I am a BIG fan of using hands-on, concrete experiences for students to increase depth of understanding. And in my experience one of the toughest areas for students in MS and HS is equations involving variables. In TX students experience variables as a placeholder for an unknown number beginning as early as 3rd grade. In fact, the standards specifically mention variables in grades 4 & 5. In middle school, students work to solve for the variable's value using algebra skills rather than problem solving methods. While reasoning through 1-step, 1-variable equations in grade 6 should help reinforce the concept of working backward through the order of operation steps, often students hit a concept wall in 7th & 8th grade, perhaps due to the abstractness of variables and multiple steps in the solution process. When teachers scaffold students using concrete experiences, students are better equipped to understand the abstract presentation of equations and how to solve them using inverse operations. The focus of this newsletter is to provide some ideas on how to make equations and variables, and their operations, more visually evident for students using tiles or similar manipulatives.
A Few Teaching Thoughts
- Start with the concrete, no matter what level students are in, including upper high school.
- Build in time for physical to pictorial modeling. This is an important step to getting students to form images in their mind when the models are gone.
- Include mathematical connections to the procedures. If we don’t show and talk about the connections, students may not make them. This includes things like multiplying through by a negative, and adding zero pairs.
- Link the pictorial to the abstract mathematics going on behind the scenes.
Develop operations with variable expressions in the same way, building from concrete to pictorial to abstract (absence of models)
- Solving 1-, 2-, and multiple step equations, even those with variables on both sides.
- Extend equations to inequalities using the same processes.
- Adding and subtracting with polynomial expressions.
- Multiplying monomials and binomial expressions.
- Factoring and/or dividing binomials and trinomials.
- Extending division of polynomials to solving using completing the square.
I appreciate that you took the time to do some hands on math with me today. Be sure to check out other newsletters and my pre-made Math Cut Ups activities at www.mathcutups.com. And reach out anytime!
Kelli D. Mallory, Ed.D.