At lunch the other day a geometry teacher told me that the current sophomores know that absolute value means distance from zero. This was very exciting news since solving absolute value equations was something those same kids consistently struggled with last year. Every time a student forgot how to solve an absolute value equation or inequality I asked them what absolute value meant and told them to draw that on a number line. Every time. It was so repetitive. We did an absolute value problem weekly as a warm up for months. I kept at it. Apparently, they learned it.
This sort of repetition works for a definition. A basic concept. Can we identify several of these building blocks in our courses? If so, we could build in repetition of these ideas (I'm thinking spiraled homework) and narrow our focus. If students can learn these they will experience greater success with higher level tasks. This is not to say that rote practice should come before application but to say that it would be helpful to have a short list of key ideas we can ask students to think back to during problem solving tasks.
- Solve equations by combining like terms, using opposite operations for opposite expressions.
- The solution to an inequality is a region.
- Absolute value means distance from zero.
- Lines are defined by a starting point plus the rate of change.
- Exponentials are defined by a starting point times the rate.
- An exponent tells you how many factors of the base (if you forget the rules, write out the expanded form)
- Multiply (and factor) polynomials using the distributive property.
- Function transformations (adding shifts, multiplying stretches, negative reflects)
- Undefined values are formed by dividing by zero, even roots of negatives, and logs of zero or negatives.
- You always have multiple options when solving a problem, choose strategically (aka, child please, use the graphing calculator wisely!)
Can you phrase any of these better? Which would you de-emphasize? What would you add to the list? What are the building blocks of other courses?