Time Frame: Approximately five weeks
Unit Description
In this unit, linear equations are considered in tandem. Solutions to systems of two linear equations are represented using graphical methods, substitution, and elimination. Matrices are introduced and used to solve systems of two and three linear equations with technology. Heavy emphasis is placed on the reallife applications of systems of equations. Graphs of systems of inequalities are represented in the coordinate plane.
Student Understandings
Students state the meaning of solutions for a system of equations and a system of inequalities. In the case of linear equations, students use graphical and symbolic methods of determining the solutions. Students use methods such as graphing, substitution, elimination or linear combinations, and matrices to solve systems of equations. In the case of linear inequalities in two variables, students to see the role played by graphical analysis.
Guiding Questions

Can students explain the meaning of a solution to a system of equations or inequalities?

Can students determine the solution to a system of two linear equations by graphing, substitution, elimination, or matrix methods (using technology)?

Can students use matrices and matrix methods by calculator to solve systems of two or three linear equations Ax = B as x = A^{1}B?

Can students solve realworld problems using systems of equations?

Can students graph systems of inequalities and recognize the solution set?
Unit 5 GradeLevel Expectations (GLEs)
GLE #

GLE Text and Benchmarks

Algebra

11.

Use equivalent forms of equations and inequalities to solve reallife problems (A1H)

12.

Evaluate polynomial expressions for given values of the variable (A2H)

14.

Graph and interpret linear inequalities in one or two variables and systems of linear inequalities (A2H) (A4H)

15.

Translate among tabular, graphical, and algebraic representations of functions and reallife situations (A3H) (P1H) (P2H)

16.

Interpret and solve systems of linear equations using graphing, substitution, elimination, with and without technology, and matrices using technology (A4H)

Geometry

23.

Use coordinate methods to solve and interpret problems (e.g., slope as rate of change, intercept as initial value, intersection as common solution, midpoint as equidistant) (G2H) (G3H)

Patterns, Relations, and Functions

39.

Compare and contrast linear functions algebraically in terms of their rates of change and intercepts (P4H)

Activity 1: Systems of Equations (GLEs: 15, 16, 23)
Materials List: paper, pencil, Graphing Systems of Equations BLM, graphing calculator
Use the Graphing Systems of Equations BLM to work through this activity with students.
Have students read the scenario on the BLM to visualize two people are walking in the same direction at different rates, with the faster walker starting out behind the slower walker. At some point, the faster walker will overtake the slower walker.
Suppose that Sam is the slower walker and James is the faster walker. Sam starts his walk and is walking at a rate of 1.5 mph, and one hour later James starts his walk and is walking at a rate of 2.5 miles per hour.
Ask the students how to use graphs to determine where and when James will overtake Sam. Review with the students the distance = rate time relationship and guide them to the establishment of an equation for both Sam and James (Sam’s equation should be , and James’ equation should be ). Have students graph each equation and find the point of intersection (2.5, 3.75).
Lead the students to the discovery that two and onehalf hours after Sam started, James would overtake him. They both would have walked 3.75 miles. Show the students that the goal of the process is to find a solution that makes each equation true, and that is the solution to the system of equations. Lead students to write a definition of a system of equations.
Continue using the BLM to present reallife examples to show when a system of equations might have no solution or many solutions. Give the students a number of problems involving 2 2 systems of equations, and have them use a graphing calculator to solve them graphically. Emphasize that the solution of a system is the point(s) where the graphs intersect and that the point(s) is (are) the common solution(s) to both equations.
Using an algebra textbook as a reference, provide opportunities for students to practice solving systems of equations by graphing. Include systems with one solution, no solutions, and infinite number of solutions.
Activity 2: Battle of the Sexes (GLEs: 11, 15, 16, 23, 39)
Materials List: paper, pencil, Battle of the Sexes BLM, graphing calculator
Have students use the Battle of the Sexes BLM to complete this activity. The BLM provides students with the following Olympic data of the winning times for men and women’s 100meter freestyle. Have students create scatter plots and find the equation of the line of best fit for each set of data either by hand or with the graphing calculator(men: y = 0.167x + 64.06, women: y = 0.255x+77.23). Have students find the point of intersection of the two lines and explain the significance of the point of intersection. (The two lines of best fit intersect leading to the conclusion that eventually women will be faster than men in the 100Meter Freestyle.) Also have students compare the two equations in terms of the rates of change. (i.e. How much faster are the women and the men each year?)
Activity 3: Substitution (GLEs: 11, 12, 15, 16, 23, 39)
Materials List: paper, pencil, graph paper, calculator
Begin by reviewing the process for solving systems of equations graphically. Inform the students that it is not always easy to find a good graphing window that allows the determination of points of intersection from observation. Show them an example of a system that is difficult to solve by graphing. Explain that there are other methods of finding solutions to systems and that one such method is called the substitution method. The following example might prove useful in modeling the substitution method.
Alan Wise runs a red light while driving at 80 kilometers per hour. His action is witnessed by a deputy sheriff, who is 0.6 kilometer behind him when he ran the light. The deputy is traveling at 100 kilometers per hour. If Alan will be out of the deputy’s jurisdiction in another 5 kilometers, will he be caught?
Lead the students through the process of determining the system of equations that might assist in finding the solution to the problem. Using the relationship distance = rate time, where time is given in hours and distance is how far he is from the traffic light in kilometers, show the students that Alan’s equation can be described as . The equation for the deputy then would be . Show the students that the right member of the deputy’s equation can be substituted for the left member of Alan’s equation to achieve the equation . Solve the equation for t, and a solution of 0.03 would be determined. Substituting back into either or both of the equations, the value of d will be found to be 2.4 kilometers. The point common to both lines is (0.03, 2.4). Because the 2.4 kilometers is less than 5, Alan is within the deputy’s jurisdiction and will get a ticket.
Have students use splitpage notetaking (view literacy strategy descriptions) as the students work through the process of substituting to solve a system of equations. They should perform the calculations on the left side of the page and write the steps that they follow on the right side of the page. A sample of what splitpage notetaking might look like in this situation is shown below.
2x + y = 10
5x – y = 18

Solve one equation for either x or y.

2x + y = 10
2x 2x
y = 10 – 2x

Substitute that equation into the other equation for the solved variable

5x – (10 – 2x) = 18

Solve for the remaining variable

5x – 10 + 2x = 18
7x – 10 = 18
+ 10 + 10
7x = 28
x = 4

Substitute your answer for the variable in either of the original equations

2(4) + y = 10

Solve for the remaining variable

8 + y = 10
8 8
y = 2

Answer is _(4, 2)________

Using an algebra textbook as a reference, provide additional practice problems where the students can use the substitution method to solve systems. Work with students individually and in small groups to ensure mastery of the process. Demonstrate for students how they can review their notes by covering information in one column and using the information in the other try to recall the covered information. Students can quiz each other over the content of the splitpage notes in preparation for quizzes and other class activity.
Activity 4: Elimination (GLEs: 11, 12, 15, 16, 23, 39)
Materials List: paper, pencil, calculator
Begin by reviewing the process for solving systems of equations graphically and by substitution. Inform the students that there is another method of solving systems of equations that is called elimination. Write an equation and review the addition property of equality. Show that the same number can be added to both sides of an equation to obtain an equivalent equation. Then introduce the following problem:
A newspaper from Central Florida reported that Charles Alverez is so tall he can pick lemons without climbing a tree. Charles’s height plus his father’s height is 163 inches, with a difference in their heights of 33 inches. Assuming Charles is taller than his father, how tall is each man?
Work with the students to establish a system that could be used to find Charles’s height. Let x represent Charles’s height and y represent his father’s height and write the two equations and . Show the students that the sum of the two equations would yield the equation , which would indicate that Charles’ height is 98 inches (8 ft. 2 in.) tall. Through substitution, the father’s height could then be determined.
Have students use splitpage notetaking (view literacy strategy descriptions) as they work through the process of using elimination to solve a systems of equations. They should perform the calculations on the left side of the page and write the steps that they follow on the right side of the page. A sample of what splitpage notetaking might look like in this situation is shown below. Again, remember to encourage students to review their completed notes by covering a column and prompting their recall using the uncovered information in the other column. Also allow students to quiz each other over the content of their notes.
4x – 3y = 18
3x + y = 7

Make the coefficients of either x or y opposites of each other by multiplying the entire equation

3(3x + y) = 7(3)
9x + 3y = 21

In this equation, this multiplication will make the y’s opposites of each other

4x – 3y = 18
9x + 3y = 21
13x = 39

Add the two equations together eliminating one of the variables

x = 3

Solve for the variable

4(3) – 3y = 18

Substitute your answer for the variable in either of the original equations

12 – 3y = 18
12 12
3y = 6
y = 2

Solve for the remaining variable

(3, 2)

Answer

Continue to show examples that use the multiplication property of equality to establish equivalent equations where like terms in the two equations would add to zero and eliminate a variable. Use an algebra textbook to provide opportunities for students to practice solving systems of equations using elimination including realworld problems.
Activity 5: Supply and Demand (GLEs: 11, 15, 16, 23)
Materials List: paper, pencil, blackline masters from NCTM website (see link below), calculator
This activity can be found on National Council of Teachers of Mathematics website (http://illuminations.nctm.org/index_d.aspx?id=382). Blackline masters can be printed from the website for student use. Students investigate and analyze supply and demand equations using the following data obtained by the BurgerRama restaurant chain as they are deciding to sell a cartoon doll at its restaurants and need to decide how much to charge for the dolls.
Selling Price of
Each Doll

Number Supplied
per Week per Store

Number Requested
per Week per Store

$1.00

35

530

$2.00

130

400

$4.00

320

140

Have students plot points representing selling price and supply and selling price and demand on a graph. Have students estimate when supply and demand will be in equilibrium. Then have students find the equation of each line and solve the system of equations algebraically to find the price in exact equilibrium. (, price in equilibrium, $3.20)
In their math learning logs (view literacy strategy descriptions) have students respond to the following prompt:
Explain the reasons why supply and demand must be in equilibrium in order to maximize profits. How does using a system of equations help us to find the price in equilibrium? Do you believe that being able to solve a system of equations would be a good skill for a business owner to have? Justify your opinion.
Have students share their answers with the class and conduct a class discussion of the accuracy of their answers.
Activity 6: Introduction to Matrices (GLE: 16)
Materials List: paper, pencil, Introduction to Matrices BLM, graphing calculator
This activity provides an introduction to the use of matrices in reallife situations and provides opportunities for students to be familiarized with the operations on matrices before using them to solve systems of equations. Guide students through the activity using the Introduction to Matrices BLM
The BLM provides students with the following charts of electronic sales at two different store locations:
Store A Store B

Jan.

Feb.

Mar.


Jan.

Feb.

Mar.

Computers

55

26

42

Computers

30

22

35

DVD players

28

26

30

DVD players

12

24

15

Camcorders

32

25

20

Camcorders

20

21

15

TVs

34

45

37

TVs

32

33

14

Explain to students that these two charts can be arranged in a rectangular array called a matrix. The advantage of writing the numbers as a matrix is that the entire array can be used as a single mathematical entity. Have the students write the charts as matrix A and matrix B as such:
Discuss with students the dimensions of the matrices. (Both matrices are 4 3 matrices because they have 4 rows and 3 columns) Tell students that each matrix can be identified using its dimensions (i.e.,). Provide examples of additional matrices for students to name using the dimensions.
Ask students how they might find the total sales of each category for both stores.
Have students come up with suggestions and lead them to the conclusion that when adding matrices together, they should add the corresponding elements. Lead them to discover that two matrices can be added together only if they are the same dimensions. Provide a question for subtraction such as: How many more electronic devices did Store A sell than Store B?
Also provide a question for scalar multiplication such as this: Another store, Store C, sold twice the amount of electronics as Store B. How much of each electronic device did it sell? (Scalar multiplication is multiplying every element in Matrix B by 2)
All of the operations in this activity should be shown using paper and pencil and using a graphing calculator.
Using an algebra textbook as a reference, provide students with other examples of reallife applications of matrices and have them perform addition, subtraction, and scalar multiplication.
Activity 7: Multiplying matrices (GLE: 16)
Materials List: paper, pencil, Matrix Multiplication BLM, graphing calculator
Use the Matrix Multiplication BLM to guide students through this activity. The BLM provides students with the following charts of Tshirt sales for a school fundraiser and the profit made on each shirt sold.
Number of shirts sold Profit per shirt

Small

Medium

Large


Profit

Art Club

52

67

30

Small

$5.00

Science Club

60

77

25

Medium

$4. 25

Math Club

33

59

22

Large

$3.00

Have students write a matrix for each chart. Then have them discuss how to calculate the total profit that each club earned for selling the Tshirts. As students come up with ways to calculate, lead them to the process of multiplying two matrices together. For example:
Provide students with one more example for them to try using pencil and paper. Then have them use the graphing calculator to multiply matrices of various dimensions. Provide students with examples that cannot be multiplied, and have them discover the rule that in order to multiply two matrices together, their inner dimensions must be equal.
Activity 8: Solving Systems of Equations with Matrices (GLE: 16)
Materials List: paper, pencil, Solving Systems of Equations Using Matrices BLM, Word Grid BLM, graphing calculator
Use the Solving Systems of Equations Using Matrices BLM to guide students through this activity. Have students multiply the following two matrices: The result is .
Discuss with students that if they are given then the following system of equations would result: .
Conversely, any system of equations can be written as a matrix multiplication equation.
Using technology, matrices provide an efficient way to solve equations, especially multiple equations having many variables. This is true because in any system of equations written as matrix multiplication, Ax = B, the equation can be solved for x as, where matrix A is the coefficient matrix, , and matrix B is the constant matrix, . Use the questions and statements on the BLM to lead students to the conceptual understanding of the reason for using [A]^{1}[B] to solve systems of equations using matrices on the graphing calculator.
Have students enter matrix A and matrix B into the calculator and type [A]^{1}[B] on the home screen. The resulting matrix will be which means x = 4 and y = 4. Repeat this activity with 3 x 3 systems of equations.
Have students use a modified word grid (view literacy strategy descriptions) to determine how to find whether a system of equations has one solution, no solution, or an infinite number of solutions. A word grid provides students with an organized framework for learning through analysis of similarities and differences of key features among a related group of terms or concepts. Give students the Word Grid BLM. Guide the students to fill in the grid with information about how they can tell if a system of equations has the given number of solutions when using each solution method.
Once the grid is complete, quiz students on the similarities and differences of determining the number of solutions using each of the solution methods. Promote a discussion of how the word grid could be used as a study tool to determine the number of solutions of a system of equations.
Activity 9: Systems of Inequalities (GLE: 14)
Materials List: paper, pencil, graph paper, colored pencils
Review graphing inequalities in two variables. Present the following problem to students:
Suppose you receive a $120 gift certificate to a music and book store for your birthday. You want to buy some books and at least 3 CDs. CDs cost $15 and books cost $12. What are the possible ways that you can spend the gift certificate?
Have students use a system of inequalities to find the possible solutions and to graph the three inequalities for the problem. () Have them use different colored pencils or different shading techniques for each inequality. Ask students to explain the significance of the overlapping shaded region. Have them give the possible ways that they can spend the gift certificate.
Provide students with other realworld problems that can be solved using systems of linear inequalities.
Activity 10: Name that solution (GLE: 14)
Materials List: paper, pencil, transparency of any system of inequalities, large note cards
Divide students into groups of 3 or 4. Show students the graph of a system of inequalities on a coordinate grid transparency. Give each group a set of 4 cards, one with the correct system of inequalities, one with each inequality that makes up the system, and one with the word none on it. Call out ordered pairs and let each group decide if that ordered pair is a solution to the system, to either inequality, or to none of them. When a group consensus is reached, have one person from each group hold up the card with the correct answer.
Sample Assessments
General Assessments 
Portfolio assessment: On the first day of the new unit, give the student an application problem that can be solved using a system of equations. As each new method of solving systems of equations is introduced, the student will solve the problem using the method learned.

The student will solve constructed response items, such as this:
Prestige Car Rentals charges $44 per day plus $.06 per mile to rent a midsized vehicle. Getaway Auto charges $35 per day plus $.09 per mile for the same car.

Write a system of linear equations representing the prices for renting a car for one day at each company. Identify the variables used. (Prestige: , Getaway: )

Solve the system of equations graphically and algebraically. (, )

Suppose you need to rent a car for a day. Which company would you rent from? Justify your answer. (Prestige, if you were driving more than 300 miles and Getaway, if you were driving less than 300 miles.)

The student will solve a 2 2 or 3 3 system of equations using a graphing calculator and check the solution by hand.

The student will create a system of inequalities whose solution region is a polygon.

The student will complete entries in his/her math learning logs using such topics as:

Describe four methods of solving systems of equations. When would you use each method?

What is the purpose of using multiplication as the first step when solving a system using elimination?

Describe two ways to tell how many solutions a system of equations has.

Describe a linear system that you would prefer to solve by graphing. Describe another linear system that you would prefer to solve using substitution. Provide reasons for your choice.

How is solving a system of inequalities like solving a system of equations? How is it different?

The student will pose and solve problems that require a system of two equations in two unknowns. The student will be able to solve the system using any of the methods learned.
ActivitySpecific Assessments

Activity 2: The student will solve constructed response items such as this:
The table shows the average amounts of red meat and poultry eaten by Americans each year.

Year

1970

1975

1980

1985

1990

Red meat

152 lb

139 lb

146 lb

141 lb

131 lb

Poultry

48 lb

50 lb

60 lb

68 lb

91 lb


Create scatter plots for the amounts of red meat and poultry eaten.

Find the equation of the lines of best fit. (Red meat , Poultry: )

Does the data show that the average number of pounds of poultry eaten by Americans will ever equal the average number of pounds of red meat eaten? Justify your answer. (Yes, in the year 2007)

Activity 5: The student will solve constructed response items such as this:
The data provided in the table below show the supply and demand for game cartridges at a toy warehouse.

Price

Supply

Demand

$20

150

500

$30

250

400

$50

450

200





Find the supply equation. ()

Find the demand equation. ()

Find the price in equilibrium. ($37.50)
Justify each of your answers.

Activity 10: Given the graph to a system of inequalities, the student will list three points that are solutions to the system, to each inequality, and to none of the inequalities.
Algebra 1
Unit 6: Measurement
Time Frame: Approximately three weeks
Unit Description
This unit is an advanced study of measurement. It includes the topics of precision and accuracy and investigates the relationship between the two. The investigation of absolute and relative error and how they each relate to measurement is included. Significant digits are also studied as well as how computations on measurements are affected when considering precision and significant digits.
Student Understandings
Students should be able to find the precision of an instrument and determine the accuracy of a given measurement. They should know the difference between precision and accuracy. Students should see error as the uncertainty approximated by an interval around the true measurement. They should be able to calculate and use significant digits to solve problems.
Guiding Questions

Can students determine the precision of a given measurement instrument?

Can students determine the accuracy of a measurement?

Can students differentiate between what it means to be precise and what it means to be accurate?

Can students discuss the nature of precision and accuracy in measurement and note the differences in final measurement values that may result from error?

Can students calculate using significant digits?
Unit 6 GradeLevel Expectations (GLEs)
GLE #
 GLE Text and Benchmarks  Number and Number Relations 
4.

Distinguish between an exact and an approximate answer, and recognize errors introduced by the use of approximate numbers with technology (N3H) (N4H) (N7H)

5.

Demonstrate computational fluency with all rational numbers (e.g., estimation, mental math, technology, paper/pencil) (N5H)

Measurement

17.

Distinguish between precision and accuracy (M1H)

GLE #
 GLE Text and Benchmarks 
18.

Demonstrate and explain how the scale of a measuring instrument determines the precision of that instrument (M1H)

19.

Use significant digits in computational problems (M1H) (N2H)

20.

Demonstrate and explain how relative measurement error is compounded when determining absolute error (M1H) (M2H) (M3H)

21.

Determine appropriate units and scales to use when solving measurement problems (M2H) (M3H) (M1H)

