Unit 2, Lesson 18
Asking about Solving Systems (Optional)
Algebra 1
18.1
Warm-up
Standards Alignment
Building On
Addressing
HSA-CED.A.4
Rearrange formulas to highlight a quantity of interest, using the same reasoning as in solving equations. For example, rearrange Ohm's law to highlight resistance .
Materials
NoneActivity Narrative
This Math Talk focuses on finding the slope of a line from a linear equation. It encourages students to think about how to rearrange equations to find slope and to rely on the structure of slope-intercept form to mentally solve problems. The strategy elicited here will be helpful later in the lesson when students determine the number of solutions for a system of equations.
To find the slope, students need to look for and make use of structure (MP7).
Launch
Tell students to close their books or devices (or to keep them closed). Reveal one problem at a time. For each problem:
- Give students quiet think time, and ask them to give a signal when they have an answer and a strategy.
- Invite students to share their strategies, and record and display their responses for all to see.
- Use the questions in the Activity Synthesis to involve more students in the conversation before moving to the next problem.
Keep all previous problems and work displayed throughout the talk.
Action and Expression: Internalize Executive Functions. To support working memory, provide students with sticky notes or mini whiteboards.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Activity
NoneMentally find the slope of each linear equation.
Activity Synthesis
To involve more students in the conversation, consider asking:
- “Who can restate ’s reasoning in a different way?”
- “Did anyone use the same strategy but would explain it differently?”
- “Did anyone solve the problem in a different way?”
- “Does anyone want to add on to ’s strategy?”
- “Do you agree or disagree? Why?”
- “What connections to previous problems do you see?”
MLR8 Discussion Supports. Display sentence frames to support students when they explain their strategy. For example, “First, I _____ because . . . .” or “I noticed _____ so I . . . .” Some students may benefit from the opportunity to rehearse, with a partner, what they will say, before they share with the whole class.
Advances: Speaking, Representing
Advances: Speaking, Representing
18.2
Activity
Optional
Instructional Routines
NoneMaterials
NoneActivity Narrative
This optional activity gives students another opportunity to apply what they learned about the features of systems of linear equations with one solution, zero solutions, and many solutions. Plan to use this activity if students need additional time to recognize the importance of slope and intercept in determining the number of solutions for a system.
To answer the first question (a system with one solution), students could write a second equation with randomly chosen parameters. Answering the second and third questions, however, relies on an understanding of what "zero solutions" and "infinitely many solutions" mean and how these conditions are visible in a pair of equations and graphically.
For example, students could reason that in a system with no solutions:
- The two equations have the same variable expressions on one side but different numbers on the other side, and then write a second equation accordingly. For instance, if is equal to 10, it cannot also be equal to 4, so and would form a system with no solutions.
- The graphs of the equations have the same slope, but they cross the vertical axis at different points. Rewriting in the form of gives . A second equation with a coefficient of for but a different constant would have a graph that is parallel to the graph of the first equation, forming a system with no solutions.
Regardless of the form that students use to write the second equation, they need to choose the parameters strategically to achieve a system with the desired number of solutions. The work here prompts students to look for and make use of structure (MP7).
18.3
Activity
Instructional Routines
MLR4: Information Gap CardsMaterials
To Copy (from Blackline Masters)
Linear Systems Cards
Activity Narrative
This activity gives students an opportunity to determine and request the information needed to write and solve a system of linear equations.
The Information Gap structure requires students to make sense of a problem by determining what information is necessary, and then to ask for information that they need to solve it. This may take several rounds of discussion if their first requests do not yield the information they need (MP1). It also allows them to refine the language they use and to ask increasingly more precise questions until they get the information they need (MP6).
This activity uses the Information Gap math language routine, which facilitates meaningful interactions by positioning some students as holders of information that is needed by other students, creating a need to communicate.
Launch
Tell students that they will set up and solve systems of equations. Display, for all to see, the Information Gap graphic that illustrates a framework for the routine.
Remind students of the structure of the Information Gap routine, and consider demonstrating the protocol if students are unfamiliar with it. There is an extra set of cards available for demonstration purposes.
Arrange students in groups of 2 or 4. If students are new to the Information Gap routine, allowing them to work in groups of 2 for each role supports communication and understanding. In each group, give a problem card to one student and a data card to the other student. After reviewing their work on the first problem, give students the cards for a second problem, and instruct them to switch roles.
Action and Expression: Internalize Executive Functions. Check for understanding by inviting students to rephrase directions in their own words. Keep a display of the Information Gap graphic visible throughout the activity, or provide students with a physical copy.
Supports accessibility for: Memory, Organization
Supports accessibility for: Memory, Organization
Lesson Synthesis
To highlight the important concepts of the lesson, discuss:
- “How are the slopes helpful in determining the number of solutions for a system of linear equations?” (If the two lines have the same slope, they are either parallel and the system has no solution or they are the same line and have infinitely many solutions. If the slopes are different, the two lines will intersect at a point and have a single solution.)
- “To determine the number of solutions, would you rather have the equations in slope-intercept form, , or standard form, ?” (I would rather have the equations in slope-intercept form because I can see the slope and vertical intercept which, together, determine the number of solutions for the system.)
Student Lesson Summary
When looking at a system of equations, it is often helpful to first determine how many solutions it has. We can use the slopes of the lines to determine whether there is exactly 1 solution or not. If the slopes are different, that means the lines aren’t parallel and don’t coincide, so they must intersect. If the slopes are the same, we still have to figure out if they are parallel or coincide. To determine whether there are infinitely many or zero solutions, substitute an -value in each equation to see if they have the same associated -value or not. The vertical intercept, where , is an easy point to check.
Although these checks can be done in any form that the linear equations take, it can be helpful to arrange them into similar forms to make the comparison easier. For example, if each equation is rearranged into slope-intercept form, , the slope, , and vertical intercept, , are immediately available.