Shouting for Equivalence
The following reactions are no longer a surprise to me as I begin the year with my new 6th grade math students:
In my years teaching, these interactions are common. I have come to expect them, I am just now more prepared to address them.
Shifts in the Standards
My 6th graders have come to me with a concrete rule for equivalence. Students have come to understand the equal sign as representative of an answer, rather than a relationship that shows two expressions as equivalent. I have always known that students should be able to identify equivalent expressions, but with the specific shift in the standards toward understanding equivalence I find myself now directly addressing it.
Apply the properties of operations to generate equivalent expressions. For example, apply the distributive property to the expression 3 (2 + x) to produce the equivalent expression 6 + 3x; apply the distributive property to the expression 24x + 18y to produce the equivalent expression 6 (4x + 3y); apply properties of operations to y + y + y to produce the equivalent expression 3y.
Identify when two expressions are equivalent (i.e., when the two expressions name the same number regardless of which value is substituted into them). For example, the expressions y + y + y and 3y are equivalent because they name the same number regardless of which number y stands for.
Teaching equivalence happens throughout the year beginning with number sense and moving up and building to work with expressions. One of my math trademarks to address equivalence is a call and response chant, where I yell “SAME VALUE,” students respond with, “DIFFERENT FORM.” It can be pretty obnoxious, but hey, it sticks. You can watch it below (click here for a longer clip):
Other ways I reinforce equivalence:
- Have students sort various expressions into piles, each pile showing expressions with equal value
- Let students be creative (and sometimes ridiculous), have them write their own equivalent expressions on the white board (see long clip of video).
- Have students create expressions that have a “target value” using predetermined numbers and operations.
- Have students find their “equivalence partner/group” by matching expressions with equal values.
- Analyze sample work that contain errors, some of which include mistakes in generating equivalent expressions
While I am reminded of the importance of this skill every year, the addition of the Common Core State Standards only confirms and highlights my belief that students’ understanding of equivalence is essential. My students will need too know if equations are true or if various expressions are equal in value as they become more strategic problem-solvers and engage algebraic expressions more deeply in the coming years. They will gain a deeper understanding of equations and equality if I push them to confirm or deny equivalence while also requiring them to follow a set o logical steps to simplify and rewrite numerical and algebraic expressions. Ultimately, I think through a year-long, continually focus on “Same Value, Different Form,” my students will become more flexible and successful as mathematicians.