Speeding Up and Slowing Down on a Graph looking at Velocity and Acceleration

We all understand speed intuitively. Velocity and acceleration are more mathy and harder for many students to comprehend, especially on a graph. Even harder is the idea of when an object is slowing down or speeding up. In this post I attempt to unpack both.

Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling (think of backing up). Both cars below are traveling at equivalent speeds.

Velocity can be graphed (the red curve below). Here is information found in the graph.

  • Where the graph is above the x-axis (positive) the car is traveling to the right. Below is negative which indicates the car is traveling to the left.
  • The 2 points on the x-axis indicate 0 velocity meaning the car stops (no speed).
  • The blue lines are tangent lines with slope equal to acceleration (the derivative of velocity). If the line is going down from left to right, the acceleration is negative. Up, positive.velocity graph with tangents

Below is an example of using prior knowledge and visuals to help make sense of speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. When the car is above the x-axis, it has positive velocity. Below, negative. When the car is headed down, the acceleration is negative. Up, positive.

Now for speeding up or slowing down.

  • Think of a bank account balance.
    • If you have a positive balance and a transaction that is negative, Your balance is moving towards 0.
    • Suppose you have a negative balance in your bank account and you have a positive transaction. Your balance is moving towards 0.
  • As the car moves towards a stop sign it will slow down. The speed is getting closer to 0.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0, like the bank balance analogy.
    • Car at top left has positive velocity and negative acceleration so the change is going opposite of the direction. The car is moving towards 0.
  • As the car moves away from a stop sign it speeds up. The speed is moving away from 0.
    • Positive bank balance, positive transaction means balance is moving away from 0.
    • Negative bank balance, negative transaction means balance is moving away from 0.
speeding up and slowing down

Scaffolding Higher Level Math Topics

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The graph shown involves derivatives – a calculus level topic. Before getting into this heavier mathy stuff, consider the title of this post and the other content presented on this blog. Making math accessible to all students is not a special ed or a low level math thing. It is a learning thing. This artifact is what I drew to explain the math concept to a student in calculus to help her grasp the concept as well as the steps. The following are strategies used.

  • color coding – each of the 4 sections written in different color
  • connecting to prior knowledge – the concept of velocity was presented in terms of a car’s speed and direction (forward or backing up)
  • chunking – the problem was broken into parts and presented as parts before exploring the whole
  • multiple representations – the function was represented with a graph, data (1, 2, 3, 4, 5) and a picture (at the bottom)

As for the mathy stuff, the concept of velocity was address by its two parts: speed (increasing or decreasing) and direction (positive or negative). The graph was broken into the following parts: decreasing positive, decreasing negative, increasing negative and increasing positive. Each part was presented with possible y-values (data) and the sign. The most intuitive part is increasing positive which is a car going forward and speeding up.

I find that when I provide intervention, this approach especially by addressing conceptual understanding is effective as the students respond well.

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