This post presents a scaffolded and meaning making approach to exponents that are 0 or negatives.

Handout

The slide show below presents all 4 pages.

The handout starts with an initiation to preview the prerequisites for what is presented in the lesson. It also introduces a chart that will be used for discovery.

Page 2 presents a discovery activity of following a pattern of dividing by 2 down to the 0 exponent. The concept of exponents is presented as the number of occurrences of the base. This leads to the idea of a 0 exponent indicating the base is no longer present, but there is still 1.

Similarly, on the 3rd page the pattern of dividing continues into negative exponents to show the resulting fractions. The negative exponents are then presented as reciprocals.

For terms with multiple factors (e.g., 5x vs just x) the students are presented steps to write the factors separately. This unpacks the reason why the negative exponent acts only on one of the factors (unless both are grouped with parentheses).

I had an interesting discussion through a Facebook post recently regarding concepts vs skills. I want to share some information I have gathered regarding this topic. I do so, because there were a substantial number of teachers advocating for skill based learning. I hope to initiate some meaningful discussion.

Below left is a photo of an information processing model presented in a graduate level course on learning I took at UCONN. A key element I want to highlight is that information is more effectively processed if the information is meaningful. A theory behind this is Gestalt Theory in which the brain want to make information meaningful or organize it, e.g., the closure model in which our brains complete the triangle in the middle of the circle portions.

The meaning underlying math skills originates in the concepts. Below are the definitions for both, with the concepts being the “how or why” underlying the skills which are the “what to do” part.

I am not arguing that skills are unimportant or that rote practice is wrong. My position is that the concepts should drive the process. Here is a cartoon I think highlights the challenges with students having only skill based knowledge for topics that have important underlying concepts. I witnessed this first hand as a college adjunct instructor and found that a substantial number of students only understood slope by its formula. I also see a substantial number of students receiving special ed services who are taught at a skill level only to allow for progress. Often this is challenging for them when they have working memory or processing issues.

I will summarize in my own words an interpretation an article I read on the definition of Math, which stated there is no singular definition. The following was a theme that appeared to emerge. Math is a set of quantitative related ideas that can help explain the phenomena and the world. The mathematical symbols are used to represent these ideas. There are different ways to represent these ideas, e.g., we represent functions with tables, graphs, and equations. Formal proofs in Western Civilization are not the same a those in the East. Computer based proofs are not fully accepted by many math experts.

Technology has provided amazing ways to represent mathematical ideas. The most genius approach I have encountered is Dragonbox. The image below shows their initial representation of an equation through their algebra app. It develops the concept and the skills simultaneously.

Below is a list of some algebra 1 topics and some of the associated concepts. These are largely derived from math sources but include some massaging by me. I am happy to hear the working definitions of others.

The scatterplot above is an approach I use to introduce systems of equations. Here is the process I use. (Note: I have found that students like math associated with buying a car – relevant, real life application for them.)

In my class, students would have seen a scatterplot with mileage and price for a single car. I explain that we will now compare two cars.

To review, in a do now or initiation at the start of class I would have one group generate a scatterplot for the Toyota Camry data and the other groups, Mustang (Excel sheet for all of this note: this data is old). Then they would share with each other

We would revisit the relationship shown and revisit the idea of depreciation.

I show a Camry and Mustang and ask two questions: Which car do you think costs more brand new? Which do you think depreciates faster and why?

Then I show them the scatterplot above and ask which car has higher dots at the far left? Explain what this means (Mustangs start off with a higher price). Then I ask about the dots at the far right.

The students are then asked to estimate when the cars have approximately the same value.

Then I present scatterplot below, with lines of best fit (trend lines) and they are asked the same question. We estimate the specific mileage and price and write as an ordered pair.

Finally, I explain that this is known as a system of equations and the ordered pair is the THE solution. The entire unit will focus on finding an ordered pair as a solution.