Strategic Solving
5 min
Teacher Prep
Setup
Narrative
The purpose of this activity is for students to begin building linear equations and solving them. They use the familiar context of polygon perimeter to find the values of x that give 2 shapes the same perimeter.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time, and then 2 minutes to discuss their solutions with a partner.
If necessary, remind students how to find the perimeter of a shape. Instruct groups to explain to each other how they came up with expressions and an equation to represent the situation.
Student Task
The triangle and the square have equal perimeters.
- Find the value of x.
- What is the perimeter of each of the figures?
Sample Response
-
x=16. Sample reasoning: Since the perimeters are equal, the perimeter of the triangle must equal the perimeter of the square. The perimeter of the triangle is 2x+2x+(x−8), which is 5x−8, and the perimeter of the square is 4 times the side length, or 4(x+2).
\begin{align} 5x-8&=4(x+2) \\ 5x-8&=4x+8 \\ 5x &=4x+16 \\ x &=16 \end{align} - P=72. Sample reasoning: Since the perimeters are equal, we can use either expression to find the perimeter. From the square: 4(16+2)=72.
Activity Synthesis (Teacher Notes)
Ask groups to share their strategies for solving the question. Here are some questions for discussion:
- “What expression represents the perimeter of the triangle? The perimeter of the square?” (The expression for perimeter of the triangle is 5x−8, and for 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.)
Standards
- 8.EE.7·Solve linear equations in one variable.
- 8.EE.C.7·Solve linear equations in one variable.
10 min
20 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Strategic Solving
5 min
Teacher Prep
Setup
Narrative
The purpose of this activity is for students to begin building linear equations and solving them. They use the familiar context of polygon perimeter to find the values of x that give 2 shapes the same perimeter.
Launch
Arrange students in groups of 2. Give students 2 minutes of quiet think time, and then 2 minutes to discuss their solutions with a partner.
If necessary, remind students how to find the perimeter of a shape. Instruct groups to explain to each other how they came up with expressions and an equation to represent the situation.
Student Task
The triangle and the square have equal perimeters.
- Find the value of x.
- What is the perimeter of each of the figures?
Sample Response
-
x=16. Sample reasoning: Since the perimeters are equal, the perimeter of the triangle must equal the perimeter of the square. The perimeter of the triangle is 2x+2x+(x−8), which is 5x−8, and the perimeter of the square is 4 times the side length, or 4(x+2).
\begin{align} 5x-8&=4(x+2) \\ 5x-8&=4x+8 \\ 5x &=4x+16 \\ x &=16 \end{align} - P=72. Sample reasoning: Since the perimeters are equal, we can use either expression to find the perimeter. From the square: 4(16+2)=72.
Activity Synthesis (Teacher Notes)
Ask groups to share their strategies for solving the question. Here are some questions for discussion:
- “What expression represents the perimeter of the triangle? The perimeter of the square?” (The expression for perimeter of the triangle is 5x−8, and for 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.)
Standards
- 8.EE.7·Solve linear equations in one variable.
- 8.EE.C.7·Solve linear equations in one variable.