Analyze, Model Real-World Problems: Supply & Demand: Expenses V Income

• MA-HS-AT-S-EI6: Students will solve systems of two linear equations in two variables.
• MA-HS-AT-S-PRF6: Students will interpret representations of functions of two variables.
• MA-HS-AT-S-EI12: Students will relate a solution of a system of two linear equations in two variables and the graphs of the corresponding lines.
• MA-HS-AT-S-VEO1: Students will write expressions, equations, inequalities and relations in equivalent forms.
• MA-HS-AT-S-EI2: Students will draw reasonable conclusions about a situation being modeled.

• Journal /Admit Slip-“What is a linear equation? How can a linear function model a real-world problem?”
• Using journal/admit slips discuss linear equations; emphasize constant rate of change/slope, y-intercept, and applications of each
• Students generate different examples of linear equations and discuss characteristics

• Students listen to a “read aloud” of real-world problem
• Students represent the problem algebraically, graphically while making notes in the double entry journal. working example on left column reacting to transformations on right hand column
• Students analyze the graphical representation of the problem
• Identify different parts of a linear system using the example
• Students identify/analyze different types of linear systems graphically

• “Describe a linear system that you have experience with or describe the characteristics that make expenses compared to income a linear system?” or “How does a linear system differ from a linear equation?”

• Group conversations and questions
• Double entry journals
• Reflections

RESOURCES &

MATERIALS

• Real-world examples of linear system problems
• Graphing overhead/Graph on smartboard
• Double Entry Journal template
• Suggested: Graphing Calculator
• Suggested Option Computer Resources

2008 Hybrid Algebra Research Project

Quilt Company: Expenses vs. Income

Simple Interest Accounts

Acid Solutions

(Overhead)

2008 Hybrid Algebra Research Project

• MA-HS-AT-S-EI6: Students will solve systems of two linear equations in two variables.
• MA-HS-AT-S-EI12: Students will relate a solution of a system of two linear equations in two variables and the graphs of the corresponding lines.
• MA-HS-AT-S-VEO1: Students will write expressions, equations, inequalities and relations in equivalent forms.
• MA-HS-AT-S-EI5: Students will solve an equation involving several variables for one variable in terms of the others.

• Admit Slip- “Write the equation 2x + y = 4 in terms of one of the variables, explain why you chose to solve for the variable you chose.”or “Explain how in the function f(x) = 2x – 3, f(x) is written in terms of x.”
• Review admit slip in class to set the stage for substitution
• Review vocabulary words necessary for understanding/representing equations in multiple forms.

• Students discuss an example system that is represented graphically, creating the equations from the graph.
• Students rewrite the equations in standard form with integer coefficients
• Student pairs discuss: why/how multiple representations of the equations are the same/different, reasons for multiple representations of the equations, value of representing an equation in terms of one variable, whole class share-out
• Solve the system using substitution allowing students to lead the process where they are able
• Student pairs discuss the process of solving a system using substitution, whole class share-out
• Solve another system using substitution with the help of the class
• Student pairs solve a system using substitution
• Students discuss different systems that might create special issues

• “For the system 2x – 3y = 6 and y + 2x = 8, in which equation and for which variable would you solve to start the substitution method process? Why? (You do not have to actually solve)

• Group conversations and questions
• Double entry journals
• Reflections

RESOURCES &

MATERIALS

• Examples resource
• Double Entry Journal template

2008 Hybrid Algebra Research Project

Linear Systems Examples for use with Substitution

2008Hybrid Algebra Research Project

• MA-HS-AT-S-EI5: Students will solve an equation involving several variables for one variable in terms of the others.
• MA-HS-AT-S-VEO1: Students will write expressions, equations, inequalities and relations in equivalent forms.
• MA-HS-AT-S-EI6: Students will solve systems of two linear equations in two variables.

• Journal/Admit slip: Starter “problem of the day” y = 2x, 6x – 3y = 0, -2x + y = 0, when you come in for the POD, describe the relationships between these equations.
• Discuss student journal/problem of the day to begin creating a list of characteristics that help determine what students already understand about equivalent equations.
• Have students perform any mathematical manipulations that they can to prove their observations.

• Have students work through problem #?of the warm-up, and provide them the answers for the other three problems. (Solutions: 1 (4,4), 2 (3, 4), 3 (7,6), 4 (3,7))
• Work through lesson using attached note-taking guide/double entry organizer. Set up the expectation that there will be a class discussion at the end of the lesson around the five text-codes used for note-taking or the ‘right hand’ column from the dbl. entry organizer
• Discussion about main points summarizing of material from the notes in whole group setting.
• Intentional modeling of notes using LCD display
• If conversation stalls, take conversation to specific points in the note-taking tool or to text-codes (i.e. Using your text codes, what did you identify as important? what did you identify as making connections to what we’ve studied previously? what questions did you have?)

• Using the big ideas sheet, students capture their understanding of the big ideas for the unit, in an on-going dialogue with the teacher about what they understand about linear systems

• Journal/Admit slip
• Note-taking device

• Group conversations
• Reflections/big ideas- collect to provide feedback and hand back the next day
• HW problems 1-5 of the homework

RESOURCES &

MATERIALS

• Individual student note-taking guide
• Homework problems (printed copies for those unable to access from internet)
• Extension Optional Computer Resource (ties into the standard form of the equation)

• MA-HS-AT-S-EI5: Students will solve an equation involving several variables for one variable in terms of the others.
• MA-HS-AT-S-VEO1: Students will write expressions, equations, inequalities and relations in equivalent forms.
• MA-HS-AT-S-EI6: Students will solve systems of two linear equations in two variables.

• Journal/Admit slip: Starter “problem of the day” Look through the Big Ideas feedback and make notes of any points that you need to reconsider/reflect on/explore”
• Find the equations of lines from the graphed system of linear equations. See Attachment A

• Review Homework having selected students work the problems on the board showing their work, (if smartboard/pentab available be sure to have two students show their work on the computer so the problems can be saved as examples for later review
• Think aloud as students go through the explanation of their problem, (i.e. “Oh I’m glad you showed us that I didn’t catch that the coefficients of the x term were opposite of each other because it wasn’t in standard form correctly. I have to be sure to look for that.”)

• Work through lesson using attached note-taking guide/double entry organizer. Set up the expectation that there will be a class discussion at the end of the lesson around the five text-codes used for note-taking or the ‘right hand’ column from the dbl. entry organizer
• Discussion about main points summarizing of material from the notes in whole group setting.
• Intentional modeling of notes using LCD display or overhead with a transparency.
• If conversation stalls, take conversation to specific points in the note-taking tool or to text-codes (i.e. Using your text codes, what did you identify as important? what did you identify as making connections to what we’ve studied previously? what questions did you have?)
• Assign HW 1, 3, 5**, 15- (**Great problem to explore to see if students solved in more than one way)

• Using the big ideas sheet, students capture their understanding of the big ideas for the unit, in an on-going dialogue with the teacher about what they understand about linear systems, focusing on linear combination today

• Journal/Admit slip
• Note-taking device

• Group conversations
• Reflections/big ideas- collect to provide feedback and hand back the next day
• HW

RESOURCES &

MATERIALS

• Individual student note-taking guide
• Homework problems (printed copies for those unable to access from internet)
• Extension Optional Computer Resource (ties into the standard form of the equation), (additional notes/examples, more direct instruction oriented)

2008Hybrid Algebra Research Project

2008Hybrid Algebra Research Project

Attachment A

2008Hybrid Algebra Research Project

• MA-HS-AT-S-EI5: Students will solve an equation involving several variables for one variable in terms of the others.
• MA-HS-AT-S-VEO1: Students will write expressions, equations, inequalities and relations in equivalent forms.
• MA-HS-AT-S-EI6: Students will solve systems of two linear equations in two variables.

• Journal/Admit slip: “Using the Big Idea sheet put together a list of the main points about linear systems that you’ve learned so far; work with a partner to identify points that you both felt were important, points that only one of you found important, and any questions that you might have about systems still rolling around in your head.”
• Discuss student journal/problem of the day to begin creating a list of characteristics that help determine which methods to utilize to solve the system

• Review HW being sure to monitor your time. this is a packed lesson.
• Students work in small groups on chart paper to solve problems
• Students solve 6-10 linear systems using various methods
• Students work individually for 15-20 minutes, on the chart paper, all working the same problems
• If you have students who can’t work all the problems be sure to assign them a smaller load making sure that they get any special problems you feel would be beneficial for them.
• Provide students the solutions key, they may now work together, ask them to check the problems they missed, and continue working for another 5-10 minutes,
• Students choose the 2 problems that were hardest for them to solve and record what was hard about that particular problem (again you may need to model for them through a think aloud- it would be good to set this up during the review of homework, and continue your thinking now on a new problem)
• As a group the students nominate the 2-3 hardest problems for the group to solve, and what about those problems created issues for everyone
• Record findings on the board/overhead/computer for sharing later

• Students discuss which methods to use to most efficiently solve each system before solving (it’s ok if they switch methods after they get started if an issue arises during the process)
• Students discuss problems that presented special issues, i.e. inconsistent, equations with decimal/fraction coefficients, hard to solve and why
• Students share their thinking about which systems presented special issues for them as mathematicians or systems that presented unique issues for them

• Capture the conversation on chart paper/Computer projection so students can use as a study guide

• Group conversations and questions
• Chart paper with problems solved on it
• Journal/Admit slip
• Reflections

RESOURCES &

MATERIALS

• Individual student worksheet
• Chart paper
• Systems problems of various forms
• Extension Optional Computer Resource

2008Hybrid Algebra Research Project

Linear Systems Problems for use with Group Work

1. x= 3 + y

1. 6x + 2y = 8

1. x + y = 4
   x + 2y = -4

1. 0.8x+2.1y = 1.8

1. -3x +2y = -1
   5x– 3y = 4

1. 2x + 9y = 22
   3y – 4x= 9

1. 2x – 3y = -1.6

1. 6x + 2y = 1

1. 3x – 9y = 7

1. x = -2

1. 2x – 3y = -6
   5x + 6y = 39

Solutions key:

1. (2, -1)

1. (Infinite solutions)

1. (12, -8)

1. (-3, 2)

1. (5, 7)

1. (3½,1⅔)

1. (10.3, 7.4)

1. (Inconsistent, no solution)

1. (2, 2)

1. (21⅔, -6⅓)

1. (-2, 2)

1. (3, 4)

2008 Hybrid Algebra Research Project