Mathematics – Algebra 2
Unit of Study: Systems of Equations
Second Grading Period: Weeks 1- 3 (14 days) CURRICULUM OVERVIEW
Big Idea / Unit RationaleA wide array of methods and tools is available to solve systems of equations and inequalities including algebraic, graphical, tabular and technological. / Solutions to systems of linear equations and inequalities are in demand continuously in business and industry from finance to logistics. Linear algebra is a popular course at schools emphasizing business and economics.
TEKS / TEKS Specificity - Intended Outcome
Concepts / 2A.2 Foundations for Functions. The student understands the importance of the skills required to manipulate symbols in order to solve problems and uses the necessary algebraic skills to simplify algebraic expressions and solve equations and inequalities in problem situations. The student is expected to:
2A.2 (A) use tools including factoring and properties of exponents to simplify expressions and to transform and solve equations.
2A.3 Algebra and Geometry. The student formulates systems of equations and inequalities from problem situations, uses a variety of methods to solve them, and analyzes the solutions in terms of the situation. The student is expected to:
2A.3 (A) analyze situations and formulate systems of equations in two or more unknowns or inequalities in two unknowns to solve problems;
2A.3 (B) use algebraic methods, graphs, tables, or matrices, to solve systems of equations or inequalities; and
2A.3 (C) interpret and determine the reasonableness of solutions to systems of equations or inequalities for given contexts. / ” I CAN” statements highlighted in yellow should be displayed for students.
I can :
· use matrices and graphing calculators to solve systems of linear equations (2A.2A)
· formulate equations or inequalities in two or more unknowns from analyzing a given situation (2A.3A)
· use several methods to solve systems of equations or inequalities (2A.3B)
· determine whether a solution to a system of equations or inequalities is reasonable in the context of the given situation (2A.3C)
Evidence of Learning
· Given a system of linear equations, the student will demonstrate an ability to solve them with matrices and a graphing calculator at least 80% of the time.
· Given an appropriate situation, the student will analyze it to formulate a system of equations in two or more unknowns or inequalities at least 80% of the time.
· Given a system of equations or inequalities, the student will demonstrate an ability to use several methods and tools to solve them at least 80% of the time.
· Given a solution to a system of equations or inequalities, the student will determine its reasonableness in the context of the given situation at least 80% of the time.
Mathematics – Algebra 2
Unit of Study: Systems of Equations
Second Grading Period: Weeks 1- 3 (14 days) CURRICULUM GUIDE
Essential Questions / Essential Pre-requisite Skills· What are the various methods used to solve systems of linear equations?
· Do all the solution methods work for all systems of linear equations?
· What are the various methods used to solve systems of linear inequalities?
· Do all the solution methods for systems of linear equations work for systems of linear inequalities? / Algebra 1
· analyze situations and formulate systems of linear equations in two unknowns to solve problems (A.8A)
· solve systems of linear equations using concrete models, graphs, tables, and algebraic methods (A.8B)
· interpret and determine the reasonableness of solutions to systems of linear equations (A.8C)
Geometry
· select an appropriate representation (concrete, pictorial, graphical, verbal or symbolic) in order to solve problems (G.4)
· use numeric and geometric patterns to develop algebraic expressions representing geometric properties (G.5A)
· use properties of transformations and their compositions to make connections between mathematics and the real world, such as tessellations (G.5C)
The Teaching Plan
Instructional Model & Teacher Directions
The teacher will… / So students can…
Day 1 – Introduction to Systems
Engage/Explore (5 minutes)
· Have students in pairs discuss the word formulate and how it relates to algebra class. How is it used outside of math class?
Explain (30 minutes)
· Distribute Introduction to Systems Notes.
· Direct the students in pairs to appreciate the first two steps of solving systems – picking variables and the formulation of equations.
· Allow no more than 10 minutes for this activity which emphasizes the mechanics of the process.
· Distribute Introduction to Systems Assignment.
· Direct the students in pairs to master the first two steps of solving systems – picking variables and the formulation of the two equations – by one student working odd problems and one working the even problems. Explain their results to each other after about 10-12 minutes.
· Allow no more than 25 minutes for this activity which emphasizes the math content of the formulation process.
Elaborate (5 minutes)
· Have students journal what they learned from this activity.
Evaluate
· Monitor student activity throughout the class session.
· Consider the option of collecting the activity for either a grade or a completion check. / Day 1 – Introduction to Systems
Engage/Explore
· Discuss the word formulate with a partner. (2A.3A)
Explain
· Complete with a partner the systems of equations introduction activity. (2A.3A)
· Complete with a partner the systems of equations refinement activity. (2A.3A)
· After completing your half, explain your half to your partner and vice versa. (2A.3A)
Elaborate/Evaluate
· Journal the learning points of this activity and how you rate your understanding of equation formulation. (2A.3A)
Day 2 – Solving Systems by Graphing
Engage/Explore (5 minutes)
· Have students put the following 2 equations in their graphing calculator and determine the point of intersection through any method :
y1 = x + 2
y2 = -x – 2
Explain (30 minutes)
· Distribute Systems of Equations Notes
· Direct the students in pairs to perform the culmination steps of solving systems by graphing the two equations.
· Allow no more than 10 minutes for this activity which covers all aspects of the manual graphing process including interpretation
· Distribute Systems of Equations: Graphing Method.
· Direct the students in pairs to master the use of the graphing calculator to solve systems of linear equations – by one student working odd problems and one working the even problems. Explain their results to each other after about 10-12 minutes.
· Allow no more than 25 minutes for this activity which emphasizes the graphing method of solving systems of linear equations.
Elaborate (5-10 minutes)
· Have the students journal what they learned from this activity.
Evaluate
· Monitor student activity throughout the class session.
· Consider the option of collecting the activity for either a grade or a completion check. / Day 2 – Solving Systems by Graphing
Engage/Explore
· Graph the two equations given on the graphing calculator and “solve” through any method (2A.3B)
Explain
· Complete the first activity with a partner by manually graphing the system of equations. (2A.3A, 2A.3B)
· Complete with a partner the systems of equations graphing calculator activity. (2A.3A, 2A.3B)
· After completing your half, explain your half to your partner and vice versa (2A.3A, 2A.3B)
Elaborate/Evaluate
· Journal the learning points of this activity and how you rate your understanding of equation formulation. (2A.3A, 2A.3B)
Day 3 – How You Gonna Call?
Engage/Explore (5 minutes)
· Ask how many students have cell phones. Ask how much they know about how the monthly bill works. Also ask how much the cell phone costs each month and whether the student pays any part of the bill.
Explain (25-30 minutes)
· Distribute the How You Gonna Call? Activity.
· Have the students complete the activity in pairs.
Evaluate (5-10 minutes)
· Have a class discussion on how meaningful this activity is – involves phones and money!!
· Have students journal how confident they are in solving systems of equations and setting up the equations with a valid formulation. / Day 3 – How You Gonna Call?
Engage/Explore
· Answer teacher’s questions related to personal cell phone usage.
Explain
· Complete the systems of equations activity involving cell phone payment plans (2A.3A, 2A.3B, 2A.3C)
Evaluate
· Participate in the class discussion on how “fun” this activity is – phones and $$$
· Journal your confidence level in solving systems of linear equations
Day 4 – Solving Systems by Substitution
Engage/Explore (5 minutes)
· Have students in pairs discuss the word substitution and how it relates to algebra class. How is it used outside of math class?
Explain (35 minutes)
· Distribute Solving Systems using Substitution Notes.
· Direct the students in pairs to use yet another method of solving systems of linear equations. Split the problems, half to each partner.
· Allow no more than 15 minutes for this activity.
· Distribute Solving Systems using Substitution Assignment activity.
· Direct the students in pairs to master the substitution method of solving systems by one student working odd problems and one working the even problems. Explain their results to each other after about 12-15 minutes.
· Allow no more than 20 minutes for this activity which includes formulation of equations.
Elaborate (5 minutes)
· Have students journal what they learned from this activity.
Evaluate
· Monitor student activity throughout the class session.
· Consider the option of collecting the activity for either a grade or a completion check. / Day 4 – Solving Systems by Substitution
Engage/Explore
· Discuss the word substitution with a partner.
Explain
· Complete with a partner the activity introducing the substitution method of solving systems of linear equations – one student does evens, one does odds. (2A.3A, 2A.3B, 2A.3C)
· Complete with a partner the activity using the substitution method of solving systems of linear equations derived from word problems– one student does evens, one does odds. (2A.3A, 2A.3B, 2A.3C)
Elaborate/Evaluate
· Journal what you learned about using substitution to solve systems of linear equation (2A.3A, 2A.3B, 2A.3C)
Day 5 – Solving Systems by Substitution cont’d
Engage/Explore (10-15 minutes)
· Ask the students if they are involved in any school fund-raising activities. Ask for details – money raised, how, how many students involved, how much time from planning to execution.
Explain (15-20 minutes)
· Distribute The Substitution Method to the students.
· Have the students in pairs complete this activity which runs the students from a given situation and walks them through the problem solving process step-by-step and thought-by-thought.
Elaborate (10 minutes)
· Ask students for problem or challenge areas in preparation for the quiz in the next class session covering:
formulating equations
solving systems by graphing
solving systems by substitution / Day 5 – Solving Systems by Substitution cont’d
Engage/Explore
· Discuss any school fund-raising activities that you are a part of.
Explain
· Direct student to complete the Substitution activity (2A.3A, 2A.3B, 2A.3C)
Elaborate
· Have questions ready for the quiz review
Day 6 – Quiz over Solving Systems of Equations (Formulating and Substitution)
Explain
· Review any homework or general questions from students
Evaluate
· Administer quiz ensuring alignment with guiding TEKS / Day 6 – Quiz over Solving Systems of
Equations (Formulating and Substitution)
Explain
· Get any lingering concerns or questions answered
Evaluate
· Take quiz (2A.3A, 2A.3B, 2A.3C)
Day 7 - Solving Systems by Elimination
Engage/Explore (15 minutes)
· Go over the previous day’s quiz results
· Have students in pairs discuss the word elimination and how it relates to algebra class. How is it used outside of math class?
Explain (30 minutes)
· Distribute Solving Systems by the Simple Elimination Method Notes.
· Direct the students in pairs to complete the activity which delineates step-by-step the elimination method of solving systems of linear equations
· Allow about 10 minutes for this activity which emphasizes the basics of the process.
· Distribute Solving Systems of Equations by the Elimination Method Assignment.
· Direct the students in pairs to master the elimination method of solving systems of linear equations – by one student working odd problems and one working the even problems. Explain their results to each other after about 10-15 minutes.
· Allow no more than 20 minutes for this activity which emphasizes the math content of the formulation process.
Evaluate
· Monitor student activity throughout the class session.
· Consider the option of collecting the activity for either a grade or a completion check. / Day 7 - Solving Systems by Elimination
Engage/Explore
· Go over quiz results
· Discuss the word elimination with a partner
Explain
· Complete with a partner the two activities related to the elimination method of solving systems of linear equations (2A.3A, 2A.3B, 2A.3C)
Day 8 – Solving Systems using Matrices
Engage/Explore (10 minutes)
· Have students in pairs discuss the word matrix and how it relates to algebra class. How is it used outside of math class? Is matrix on your graphing calculator?
Explain (25 minutes)
· Distribute Systems of Equations using Matrices.
· Direct the students in pairs to learn the matrices method of solving systems of linear equations – by one student working odd problems and one working the even problems. Explain their results to each other after about 10-15 minutes.
· Monitor that students understand from the handout how to enter and use matrices on their graphing calculator.
Elaborate (5-10 minutes)
· Direct all students individually to solve problem #1 on the handout by the substitution method and then the elimination method. Does either of these methods resemble the matrix method?
Evaluate (time allowing)
· Consider the option of collecting the activity for either a grade or a completion check.
· If time allows, have the students journal on how they like the matrix method compared to the other two algebraic methods and why. / Day 8 – Solving Systems using Matrices
Engage/Explore
· Discuss the word matrix with a partner
Explain
· Read the handout how to use the matrix method of solving systems of equations. (2A.2A, 2A.3A, 2A.3B, 2A.3C)