Making Butterflies from Recycled Materials - Earth Day Ideas

When one of my granddaughters was in kindergarten, she came home one day with egg carton caterpillars.  I know most of us have made one of these in our lifetime, but to her, this was the best craft ever!

She told me that her teacher was raising butterflies in her classroom, and soon the butterflies would hatch.  Anticipation and excitement reigned until the day she came out of school telling everyone that one of the butterflies had hatched.  However, much to her chagrin, the teacher was going to let it go.  My granddaughter just couldn't understand why or how her teacher could do that!

But, here is the good part!  She got to make a cocoon out of a toilet paper cylinder.  She covered it by gluing on white cotton balls.  Then she made a butterfly out of tissue paper and a small plastic bag tie.  She put the butterfly inside the cocoon and then pretended to have the butterfly hatch!  This was done over and over and over until the cocoon was no more.  Luckily, I was able to get pictures before both were literally destroyed!

Now, what does all of this have to do with math?  I contemplated all the ways to use recycled products to make items for the classroom.  Thus Trash to Treasure was created. It is full of art ideas, fun and engaging mini-lessons as well as cute and easy-to-construct crafts all made from recycled or common, everyday items.

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Find out more than 14 ways to use milk lids for math. Did you know that you can practice math facts using clear plastic containers? Learn how to take two plastic plates and turn them into angle makers. How about using two plastic beverage lids to make card holders for kindergartners or for those whose hands are disabled? Discover ten ways to use carpet squares as well as nine ways to use old calendars. How about playing hop scotch on old carpet squares? Were you aware that butter tubs can become an indoor recess game to practice addition or multiplication facts? These are just a few of the fun and exciting activities that use recycled items found in this resource entitled Trash to Treasure.

Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK - 4th grade classroom, and the visual and/or kinesthetic learners will love them.

Earth Day is April 22nd. How Will You Celebrate?

Earth Day began in 1970, when Gaylord Nelson, a U.S. Senator from Wisconsin, wanted nation-wide teaching on the environment. He brought the idea to state governors, mayors of big cities, editors of college newspapers, and to Scholastic Magazine, which was circulated in U.S. elementary and secondary schools.

Eventually, the idea of Earth Day spread to many people across the country and is now observed each year on April 22nd. The purpose of the day is to encourage awareness of and appreciation for the earth's environment. It is usually celebrated with outdoor shows, where individuals or groups perform acts of service to the earth. Typical ways of observing Earth Day include planting trees, picking up roadside trash, and conducting various programs for recycling and conservation.

Symbols used by people to describe Earth Day include: an image or drawing of planet earth, a tree, a flower or leaves depicting growth or the recycling symbol. Colors used for Earth Day include natural colors such as green, brown or blue.

The universal recycling symbol as seen above is internationally recognized and used to designate recyclable materials. It is composed of three mutually chasing arrows that form a Mobius strip which, in math, is an unending single-sided looped surface. (And you wondered how I would get math in this article!?!) This symbol is found on products like plastics, paper, metals and other materials that can be recycled. It is also seen, in a variety of styles, on recycling containers, at recycling centers, or anywhere there is an emphasis on the smart use of materials and products.

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Inspired by Earth Day, Trash to Treasure is a FREE resource. In it, you will discover how to take old, discarded materials and make them into new, useful, inexpensive products or tools for your classroom. Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK-3 classroom, and the visual and/or kinesthetic learners will love them.

To download the free version, just click under the cover page on your left.

Techniques for Remembering the Slope for Vertical and Horizontal Lines

I work in the Math Lab at the community college where I also teach. Last week, I had two College Algebra students who were having difficulty with slope.  They knew the equation y = mx + b, but were unsure when it came to horizontal or vertical lines. By the way, they were using their graphing calculators which I made them put away. (The book said no calculators.) I feel that if they construct the lines themselves, it puts a visual image into their brain much better than if the calculator does it for them. Sure enough, one of the sections in their math books gave the picture of the line from which they had to write the equation. They were amazed that I could just look at a graph and know the slope, give the equation, etc. When I taught high school math, my students couldn't use a graphing calculator until the middle of this particular chapter as I wanted them to physically draw the lines.

First, for those who have no idea what I am talking about, slope is rise over run.  Rise is how far a line goes up, and run is how far a line goes along.  At the right, the line goes up 3 and has a run 5; therefore, the slope is 3/5.  Rise/Run (Rise divided by Run) gives us the slope of the line.

When a line is horizontal, it has no rise, only a run. So the numerator would be zero (for no rise) and the denominator would be a number such as 5 for the run.  0 ÷ 5 = 0  This is true for any horizontal line.

A vertical line is different.  It has rise, but no run; therefore there would always be a number in the numerator, but always a zero in the denominator.  Since we cannot divide by zero, the slope is considered undefined. (I do use rise over run stating that a horizontal line might have 0/5 which is equal to 0 and that a vertical line might have 3/0 is undefined because we can't divide by zero. Our college algebra book uses O/K for okay and K/O for knock out which I like, but I still think the students need to know why.)

I wanted these two students to have a picture that would help them remember the difference.  I thought of a table for the horizontal line and asked them what would happen if the legs of the table were uneven.  They agreed that the table would have slope.  Therefore, the table would have a slope of zero if the legs were even.

I then went blank.  In other words, by creative juices stopped working, and I could not think of a picture that would help them visualize undefined. Since Teachers Pay Teachers has a forum,, I asked my fellow math teachers if they had any ideas.  Here is what some of them came up with.

The Enlightened Elephant suggested using a ski slope. She talks about skiing down a "cliff", which would not be possible (although some students try to argue that they could ski down a vertical cliff) and so the slope is "undefined" because it doesn't make sense to ski down a cliff.  Skiing on a horizontal line is possible so it's slope is zero,  She also talks about uphill (positive slope) and downhill (negative slope). 

Math by Lesley Elisabeth tells her students to use "HOY VUX" (rhymes with 'toy bucks')

             Horizontal - Zero (0) slope - y = ?   

             Vertical - Undefined slope - x = ?

All horizontal lines are y =7 or y = -3 etc., and all vertical lines are x =1 or x = 6, etc. Students forget this so the acronym HOY VUX helps them to remember. Once they've mastered the slope concept in Algebra I, for the rest of the school year, for Algebra II (especially equations of asymptotes - a line that continually approaches a given curve but does not meet it at any finite distance) and even in calculus classes for tangent lines, HOY VUX is just faster and more practical. 

Animated Algebra created a video lesson on the Slope Intercept.  She has a boy skateboard down a negative slope, literally right on the graph line. Karen then shows the same boy taking an escalator up on a line that has a positive slope. Later in the lesson, she rotates the line clockwise, each movement with a click, to show the corresponding slope number to link the line to the slope.  She includes lots of other visual cues to help students focus on and pay attention to the concepts.

When Dividing, Zero Is No Hero - Why We Can't Divide by Zero

Have you ever wondered why we can't divide by zero?  I remember asking that long ago in a math class, and the teacher's response was, "Because we just can't!"  I just love it when things are so clearly explained to me. So instead of a rote answer, let's investigate the question step-by-step.

The first question we need to answer is what does a does division mean?  Let's use the example problem on the right.

1. The 6 inside the box means we have six items such as balls. (dividend) 
2. The number 2 outside the box (divisor) tells us we want to put or separate the six balls into two groups. 
3. The question is, “How many balls will be in each group?” 
4. The answer is, “Three balls will be in each of the two groups.” (quotient)

                                      

Using the sequence above, let's look at another problem, only this time let's divide by zero.

1. The 6 inside the box means we have six items like balls. (dividend) 
2. The number 0 outside the box (divisor) tells us we want to put or separate the balls into groups into no groups. 
3. The question is, “How many balls can we put into no groups?” 
4. The answer is, “If there are no groups, we cannot put the balls into a group.” 
5. Therefore, we cannot divide by zero because we will always have zero groups (or nothing) in which to put things. You can’t put something into nothing.

Let’s look at dividing by zero a different way. We know that division is the inverse (opposite) of multiplication; so………..

1. In the problem 12 ÷ 3 = 4.  This means we can divide 12 into three equal groups with four in each group.
2. Accordingly, 4 × 3 = 12.  Four groups with three in each group equals 12 things.

So returning to our problem of six divided by zero..... 

1. If 6 ÷ 0 = 0....... 
2. Then 0 × 0 should equal 6, but it doesn’t; it equals 0. So in this situation, we cannot divide by zero and get the answer of six.

We also know multiplication is repeated addition; so in the first problem of 12 ÷ 3, if we add three groups of 4 together, we should get a sum of 12. 4 + 4 + 4 = 12

As a result, in the second example of 6 ÷ 0, if six zeros are added together, we should get the answer of 6. 0 + 0 + 0 + 0 + 0 + 0 = 0 However we don’t. We get 0 as the answer; so, again our answer is wrong.

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It is apparent that how many groups of zero we have is not important because they will never add up to equal the right answer. We could have as many as one billion groups of zero, and the sum would still equal zero. So, it doesn't make sense to divide by zero since there will never be a good answer. As a result, in the Algebraic world, we say that when we divide by zero, the answer is undefined. I guess that is the same as saying, "You can't divide by zero," but now at least you know why.

If you would like a free resource about this very topic, just click under the resource title page on your right.