Jonkled

Here are extremely detailed, descriptive notes in a clear note-taking format based on the document you uploaded. Everything is broken down step by step so you can copy and paste easily. Fractions and operations are written out exactly as you requested. Section 1 – Introduction Main Question: How can modeling be used to solve equations with variables on both sides? Review: Students first reviewed the parts of an equation. New Skill: Learn to use algebra tiles to model an equation that represents a real-world problem. Section 2 – Writing an Equation with Variables on Both Sides Scenario: Javier and Bernard have savings plans. Javier: starts with 1 dollar, saves 3 dollars each week. Bernard: starts with 5 dollars, saves 2 dollars each week. Unknown Variable: Let x = number of weeks they save. Expressions: Javier’s money = 1 + 3x. Bernard’s money = 5 + 2x. Equation: Since we want to know when they have the same amount of money: 1 + 3x = 5 + 2x. Section 4 – Modeling with Algebra Tiles Tile Representations: Orange square = positive constant (1). Blue square = negative constant (−1). Orange rectangle = positive x. Blue rectangle = negative x (−1x). Modeling Left Side (1 + 3x): Drag 1 orange square tile. Drag 3 orange rectangular tiles (grouped together). Modeling Right Side (5 + 2x): Drag 5 orange square tiles (grouped). Drag 2 orange rectangular tiles (grouped). Result: Board sum matches equation → 1 + 3x = 5 + 2x. Section 6 – Transition Goal: Use algebra tiles not just to model, but to solve equations with variables on both sides. Section 7 – Identifying Parts of the Equation Equation: 3x + 1 = 2x + 5. Coefficients: Left side coefficient = 3. Right side coefficient = 2. Constants: Left side constant = 1. Right side constant = 5. Zero Pairs: Definition: A positive and negative number that cancel to 0. Example: 2 and −2 form a zero pair. Section 8 – Solving with Algebra Tiles Equation: 3x + 1 = 2x + 5. Step 1 – Isolate x terms: Cancel 2x on the right by adding 2 negative x tiles. To balance, add 2 negative x tiles on the left. Left side now has 1x. Step 2 – Isolate constants: Cancel 1 on the left by adding 1 negative unit tile. Balance by adding 1 negative unit tile on the right. Right side now has 4. Result: x = 4. Interpretation: Javier and Bernard will have the same amount of money after 4 weeks. Section 10 – Another Example Equation: 2 + 4x = −3 + 5x. Step 1 – Isolate x terms: Cancel 4x on the left by adding 4 negative x tiles. Balance by adding 4 negative x tiles on the right. Left side cancels out x terms, leaving constants. Right side has 1x remaining. Step 2 – Isolate constants: Cancel −3 on the right by adding 3 positive unit tiles. Balance by adding 3 positive unit tiles on the left. Left side becomes 5. Result: 5 = x, or equivalently x = 5. Section 12 – Summary Key Idea: Modeling with algebra tiles helps solve equations with variables on both sides. Method: Use zero pairs to eliminate like terms and isolate the variable. Section 13 – Real-World Temperature Problem Scenario: Lakota and Kiana’s cities experience falling temperatures. Lakota: starts at 1 degree, decreases 3 degrees per hour. Kiana: starts at 5 degrees, decreases 5 degrees per hour. Equation: −3x + 1 = −5x + 5. Variable: x = number of hours until temperatures are equal. Process: Step 1: Use zero pairs to cancel x terms. Step 2: Use zero pairs to cancel constants. Step 3: Divide by coefficient of x. Section 14 – Solving the Temperature Equation Equation: −3x + 1 = −5x + 5. Step 1 – Isolate x terms: Cancel −5x on the right by adding 5 positive x tiles. Balance by adding 5 positive x tiles on the left. Left side becomes 2x + 1. Step 2 – Isolate constants: Cancel +1 on the left by adding 1 negative unit tile. Balance by adding 1 negative unit tile on the right. Right side becomes 4. Step 3 – Divide by coefficient: Equation is 2x = 4. Divide both sides by 2. x = 2. Interpretation: After 2 hours, the temperatures in both cities are the same. Overall Notes Algebra Tiles Purpose: Visual tool to represent constants and variables. Zero Pairs: Used to cancel terms and simplify equations. Steps to Solve Equations with Variables on Both Sides: Isolate variable terms using zero pairs. Isolate constants using zero pairs. Divide by the coefficient of the variable. Applications: Savings problems, temperature problems, and general equations. Would you like me to also reformat these notes into a condensed study guide (like bullet-point formulas and quick steps) so you can use them for fast review before tests?