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Solving Equations (Lesson 6) OPTIONAL


The triangle and the square have equal perimeters. 

  1. Find the value of x.
  2. What is the perimeter of each of the figures?

Ask groups to share their strategies for solving the question. Consider asking some of the following questions:
  • “What expression represents the perimeter of the triangle? The perimeter of the square?” (The expression for perimeter of the triangle is 5x−8, and the perimeter of the square is 4(x+2).)
  • “What was your strategy in making an equation?” (If both perimeters are the same, we can say their expressions are equal.)
  • “What does x mean in the situation?” (It means an unknown value. None of the sides or perimeter is represented by x, so we cannot say it represents a specific thing on the figures.)
  • “Looking at the figures, are there any values that x could not be? Explain your reasoning.” (Since the triangles have sides that are 2x, x cannot be 0 or a negative value. Triangles cannot have sides with 0 or negative side lengths. Since the third side is x−8, we can use this same reasoning to realize that x must actually be greater than 8.)
  • “How does this information help when solving?” (If I make a mistake in my solution and get a value of x that is less than or equal to 8, then I know immediately that my answer is not reasonable and I can try to find my error.)
ACTIVITY: 10 minutes6.2: Predicting SolutionsInstructional RoutinesCCSS Standards: AddressingThe purpose of this activity is to shift the focus from solving an equation to thinking about what it means for a number to be a solution of an equation. Students inspect each equation, looking at the structure, the signs, and the operations in it to decide if the solution is positive, negative, or zero. Some questions are paired with another question (for example, the last two questions) so students can take advantage of their thinking from one to the next.LaunchArrange students in groups of 2.Display the equation 5x=6x for all to see.Ask students, "How might we know whether x is a positive number, negative number, or zero, without solving the equation?" (The variables can be combined into one term, but there are no constant terms. That means eventually the variable term has to equal 0, so x must be 0.)Display the equation 5x=-16.5 for all to see and ask the same question. (Without solving, we can see that a positive number of xs has to equal a negative value, so x must be a negative number.)Instruct students to inspect each equation carefully and use reasoning to answer the questions in the activity rather than trying to solve each equation for a specific value. Give 5 minutes of quiet think time, and then ask students to compare their work with their partner. For any questions they disagree on, students should work to reach an agreement.Conceptual Processing: Processing Time. Begin with a demonstration of the first equation, which will provide access for students who benefit from clear and explicit instructions.

Without solving, identify whether these equations have a solution that is positive, negative, or zero.

  1. x6=3x4
  2. 7x=3.25
  3. 7x=32.5
  4. 3x+11=11
  1. 9−4x=4
  2. -8+5x=-20
  3. -12(-8+5x)=-20