Tag Archives: car

Velocity, Acceleration, Speeding Up and Slowing Down

DISCLAIMER: This is a very mathy, math geek post but it also has value in demonstrating instructional strategies and multiple representations.

We all understand speed intuitively. Velocity is speed with a direction. Negative in this case does not indicate a lower value but simply which way an object is traveling. Both cars below are traveling at equivalent speeds.

velocity vs speed

The velocity can be graphed (the red curve below). 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). (I will address the blue line at the end of this post as to not clutter the essence of what is shown here for the lay people who are not math geeks.)velocity graph with tangents

Below is an example of using instructional strategies to help make sense of the graph and of velocity, acceleration, speeding up and slowing down.

As stated previously, the points on the x-axis indicate 0 velocity – think STOP sign. As the car moves towards a stop sign it will slow down. When a car moves away from a stop sign it speeds up.

The concept and the graph analysis are challenging for many if not most students taking higher level math. This example shows how instructional strategies are not simply for students who struggle with math. Good instruction works for ALL students!

speeding up and slowing down

It is counterintuitive that when acceleration is negative the car can be speeding up. The rule of thumb is when acceleration and velocity share the same sign (+ or -) the object is speeding up. When the signs are different the object is slowing down. This rule is shown in the graph but the stop sign makes this more intuitive.

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Conceptual Understanding Before Getting “Mathy”

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All too often math topics are introduced first with the skills and steps. This is backwards. The photo above shows how I introduced solving equations a high school student with autism using the concept as an entry point.

We discussed what was involved in buying a car, including payments (no interest) then I posed the problem seen at the top. I asked him to figure out the monthly payment. He worked out the problem, overlooking the down payment. With a minimal prompt he self corrected. I followed this by “showing him the mathy way of doing the problem.” (Seen in the bottom half of the photo). He conceptually understood why the -1,000 was the first step and x had meaning.

This is a version of CRA.

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Scaffolding for solving equations

 

 

 

Solving equations can be very abstract and¬†inaccessible¬†for students. The photo below shows a scaffolded handout that introduces students to solving two-step linear equations. The students are introduced to buying a car: down payment and monthly payment. This provides meaning for the variable (x is the monthly payment), for the constant (down payment) and coefficient (12 months – fabricated situation). This approach introduces the concept of having two steps and for which to choose. Take note of the little “PEMDAS” on the left side. I explain that solving involves using PEMDAS in reverse

 

scaffolding 2 step equations with buying a car

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