## Slope is one of the most important topics in algebra but perhaps one of the most challenging topics. One issue is that students often think of the formula as opposed to the idea of slope. The images shown below are for a handout that unpacks the idea or concept of slope.

### Handout

The images show sections of a handout. Slope is a measurement of a graphed line. It measures steepness and also indicates direction. The handout starts by drawing on prior knowledge of measurements for newborns.

Prior knowledge of steepness and up and downhill are invoked.

How to measure steepness is introduced through a focus on stairwells.

The measurement of the length and height of steps leads into the rise and run of a hill. I use “right” and “up” or “down” which focus on direction. The terms “rise” and “run” can be addressed subsequent to the introduction. This part of the handout is scaffolded to reduce task demand and focus on the concept.

Finally, the students are provided a line and apply the ratio procedure. Note: the slopes are not the same as in the prior section.

### Jamboard

An accompanying Jamboard will eventually be shared in this space.

## Intro to Slope as Rate of Change

Slope may be the most challenging concept to teach in algebra yet it is one of the most important concepts. I use the following sequence to introduce slope: rate of change, rise over run, rise over run as a rate of change. The first photo is a map of Manhattan with directions on counting city blocks. This builds on prior knowledge to introduce rise and run.

The photo below builds on the map and transitions students into coordinate planes. They are introduced to rise over run and positive and negative as indicators of the direction of a line.

The photo below combines rise and run with rate of change. The hourly wage is prior knowledge they can much more easily comprehend. A major issue is getting students to include units and to understand what units are. This would have been addressed in the previous unit on rates and proportions.

This is the handout used with a train activity in which I use battery operated trains and time them as they travel 200″. I project a stop watch on the screen as the train moves. The kids pick up right away that Percy is “slow.” As Percy is traveling I ask them how they know it is slow and get answers like “it takes a long time.” This is a concrete representation which they can draw upon as they work with the graph and calculations.

The photo below shows a scaffolded version of a Smarter Balance (Common Core assessment) test question. The original question simply shows the graph and asks for average rate of change from 0 to 20 years. Even with the scaffolding many problem areas appear: units vs variable (student wrote “value” as opposed to \$), including \$ with the 1000, finding unit rate, and even identifying the part of the graph at 0 years.