(11) Algebra 1: Solutions & Equality

Topic outline

• Course Outline

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  Essential Questions

  1. What does it mean to "Solve an equation"? How do you know if you've done it?
  2. What are some different ways to represent equality? What some pros and cons of each?

  Knowledge Goals

  1. Create Frayer diagrams for the following words: equation, variable, unknown, coefficient, solution, non-solution, solving, isolate, check.

  Insight Goals

  1.  

  Skill Goals

  1. Solve and check linear equations.
  2. Represent and solve equations in a variety of ways.

  Learning Sequence

  1.  

• Lesson 0: Prerequisite Knowledge of Equations

  If you have already mastered this lesson's content, skip it.

   

  I will...

  1. Do the Equal or Not? exercises, making sure I can do it all by hand.
  2. Learn to check my work above using Symbolab or Microsoft Math Solver.
  3. Practice explaining, from scratch, the meaning of the equals sign and the not equals sign, and what it means to solve, simplify, and interpret equations.

   

  I can...

  1. Correctly interpret and use the equals sign and the not equals sign.
  2. Solve and simplify basic equations and judge them accurately as true or false. 
  3. Explain from scratch all of the above based on examples that I create.

   

  As assessed by...

  [coming soon]

• Lesson 1: Modeling and Solving Equations

  I will...

  1. Solve and check the Noah's Ark problem.
  2. Solve and check Visual Balance Problems. I will represent my thinking with equations.
  3. Solve Balance Problems.
  4. Solve and Check Equations Visually
  5. Solve Equations by Guess & Check [template] while using reasoning to iterate my guesses. [Teacher will have to create exercises.]
  6. Do Equations and Solutions Exercises, Levels 1-4 [work-in-progress]. If this is too easy, I will create more challenging exercises for myself with this template.
  7. Do model equations linearly. [Under construction.]
  8. Draw an equation clock in which each hour number is represented by an equation for which that hour number is a solution. E.g. ten o'clock below

   

  

   

  So I can...

  1. Translate between visuals and setup equations for balance problems of forms x+b=c, ax=c, ax+b=c, and all jumbled variations thereof, where all variables are non-negative integers and easy to visualize.
  2. Solve the balance problems with logic and guess-and-check.
  3. Check my work visually, symbolically, and with software.
  4. Show by example what the following words mean: equation, solution, non-solution, check.

   

  As assessed by...

  [coming soon]

• Lesson 2: Two Instance and Fractional Equations

  I will...

  [coming soon]

   

  So I can...

  1. Translate between visuals and setup equations for balance problems of forms ax+b=cx+d and and all jumbled variations thereof, where all variables are non-negative integers, some of which are easy to visualize.
  2. Solve all equations of the form ax+b=cx+d and all jumbled variations thereof involving fractions that are easy to visualize and draw, by guess and check.
  3. Create my own examples of linear equations to solve.
  4. Check my work visually, symbolically, and with software.

   

  As assessed by...

  [coming soon]

• Lesson 3: Visual Solving & Preservation of Equality

  I will...

  1. Solve and check balance problems visually by isolation (and ignore the math symbols for now) in Solve & Check - Example & Exercises.
  2. Solve balance problems through isolation that I have to draw myself. Boxes & Circles, Levels 4, 5, and 6.
  3. Create my own solve and check problems with the Isolate then Check [template]
  4. For each of addition, subtraction, multiplication, and division, create an example and non-example of a manipulation that preserves equality.

   

  So I can...

  1. Solve visually represented equations visually by isolating the unknown.
  2. Explain how doing the same thing to both sides preserves equality.

   

  As assessed by...

  [coming soon]

   

• Lesson 4: Symbolic Solving & Preservation of Equality

  I will...

  1. See how well I understand preservation of equality in
     1. This Twitter thread [Focus on the 3 highest webs on the left.]
     2. JUMP 7.1 Scale Diagrams
  2. Drill solving equations.
     1. Example:
     2. Level 1
     3. Level 2
     4. Level 3
     5. Level 4
     6. Level 5
     7. More advanced: Solving & Checking Linear Equations
  3. Do JUMP 6 Solving Symbolicallyexercises.
  4. Create my own problems and solutions with this balance template.

   

  So I can...

  1. Describe all my previous visual solutions symbolically, including every step and check, using correct math terminology.
  2. Solve and check linear equations symbolically using more than 1 method.
  3. Distinguish manipulations that do and do not preserve equality.
  4. Distinguish manipulations between the useful and the non-useful.

   

  As assessed by...

  [coming soon]

• Lesson 5: Solving with Inverse Operations

  I will...

  1. Solve the "Think of a Number" trick and be able to explain it to others. Create my own variations on the trick and perform them.
  2. Solve any previous equations from previous lessons using inverse operations.

   

  So I can...

  1. Solve and check linear equations using inverse operations or the same manipulation to both sides.
  2. Show these methods' equivalence.
  3. Use both methods to solve magic tricks.

   

  As assessed by...

  [coming soon]

• Lesson 6: Beyond Visuals

  I will...

  [coming soon]

   

  So I can...

  1. Solve equations of the form ax + b = cx + d involving rational numbers and numbers that are difficult to visualize. I can solve them in multiple ways.
  2. "Look ahead" to determine when solutions exist and when they do not.
  3. Set up and solve equations in applied situations.

   

  As assessed by...

  [coming soon]