# Solving Systems with Elimination (Lesson 6.3)

###### Unit 0: Prerequisites

**Day 1:**The Cartesian Plane**Day 2:**Equations of Circles**Day 3:**Solving Equations in Multiple Representations**Day 4:**Reasoning with Formulas**Day 5:**Quiz 0.1 to 0.4**Day 6:**Linear Relationships**Day 7:**Reasoning with Slope**Day 8:**Set Notation**Day 9:**Quiz 0.5 to 0.7**Day 10:**Unit 0 Review**Day 11:**Unit 0 Test

###### Unit 1: Functions

**Day 1:**Functions and Function Notation**Day 2:**Domain and Range**Day 3:**Rates of Change and Graph Behavior**Day 4:**Library of Parent Functions**Day 5:**Transformations of Functions**Day 6:**Transformations of Functions**Day 7:**Even and Odd Functions**Day 8:**Quiz 1.1 to 1.6**Day 9:**Building Functions**Day 10:**Compositions of Functions**Day 11:**Inverse Functions**Day 12:**Graphs of Inverse Functions**Day 13:**Piecewise Functions**Day 14:**Quiz 1.7 to 1.11**Day 15:**Unit 1 Review**Day 16:**Unit 1 Test

###### Unit 2: Polynomial and Rational Functions

**Day 1:**Connecting Quadratics**Day 2:**Completing the Square**Day 3:**Polynomials in the Short Run**Day 4:**Polynomials in the Long Run**Day 5:**Review 2.1-2.4**Day 6:**Quiz 2.1 to 2.4**Day 7:**Factor and Remainder Theorem**Day 8:**Factor and Remainder Theorem**Day 9:**Complex Zeros**Day 10:**Connecting Zeros Across Multiple Representations**Day 11:**Intro to Rational Functions**Day 12:**Graphing Rational Functions**Day 13:**Quiz 2.5 to 2.9**Day 14:**Unit 2 Review**Day 15:**Unit 2 Test

###### Unit 3: Exponential and Logarithmic Functions

**Day 1:**Exponential Functions**Day 2:**Graphs of Exponential Functions**Day 3:**Compound Interest and an Introduction to "e"**Day 4:**Review 3.1-3.3**Day 5:**Quiz 3.1 to 3.3**Day 6:**Logarithmic Functions**Day 7:**Graphs of Logarithmic Functions**Day 8:**Logarithm Properties**Day 9:**Solving Exponential and Logarithmic Equations**Day 10:**Quiz 3.4 to 3.7**Day 11:**Exponential and Logarithmic Modeling**Day 12:**Unit 3 Review**Day 13:**Unit 3 Test

###### Unit 4: Trigonometric Functions

**Day 1:**Right Triangle Trig**Day 2:**Inverse Trig Ratios**Day 3:**Radians and Degrees**Day 4:**Unit Circle**Day 5:**Unit Circle**Day 6:**Other Trig Functions**Day 7:**Review 4.1-4.6**Day 8:**Quiz 4.1 to 4.6**Day 9:**Graphing Sine and Cosine**Day 10:**Transformations of Sine and Cosine Graphs**Day 11:**Graphing Secant and Cosecant**Day 12:**Graphing Tangent and Cotangent**Day 13:**Quiz 4.7 to 4.10**Day 14:**Inverse Trig Functions**Day 15:**Trigonometric Modeling**Day 16:**Trigonometric Identities**Day 17:**Unit 4 Review**Day 18:**Unit 4 Review**Day 19:**Unit 4 Test

###### Unit 5: Applications of Trigonometry

**Day 1:**Law of Sines**Day 2:**The Ambiguous Case (SSA)**Day 3:**Law of Cosines**Day 4:**Area and Applications of Laws**Day 5:**Vectors**Day 6:**Review 5.1-5.5**Day 7:**Quiz 5.1 to 5.5**Day 8:**Polar Coordinates**Day 9:**Equations in Polar and Cartesian Form**Day 10:**Polar Graphs Part 1**Day 11:**Polar Graphs Part 2**Day 12:**Review 5.6-5.9**Day 13:**Quiz 5.6 to 5.9**Day 14:**Parametric Equations**Day 15:**Parametric Equations (With Trig)**Day 16:**Unit 5 Review**Day 17:**Unit 5 Test

###### Unit 6: Systems of Equations

**Day 1:**What is a Solution?**Day 2:**Solving Systems with Substitution**Day 3:**Solving Systems with Elimination**Day 4:**Review 6.1-6.3**Day 5:**Quiz 6.1 to 6.3**Day 6:**Solving Systems in 3 Variables**Day 7:**Solving Systems in 3 Variables**Day 8:**Partial Fractions**Day 9:**Unit 6 Review**Day 10:**Unit 6 Test

###### Unit 7: Sequences and Series

**Day 1:**Introducing Sequences**Day 2:**Using Sequences and Series to Describe Patterns**Day 3:**Arithmetic Sequences and Series**Day 4:**Review 7.1-7.2**Day 5:**Quiz 7.1 to 7.2**Day 6:**Geometric Sequences and Finite Series**Day 7:**Infinite Geometric Sequences and Series**Day 8:**Proof by Induction**Day 9:**Proof by Induction**Day 10:**Quiz 7.3 to 7.5**Day 11:**Unit 7 Review**Day 12:**Unit 7 Test

###### Unit 8: Limits

**Day 1:**What is a Limit?**Day 2:**Evaluating Limits Graphically**Day 3:**Evaluating Limits with Direct Substitution**Day 4:**Evaluating Limits Analytically**Day 5:**Evaluating Limits Analytically**Day 6:**Review 8.1-8.4**Day 7:**Quiz 8.1 to 8.4**Day 8:**Continuity**Day 9:**Continuity**Day 10:**Intermediate Value Theorem**Day 11:**Intermediate Value Theorem**Day 12:**Review 8.5-8.6**Day 13:**Quiz 8.5 to 8.6**Day 14:**Limits at Infinity**Day 15:**Unit 8 Review**Day 16:**Unit 8 Test

###### Unit 9: Derivatives

**Day 1:**Introduction to Derivatives**Day 2:**Average versus Instantaneous Rates of Change**Day 3:**Calculating Instantaneous Rate of Change**Day 4:**Calculating Instantaneous Rate of Change**Day 5:**The Derivative Function**Day 6:**The Derivative Function**Day 7:**Review 9.1-9.3**Day 8:**Quiz 9.1 to 9.3**Day 9:**Derivative Shortcuts**Day 10:**Differentiability**Day 11:**Connecting f and f’**Day 12:**Connecting f and f’**Day 13:**Review 9.4-9.6**Day 14:**Quiz 9.4 to 9.6**Day 15:**Derivatives of Sine and Cosine**Day 16:**Product Rule**Day 17:**Quotient Rule**Day 18:**Review 9.7-9.9**Day 19:**Quiz 9.7 to 9.9**Day 20:**Unit 9 Review**Day 21:**Unit 9 Test

###### Unit 10: (Optional) Conic Sections

**Day 1:**Intro to Conic Sections**Day 2:**Defining Parabolas**Day 3:**Working with Parabolas**Day 4:**Quiz 10.1 to 10.3**Day 5:**Defining Ellipses**Day 6:**Working with Elllipses**Day 7:**Defining Hyperbolas**Day 8:**Working with Hyperbolas**Day 9:**Quiz 10.4 to 10.7**Day 10:**Unit 10 Review**Day 11:**Unit 10 Test

#### Learning Targets

Explain the method of elimination using scaling and comparison

Determine the conditions that result in dependent, independent, and inconsistent systems

Connect contextual, graphical, and analytical representations of dependent, independent, and inconsistent systems

Tasks/Activity | Time |
---|---|

Activity | 20 minutes |

Debrief Activity | 10 minutes |

Important Ideas | 5 minutes |

Check Your Understanding | 15 minutes |

#### Activity: How much for a Bagel and Cream Cheese?

#### Lesson Handouts

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#### Answer Key

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#### Experience First

In this lesson students look at various Panera orders to determine the price of a tub of cream cheese and a bagel. Questions 1-3 introduce the context of Panera Bread orders and conceptualize the idea of a solution as well as the number of solutions. Most students use their own experiences at bakeries to suggest a reasonable price for a bagel or tub of cream cheese, and then solve for the remaining variable. Students realize in question 1 that having one order is insufficient to determine the cost of each order. Choosing any price of bagel would allow students to solve for the necessary price of a tub of cream cheese, or vice versa.

In question 2, students must identify the fact that the number of bagels and tubs of cream cheese purchased by Kelly is exactly half the number of bagels and tubs of cream cheese purchased by Peyton, so it follows that Kelly’s order should be exactly 1⁄2 the cost of Peyton’s order. Students learn that scaled versions of an equation produce an equivalent equation, which is still a true equation, but it does not provide additional information that would allow us to narrow down the cost of a bagel or tub of cream cheese.

Finally, in question 4, students receive Carter’s order which is an independent equation. The question is worded intentionally so they will compare Carter’s order to twice Peyton’s order. The difference in price between twice Peyton’s order and Carter’s order must be the price of 3 bagels, since otherwise the orders are the same! This is the idea of elimination--scaling the equations so that the only difference in price can be attributed to one variable. It’s important that students understand this conceptually instead of just going through the rote procedure of multiplying equations by a scalar and then adding or subtracting equations. While students leave Algebra 2 feeling pretty confident using elimination as a strategy, we want students to be able to connect this method with important ideas about equivalence.

#### Formalize Later

This experience reveals the big idea of elimination: creating equivalent equations by scaling so that the only remaining difference between the two equations can be attributed to one variable. The debrief should push students to articulate why they would compare Carter's order to twice Peyton's order. The debrief will also add the vocabulary words of "dependent system" and tie this to a system having infinitely many solutions and equations in the system being equivalent.

The Important Ideas section ties together graphical and analytical representations of dependent, independent, and inconsistent systems. Students should be able to reason about systems of linear equations from the perspective of slopes and y-intercepts, as well as equivalent equations and scalar multiples. Questions 3 and 4 on the Check Your Understanding encourage students to strategically assess what conditions are needed to classify a system as independent, dependent, or inconsistent. This understanding is a critical piece of the open middle task on day 5.