Lesson and Assessments - What is a variable?

Progression of Skills (outline)

Before the lesson: 

• Students should review basic skills involving variables, including an understanding that coefficients of variables indict multiplication (i.e. 5x means 5 times x) and evaluating an expression when a variable has a value (i.e. if k is 2, then 3k is 6) 

During the lesson:

• This toolkit lesson focuses on developing a conceptual understanding of variables, including that a variable represents an unknown quantity, a variable represents a varying quantity (a number that can change), and that related numbers that change together are variables.

After the lesson: 

• Use their strong concept of variables to begin solving algebraic problems and assigning variable meaning in word problems

Lesson Task 1:

• Using slide 1 of the discussion questions, start by leading a discussion with your students about variables. 
  • This discussion can take many forms depending on the number of students. For small groups lead a conversation, for larger groups have students work in teams to come up with answers to the questions that they all agree with.
• Pass out the “Variables: Sometimes, Always, Never” note catcher. Walk the students through the example statement, the correct answer, and the reasoning. Have students work individually to complete the rest of the problems. 
  • Emphasize that the reasoning is more important than getting the sometimes, always, or never correct.
• Designate three parts of the room as “Sometimes,” “Always” and “Never.” Read a statement aloud, and have students move to the part of the room that they agree with. 
• Have students in each part of the room discuss their reasoning, then have one student from each group share out. Once students have shared, people can move to different parts of the room if they changed their mind. 
  • The goal of this is to reach a consensus where all students are eventually in one part of the room. 
  • If you would prefer for students to not move around, you can provide students with yellow, green, and red pieces of paper to represent “sometimes,” “always” and “never” respectively. 
• Repeat this process for each statement.
• Once finished, have students use a colored pencil to correct any mistakes they made on the note catcher. 
• Progress monitoring: Collect the note catcher to assess student understanding

Anticipate Misconceptions:

• Variables can be viewed as labels (i.e. in the expression 5S, S represents students rather than the number of students)
• Variables can be ignored (i.e. if asked to add 4 to n+2, they would erroneously get 6)
• Variables represent a specific unknown (i.e. The variable t can only have 1 value rather than many values)

Lesson Task 2:

• Set up a long number line across a room with the center labeled 0. (It is not necessary to put any other points on the number line). Print and cut out the variable cards in advance. 
• Pass out a variable card to each student and have them put themselves in the correct order in the number line. (NOTE: This task requires some trust and vulnerability. It is important to build a sense of community before engaging in this task as written. If the math community has not been established,  you could make it more straightforward you could start by asking students to line up in order if given t=1.)
  • As a differentiation strategy, use discretion in assigning variable cards (do not assign them randomly) 
  • For large groups or students who need extra support, have pairs share a variable card
  • Students may realize that there is not one correct order because the variables could have multiple values. If they struggle to come to this conclusion you can ask questions to help them realize this such as:
    • How do you know that you belong there?
    • Is this the only correct answer?
    • What does the variable t mean? What does it represent?
• Once students have realized that there is not one correct answer, ask the students to line up if t is 1. Once they have settled, have students go down the line and explain their reasoning. Make sure that all the students agree with everyone's placement in line.
• Now, ask them to line up if t is 4. Once they move, have the students discuss if they are standing next to the same people or not. Do the same thing for if t is negative 2.
• Repeat this procedure a few more times with students coming up with numbers for t to be. Ask the students to pay attention to expressions for which the order never changes, such as t and t+4. Have students reflect on why some expressions change order and some do not.
• Give students the exit ticket so they can reflect on their learning. 
• Progress monitoring: Collect the exit tickets to assess student knowledge.

Lesson Task 3:

• What situations can you create an equation with a variable for?

• Diana L. Moss, Jennifer A. Czocher, & Teruni Lamberg. (2018). Frustration with Understanding Variables is Natural. Mathematics Teaching in the Middle School, 24(1), 10–17. https://doi.org/10.5951/mathteacmiddscho.24.1.0010
• Sarah B. Bush, Karen S. Karp. (2013) Prerequisite algebra skills and associated misconceptions of middle grade students: A review. The Journal of Mathematical Behavior, 32(3) 613-632. https://doi.org/10.1016/j.jmathb.2013.07.002.
• Knuth, E. J., Alibali, M. W., McNeil, N. M., Weinberg, A., & Stephens, A. C. (2005/02//). Middle school students' understanding of core algebraic concepts: Equivalence & Variable1. ZDM.Zentralblatt Fuer Didaktik Der Mathematik, 37(1), 68-76. https://doi.org/10.1007/BF02655899
• Joan Lucariello, Michele T. Tine, Colleen M. Ganley. A formative assessment of students’ algebraic variable misconceptions. The Journal of Mathematical Behavior, 33, 30-41. https://doi.org/10.1016/j.jmathb.2013.09.001.

• Interpreting Equations lesson from the Mathematical Assessment Project  
• Interpreting Equations is licensed under CC BY-NC-ND 3.0