Situations that sometimes, always, or never happen.

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SWBAT determine when an equation has no solution, one solution, or an infinite amount of solutions.

Big Idea

Exploring ideas or relating ideas to real life situations improves learning. Students in this lesson learn to recognize the nature of the solutions to an equation.

Launch (Exploration task)

15 minutes

The lesson begins with a challenging Launch task (Launch Exploration). This means that the productive struggle will begin early in the lesson. Because of this, as I walk around I motivate everyone to try and come up with solutions independently, without relying on help from another student or from me. When I see a student reaches a correct answers, I will also ask that he/she be discreet, so others can arrive at answers on their own (See my Reflection Allowing others to work independently).

As students work, I walk through giving careful guidance, especially helping those who seem not to have a clue of where to start. But I motivate them to persevere and those who do, will soon find that the answers can be figured out substituting values for the variables and solving for x.

Students may not know how to deal with parameters a, b, c, and d because algebra is still relatively new to them, they may be uncomfortable with sentences that relate variables or expressions.  If I see that struggling isn't leading anywhere, only then I resort to peer collaboration asking those students who are making progress to help out those that are not. By trial and error, students should eventually come up with example equations that satisfy all three scenarios:

  1. There is exactly one solution any time a ≠ c
  2. There is no solution when a = c and b ≠ d
  3. There are an infinite number of solutions when a = c and b = d

I end the Launch section by asking volunteers to demonstrate and explain their work on the board for each case.




New Info

20 minutes

A common belief among students is the idea that every math problem has one answer. The Launch task should raise questions about whether or not this is true. Algebra can be used to model real situations where there is no possible solution, one solution, or an infinite number of solutions.

The New Info 2 Situations slides present problems involving real world situations.  My plan is to show one slide at a time asking the class to read the task and then work in groups to solve it. I allow students to use any method to find the solution and walk around giving guidance. Some will try graphing while others will use algebraic methods or trial and error. I encourage students to try solving algebraically hinting that the variable be the number of years worked. In this problem, students should find that the two salaries will never be equal. Job 1 will always pay higher. After I see that most students discover that there is no solution, I ask students that have solved algebraically to write their work on the board. Setting the equations equal to each other, students will find that the variable is eliminated when solving, and the quest for the solution is frustrated. At this point I ask if any other students tried to solve it differently and explain their results. I expect students to say the two jobs will never earn the same salary and that job 1 will always pay more. 

I then show slide 2 and ask students again to read carefully, study the diagram and work together to solve the problem. In this case students will almost certainly write two equations equal to each other and solve algebraically. Once done, I call on a student to write their work on the board beside the work of the problem 1. Again, the variable will be eliminated, but this time ending with a true equation, 7 = 7. I then ask that students try and substitute different values for x in the task. They all should find that the perimeters are the same no matter what value substituted. 

To finalize this section, I ask that students observe both problems written on the board and compare these, especially looking at the final sentence in both problems. They should be able to see when an equation has no solution and when there is an infinite amount of solutions. 






15 minutes

Next, I hand out a copy of the Application worksheet to each student. The students will continue to work in pairs. On this worksheet, I ask that they show all work.

In Question 1, students could refer to the Launch section of the lesson to answer each part. 

Question 2 reinforces the idea of no solution when an equation ends with a false statement and variable is eliminated. 

Question 3 is a similar problem to the first slide in the "New Info" section where there is no solution to the situation. Students should set up an equation a

Question 4: The equation has one solution.

Problem 5, part b asks that students change the problem such that its solution is not feasible for the problem. Students should understand that equations may have one solutions, but in real world situations, this solutions may not be possible. If students have difficulty with this question, I may ask to think about what numbers are not possible if we are dealing with number of people.  Students will probably recognize that negative numbers are not feasible for the problem and change the values around to get a negative solution for x.


Closure (Generalization)

5 minutes

As we try to generalize what we have explored during this lesson, I want my students to do most of the talking (or writing).

For this lesson, and after the application problems are reviewed, I ask students to write a paragraph in the back of their Application Worksheet. The paragraph should cover two things:

1) What did we learn today? 

2) If you had to make a very brief "cheat slip" for a quiz on today's lesson, what would you write?

Like all closure activities, the students' responses will tell me if how to proceed following this lesson. 


This Homework assignment is good practice. Students will soon know before solving, the nature of the solutions to an equation. 

Common error to be expectedIn an equation like  3x + 5 = -3x - 6, students may "jump the gun" and state that the equation has no solution when in fact it does. The see the opposite sign of the coefficients 3 and -3 and think that the variable will be eliminated when in fact you end up with 6x or -6x on one of the sides. I always warn students of this common error.