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. 

Salt Water Painting

Make a batch of super saturated salt water*.

Fill a shallow pan with the salt water.  Cut pieces of paper smaller than the pan, any shape you like.

Drip food coloring onto the water and swirl with a toothpick.


I'm not sure why the red dye looks so metallic in the photos...


Quickly place a sheet of paper on top of the coloring.  When the paper is wet, pick it up and lay flat to dry.

The salt water is much denser than the food coloring, so the coloring floats on the surface, at least long enough to complete this project.  Given time, the food coloring will disperse in the salt water.

*Fill a jar part way with hot water.  Add some salt and stir.  Keep adding salt and stirring until no more salt dissolves (you see the salt sinking to the bottom).  Let it sit for a few hours.  Then pour off the clear water - the super-saturated salt water solution.

Denisty: Candy Density

This is really just another way to get your students to practice measuring and calculating density.  But, it involves the use of candy, so it's more interesting than your run-of-the-mill density calculation.

The original version of this activity uses Whoppers, Lemon Heads and jelly beans.  You could certainly modify that to use whatever you have available and/or is on sale.  However, you will want to make sure you have a variety of densities present in your candy selection.  The Whoppers are nice, because they will float.

You'll want students to use 3 pieces of each candy, so they can average their data.  (This could be done in groups, to save time and candy usage).

First have students find the mass of each piece of candy.  Then they'll find the volume via water displacement.  Because the Whopper floats, they'll need to use the tip of a pencil to push it down, so it's just below the water surface.  Finally, they can calculate the density.

If you wish to take it a step further....
After calculating the density of individual pieces of candy, have them calculate the density of all three pieces at the same time.  The mass and the volume will each be lager than they were for the individual pieces, but the density will remain the same (assuming all measurements and calculations are made accurately).  It's a good opportunity to remind students that density is an intensive property, not dependent upon the amount present.  And, by making it a hands-on reminder, your students are more likely to remember it!

Density: Sink or Float: Candy Edition

Same idea as the original Sink or Float activity, but it uses candy instead.  Again, it's good for the preschool and early elementary set.  (I've got a candy density activity for older students coming tomorrow). 

Gather an assortment of candies.

Hypothesize which candies will sink and which will float.  Divide the candy into appropriate piles. 

Then test!  You may want to test the candy first with the wrapper on and then with it off - why do some candies float when they're in the wrapper, but sink when they're unwrapped? 

After seeing which candies float and which sink, you may want to slice or break some of them open to look at the inside.  See if there are any clues to help you figure out what some float!

The Ivory Soap "Explosion"

I absolutely love it when I learn a new science trick.  At this point, I've seen quite a few, and while there are always new-and-improved versions out there, it's not too common for me to come across something brand new.  Which is one of the reasons I love this demonstration (and it's just SO cool).  It's apparently a well-known demonstration, but I've missed it up 'til now.  (And that's another reason why I make a point of sharing the "classic" science experiments that so many have already seen - everyone has to learn about them for the first time, some time). 

On with the demonstration....

Begin with a bar of Ivory soap (or you may want to use a sliver of soap.... you'll see what I mean). 

Make the appropriate observations of the soap. 

Place the soap on a microwave-safe plate. 

Make a hypothesis* about what will happen when the soap is heated in the microwave.

Now heat the soap in the microwave - set the time for 2 minutes, but keep an eye on it (you'll be doing that any way, trust me).

Observe the soap carefullly.

The soap will expand to a huge volume.  If you use whole bar of soap, it will nearly fill the microwave!  Great wow factor! 



It deflated a little at this point, because my camera's batteries died at this point and I had to wait for them to recharge.



Why does this happen....

Remember when the Ivory floated because it had more air in it than the "other" soap?  When the Ivory is heated, the soap softens and the air bubbles expand.

How can you use this in your science class?

1. A follow-up to the previously mentioned density experiment.
2. A discussion of gas laws (Charles Law, specifically) - when a gas is heated, its volume will increase. 
3. A lesson on physical and chemical changes.  Explosions are chemical changes by definition.  This demonstration looks like explosion, but it's not.  It's just a physical change. 

*A funny story - I told my 5 year old that we were going to do a science experiment after dinner.  He asked what we were going to do and all I would tell him is that we were using soap.  Then I asked if he had any hypotheses about what would happen to the soap (knowing absolutely nothing about what we were going to do to it) and he said "It's going to explode."  I think he was a little surprised at how close to right he was!

Density: Ivory Soap vs. "Other" Soap

Another simple demonstration involving density....

Show the students a bar of Ivory soap and a bar of any other brand of soap.  Ask for predictions about what will happen when the bars are placed in a tub of water.  Float or sink?  Will the both do the same thing, or will they do different things? 

Then put the bars in the water, using as much flair as you deem necessary.

The Ivory soap will float, but the other soap will sink. 

Of course, now the work begins and it's time to hypothesize why the difference.

I'm sure your students will come up with all sorts of possibilities (the more the better), but the reality is that Ivory whips more air into their soap a they're making it.  More air pockets = lower density. 

You could have the students calculate the density of each of the bars.  If you're using nice rectangular bars, the volume can be calculated using dimensional measurements.  If you're using a funky shaped bar of soap, as this bar of Zest is, you'll need to use water displacement to get the volume.

Density: The Sugar Density Column

Did you know that you can change the density of water by adding sugar to it?  Did you know that you can actually create layers of sugar water that have different densities? 

I've seen the Flinn Version, How Sweet It Is, numerous times, and it's very cool. Unfortunately, it's not very practical for me.  I'm not in a classroom, so I don't have balances readily available, nor do I have access to ring stands and separatory funnels.  I've always figured I could find a way to recreate the experiment to make it work for me, but I've just never made it that far down my list of things to do.

And then I found this version, which does not require any of the aforementioned equipment.  (I have adapted it slightly to include two additional colors).  And, if you're just planning to make the solutions ahead for demonstration purposes, it's faster.  In fact it's perfect for doing at home.  It's also simple enough for young students to help with. 

Now, before I go on to show you how simple it is, let me point out that for older students the Flinn version may be superior:
1 - It's always good to practice using equipment to make accurate measurements.
2 - In the Flinn version, students find the mass of the sugar, which allows them to calculate the actual density of each solution.  You could also have them calculate the sugar concentrations.

Enough talking, on with the fun!

Line up 5 glasses.  Add sugar to the glasses as follows:

Glass 1: no sugar
Glass 2: 1 tablespoon
Glass 3: 2 tablespoons
Glass 4: 3 tablespoons
Glass 5: 4 tablespoons
Glass 6: 5 tablespoons

Add 4 tablespoons of water to each glass and stir to dissolve the sugar.  Make sure the sugar in each glass is completely dissolved.  If you need to add water to one glass, you'll need to add an equal amount of water to each of the other glasses.

Add food coloring to the glasses, a total of 2-3 drops per glass, as follows:

Glass 1: red
Glass 2: red + yellow
Glass 3: yellow
Glass 4: green
Glass 5: blue
Glass 6: blue + red

To make the column:

Pour the purple solution into a tall, colorless glass (or a graduated cylinder if you have one). 

Hold a spoon over the glass, near the top of the purple solution, and pour the blue solution slowly over the back of the spoon.  This technique will minimize the mixing of solutions.

Using the same technique, add the remaining solutions in the following order: green, yellow, orange and red.

As you can see, I haven't yet perfected the pouring technique, but it isn't completely muddled either.  I think I could have gotten a better rainbow if I had tried again immediately after doing this one, but I decided it wasn't worth using a bunch more sugar just to capture a better photo.  That said, I think my rainbow looked better than the above photo shows - I just couldn't get the light right to show all the colors. 

If you're doing this as a demonstration, you very well may want to make sure you have enough solutions to give yourself a test run before the actual assembly.

Density: Penny Boats

How can you make a penny float?

Each student gets a 6" square of aluminum foil, with which he/she designs a boat that can carry a penny payload. 

The goal is carry a larger payload than anyone else.

There are several ways to present this activity:

--Each student gets one square of foil to make their boat and they compete with that original boat.
--Each student gets one square of foil, as well as access to water and pennies to test.  They then get a second square of foil to make their competition boat.
--Each student gets one square of foil, as well as access to water and an alternate payload.  They then get a second square of foil to make their competition boat.
--Each student gets a square of paper to use for design purposes.  They then get a square of foil to make their competition boat.

Depending upon how you choose to present this, you can make it a single day activity or a two day activity.

Does Clay Float?

Ask your students if they think clay will float or sink in water. 

Take a lump of clay and drop it into a beaker of water.  It sinks.

Give each student a similarly sized lump of clay and challenge the students to come up with a shape that will allow the clay to float.

Density: Sink or Float?

In its most basic form, preschool students can complete this activity, but it can be a worthwhile experience for elementary students and can be made a bit more challenging for middle school students.

Provide students with an assortment of objects (make sure some float and some sink - some good floaters are aluminum foil, ping pong balls, candles, eye droppers).

Students hypothesize which objects will float and which will sink.  Older students can write down their hypothesis, while younger students can simply divide the objects into two piles.

Students then test the objects in a tub of water.

Older students could be challenged to calculate the actual density of each object. 

How Does That Work: Magic Marble

A "marble" that hovers right in the middle of the cup.  Can your students explain it? 
{It's all about density and immiscibility}

Fill a cup between 1/3 and 1/2 full of water.  Add a drop of two of food coloring.

Use a dropper to place a couple of blobs of cooking oil on the water.

Tilt the cup and pour in rubbing alcohol.  (You want to pour the alcohol down the side of the cup, so you don't mix the water and alcohol and also so you don't disturb your oil blobs).

Add a drop or two of food coloring to the top of the cup, if needed.

[If you wish for something a bit more permanent, create your magic marble in a jar and cover with the lid.  


To keep your "marble" hovering in the liquid, don't shake or disturb the cup.  If you're curious what happens if the alcohol and water are mixed, make a hypothesis and have at it!


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How Does That Work is a series of products and demonstrations that you can present to your students and challenge them to explain the science of how they work. Make sure you decide ahead of time what you'll accept as a valid explanation - can it be printed straight off the internet, written in the student's own words, or does the student need to be able to explain it to you conversationally? What will a valid explanation earn the student - a prize, extra credit, a feeling of goodness?

Layered Water

This is a lovely demonstration of the way water's density differs with temperature and convection currents (you don't actually see the currents, but you see the end result).

Allow a pitcher of blue-colored water to cool in the refrigerator overnight. 

At demonstration time, prepare a pitcher of hot tap water.  Color this water yellow.

Fill a jar (or cup) all the way with blue water.  Fill an identical jar all the way with yellow water.

Place an index card on top of the blue jar.  Carefully turn the jar over and set it on top of the yellow jar - make sure the rims line up.

Ask for hypotheses as to what will happen when you slide the card out.  Slide the card out -- the blue water sinks, mixing with the yellow, creating green water.

Now try again....
Prepare the jars in the same way.  But, this time, place the index card on the yellow jar, and place the yellow jar on top of the blue jar.

Remove the card and watch.....

You'll get a little green water right at the interface, but the yellow and blue water will mostly remain separate.


Why did you get two different results? 
Cold water is denser than hot water - it sinks.  When the cold water was on top of the warm, it sank to the bottom of the vessel, mixing with the warm, as evidenced by the mixing of colors.

When the cold water was on the bottom, it was content to stay right there.  Just a little mixing occurs right where the two temperatures meet.  What do you think would happen if you allowed it to sit for awhile?  Would the colors remain separate, or would they eventually mix?

Density Meltdown

This one is so very simple, but you can learn several things about density while watching.  

Fill a jar with vegetable oil.  Drop in an ice cube.  Watch what happens.

PS My apologies for not using an ice cube made of colored water.  But I think you can still see what's going on.