Developmental Math Instructor Guide€¦ · move on to graphing systems of linear inequalities...

Developmental Math – An Open Curriculum

Instructor Guide

14.1

Unit 14: Systems of Equations and Inequalities

Learning Objectives 14.2

Instructor Notes The Mathematics of Systems of Equations and

Inequalities

Teaching Tips: Challenges and Approaches

Additional Resources

14.4

Instructor Overview Tutor Simulation: Getting the Mixture Right

14.12

Instructor Overview Puzzle: Apples and Oranges

14.13

Instructor Overview Project: Nutty Economics

14.15

Common Core Standards 14.25

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Unit 14 – Table of Contents

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Instructor Guide

14.2


Lesson 1: Graphing Systems of Equations and Inequalities

Topic 1: Graphing Systems of Linear Equations

Learning Objectives

Solve a system of linear equations by graphing.

Determine whether a system of linear equations is consistent or inconsistent.

Determine whether a system of linear equations is dependent or independent.

Determine whether an ordered pair is a solution of a system of equations.

Solve application problems by graphing a system of equations. Topic 2: Graphing Systems of Inequalities

Learning Objectives

Solve a system of linear inequalities by graphing.

Determine whether an ordered pair is a solution of a system of inequalities.

Solve application problems by graphing a system of inequalities.

Lesson 2: Algebraic Methods to Solve Systems of Equations

Topic 1: The Substitution Method

Learning Objectives

Solve a system of equations using the substitution method.

Recognize systems of equations that have no solution or an infinite number of solutions.

Solve application problems using the substitution method.

Topic 2: The Elimination Method

Learning Objectives

Solve a system of equations when no multiplication is necessary to eliminate a variable.

Solve a system of equations when multiplication is necessary to eliminate a variable.

Recognize systems that have no solution or an infinite number of solutions.

Solve application problems using the elimination method.

Unit 14 – Learning Objectives


Instructor Guide

14.3

Lesson 3: Systems of Equations in Three or More Variables

Topic 1: Solving Systems of Three Variables

Learning Objectives

Solve a system of equations when no multiplication is necessary to eliminate a variable.

Solve a system of equations when multiplication is necessary to eliminate a variable.

Solve application problems that require the use of this method.

Recognize systems that have no solution or an infinite number of solutions.


Instructor Guide

14.4


Instructor Notes

The Mathematics of Systems of Equations and Inequalities

This unit extends the skills learned from plotting and analyzing individual equations and

inequalities into working with linear systems. Students will learn three techniques for solving

these systems—graphing, substitution, and elimination. They’ll also see how to use these

methods to analyze application problems.

Teaching Tips: Challenges and Approaches

Students are very familiar with finding a single number or a range of numbers when they solve

mathematical problems. In this unit, they'll be asked for solutions that are ordered pairs (x, y)

instead. This will be a conceptual hurdle for many of them. In order to help students build an

intuitive feel for the concept of a linear system, we start the unit by using graphing to explore

systems. Just a glance at the graph of a system reveals if the lines cross or cover one another.

We then move on to the substitution method, which many students will also be able to

understand and use without much difficulty. The idea is fairly easy to grasp, and once the

substitution is performed, all they have to do is follow the familiar steps for solving equations

and inequalities.

But students are likely to struggle with elimination (or addition method) as a means for solving

linear systems. We suggest teaching this technique last, after graphing (easiest to understand),

and substitution (most familiar mathematics).

Graphing Systems

Students learned how to graph single equations on the coordinate plane in the previous unit, so

they'll have little trouble adding a second line. The sight of two (or more) equations on a graph

conveys the idea of a linear system more clearly than words or symbols. It also helps students

understand the three possible scenarios for the solutions of linear systems:

Unit 14 – Instructor Notes


Instructor Guide

14.5

[From Lesson 1, Topic 1, Presentation]

The vocabulary that describes these different scenarios uses familiar words—

consistent/inconsistent and dependent/independent—in unfamiliar ways. This can be confusing,

so you may need to spend extra time discussing what these terms mean mathematically.

Once students are comfortable solving systems of equations with graphing, it is effective to

move on to graphing systems of linear inequalities rather than introducing other methods for

working with equations. This will strengthen their grasp of the systems concept. When graphing

inequalities, use colors to help students see where the shaded areas overlap:


Instructor Guide

14.6

[From Lesson 1, Topic 2, Topic Text]

Students tend to forget to draw dashed boundary lines when the inequality is “less than” or

“greater than”. You can help them realize the importance of this distinction by having them find

points in all four of the colored regions and on the two boundary lines of graphs like the one

above, and checking those coordinates in the inequalities.

Substitution

Most students can easily look at the graph of a system and see if the lines cross or cover one

another. It's then an easy step for them to find the solution if the problem is “nice”—the ordered

pair lies right on an intersection on the grid. But what happens if the solution isn’t easy to read

from the graph? This is the motivation behind knowing other ways to solve systems of

equations.

Solving systems of equations by substitution is probably the easiest algebraic method for

students to understand. The idea is fairly easy to grasp, and once the actual substitution is

performed, all they have to do is follow the familiar steps for solving equations and inequalities.

Without a graph to look at, students may need to be reminded that sometimes lines don't cross

and systems have no solution. Consider this example:


Instructor Guide

14.7

From Lesson 2, Topic 1, Topic Text]

Many students get to -8 = 4 and think the solution is (-8, 4). They'll catch this mistake on their

own and learn from it if you require that all ordered pair solutions be plugged into either equation

to see if they work. You can also remind them of the possibility of an infinite number of

solutions by asking them to solve the system above if the second equation is 10x – 2y = -8

instead of 10x – 2y = 4. Make sure that your students understand that when solving a system of

equations algebraically instead of graphically, there are still three possibilities: one solution, no

solution, and an infinite number of solutions.

Elimination

Students are very likely to struggle with elimination as a method for solving linear systems, but

it's important they understand this method as it will be used in later mathematics, such as

matrices, vectors, linear programming, and so forth.

Students like the elimination method when adding two equations together automatically

eliminates one of the variables. Work a number of these simple problems so students get

comfortable with the method.

Then show your students a system where neither of the variables cancels out. Here's an

example:


Instructor Guide

14.8

From Lesson 2, Topic 2, Topic Text]

Ask your students how to change one of the equations so that they could simply add the two

equations together and one of the variables would be eliminated. Hopefully someone will

suggest that if the y term in the second equation were -4y, the equations could be added and

the y’s would be eliminated. Try to get them to say that the second equation needs to be

multiplied by -4 in order to eliminate the y terms.

Once they've gotten the idea of rewriting one of the equations, ask how they would eliminate the

x variable instead of the y. Understanding this will be crucial for solving harder systems of

equations in the future.

When solving by elimination there are two specific areas students will likely find difficult:

choosing factors and keeping track of the changing equations. Choosing the appropriate factors

to multiply the equations takes both intuition and practice. It is a good idea to present students

with a number of systems with two equations and ask them how they would solve it using

elimination. In the beginning it isn’t necessary for them to even solve the system. They just

need the practice of finding what to multiply the equations by.

The other stumbling block, keeping track of what happens to equations as they are manipulated,

can be reduced by encouraging students to label their work as they go. The following layout

may be helpful:


Instructor Guide

14.9

Solve by elimination 2 3 8

3 4 46

x y

x y

Operations Equations Labels

2 3 8x y

3 4 46x y

equation 1

equation 2

multiply equation 1 by -3 6 9 24x y new equation 1

multiply equation 2 by 2 6 8 92x y new equation 2

add new equations

6 9 24x y

6 8 92x y

17 68y

simplify 4y

substitute value of y into

original equation 1 2 3(4) 8x

solve for x 10x

write x and y as co-

ordinates , 10,4x y

Solution to the

System

One common mistake that students make is that they multiply the variable terms by a factor but

forget to multiply the constant term after the equals sign by that same factor. In order to catch

this type of mistake, encourage your students to check their solution in both of the original

equations.

Three Equation Systems

Students studying intermediate algebra need to know how to solve systems with three

equations and three variables. The technique is the same as solving a system with two

equations, except that it is more involved and students are more apt to make mistakes.

Labeling the equations during this process is even more important here than it was with two

equation systems.


Instructor Guide

14.10

Students may try to eliminate the x-variable in one pair of equations and the z-variable in

another pair. Make sure they understand that they must eliminate the same variable in both

pairs of equations.

Be aware of the fact that one system of three equations with three unknowns may take a long

time for a student to work through. Because of the complexity of these problems, we suggest

giving problems that have only integer values for each of the variables in the solution. Then

when students get a fraction for one of the variables, they'll know something isn’t correct.

Usually it is easier to go back and start the problem from the beginning rather than looking back

to try and find the error (sometimes the actual error was one of copying the problem down

incorrectly).

Students know that they should check their answer and many will. Unfortunately, they will only

check it in one of the three equations. Explain to them it is possible to have their solution work

in one of the equations but not in the other two. They should be checking their solution in all

three given equations.

Keep in Mind

Although students are quite familiar with algebraic statements by this point in the course,

understanding and solving systems of multiple equations and inequalities can still be difficult.

One thing that is useful to students is to take a simple system such as

x + y = 5

x − y = 3

and solve this using all three methods of graphing, substitution, and elimination. It should be

noted that more than likely the students will know that the solution is (4, 1) without using any of

these methods! This is when you explain that it is necessary to know these methods so that

they can be used when the solution isn’t so readily apparent.

Most of the material in this unit has been geared to both beginning and intermediate students.

More difficult examples and problems included for intermediate students could be used to

challenge the beginning algebra student. However, the topics of graphing systems of

inequalities and solving systems of equations in three variables are not appropriate for

beginners.

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.


Instructor Guide

14.11

Solving systems by graphing can be found at www.wolframalpha.com. At the prompt students

can type in an equation and its graph will appear. If they input multiple equations, both the

graph and the solution to the system of equations are given. This also will work for inequalities

and systems of inequalities, but the resulting graphs do not necessarily display the familiar x-

and y- axes and dotted lines.

Practice solving two equations with two unknowns using the substitution and elimination

methods at http://www.mathsnet.net/algebra/d11.html (get additional problems on this site by

clicking on “change” or “more on this topic”).

More practice solving application problems (mixture and distance) can be found at

http://www.ltcconline.net/greenl/java/BasicAlgebra/MoneyProblems/MoneyProblems.html and

http://www.ltcconline.net/greenl/java/BasicAlgebra/DistanceRateTime/DistanceRateTime.html.

Students can enter the coefficients of x, y, and z at

http://alumnus.caltech.edu/~chamness/equation/equation.html in order to solve systems of

equations with three unknowns.

Summary

This unit teaches students how to solve systems of linear equations using graphing,

substitution, and elimination. Students will also learn how to represent systems of inequalities

on the coordinate plane. Finally, they’ll practice using these techniques to solve application

problems. To help students understand the idea of systems, we suggest beginning with

graphing. To help them become skilled at substitution and elimination, show them how to label

and organize problems and make sure they get plenty of practice.

http://www.wolframalpha.com/

http://www.mathsnet.net/algebra/d11.html

http://www.ltcconline.net/greenl/java/BasicAlgebra/MoneyProblems/MoneyProblems.html

http://www.ltcconline.net/greenl/java/BasicAlgebra/DistanceRateTime/DistanceRateTime.html

http://alumnus.caltech.edu/~chamness/equation/equation.html


Instructor Guide

14.12


Instructor Overview

Tutor Simulation: Getting the Mixture Right

Purpose

This simulation allows students to demonstrate their ability to write and solve systems of

equations in order to understand real world problems. Students will be asked to apply what they

have learned to solve a problem involving:

Writing and solving equations

Writing systems of equations

Solving systems of equations

Problem

Students are presented with the following problem:

A merchant and a scientist are both conducting experiments.

The merchant is blending two different types of teas and wants to come up with a combination

that is priced to sell. The scientist is mixing different acid solutions to come up with a desired

result.

You will work through each of the problems to determine the right mixtures to get the correct

combinations that the merchant and scientist need.

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.

Unit 14 – Tutor Simulation


Instructor Guide

14.13


Instructor Overview

Puzzle: Apples and Oranges

Objectives

Apples and Oranges is a puzzle that represents systems of equations visually, by grouping

objects in various combinations. To solve the puzzles, students must translate the images into

two or more equations, and then use substitution or elimination to find the value of each

variable. Students will need to understand how to write and analyze systems of equations to

succeed.

Figure 1. Apples and Oranges uses fruits and other objects to test a student's grasp of techniques for

solving systems of equations.

Description

This puzzle has three levels of difficulty. In each level, multiple objects are arranged in various

combinations. Players must write an equation that represents each grouping, and then use the

resulting system of equations to solve for each variable. When the player clicks on the right

answer, the algebraic equations underlying the problem and the solution are shown.

Unit 14 – Puzzle


Instructor Guide

14.14

At the easy level there are three kinds of objects (apples, oranges, and cherries) grouped into

two arrangements. The medium level also presents a two equation system, but the three objects

(opals, emeralds, and rubies) are grouped in a more complicated way. At the hard level, there

are four objects (rockets, spaceships, ray guns, and asteroids) arranged into a system of three

equations.

Players score points for solving each puzzle correctly. The ten puzzles at each level are

scripted, but they are sufficiently difficult to solve visually that there is opportunity for replay.

Apples and Oranges is suitable for both individual play and group learning in a classroom

setting.


Instructor Guide

14.15


Instructor Overview

Project: Nutty Economics

Student Instructions

Introduction

Linear systems of equations and inequalities are used widely in the fields of business and

economics to maximize profit, minimize cost, and balance the production and consumption of

goods. In fact, there are entire careers in Production and Operations Management that are built

around this skill.

Task

In this project you will play the role of consultant to a retailer of nuts. You will apply your

knowledge of systems of linear equations and inequalities to balance the production of two

types of goods in such a way that the profit will be maximized.

Instructions

Work with at least one other person to complete the following exercises. Solve each problem in

order and save your work as you progress as you will create a professional presentation in the

end.

First Problem – Defining the Quantities: The Golden Nut specialty store has obtained 60 lbs. of cashews and 50 lbs. of premium almonds at a good price from its supplier. Since the almonds are more expensive and more of a delicacy than the cashews, it is better to sell the cashews mixed with almonds in order to get a better price for them. The storeowner decides to sell two products: Hearty Nut Mix and Premium Blend. The company hires you to determine how much of each product they should mix together to maximize their profit.

o If each pound of Hearty Nut Mix contains 4 ounces of almonds and 12 ounces of cashews, write an expression that represents the number of pounds of each nut in x pounds of the mix. [Hint: What fraction of the pound are cashews, and what fraction are almonds?]

o If each pound of Premium Blend contains 8 ounces of almonds and 8 ounces of cashews, write an expression that represents the number of pounds of each nut in y pounds of the mix.

Unit 14 – Project


Instructor Guide

14.16

o Complete the following table and use it to write an expression that represents the total number of pounds of cashews used for both mixes together. Then, write an expression for the total number of almonds used for both mixes together.

Type of Mix No. of Pounds of

Mix

No. of Pounds of

Cashews

No. of Pounds of

Almonds

Hearty Nut Mix x

Premium Blend y

o Finally, your market research shows that since the Premium Blend is much more popular, you can afford to charge quite a bit more for it, obtaining a profit of $0.45 per pound for it while you can only earn a profit of $0.30 per pound for the Hearty Nut Mix. Write an expression for the amount of profit that you would make from selling x pounds of Hearty Nut Mix and y pounds of Premium Blend.

Second Problem – Constraints: It might seem like selling all Premium Blend, since it obtains a better profit, might give the most profit. But this is not necessarily so. Explain why. [Hint: Remember that we are not going to sell cashews or almonds by themselves at this point.]

o You have a limited amount of each type of nut. This places a constraint on the

problem that we must account for. Write an inequality that expresses the fact that the total number of cashews cannot be greater than 60 lbs. and the total number of almonds cannot be greater than 50 lbs. [Hint: Use the table from the first problem above.]

o Write down two other inequalities that represent constraints on the variables x and y. [Hint: What types of values for x and y make sense considering that they represent pounds? These will be simple inequalities.]

Third Problem – Possibilities: The next step in the process is to graph the inequalities and determine the points that satisfy all of them at once, because these are the only values of x and y that we can realistically use with our supply of nuts. Sketch all four inequalities on the same graph and shade in the region that represents those points that satisfy all of the inequalities.

Fourth Problem – Maximizing Profit: We are now ready to determine the maximum profit.


Instructor Guide

14.17

o The points in the region you graphed represent all possible x and y values that we would be allowed to use to maximize our profit. This is a lot of points to check just to find the maximum profit! Fortunately, we can eliminate many of them. It turns out that both the maximum and minimum profit will occur on the edges of this region (in fact, on the extreme points of the edge—the corners). Therefore, compute the x and y coordinates of each of the corners of your region. This will require you to solve some systems of equations.

o Since we know the maximum (and minimum) profit must occur at one of these corners, plug the x and y values for each corner into the expression for profit from the first problem, and record your answers in the table below. Then, identify the maximum profit and tell how many pounds of each mix should be made. [Note: The table below may contain more rows than you need to complete the problem.]

Expression for Profit:_________________________________

Corner (x,y) Profit

o By completing the following chart, determine how many nuts are left unmixed.


Instructor Guide

14.18

Mix Pounds Made Pounds of Cashews

in the Mix

Pounds of Almonds

in the Mix

Hearty Nut Mix

Premium Blend

Total Pounds of

Cashews Used=

Total Pounds of

Almonds Used=

Conclusions

Compare your expressions, computations, and the graph with those from another group. Work

to make sure that your explanation is clear and concise.

Prepare a final report that clearly identifies and explains

the quantities involved in the problem,

the expression for profit,

the constraints,

the region of possible x and y values,

the possible combination of mixes that could result in maximum profit, and

the maximum profit and the number of pounds of each mix that should be made.

Finally, present your solution to the class.

Instructor Notes

Assignment Procedures

Problem 1

Since three quarters of the mix are cashews and one quarter is almonds, then there are 3

x4

pounds of cashews and 1

x4

pounds of almonds in x pounds of the mix.


Instructor Guide

14.19

Since each type of nut comprises half of the mix, then there are 1

y2

pounds of cashews and

1y

2 pounds of almonds in y pounds of the mix.

The answers are in the table below. Accordingly, then, the total number of pounds of cashews

used is 3 1

x+ y4 2

, and the total number of pounds of almonds used is 1 1

x+ y4 2

.

Type of Mix No. of Pounds of

Mix

No. of Pounds of

Cashews

No. of Pounds of

Almonds

Hearty Nut Mix x 3

x4

1

x4

Premium Blend y 1

y2

1

y2

The profit would be 0.30x+0.45y.

Problem 2

If we did sell all Premium Blend, we would have an excess of cashews (10 lbs.) that we didn’t

sell in a mix. The problem states that we obtain a better price for them when they are mixed

with almonds. We could possibly increase our profit by making some Hearty Nut Mix in order to

sell more of our cashews in the mixes. This is a subtle point that requires some good intuition

on the part of the student. In the end, whether this would actually increase profit depends

largely on how different the profit margin is between the two mixes. If the profit on the Premium

Blend is significantly higher than that for the Hearty Nut Mix, then it may pay to sell all Premium

Blend and leave some cashews unmixed.]

Using the results in the table above, we find that 3 1

x+ y 604 2

and 1 1

x+ y 504 2

.

Since neither x nor y can be negative, we obtain x 0 and y 0 .

Problem 3

The region is shown in the figure below. In the field of Linear Programming, this region would be

called the feasible region, since it contains all allowable x and y values.


Instructor Guide

14.20

Problem 4

The coordinates are (x,y)=(0,0), (80,0), (0,100), and (20,90).

The answers are in the table below. The maximum profit attainable is $46.50, and we should

make 20 pounds of the Hearty Nut Mix and 90 pounds of the Premium Blend.

As an optional exercise that ties in with the first question under the second problem, you may

wish to have students re-examine the total profit if the profit on the Premium Blend is above

$0.60 per pound.


Instructor Guide

14.21

Expression for

Profit:__0.30x+0.45y______________________

Corner (x,y) Profit

(0,0) $0.00

(80,0) $24.00

(0,100) $45.00

(20,90) $46.50

As shown in the table below, we efficiently use all of each type of nut.

Mix Pounds Made

Pounds of

Cashews in the

Mix

Pounds of

Almonds in the

Mix

Hearty Nut Mix 20 3

20 154 pounds

120 5

4 pounds

Premium Blend 90 1

90 452 pounds

190 45

2 pounds

Total Pounds of

Cashews Used=

60 pounds

Total Pounds of

Almonds Used=

50 pounds


Instructor Guide

14.22

Recommendations

have students work in teams to encourage brainstorming and cooperative learning.

assign a specific timeline for completion of the project that includes milestone dates.

provide students feedback as they complete each milestone.

ensure that each member of student groups has a specific job.

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.

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.

http://www.moodle.org/

http://www.wikispaces.com/site/for/teachers

http://pbworks.com/content/edu+overview

http://www.docs.google.com/

http://why.openoffice.org/


Instructor Guide

14.23

Rubric

If you had your students complete the optional exercise suggested in the fourth problem, you

may wish to add to the following rubric a requirement that students explain, intuitively, why it

makes sense that selling all Premium Blend and leaving some cashews unmixed would be

advantageous in that case.

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 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.

• 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 for the problem.

• Your project contains low quality graphics and colors that do not add interest to the project.


Instructor Guide

14.24

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.


Instructor Guide

14.25


Common Core Standards

Unit 14, Lesson 1, Topic 1: Graphing Systems of Linear Equations

Grade: 8 - Adopted 2010

STRAND / DOMAIN CC.MP. Mathematical Practices

CATEGORY / CLUSTER MP.1. Make sense of problems and persevere in solving them.

STRAND / DOMAIN CC.8.EE. Expressions and Equations

CATEGORY / CLUSTER Analyze and solve linear equations and pairs of simultaneous

linear equations.

STANDARD 8.EE.8. Analyze and solve pairs of simultaneous linear equations.

EXPECTATION 8.EE.8(a) Understand that solutions to a system of two linear equations in

two variables correspond to points of intersection of their graphs,

because points of intersection satisfy both equations

simultaneously. (SBAC Summative Assessment Target: 1.06, 2.02,

3.03, 4.01)

EXPECTATION 8.EE.8(b) Solve systems of two linear equations in two variables

algebraically, and estimate solutions by graphing the equations.

Solve simple cases by inspection. For example, 3x + 2y = 5 and 3x +

2y = 6 have no solution because 3x + 2y cannot simultaneously be

5 and 6. (SBAC Summative Assessment Target: 1.06, 2.02, 3.03,

4.01)

EXPECTATION 8.EE.8(c) Solve real-world and mathematical problems leading to two linear

equations in two variables. For example, given coordinates for two

pairs of points, determine whether the line through the first pair

of points intersects the line through the second pair. (SBAC

Summative Assessment Target: 1.06, 2.02, 3.03, 4.01)

Grade: 9-12 - Adopted 2010



STRAND / DOMAIN CC.A. Algebra

CATEGORY / CLUSTER A-CED. Creating Equations

STANDARD Create equations that describe numbers or relationships.

Unit 14 – Correlation to Common Core Standards

Learning Objectives


Instructor Guide

14.26

EXPECTATION A-CED.3. Represent constraints by equations or inequalities, and by systems

of equations and/or inequalities, and interpret solutions as viable

or nonviable options in a modeling context. For example,

represent inequalities describing nutritional and cost constraints

on combinations of different foods.


CATEGORY / CLUSTER A-REI. Reasoning with Equations and Inequalities

STANDARD Solve systems of equations.

EXPECTATION A-REI.6. Solve systems of linear equations exactly and approximately (e.g.,

with graphs), focusing on pairs of linear equations in two

variables.



STANDARD Represent and solve equations and inequalities graphically.

EXPECTATION A-

REI.11.

Explain why the x-coordinates of the points where the graphs of

the equations y = f(x) and y = g(x) intersect are the solutions of the

equation f(x) = g(x); find the solutions approximately, e.g., using

technology to graph the functions, make tables of values, or find

successive approximations. Include cases where f(x) and/or g(x)

are linear, polynomial, rational, absolute value, exponential, and

logarithmic functions.

Unit 14, Lesson 1, Topic 2: Graphing Systems of Inequalities






linear equations.












Instructor Guide

14.27










STANDARD Represent and solve equations and inequalities graphically.

EXPECTATION A-

REI.12.

Graph the solutions to a linear inequality in two variables as a half-

plane (excluding the boundary in the case of a strict inequality),

and graph the solution set to a system of linear inequalities in two

variables as the intersection of the corresponding half-planes.

Unit 14, Lesson 2, Topic 1: The Substitution Method






linear equations.







4.01)












Instructor Guide

14.28












variables.

Unit 14, Lesson 2, Topic 2: The Elimination Method






linear equations.







4.01)













Instructor Guide

14.29









EXPECTATION A-REI.5. Prove that, given a system of two equations in two variables,

replacing one equation by the sum of that equation and a multiple

of the other produces a system with the same solutions.



variables.

Unit 14, Lesson 3, Topic 1: Solving Systems of Three Variables






linear equations.







4.01)











Instructor Guide

14.30













variables.

Developmental Math Instructor Guide€¦ · move on to graphing systems of linear inequalities...

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