Showing posts with label Math in Science. Show all posts
Showing posts with label Math in Science. Show all posts

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. 

Let Us Weigh Lettuce

An easy lesson in measuring mass, collecting data, graphing (if you wish), percentages and plants. And a great experiment to start at the beginning of the school year.

You'll need a leaf of lettuce and a balance.  The precision of an electronic balance is nice for this particular activity, if you have one available.

If you have a balance that can remain dedicated to this activity, you can place the lettuce leaf right on it.  Record the mass.  Each day when the students come to class, they should record the mass of the lettuce.  Continue recording the mass every day for a month.

[If you cannot dedicate a balance to the activity, you'll need to first find the mass of a weighing paper.  Record that, then place the lettuce on the weighing paper and record that mass.  Lift the paper with the lettuce on top and keep in a safe place while the balance is being used elsewhere.  Return the paper and lettuce to the balance each day to find the mass.  You'll have to subtract the mass of the weighing paper from each measurement to get the mass of the lettuce.]

Once you've collected all the data, you can graph it if you wish.  Is the water lost at the same rate throughout the month or does it change?

You can also determine how much of lettuce (by mass) is water.

Mass of lettuce at start - Mass of lettuce at end = Mass of water

(Mass of water / Mass of lettuce at start) * 100 = % of lettuce mass that was water

If you've caught your students' attention with this one, you can proceed to follow the same procedure to find the water content in other items.  Maybe your students will want to compare the water content in different types of lettuces or different types of leaves or different types of fruits or vegetables.  Lots of possibilities - you could have something going every month of the school year!

Moles: Challenge!

After discussing what a mole is (6.02x10^23 things), challenge students to bring in a mole of something.

Some examples to get you started...

A mole is...
...58 g of salt
...18 g of water


This is a good activity to do for Mole Day (celebrated October 23 - sorry to be late in sharing, you'll have to save it for next year!).

I would leave this as an extra credit opportunity for my students, as it's really beyond some of them. But, if you work with older or higher level students, go ahead and make them all do it!

Weather: How Much Air is Pushing on You?

The air all around you is filled with molecules, all of which exert pressure on you. 

Picture this...
You're standing upright.  Rising straight up from your head, into the furthest reaches of the atmosphere, is a column.  This column is filled with air molecules.  While the effect of each individual molecule is miniscule, their combined effect is a force with which to be reckoned.  How much atmospheric weight do you think your head has to support?  Go ahead, take a guess...

First we need to find out how large your head is.  For the purposes of this activity, we're going use inches so we can get an answer in pounds.  It's rather un-scientific of us, but it will provide us (in the U.S.) with the greatest understanding.

Back to your head.... find the circumference of your head, using either a fabric measuring tape or a length of string that you then lay against a meter stick.  I come up with 22 inches.

Now you'll need to do some math to find the radius.  Circumference is equal to 2 x pi x radius.  So, to get the radius, you'll need to divide the circumference by pi and then divide that number by 2.  For me, it's 3.5 inches.

Now you'll use the radius to find the area of the top of your head.  Area is equal to pi x radius x radius.  For me it's 38.47 square inches. 

Atmospheric pressure at sea level is 14.7 psi (that's pounds per square inch), and while I don't live exactly at sea level, that number will work well enough for our purposes.  So, the area of my head multiplied by atmospheric pressure gives me the weight of air pushing on my head.  In my case, it's 38.47 square inches x 14.7 psi = 565 pounds. 

Pretty unbelievable, isn't it?  But it's true.  We aren't aware of it because we're used to it, we've never known anything different.  And we aren't crushed by that force because there are fluids inside our body exerting pressure that keeps things balanced.  Those air molecules are pushing on all sides of your body, not just on top of your head, which also helps keep things balanced.

If you're interested, atmospheric pressure in Denver, with an approximate altitude of 1 mile, is 12.2 psi.  You might want to have your students determine how much the atmospheric weight changes as they go from sea level to 1 mile.

Air Pressure: Suction Cup Drink Holder

Education Innovations sells these neat rings that work like a suction cup to hold your drink. 

You slide a can or bottle into the ring, set it on a smooth, flat surface and it sticks! 

While not terribly expensive, it's simple enough to make your own version.  You'll need a flexible, stretchy material.  I've found that the plastic jar grippers, sometimes given away by companies, work well.

The original Lil' Suctioner has a radius of just over 2 inches.  While I wouldn't go smaller than that, it certainly wouldn't hurt if yours was bigger. 

Cut about a one inch hole in the center of your material. 

Slip the material over a can or bottle and test it out for yourself. 

By the way, the Lil' Suctioner includes some air pressure facts, including the weight of the atmosphere pushing down on it (~221 lbs).  But, there's no reason you can't figure it out for your own drink holder.  If it's a circle, measure the radius; if it's a rectangle, measure the length of the sides.  Make all measurements in inches. 

Calculate the area of your material (for a circle: pi x radius x radius; for a rectangle: side x side) and multiply that area by 14.7 psi (pounds per square inch).  That will give you the number of pounds of the atmosphere pushing down on your drink holder, of in other words, the pounds of force you'll have to exert to lift up your drink.

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!

Body Systems: Cardiovascular System: Beat Your Heart

How much work does your heart really do?  The numbers are quite staggering... figure them out for yourself.


Part I: How many heartbeats in your lifetime?
Determine your pulse: count the number of beats in 15 seconds.

Multiply by 4 to get the number of beats her minute.  (Average is between 70 and 90).

Now, use that number to determine how many times your heart will beat, assuming an average lifespan of 78 years.


Part II: How Much Blood?
1/4 cup (2 oz.) of blood is pumped per contraction.  Using the number of beats per minute you calculated above, determine how much blood is pumped in one minute.  (Average is 20 cups = 5 quarts).

How many quarts of blood are pumped in a day?

How many gallons are pumped in a day?

Remember: 4 cups = 1 quart; 4 quarts = 1 gallon



Part III: How Much Does Your Blood Weigh?
About 10% of your weight is blood.

How many pounds of blood are in your body?

Metric System: Magic Number Lab

This activity provides students with a first-hand opportunity to experience one of the benefits of the metric system.

Materials:
Stop watch
Pencil

Procedure:
Have your partner time how long it takes you to do all four problems in one column.
Time your partner doing the same four problems.
Complete the remaining columns in the same way.
Come to the front table and check each other’s papers after you’ve both finished all four columns.

The Problems:
Column 1: 13x5
462x5
87x5
78,956x5

Column 2: 13x12
462x12
87x12
78,956x12

Column 3:
13x100
462x100
87x100
78,956x100

Column 4: 13x10
462x10
87x10
78,956x10

For each column, record the time (in seconds) it took to complete the column and the number of problems you solved incorrectly.

Discussion questions:
Which number could you multiply by the fastest?
With which number did you make the fewest mistakes?
Explain what this activity had to do with the metric system.

Gravity: Weight on Different Planets

Because of the varying sizes and compostion of the planets, each planet has a different amount of gravitational pull. A stronger gravitational pull means that objects are being pulled toward the center of the planet with greater force. The end result is that the object weighs more.

(remember... weight is the measure of gravitational pull on an object, mass is the amount of "stuff" an object is made up of - that doesn't change)

In short:
The larger, more massive the planet, the more gravitational pull, the more something weighs.

The smaller, less massive the planet, the less gravitational pull, the less something weighs.


To help give students a feel for these differences, I created these:

(I've got a whole set, this is just a representative sample!)

I used this calculator (very cool - have your students play around with it to find their weight on other planets) and an Earth weight of 50 g.

The calculator gave me the following weights:
Mercury: 18.9 g
Venus: 45.3 g
Mars: 18.8 g
Jupiter: 118.2 g
Saturn: 45.8 g
Uranus: 44.4 g
Neptue: 56.2 g
Pluto: 3.3 g

I then created these containers, by taping two cups, filled with an appropriate amount of stuff, together.

It's probably not the best container, but it met these requirements:
1 - Light enough to account for the weight on Pluto when empty.
2 - Opaque - I didn't want students to see through the container. I wanted it to look like they were all the same thing, they just weighed different.
3 - I had them on hand.

I believe I mostly used dried peas/beans for the filling. Jupiter may have a few pennies or other more dense weight thrown in to bulk it up!

If you have a dense material on hand (lead, or something of that sort), you could use film canisters, which would be sturdier than my creation.

Mining: Birdseed Mining


This is based on an activity from the Women in Mining website. However, it not currently listed as one of their activities.

You'll need:
Bird seed mix (The mix will need to include sunflower seeds, and at least two other types of seeds. You'll need approximately 20 mL of birdseed for each pair of students.)

Small beads - "seed beads" - blue, gold and silver

50 mL beakers - 1 per pair of students

Prep Work:
For every 300 mL of birdseed, add 7 gold beads, 13 silver beads and 25 blue beads.

The activity:
Each pair of students is given 15-20 mL of prepared bird seed mixture.

Students search through the mixture and separate out (i.e. "mine") the sunflower seeds, millet, and beads, making piles of each.



Students count and record each pile of seed: sunflower seeds, millet, blue, gold, silver beads, and everything else (doesn't need to be separated, just counted).

Now for the math...
I typically provide my students with a data table (sometimes it's drawn on the board and they copy it into their notebooks, other times it's provided to them on a piece of paper).

I've tried to recreate said data table here, but have failed. Miserably.

So, here is a picture of said data table:
(I can't figure out why the photos is being rotated... that's just the way my day is going... I'm sorry, you'll have to turn your head, until I can figure out how to fix it).


Hopefully the picture makes things clear, but in case it doesn't, here is an explanation:

A dollar value is assigned to each of the mined substances:

Gold beads = Gold = $5
Silver beads = Silver = $4
Blue beads = Copper = $3
Sunflower seeds = Iron = $2
Millet = Lead/Zinc = $1
Other = Waste = n/a

Students will calculate the value of their mined materials by multiplying the number of pieces of each material by the value of the respective material.

Adding together these values will give students their Gross Income.

In addition, students will determine their expenses by multiplying the total number of materials (including waste - it costs money to dig up waste too!) by 0.23.

Gross Income - Expenses = Net Income (or Loss)

In addition, students can determine their % profit (or loss) with the following formula:

[Net Income (or Loss) / Gross Income] x 100

(In words: net income divided by gross income; that number multiplied by 100)
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