# Solving Linear Equations by Balancing (Lesson 3.3)

###### Unit 1: Generalizing Patterns

**Day 1:**Intro to Unit 1**Day 2:**Equations that Describe Patterns**Day 3:**Describing Arithmetic Patterns**Day 4:**Making Use of Structure**Day 5:**Review 1.1-1.3**Day 6:**Quiz 1.1 to 1.3**Day 7:**Writing Explicit Rules for Patterns**Day 8:**Patterns and Equivalent Expressions**Day 9:**Describing Geometric Patterns**Day 10:**Connecting Patterns across Multiple Representations**Day 11:**Review 1.4-1.7**Day 12:**Quiz 1.4 to 1.7**Day 13:**Unit 1 Review**Day 14:**Unit 1 Test

###### Unit 2: Linear Relationships

**Day 1:**Proportional Reasoning**Day 2:**Proportional Relationships in the Coordinate Plane**Day 3:**Slope of a Line**Day 4:**Linear Equations**Day 5:**Review 2.1-2.4**Day 6:**Quiz 2.1 to 2.4**Day 7:**Graphing Lines**Day 8:**Linear Reasoning**Day 9:**Horizontal and Vertical Lines**Day 10:**Standard Form of a Line**Day 11:**Review 2.5-2.8**Day 12:**Quiz 2.5 to 2.8**Day 13:**Unit 2 Review**Day 14:**Unit 2 Test

###### Unit 3: Solving Linear Equations and Inequalities

**Day 1:**Intro to Unit 3**Day 2:**Exploring Equivalence**Day 3:**Representing and Solving Linear Problems**Day 4:**Solving Linear Equations by Balancing**Day 5:**Reasoning with Linear Equations**Day 6:**Solving Equations using Inverse Operations**Day 7:**Review 3.1-3.5**Day 8:**Quiz 3.1 to 3.5**Day 9:**Representing Scenarios with Inequalities**Day 10:**Solutions to 1-Variable Inequalities**Day 11:**Reasoning with Inequalities**Day 12:**Writing and Solving Inequalities**Day 13:**Review 3.6-3.9**Day 14:**Quiz 3.6 to 3.9**Day 15:**Unit 3 Review**Day 16:**Unit 3 Test

###### Unit 4: Systems of Linear Equations and Inequalities

**Day 1:**Intro to Unit 4**Day 2:**Interpreting Linear Systems in Context**Day 3:**Interpreting Solutions to a Linear System Graphically**Day 4:**Substitution**Day 5:**Quiz Review 4.1 to 4.3**Day 6:**Quiz 4.1 to 4.3**Day 7:**Solving Linear Systems using Elimination**Day 8:**Determining Number of Solutions Algebraically**Day 9:**Graphing Linear Inequalities in Two Variables**Day 10:**Writing and Solving Systems of Linear Inequalities**Day 11:**Quiz Review 4.4 to 4.7**Day 12:**Quiz 4.4 to 4.7**Day 13:**Unit 4 Review**Day 16:**Unit 4 Test

###### Unit 5: Functions

**Day 1:**Using and Interpreting Function Notation**Day 2:**Concept of a Function**Day 3:**Functions in Multiple Representations**Day 4:**Interpreting Graphs of Functions**Day 5:**Review 5.1-5.4**Day 6:**Quiz 5.1 to 5.4**Day 7:**From Sequences to Functions**Day 8:**Linear Functions**Day 9:**Piecewise Functions**Day 10:**Average Rate of Change**Day 11:**Review 5.5-5.8**Day 12:**Quiz 5.5 to 5.8**Day 13:**Unit 5 Review**Day 14:**Unit 5 Test

###### Unit 6: Working with Nonlinear Functions

**Day 1:**Nonlinear Growth**Day 2:**Step Functions**Day 3:**Absolute Value Functions**Day 4:**Solving an Absolute Value Function**Day 5:**Review 6.1-6.4**Day 6:**Quiz 6.1 to 6.4**Day 7:**Exponent Rules**Day 8:**Power Functions**Day 9:**Square Root and Root Functions**Day 10:**Radicals and Rational Exponents**Day 11:**Solving Equations**Day 12:**Review 6.5-6.9**Day 13:**Quiz 6.5 to 6.9**Day 14:**Unit 6 Review**Day 15:**Unit 6 Test

###### Unit 7: Quadratic Functions

**Day 1:**Quadratic Growth**Day 2:**The Parent Function**Day 3:**Transforming Quadratic Functions**Day 4:**Features of Quadratic Functions**Day 5:**Forms of Quadratic Functions**Day 6:**Review 7.1-7.5**Day 7:**Quiz 7.1 to 7.5**Day 8:**Writing Quadratics in Factored Form**Day 9:**Solving Quadratics using the Zero Product Property**Day 10:**Solving Quadratics Using Symmetry**Day 11:**Review 7.6-7.8**Day 12:**Quiz 7.6 to 7.8**Day 13:**Quadratic Models**Day 14:**Unit 7 Review**Day 15:**Unit 7 Test

###### Unit 8: Exponential Functions

**Day 1:**Geometric Sequences: From Recursive to Explicit**Day 2:**Exponential Functions**Day 3:**Graphs of the Parent Exponential Functions**Day 4:**Transformations of Exponential Functions**Day 5:**Review 8.1-8.4**Day 6:**Quiz 8.1 to 8.4**Day 7:**Working with Exponential Functions**Day 8:**Interpreting Models for Exponential Growth and Decay**Day 9:**Constructing Exponential Models**Day 10:**Rational Exponents in Context**Day 11:**Review 8.5-8.8**Day 12:**Quiz 8.5 to 8.8**Day 13:**Unit 8 Review**Day 14:**Unit 8 Test

#### Learning Targets

Use comparative reasoning with constant and variable terms to find solutions to linear equations.

Understand the solution to an equation as the equalizer that makes two variable expressions equal.

Justify the process of solving a linear equation symbolically or with a semi-concrete representation (such as a bar model).

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

Activity | 20 minutes |

Debrief Activity with Margin Notes | 15 minutes |

QuickNotes | 5 minutes |

Check Your Understanding | 10 minutes |

#### Activity: Whose Gift is Better?

#### Lesson Handouts

##### Media Locked

##### Media Locked

#### Answer Key

##### Media Locked

#### Experience First

While yesterday’s lesson had students using context to isolate the impact of the variable term, today students return to the idea of balancing and relational reasoning to solve equations that could have variables on both sides.

Todd and Tiffany’s gift each have a known component (cash) and an unknown component (gift card value). In questions 1 and 2, students decide if the allocation of cash and gift cards is fair, which here we define as the two twins getting a gift of equal monetary value. Students may have their own opinions about the value of a gift card over cash, but ultimately, the value of each gift depends on how much money is on each gift card!

By knowing that grandma spent the same amount of money on each gift, students can reason that the extra gift card must have the same value as Todd’s extra $15. That’s the only way the gifts could have the same value!

In question 4 students use similar relational reasoning. Tiffany has 5 more dollars but Todd has 4 extra boxes of candy. How is it possible that the two gifts are really “equal”? Only if the 4 extra boxes of candy have the same value as the $5, meaning that each box of candy is worth $1.25.

In question 5, Tiffany gets 5 more dollars than Todd but they both get the same number of socks (assume they are identical socks of equal price). Is it possible that the gifts have equal value? No! The socks are identical and “balance” each other out, but Tiffany got $5 more. There is nothing that Todd got that can make up for those extra $5. There is no value of socks that could make the two gifts equal (again, assuming the socks are identical and of equal price).

Monitoring Questions:

- What is the same about their two gifts? What is different?
- What does grandma mean when she says the gifts have equal value?
- What if Todd got $10 and 1 pair of socks? Is it possible for their grandma to have spent the same amount on each gift?
- What if Todd got $10 and 3 pairs of socks? Is it possible for their grandma to have spent the same amount on each gift?
- Do you notice anything about the value of the cash and the quantities of the other item in these problems that make it possible for the gifts to have equal value?

#### Formalize Later

We introduce the bar model today as a problem solving technique that helps students visualize the solution to an equation. Bar models can be used when variables are on one side or both sides of the equation, but are best introduced with positive coefficients and positive constant terms. The visual allows students to see what each variable term balances with or is responsible for on the other side of the equation. The bar model works on length, so two expressions that are equivalent will have the same length. Actually seeing how the variable terms and constant terms line up with each other is powerful for isolating the impact of the variable and determining its value.

You or your students may find the bar model to be tedious to draw each time. The goal is not that students always draw out a bar model every time they see a linear equation, but we do want to avoid rushing to an algorithm students do not yet understand. The use of a visual model allows for differentiation. Some students may need to actually draw the bar model only 2 or 3 times before they can start thinking about the principle in their heads (4 extra x’s must make up for the 20 extra unit lengths, each x must be 5). Other students can use the bar model as a reasoning tool for much longer before they switch to the mental model. Since students generalize strategies at their own pace, students should feel free to use any model for as long as it is useful for them.