9 22.68 3.78


1.28#

8 3 16.2 19.44 3.24

9 2.8 15.12 18.14 3.02

10 2.6 14.04 16.85 2.81

11 2.3 12.42 14.90 2.48

12 2.2 11.88 14.26 2.38

The answers for each group may vary. I found the average of the differences for the first five hours since the drug was injected. Students may have proceeded in a different manner. The average in the differences for the established drug was 3.79 mg and for the trial drug was 12.6 mg.

Established Drug Statement:

For each 30 pounds above 150 pounds, I would increase the medicine by 3.79 milligrams.

Trial Drug Statement:

For each additional 30 pounds above 150 pounds, I would increase the medicine by 12.6 milligrams.

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.


1.29#

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/

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 Presentation/Communication

4

• 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.

• 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.

3

• 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.

• 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.

2 • 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

• Your project is hard to follow because the material is presented in a manner that jumps around between unconnected topics.


1.30#

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.

• 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.

1

• 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.

• 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.


1.31#


Common Core Standards #

Unit#15,#Lesson#1,#Topic#1:#Introduction#to#Rational#Expressions#

Grade:#9K12#K#Adopted#2010#STRAND'/'DOMAIN' CC.A.' Algebra#

CATEGORY'/'CLUSTER'

A3APR.' Arithmetic#with#Polynomials#and#Rational#Functions#

STANDARD' '' Rewrite#rational#expressions.#

EXPECTATION' A3APR.6.' Rewrite#simple#rational#expressions#in#different#forms;#write#a(x)/b(x)#in#the#form#q(x)#+#r(x)/b(x),#where#a(x),#b(x),#q(x),#and#r(x)#are#polynomials#with#the#degree#of#r(x)#less#than#the#degree#of#b(x),#using#inspection,#long#division,#or,#for#the#more#complicated#examples,#a#computer#algebra#system.#

STRAND'/'DOMAIN' CC.F.' Functions#

CATEGORY'/'CLUSTER'

F3IF.' Interpreting#Functions#

STANDARD' '' Interpret#functions#that#arise#in#applications#in#terms#of#the#context.#

EXPECTATION' F3IF.5.' Relate#the#domain#of#a#function#to#its#graph#and,#where#applicable,#to#the#quantitative#relationship#it#describes.#For#example,#if#the#function#h(n)#gives#the#number#of#personKhours#it#takes#to#assemble#n#engines#in#a#factory,#then#the#positive#integers#would#be#an#appropriate#domain#for#the#function.#

'' '' ## ##

Unit#15,#Lesson#1,#Topic#2:#Multiplying#and#Dividing#Rational#Expressions#


CATEGORY'/'CLUSTER'




Unit 15 – Correlation to Common Core Standards

Learning Objectives


1.32#

EXPECTATION' A3APR.7.' (+)#Understand#that#rational#expressions#form#a#system#analogous#to#the#rational#numbers,#closed#under#addition,#subtraction,#multiplication,#and#division#by#a#nonzero#rational#expression;#add,#subtract,#multiply,#and#divide#rational#expressions.#


CATEGORY'/'CLUSTER'




'' '' ## ##

Unit#15,#Lesson#1,#Topic#3:##Adding#and#Subtracting#Rational#Expressions#


CATEGORY'/'CLUSTER'




EXPECTATION' A3APR.7.' (+)#Understand#that#rational#expressions#form#a#system#analogous#to#the#rational#numbers,#closed#under#addition,#subtraction,#multiplication,#and#division#by#a#nonzero#rational#expression;#add,#subtract,#multiply,#and#divide#rational#expressions.#


CATEGORY'/'CLUSTER'




'' '' ## ##

Unit#15,#Lesson#1,#Topic#4:##Complex#Rational#Expressions#


CATEGORY'/' A3APR.' Arithmetic#with#Polynomials#and#Rational#Functions#


1.33#

CLUSTER'




CATEGORY'/'CLUSTER'




'' '' ## ##

Unit#15,#Lesson#2,#Topic#1:##Solving#Rational#Equations#and#Applications#


CATEGORY'/'CLUSTER'




STRAND'/'DOMAIN' CC.A.' Algebra#

CATEGORY'/'CLUSTER'

A3REI.' Reasoning#with#Equations#and#Inequalities#

STANDARD' '' Understand#solving#equations#as#a#process#of#reasoning#and#explain#the#reasoning.#

EXPECTATION' A3REI.2.' Solve#simple#rational#and#radical#equations#in#one#variable,#and#give#examples#showing#how#extraneous#solutions#may#arise.#


CATEGORY'/'CLUSTER'





1.34#

'' '' ## ##

Unit#15,#Lesson#3,#Topic#1:##Rational#Formulas#and#Variation#


CATEGORY'/'CLUSTER'

A3CED.' Creating#Equations#

STANDARD' '' Create#equations#that#describe#numbers#or#relationships.#

EXPECTATION' A3CED.4.' Rearrange#formulas#to#highlight#a#quantity#of#interest,#using#the#same#reasoning#as#in#solving#equations.#For#example,#rearrange#Ohm's#law#V#=#IR#to#highlight#resistance#R.#

STRAND'/'DOMAIN' CC.A.' Algebra#

CATEGORY'/'CLUSTER'

A3REI.' Reasoning#with#Equations#and#Inequalities#

STANDARD' '' Understand#solving#equations#as#a#process#of#reasoning#and#explain#the#reasoning.#

EXPECTATION' A3REI.1.' Explain#each#step#in#solving#a#simple#equation#as#following#from#the#equality#of#numbers#asserted#at#the#previous#step,#starting#from#the#assumption#that#the#original#equation#has#a#solution.#Construct#a#viable#argument#to#justify#a#solution#method.#


CATEGORY'/'CLUSTER'

F3LE.' Linear,#Quadratic,#and#Exponential#Models#

STANDARD' '' Construct#and#compare#linear#and#exponential#models#and#solve#problems.#

EXPECTATION' F3LE.1.' Distinguish#between#situations#that#can#be#modeled#with#linear#functions#and#with#exponential#functions.#

GRADE'EXPECTATION'

F3LE.1(a)' Prove#that#linear#functions#grow#by#equal#differences#over#equal#intervals,#and#that#exponential#functions#grow#by#equal#factors#over#equal#intervals.#

developmental math – an open curriculum instructor guide · lesson 3: formulas and variation...

Documents