Balanced Moves
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to connect moves that keep a hanger balanced with moves that create equivalent equations. These moves include:
- Adding or subtracting the same thing on each side.
- Multiplying or dividing each side by the same number.
Launch
Give students 5 minutes of quiet work time followed by a whole-class discussion.
Student Task
Figures B, C, and D show the result of simplifying the hanger in Figure A by removing equal weights from each side.
Let x be the weight of the blue square, y be the weight of a green triangle, and z be the weight of a red circle.
- Draw an arrow from each hanger to the next and describe the move that keeps the hanger in balance near each arrow.
-
Here are some equations. Each equation represents one of the hanger diagrams. Write the matching equation on the line below each hanger diagram.
\begin{align}2(x + 3y) &= 4x + 2y\\2y&=x\\2(x+3y)+2z&=2z+4x+2y\\x+3y&=2x+y\end{align} - Describe the move that keeps the equation for diagram A equivalent to the equation for diagram B.
Sample Response
-
Remove 2 circles from each side of the hanger, remove half the shapes on each side, remove 1 triangle and 1 square on each side
-
A: 2(x+3y)+2z=2z+4x+2y
B: 2(x+3y)=4x+2y
C: x+3y=2x+y
D: 2y=x -
Subtract 2z from each side of the equation.
Activity Synthesis (Teacher Notes)
The purpose of the discussion is to make connections between moves that can be done on a hanger to keep it in balance and moves that can be done to an equation to write an equivalent equation.
Invite students to share the order of equations, then display the image for all to see.
Select students to share their descriptions of the hanger moves and corresponding equation moves, recording for all to see. Focus on the move from Hanger B to Hanger C. If students struggle to describe the move, ask them to look at the equations for an idea of what might be happening. (Half of the weight on each side is removed. One of the square-and-2-triangle patterns is removed on the left and half of the squares and half of the triangles are removed on the right.) Ask students how this is related to the move that is done on the equations. Make sure students understand that this move can be stated as “divide each side by 2” as well as “multiply each side by 21.”
Anticipated Misconceptions
Some students may be confused about how to match the equations with the parentheses to the hangers. Ask students what they know about the connections between variables and shapes. Then, ask students how many of each shape might be needed for each side of the hanger to match the equation.
Standards
- 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
- 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
15 min
10 min
Knowledge Components
All skills for this lesson
No KCs tagged for this lesson
Balanced Moves
10 min
Teacher Prep
Setup
Narrative
The purpose of this Warm-up is for students to connect moves that keep a hanger balanced with moves that create equivalent equations. These moves include:
- Adding or subtracting the same thing on each side.
- Multiplying or dividing each side by the same number.
Launch
Give students 5 minutes of quiet work time followed by a whole-class discussion.
Student Task
Figures B, C, and D show the result of simplifying the hanger in Figure A by removing equal weights from each side.
Let x be the weight of the blue square, y be the weight of a green triangle, and z be the weight of a red circle.
- Draw an arrow from each hanger to the next and describe the move that keeps the hanger in balance near each arrow.
-
Here are some equations. Each equation represents one of the hanger diagrams. Write the matching equation on the line below each hanger diagram.
\begin{align}2(x + 3y) &= 4x + 2y\\2y&=x\\2(x+3y)+2z&=2z+4x+2y\\x+3y&=2x+y\end{align} - Describe the move that keeps the equation for diagram A equivalent to the equation for diagram B.
Sample Response
-
Remove 2 circles from each side of the hanger, remove half the shapes on each side, remove 1 triangle and 1 square on each side
-
A: 2(x+3y)+2z=2z+4x+2y
B: 2(x+3y)=4x+2y
C: x+3y=2x+y
D: 2y=x -
Subtract 2z from each side of the equation.
Activity Synthesis (Teacher Notes)
The purpose of the discussion is to make connections between moves that can be done on a hanger to keep it in balance and moves that can be done to an equation to write an equivalent equation.
Invite students to share the order of equations, then display the image for all to see.
Select students to share their descriptions of the hanger moves and corresponding equation moves, recording for all to see. Focus on the move from Hanger B to Hanger C. If students struggle to describe the move, ask them to look at the equations for an idea of what might be happening. (Half of the weight on each side is removed. One of the square-and-2-triangle patterns is removed on the left and half of the squares and half of the triangles are removed on the right.) Ask students how this is related to the move that is done on the equations. Make sure students understand that this move can be stated as “divide each side by 2” as well as “multiply each side by 21.”
Anticipated Misconceptions
Some students may be confused about how to match the equations with the parentheses to the hangers. Ask students what they know about the connections between variables and shapes. Then, ask students how many of each shape might be needed for each side of the hanger to match the equation.
Standards
- 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.
- 8.EE.C·Analyze and solve linear equations and pairs of simultaneous linear equations.