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!

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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. 

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End of School: By Golly, By Gum

I didn't have any on hand for this picture, but you'll want to make sure you include a super-sugary gum in your testing (Bubblicious, Hubba Bubba, etc.)

By Golly, By Gum is one of the free sample activities from AIMS. 

Students find the mass of a pack of gum, before and after chewing.  Most of gum's mass comes from sugar, which dissolves during the chewing process, so after chewing for a bit, the mass will decrease significantly and students can calculate a ratio.  Students can then compare different brands/varieties of gum looking at both the ingredients and the mass ratio. 

My students were so excited by the novelty of being allowed to chew gum in class that they very willingly carried out the measurements and calculations, despite the impending vacation (if I remember correctly, I used it before Christmas vacation, but the mindset is the same). 


In a similar vein, Tracy, from The Science Spot, has a Bubble Gum Physics activity in which students calculate the speed with which they can chomp gum.  You could easily tie this in with By Golly, By Gum and have your students calculate their speed while doing the chewing necessary for the mass measurements.

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Speed/Velocity: Which Goes the Fastest?

Here's a fun way to practice speed calculations. 

Gather a bunch of self-propelled vehicles.  They can be wind-up or ones you have to pull back and release.

Use masking tape to make a starting line and a finish line 1 meter apart.

Wind up (or pull back) the vehicle and place it at the starting line.

Let go and start a stopwatch.  Time how long it takes for the vehicle to reach the finish line.  Record the time and determine the speed the vehicle travelled. 

The obvious unit to use in this activity is m/s, but you may want to challenge some of your students to covert it to other units: m/h, cm/s, etc. Or, really test them by having them convert to miles/hour!

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