How Do I Solve for m?

Since we were on the subject of manipulating equations this week, I thought I'd share this other tidbit with you...

I've had plenty of students who could recite F=ma and could readily solve for F, given m and a.  But, given F and a, they lacked the understanding of how to solve for m. 

This little trick solved a lot of problems (and even students capable of solving for m enjoyed using this). 

Draw a triangle and divide the triangle into three parts by drawing a T in it (see below).

Now fill in the variables.  In the case of F=ma, the F goes on the top and m and a each go in a bottom section.

To use....
Use your finger to cover the variable you're solving for an "read" off the equation. 

If you're solving for F, cover the F and you'll notice the m and a are next to each other, which means they need to be multiplied to get F.

If you're solving for m, cover the m and you'll notice that you're left with F over a, so you'll need to divide F by a get get m.

And finally, if you're solving for a, cover the a and you'll notice that you're left with F over m, so you'll need to divide F by m to get a. 


Much like the popsicle stick, this trick can work for any three variable equation like density and speed. 

As long as you can remember one iteration of the formula, you can recreate the triangle!

Manipulating Equations

In an ideal middle school classroom, all students would understand how to manipulate simple equations and be able to explain what happens to one variable when another is changed.  But, the reality of the classroom often means doing what you can to help some struggling math students work their way through equations in science class.  For those students, these simple manipulatives may just provide the crutch they need. 

This idea for these popsicle stick manipulatives, to help your students better understand what happens to the different variables in a formula, came from the Bond with James blog, and I found it through Pinterest. 

In its original form, this manipulative is used to help students better understand the ideal gas law.

But, since I never did a whole lot of instruction on the gas laws, I immediately began thinking of the equations I did use with my students that fit this pattern (i.e. three variables). 

The equations that came to mind were:
Newton's second law: F=ma
Density = mass / volume
Speed = distance / time

The manipulative is simple a popsicle stick,  labeled with m (mass), F (force) and a (acceleration).  The biggest trick is get the right letters in the right spots. 

Once the stick is set up, you can put it to work.  For our first scenario, lets say we want to know what happens to an objects acceleration if we decrease its mass, but keep the force constant. 

Because you're keeping the force constant, you'll place a finger over the F.  The stick now pivots around that point. 

Move the m end of the stick downward, to indicate a decreasing mass and observe the a end of the stick rising. 

Therefore, when the force is kept constant while the mass decreases, acceleration will increase. 

For another example....
What happens to the acceleration of an object when we keep the mass constant, but apply less force to the object?

Place your finger over the m, because mass remains constant.  Move the F downward to indicate a lessened force and observe that a also moves downward.

Therefore, when an object maintains a constant mass, but a decreasing force is applied, the acceleration will decrease. 

Here's a manipulative stick for the density equation:

It's used in the same way....
What happens to the density of an object if its mass remains constant but it's volume increases?

Place your finger over the mass, raise the volume end of the stick and observe the density end. 

If an objects volume increases without changing the mass, the objects density will decrease.      And while I don't have a picture of one.... a stick for the speed equation would have distance in the middle and speed and time on either end.      Hopefully with enough practice, your students will begin to internalize these ideas.  And when that happens, they will have a much better understanding of whether or not their answers make sense.