Truth and Equations

10 min

Narrative

The purpose of this Warm-up is to elicit the idea that a variable represents an unknown number in an equation or diagram, which will be useful when students identify solutions for equations in a later activity. While students may notice and wonder many things about these equations, the fact that the variable xx represents the number 6 is the most important discussion point.

This prompt gives students opportunities to see and make use of structure (MP7). The specific structure they might notice is that the number 6 in the first equation is replaced by the letter xx in all the other equations. They may also notice in the last three equations that replacing each xx with 6 makes the equations true. 

Launch

Arrange students in groups of 2. Display the equations for all to see. Ask students to think of at least one thing they notice and at least one thing they wonder. Give students 1 minute of quiet think time, and then 1 minute to discuss with their partner the things they notice and wonder.

Student Task

What do you notice? What do you wonder?

36=183\boldcdot6=18

3x=183 \boldcdot x=18

3x=183x=18

18=3x18=3x

Sample Response

Students may notice:

  • The numbers 3 and 18 are in all the equations.
  • Three equations have the letter xx.
  • There is no operation symbol in the last two equations.

Students may wonder:

  • What does the letter xx represent?
  • Do all of these equations show multiplication?
  • Do all of these equations represent the same relationship?
Activity Synthesis (Teacher Notes)

Ask students to share the things they noticed and wondered. Record and display their responses without editing or commentary. 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 fact that the number 6 is replaced by the letter xx does not come up during the conversation, ask students to discuss this idea. Explain that a letter used to stand in for a number is called a variable. Throughout this unit, students will have many chances to understand and use this term.

Students may wonder about the 3x3x in the last two equations since it’s likely that this is their first experience with “next to” notation. Explain that 3x3x means the same thing as 3x3 \boldcdot x, and we will frequently use this notation from now on. Explain that in the expression 3x3x, the number 3 is called the coefficient. Use this term throughout the lesson when the need naturally arises to name it.

Standards
Addressing
  • 6.EE.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
  • 6.EE.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.
  • 6.EE.B.5·Understand solving an equation or inequality as a process of answering a question: which values from a specified set, if any, make the equation or inequality true? Use substitution to determine whether a given number in a specified set makes an equation or inequality true.
  • 6.EE.B.6·Use variables to represent numbers and write expressions when solving a real-world or mathematical problem; understand that a variable can represent an unknown number, or, depending on the purpose at hand, any number in a specified set.

10 min

10 min