Solving Systems by Elimination (Part 1)
5 min
Narrative
The purpose of this Warm-up is to give students an intuitive and concrete way to think about combining two equations that are each true.
Students are presented with diagrams of three balanced hangers, which suggest that the weights on the two sides of each hanger are equal. Each side of the last hanger shows the combined objects from the corresponding side of the first two hangers. Students can reason that if 2 circles weigh the same as 1 square, and 1 circle and 1 triangle weigh the same as 1 pentagon, then the combined weight of 3 circles and 1 triangle should also be equal to the combined weight of 1 square and 1 pentagon.
Launch
Arrange students in groups of 2. Display the hanger diagrams for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder about. Give students 1 minute of quiet think time, and then 1 minute to discuss the things they notice and wonder with their partner.
Student Task
What do you notice? What do you wonder?
Sample Response
Things students may notice:
- All the hangers are balanced.
- There are circles on the left side of every hanger and none on the right side of any hanger.
- There is a triangle on the left side of the second and third hangers.
- There is a square on the right side of the first and third hangers.
- The third hanger has all the shapes from the first two hangers.
- The shapes from the left side of the first two hangers end up on the left side of the last hanger. (The same can be said about the right side.)
Things students may wonder:
- How much does each shape weigh?
- Is the circle the lightest shape?
- Which shape weighs the most?
- Why does the last hanger have many more shapes than the other two hangers?
- Why is the last hanger still balanced if it has shapes added to each side?
Synthesis
Ask students to share the things that they noticed and wondered. Record and display, for all to see, their responses without editing or commentary. If possible, record the relevant reasoning on or near the diagrams. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and to respectfully ask for clarification, point out contradicting information, or voice any disagreement.
The idea to emphasize is that the weights on each side of the third hanger come from combining the weights on the corresponding sides of the first two hangers. If no one points this out, raise it as a point for discussion. Ask students:
- “What do you notice about the left side of the last hanger? What about the right side?"
- "If we saw only the first two hangers but knew that the third hanger has the combined weights from the corresponding side of the first two hangers, could we predict whether the weights on the third hanger would balance? "Why or why not?” (Yes. We can think of it as adding the weights from the second hanger to the first one, or vice versa. If the same weight is added to each side of a balanced hanger, the hanger would still be balanced.)
Standards
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- A-REI.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.
- HSA-REI.C.5·Prove that, given a system of two equations in two variables, replacing one equation by the sum of that equation and a multiple of the other produces a system with the same solutions.