All, Some, or No Solutions

5 min

Teacher Prep
Setup
Groups of 2–4. 1 minute quiet think time followed by small-group and then whole-class discussion.

Narrative

The purpose of this Warm-up is to elicit the idea that equations can be true for all or no values, which will be useful when students explore what aspects of equations affect the number of solutions in a later activity. While students may notice and wonder many things about these equations, the number of solutions are the important discussion points.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is how more than 1 value can solve the first equation and no values make the second equation true.

Launch

Arrange students in groups of 2. Display the equations for all to see. Ask students to think of at least one thing that 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 that they notice and wonder with their partner.

Student Task

What do you notice? What do you wonder?

2t+5=2t+52t + 5 = 2t+5

n+5=n+7n+5 = n+7

Sample Response

Students may notice:

  • The first equation has the same thing on each side.
  • Lots of numbers solve the first equations.
  • Something seems wrong about the second equation.

Students may wonder:

  • Are there any values of nn that make the equation true?
  • What is the point of the first equation?
  • Are the 5s related in the context?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary for all to see. If possible, record the relevant reasoning on or near the equations. Next, ask students, “Is there anything on this list that you are wondering about now?” Encourage students to observe what is on display and respectfully ask for clarification, point out contradicting information, or voice any disagreement.

If the number of solutions does not come up during the conversation, ask students to briefly discuss this idea. Move on to the next activity when students hear that the first equation can be solved by more than 1 value for tt and the second equation does not seem to have any solutions.

Standards
Building Toward
  • 8.EE.7.a·Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form x = a, a = a, or a = b results (where a and b are different numbers).
  • 8.EE.C.7.a·Give examples of linear equations in one variable with one solution, infinitely many solutions, or no solutions. Show which of these possibilities is the case by successively transforming the given equation into simpler forms, until an equivalent equation of the form <span class="math">\(x = a\)</span>, <span class="math">\(a = a\)</span>, or <span class="math">\(a = b\)</span> results (where <span class="math">\(a\)</span> and <span class="math">\(b\)</span> are different numbers).

15 min

15 min