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

Place Value Representation

Here is an easy way to create and implement strategy to unpack place value for students (created by one of my former graduate students). I suggest using this after manipulatives and visual representations (drawing on paper) in a CRA sequence. It is hands on but it includes the symbolic representation (numbers). Hence is another step before jumping into the mathy stuff.

pace value representation just numbers.jpg

The focus can shift to money as well.

2017-06-07 18.06.49

Concepts vs Skills – Need Both

In general math is taught by focusing on the steps. Conduct a Google search for solving equations and you will see the steps presented (below). You need a video to help your student understand solving and you typically get a presenter standing at the board talking through the examples. (I’ve posted on my approach to solving equations.)

When the math is taught through the skill approach the student may be able to follow the steps but often does not understand why the steps work (below). The brain wants information to be meaningful in order to process and store it effectively.

calvin hobbs toast

To help flesh this situation out consider the definitions of concept and skills (below). Concept: An idea of what something is or how it works – WHY. Skill: Ability” to execute or perform “tasks” – DOING.

definition concept
definition skill

Here is how the concept first approach can play out. One consultation I provided involved an intelligent 10th grader who was perpetually stuck in the basic skills cycle of math (the notion that a student can’t move on without a foundation of basic skills). He was working on worksheet after worksheet on order of operations. I explained down and monthly payments then posed a situation shown at the top of the photo below. I prompted him to figure out the answer on his own. He originally forgot to pay the down-payment but then self-corrected. Then I showed him the “mathy” way of doing the problem. This allowed him to connect the steps in solving with the steps he understood intuitively, e.g. pay the $1,000 down payment first which is why the 1000 is subtracted first. Based on my evaluation the team immediately changed the focus of this math services to support algebra as they realized he was indeed capable of doing higher level math.

solving equation with conceptual understanding first

Busy Engagement vs Intellectual Engagement

Owls are symbols of intelligence but the purported reasons are based on the appearance of awareness and the deft hunting skills. It is claimed that the appearance and skill sets are confused with actual wisdom.

owl

I find a parallel between the perceived wisdom of the owls and the perceived learning of students. Through my years in education I have seen teachers praised for their student centered activities. The students may be energetic and on task by an activity which is often considered a touchstone for learning. What is often missing is independent assessment to determine actual learning.

Once I was covering a class for a teacher widely praised for his activities and multimedia activities. In the class I covered the students were taking a test. It was clear that the majority of the students were hesitant about their performance. Several were looking around, one pulled out a phone and a couple looked at other people’s paper. Very few were locked in on completing their test.

I am not suggesting that multimedia or student centered activities are ineffective. My point is that there is a perception that such activities are inherently effective and reflective of actual learning. There is a difference between being intellectually engaged and being busy. The owl deftly executes action and skill but that does not indicate higher level functioning. Conceptual understanding requires more than simply being engaged by activity. Hopefully this is food for thought.