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: Review 4.1- 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: Review 4.4- 4.7
Day 12: Quiz 4.4 to 4.7
Day 13: Unit 4 Review
Day 14: 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
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Answer Key
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Homework
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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:
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.