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Is Zero Even or Odd?

Is the number zero even or odd?  This was a question asked on the Forum page of Teachers Pay Teachers by an elementary teacher. She stated that Wikipedia had a long page about the parity of zero and that some of the explanation went a little over her head, but basically the gist was that zero is even because it has the properties of an even number. She further stated that before reading this definition, she probably would have said that zero was neither even nor odd.

Here was my reply. Zero is classified as an even number. An integer n is called *even* if there exists an integer m such that n = 2m,  and *odd* if 2m + 1. From this, it is clear that 0 = (2)(0) is even. The reason for this definition is so that we have the property that every integer is either even or odd.

In a simpler format, an even number is a number that is exactly divisible by 2. That means when you divide by two the remainder is zero. You may want your students to review the multiplication facts for 2 and/or other numbers to look for patterns.

2 x 0 =               3 x 0 =
2 x 1 =               3 x 1 =
2 x 2 =               3 x 2 =
2 x 3 =               3 x 3 =

There is always a pattern of the products. Let the students discover these patterns - Even x Even = Even, Even x Odd = Even and vice versa and Odd x Odd = Odd. Since ALL math is based on patterns, seeing patterns in math helps students to understand and remember. Now ask yourself, "Does zero fit this pattern?"

The students can also divide several numbers by 2 (including 0), allowing them to see a second way to conclude that a number is even. (The remainder of the evens is 0, and the remainder of the odds is 1).  Again, "Does zero fit this pattern?"

To demonstrate odds and evens, I like using my hands and fingers since they are always with me. Let's begin with the number two.  I start by having the students make two fists that touch each other. I then have them put one finger up on one hand and one finger up on the other hand. Then the fingers are to make pairs and touch each other. If there are no fingers left over (without a partner), then the number is even.  (see sequence below)

Let's try the same procedure using the number three. Again, begin with the two fists. (see sequence below) Alternating the hands, have the students put up one finger on one hand and one finger up on the other hand; then another finger up on the second hand.  Now have the students make pairs of fingers. Oops!  One of the fingers doesn't have a partner, (one is left over); so, the number three is odd. (I like to say, "Odd man out.")   

So, does this work for zero?  If we start with two fists, and put up no fingers then there are no fingers left over.  The fists are the same, making zero even. (see illustration below)

So the next time you are working on odd and even numbers, make it a "hands-on" activity.

Helping Students to Think Mathematically

What does it mean to think mathematically?  

It means using math vocabulary, language and symbols to describe or interpret mathematical concepts, procedures and to discover relationships among ideas.  Therefore when a student problem solves, they use previous knowledge, skills, and understanding of concepts to solve a problem.  This process might include formulating problems, applying a variety of strategies, or interpreting results.

What can we do to help our students become better mathematical thinkers?  

We can teach and model problem solving strategies.  We can remember and plan our lessons to involve the three stages of conceptual development: concrete, pictorial, and abstract. We can have the students talk or write about how they got an answer either with the class or with a partner.  We can use writing in the mathematics classroom (such as math journals) to allow the students to practice expository writing and show their understanding.  We can exhibit math word walls and have the students use the glossary in their book to write and define terms. 

We can also create a positive and safe classroom atmosphere for problem solving... 

By being enthusiastic and allowing the students to take risks without consequences.  By emphasizing the process as well as the answer, the students may be willing to try unconventional or different ways to solve the problem.  I always tell my students that there isn't just one right way to get an answer which surprises many of them.  In fact, this is one of the posters that hangs in my classroom.

As math teachers, let's continue to emphasize problem solving so that all students will acquire confidence in using mathematics meaningfully. But most of all, let's have fun while we are doing it!

$8.00 on TPT
If you are interested in having a math dictionary for your students, check out A Simple Math Dictionary. It is a 30 page student dictionary that uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference.

Defeating Negative Self-Talk

One of the biggest problems with the college students I teach is their math anxiety level. Math anxiety is the felling of tension and anxiety that interferes with the manipulation of numbers and the solving of math problems during tests. In other words - mathphobia! This is a learned condition, not something they are born with and is in no way related to how smart a student is. In my Conquering College class, we have been looking at causes for anxiety which include bad experiences, teacher and/peer embarrassment and humiliation, or being shamed by family members. We've been looking at ways to reduce math anxiety such as short term relaxation as well as long term techniques and managing negative self-talk.

For many of my students, a song is the best way of remembering. I found an old nonsense song by Roger Miller entitled, You Can't Roller Skate in a Buffalo Herd. First we read over the words. Next we watched a video on You Tube and then we actually sang the song. I replaced the words "But Ya can be happy if you've a mind to" with "But ya can be positive if you put your mind to it."

You Can’t Roller Skate in a Buffalo Herd

By Roger Miller

Ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
  Ya can’t roller skate in a buffalo herd,
*But ya can be positive if ya put your mind to it.

 Ya can’t take a shower in a parakeet cage,
 Ya can’t take a shower in a parakeet cage,
 Ya can’t take a shower in a parakeet cage,
 *But ya can be positive if ya put your mind to it.
All ya gotta do is put your mind to it,
Knuckle down, buckle down,
Do it, do it, do it!

Well, ya can’t go swimmin’ in a baseball pool,
Well, ya can’t go swimmin’ in a baseball pool,
Well, ya can’t go swimmin’ in a baseball pool,
 *But ya can be positive if ya put your mind to it.
Ya can’t change film with a kid on your back,
Ya can’t change film with a kid on your back,
Ya can’t change film with a kid on your back,
 *But ya can be positive if ya put your mind to it.

Ya can’t drive around with a tiger in your car,
Ya can’t drive around with a tiger in your car,
Ya can’t drive around with a tiger in your car,
*But ya can be positive if ya put your mind to it.
All ya gotta do is put your mind to it,
Knuckle down, buckle down,
Do it, do it, do it!

Well, ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
Ya can’t roller skate in a buffalo herd,
 *But ya can be positive if ya put your mind to it.
Ya can’t go fishin’ in a watermelon patch,
Ya can’t go fishin’ in a watermelon patch,
Ya can’t go fishin’ in a watermelon patch,
*But ya can be positive if ya put your mind to it.

 Ya can’t roller skate in a buffalo herd,
 Ya can’t roller skate in a buffalo herd,
  Ya can’t roller skate in a buffalo herd,
  *But ya can be positive if ya put your mind to it.

So how did the lesson go? Let's just say that my college students were having so much fun singing the song that the secretary had to come and shut our classroom door. And the response from the students the next day, "I can't get that song out of mind!" Maybe negative self-talk has finally met its match!

"BOO" to Fractions?

Here is a Halloween riddle: Which building does Dracula like to visit in New York City? Give up? It's the Vampire State Building!! (Ha! Ha!) Here is another riddle. Why didn’t the skeleton dance at the party? He had no body to dance with!   

Okay, so what do these riddles have to do with teaching math? I have been attempting to come up with ways for my students to recognize fractional parts in lowest terms. As you know from this blog, I have used Pattern Sticks, the Divisibility Rules, and finding Digital Root. These are all strategies my students like and use, but to be a good mathematician requires practice - something most of my students dread doing. I can find many "drill and kill" activities, but they tend to do just that, drill those who don't need it and kill those who already know how to do it. So to drill and "thrill", I created six fractional word puzzles for specific times of the year.

The one for October is Halloween Fraction Riddles. It contains eight riddles that the students must discover by correctly identifying fractional parts of words. For instance, my first clue might be: the first 2/3's of WILLOW. The word WILLOW contains six letters. It takes two letters to make 1/3; therefore, the first 2/3's would be the word WILL. This causes the students to group the letters (in this case 4/6), and then to reduce the fraction to lowest terms. The letters are a visual aid for those students who are still having difficulty, and I observe many actually drawing lines between the letters to create groups of two.

At first, I thought my students would breeze through the activities, but to my surprise, they proved to be challenging as well as somewhat tricky - just perfect for a Trick or Treat holiday. Maybe this is an activity you would like to try with your intermediate or middle school students. Just click on this link: Halloween Fraction Riddles.

Using Bloom's Taxonomy in Geometry Class

As one of their assignments, my college students are required to create a practice test using pre-selected math vocabulary. This activity prompts them to review, look up definitions and apply the information to create ten good multiple choice questions while at the same time studying and assessing the material. Since I want the questions to be more than Level 1 (Remembering) or Level II (Understanding) of Bloom's Taxonomy, I give them the following handout to help them visualize the different levels.  My students find it to be simple, self explanatory, easy to understand and to the point.

Level I - Remembering

 What is this shape called?

Level II - Understanding

Circle the shape that is a triangle.

Level III - Applying

       Enclose this circle in a square.

Level IV - Analyzing

 What specific shapes were used to draw this picture?

Level V - Evaluating

How is the picture above like a real truck?  How is it  different?

Level VI - Creating

Create a new picture using five different geometric shapes.
(You may use the same shape more than once, but you must use five
different geometric shapes.)

Using Bloom's in Math
As teachers, we are only limited by our imagination as to the activities we ask our students to complete to help them prepare for a test. However, we still need to teach and provide information so the students can complete these types of tasks successfully. With the aid of the above chart, my students create well written practice tests using a variety of levels of Bloom's. When the task is completed, my students have also reviewed and studied for their next math exam. I consider that as time well spent!

If you would like a copy of the above chart in a similar but more detailed format, it is available on Teachers Pay Teachers as a FREE resource.

Factorial Fun

Factorial is a word that mathematicians use to describe a special kind of numerical relationship. Factorials are very simple things. They are just products, indicated by the symbol of an exclamation mark. The factorial function (symbol: !) means to multiply a series of descending natural numbers. For instance, "five factorial" is written as "5!" (a shorthand method) and means 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in determining the numbers of combinations and permutations and in finding probability.

Now all of that may seem above your mathematical head, but let me introduce you to the book Anno's Mysterious Multiplying Jar by Masaichir and Mitsumasa Anno.  It is a story about one jar and what is inside it. Anno begins with the jar, which contains one island, that has two countries, each of which has three mountains. The story continues like this until 10 is reached.  The colorful pictures are arranged within borders on the page as many times as the number of objects being discussed. For instance when four walled kingdoms are introduced, four kingdoms are on the page.

The explanation of 10! in the back of the book is also very helpful. Even if children do not understand the concept being taught, they will certainly appreciate the detailed colored drawings and imaginative story! The book is best for kids who have been introduced to at least basic multiplication facts, but younger kids will enjoy counting and looking at the pictures even if the rest of it is over their heads; so, this book helps with multiplying skills as well as the mathematical concept of factorials.

You might give the students a worksheet to keep track of how many islands, rooms, etc. there are. The final question is how many jars are there. Hopefully there will some students who catch on to the factorial concept, find the pattern and discover the answer! 

Here is an example of how you might use factorials in solving a word problem.  How many different arrangements can be made with the letters from the word MOVE?  Because there are four different letters and four different spaces, this is how you would solve the problem.

____   ____   ____   ____ 
Four Possible Spaces

All four letters could be placed in the first space. Once the first space is filled, only three letters remain to fit in the second space. Once the second space is filled with a letter, two letters remain to write in the third space. Finally, only one letter is left to take the fourth and final space. Hence, the answer is a factorial (4!) = 4 × 3 × 2 × 1 = 24 arrangements.

Try some problems in your classroom. Start with an imaginary character, Cal Q. Late, who is working at an Ice Cream Store called Flavors. A hungry customer orders a triple scoop ice cream cone with Berry, Vanilla, and Bubble Gum ice cream. How many different ways could Cal Q. Late stack the ice cream flavors on top of each other?

You could answer the question by listing all of the possible orders of the three ice cream flavors from top to bottom. (Students could have colored circles of construction paper to physically rearrange.)

  1. Bubble Gum - Berry - Vanilla
  2. Bubble Gum - Vanilla - Berry
  3. Berry - Vanilla - Bubble Gum 
  4. Berry - Bubble Gum - Vanilla
  5. Vanilla - Berry - Bubble Gum
  6. Vanilla - Bubble Gum - Strawberry
Or, if we use factorials, we arrive at the answer much faster:  3! = 3 × 2 × 1 = 6

Learning about patterns and the use of factorials will stretch a students' mathematical mind.  Why not try a few problems in your classroom?  And by all means, check out Anno's Mysterious Multiplying Jar.

Using Mnemonic Techniques

In my college class entitled Conquering College, we have been working on ways to remember for tests. Of course, mnemonic devices came up. Mnemonics connect new learning to prior knowledge through the use of visual and/or acoustic cues. Such strategies assist students in remembering and recalling larger pieces of information for tests. Included in mnemonics are acronyms, initialism, acrostics, rhyme, rhythm and song and association in addition to visualization using the loci and peg systems. Let's look at four of these categories.

1) Acronyms - A word formed from the first letters of each one of the words in a phrase.
  • HOMES – The names of the 5 Great Lakes – Huron, Ontario, Michigan, Erie, Superior 
  • ROY G. BIV – The colors in a rainbow – Red, Orange, Yellow, Green, Blue, Indigo, Violet 
  • SCUBA - When you’re scuba diving, you’re using a “self-contained underwater breathing apparatus.” 

2) Acrostics – Sentences created from the first letters of key words.
  • Please Excuse My Dear Aunt Sally – for the order of operations 
Parenthesis, Exponents, Multiplication, Division, Addition, Subtraction 

    **I personally prefer the phrase: Pale Elvis Meets Dracula After School. 
  • My Very Earthly Mother Just Sliced Up Neptune.  – the planets in order from largest to smallest: 
Mercury, Venus, Earth, Mars, Jupiter, Saturn, Uranus, Neptune 

   **I particularly like this one since Earthly gives you a clue that the third planet is earth and Neptune is listed last. This means you only have to know 6.

Free Resource
3) Rhyme, Rhythm, Song – poems, limericks or silly songs – These work well for auditory learners.
  • I before E, except after C and in sounding like A as in neighbor and weigh.
  • In 1492, Columbus sailed the ocean blue.
  • Twinkle, twinkle little star; circumference is 2 π r.       (I actually sing this for my students!)

4) Association – finding a common element. The association is usually coincidental.
  • Litmus Paper: Blue = Base – both begin with “B”. 
  • Arteries: Artery = Away – both begin with “A”. 
  • The principal is my PAL. Helps to distinguish from principle. 
  • Affect = Action (a verb) Helps to separate it from effect which is a noun.
These ideas plus many more are in a free resource called Mnemonic Techniques found on Teachers Pay Teachers. All you have to do is download it!

The ROOT of the Problem - What is Digital Root and How to Use It

When students skip count, they can easily say the 2's, 5's, and 10's which translates into easy memorization of those particular multiplication facts.  Think what would happen if every primary teacher had their students practice skip counting by 3's, 4's, 6's, 7's, 8's and 9's!  We would eradicate the drill and kill of memorizing multiplication and division facts.

Since many of my college students do not know their facts, I gravitate to the Divisibility Rules.  Sadly, most have never seen or heard of them.  I always begin with dividing by 2 since even numbers are understood by almost everyone.  (Never assume a student knows what an even number is as I once had a college student who thought that every digit of a number must be even for the entire number to be even.) We then proceed to the rules for 5 and 10 as most students can skip count by those two numbers.

Finally, we learn about the digital root for 3, 6, and 9. This is a new concept but quickly learned and understood by the majority of my students. (See the definition below).

Here are several examples of finding Digital Root:

1) 123 = 1 + 2 + 3 = 6. Six is the digital root for the number 123. Since 123 is an odd number, it is not divisible by 6. However, it is still divisible by 3.

2) 132 = 1 + 3 + 2 = 6. Six is the digital root for the number 132. Since 132 is an even number, it is divisible by 6 and by 3.

3) 198 = 1+ 9 + 8 = 18 = 1 + 8 = 9. Nine is the digital root for the number 198; so, 198 is divisible by 9 as well as by 3.

4) 201 = 2 + 0 + 1 = 3. Three is the digital root for the number 201; so, 201 is divisible by 3.

The first time I learned about Digital Root was about eight years ago at a workshop presented by Kim Sutton. (If you have never been to one of her workshops - GO! It is well worth your time.) Anyway, I was beside myself to think I had never learned Digital Root. Oh, the math classes I sat through, and the numbers I tried to divide by are too munerous to mention! It actually gives me a mathematical headache. And to think, not knowing Digital Root was the ROOT of my problem!

Divisibility Rules

A teacher resource on Using the Divisibility Rules and Digital Root is available at Teachers Pay Teachers. If you are interested, just click under the resource cover on your right.

Using Number Tiles to Problem Solve in Math

Math Activities for Grades 5-8
I prefer using hands-on activities when teaching math. One of the most successful items I have used is number tiles. Because number tiles can be moved around without the need to erase or cross out an answer, I have discovered that students are more at ease and more willing to try challenging activities. There is something about not having a permanent answer on the page that allows the student to explore, investigate, problem solve, and yes, even guess.

I have created several number tile booklets, but the one I will feature today is for grades 5-8. It is a 23 page booklet containing 15 different math problem solving activities that range from addition and multiplication, to primes and composites, to exponent problems, to using the divisibility rules. Since the students do not write in the book, the pages can be copied and laminated so that they can be used from year to year. These activities may be placed at a table for math practice or as a center activity. They are also a perfect resource for those students who finish an assignment or test early. Use these activities to reteach a concept to a small group as well as to introduce a new mathematical concept to the whole class.

Free Resource
Students solve the Number Tile Math Activities by arranging ten number tiles, numbered 0-9. Most of the number tile activities require that the students use each tile only once. The number tiles can be made from construction paper, cardboard, or square colored tiles that are purchased.  (How to make the number tiles as well as storage ideas is included in the handout.) Each problem is given on a single page, and each activity varies in difficulty which is suitable for any diverse classroom. Since the students have the freedom to move the tiles around, they are more engaged and more willing to try multiple methods to find the solution. Some of the problems will have just one solution while others have several solutions. These activities are very suitable for the visual and/or kinesthetic learner.

A free version for each of my number tile resources is listed below. Just click on the link to download the freebie.

Shaving Cream And Summer Fun

Again, I am going to deviate from the subject of math and offer a fun summer activity I do with my grandchildren. It involves a can of shaving cream, cleaning rags and lawn furniture that has set out all winter.

Were you aware that there are many unusual ways to use shaving cream besides using it for shaving? Did you know that you could...

1) Clean jewelry with it? Spray it on your jewelry and use a soft “old” toothbrush to get off the grime. Rinse with water.

2) Give chrome faucets a brilliant shine? Apply the shaving cream to a sponge and rub it on the faucet. Then wipe it off with a damp cloth.

3) Easily remove paint from your hands? Rub the shaving cream onto your hands; then rinse it off with soap and water.

4) Remove carpet stains? Blot the soiled area with a damp sponge and then spray on the shaving cream. Wipe clean with a damp sponge and let the area dry. It will also work on various clothes stains.

5) Clean vinyl lawn furniture? Spray the lawn furniture with the shaving cream and wipe the grubby areas with a damp rag. Rinse when finished.

Item #5 is what I do each summer. Our lawn furniture sets out over the winter on our patio and even though it is covered, it is filthy when summer comes. I always go to the store and purchase the cheapest shaving cream I can find. (Here in Kansas, Barbasol sells for about $.89 a can. Depending on the number of grandchildren coming over, determines how many cans I purchase. This year, it was three.) No matter their age, this is one activity that they all look forward to because it is messy!

I write the child’s name on their can of shaving cream and then assign them a piece of furniture to clean. When everyone is done scrubbing and wiping, we get out the garden hose to spray off the remaining shaving cream, and frequently we end up spraying each other.

But what happens to the leftover shaving cream? I think the picture says it all!

Cold Treat for Those Hot Days of Summer

June always brings the first day of summer. I'm not sure where you live, but I live in Kansas, and each day it is getting hotter and hotter! On a hot day, when you have been outside, there is nothing better than an ice cold treat. For years, I have made homemade Popsicles, first for my children and now for my grandchildren. I thought I would share the quick and easy recipe with you. (I know this might be considered the "far side" of math, but recipes do contain measurement and sometimes, even fractions!)

Popsicles Recipe - Will make 18

1 small package of Jello (any flavor)  As you can see, my grandchildren like the Berry Blue.)
1/2 cup sugar
2 cups boiling water
2 cups cold water

Boil water. Add to the sugar and package of Jello. Stir until all the Jello is dissolved. Add the cold water and stir again.

Pour into three sets of Tupperware Popsicle Makers. If you don't have these (I don't think they sell them anymore), use Popsicle molds found in stores. or use ice cube trays.

Place in the freezer until hardened. Eat and enjoy just like my grandchildren do!

"Sum" More Quick Tricks

Sometimes, my students think, I am a magician who pulls answers out of a hat. Over the years, I have learned that mathematicians are ingenious people who are always looking for quick and easy ways to do things. Maybe that's why we now have graphing calculators and computer programs to figure taxes.
I have a friend who teaches math on the college level in North Carolina. In fact, we have been friends since 6th grade, but that's another story. When she read one of my posts, she shared a trick for quickly finding a sum. Her trick has to do with a sequence that begins with any number, with any number of terms as long as they are separated by the same amount. For instance, the series below is a six number sequence with a difference of two between each number.
Here is what you do to quickly to find the sum. Add the first and last terms. 5 + 15 = 20. Now multiply by the number of terms which in this case is 6. 20 x 6 = 120 Finally, divide by 2. So, mentally this is what it would look like.

Now, how many of you went back to add up 5 + 7 + 9 + 11 + 13 + 15? Did you get the answer of 60? Isn't it amazing!?! Maybe math teachers are magicians after all!


Quick Times - Multiplication Tricks

I am always looking for different strategies when working with my remedial college students since many of the ways they were taught to do math aren't working for them.  I came across this "Quick Times" method and thought it would be another approach I could share with my mathphobics for multiplying.  They love anything that is different, quick and makes them look astute when doing mathematics.

Let's assume we have the multiplication problem of 41 x 12.  In the Quick Times method, first start by multiplying the first digit of 41 by the first digit of 12 to get the first digit of our answer.  We then multiply the second digit of 41 by the second digit of 12 as seen below to get the last digit of our answer (the ones place).

Now we need to find the middle digit of the product.  This is done by multiplying the outside digits, then the inside digits, and adding those two products together as shown below.

This quick method will only work when multiplying two digit numbers by two digit numbers, but it does cause the students to do mental math.  My students like the challenge of doing all of the computation in their heads.  Let's try another one that is a little different.  Let's do 63 x 41.  Again we multiply the first digit of each number and then the second digit of each number to get the first digits of the answer and the last digit of the answer.

As before, multiply the outside digits, then the inside digits, and add the two products together.
Now we must put the 18 into the middle spot, but there is only room for one digit in the tens place.  YIKES!!  What do we do now?  Very easy....because we can only have one digit where the question mark is, we must regroup (carry) the one in the tens place of the 18 and then add it to the 24.

Have you figured out the final answer?  It is.....

You are probably thinking the old method works so much better, but that is only because that is the method you are use to using.  Why not try the ones below using the Quick Times method and see if you get the correct answer.  Use the old method or a calculator to check your answers or go the the answer page above.

a)  36 x 21       b)  24 x  12      c)  48 x 29       d)  59 x 18       e)  63 x 13     

Making Gifts - Not Something I Normally Do

At this time of year, we have many friends retiring or celebrating those "up-in-years" birthdays. Many of the invitations read, "No Gifts, Please." I understand at this point in our lives, we have more than we need, but it is always nice to bring something to show your friend that you care. We just attended a 70th birthday party for someone we have known for years. Not only is he our friend, but he is someone both my husband and I have taught with. I looked on Pinterest (where else?) and found several ideas that I combined. Here is what I came up with - a large birthday card that was editable!!

Here is what I purchased to complete the giant card.

  1. A large folding poster (You need heavy poster board to hold all of the candy!)
  2. A Nestle's Symphony Bar
  3. A Snickers Bar
  4. Nestle Crunch
  5. One package of EXTRA chewing gum
  6. 100 Grand Candy Bar
  7. Butterfinger
  8. Skor Candy Bar
  9. Mr. Goodbar
  10. Package of Milk Duds
  11. Package of Whoppers
I hot glued each of the candy bars or packages of candy onto the poster. I then used rubber cement to attach the phrases. I created my own phrases that sort of matched the candy, but if you are making a card, get creative and make up your own. You might even find some better candy bars or items to put on the card.

I have to say this birthday card was a real "hit" and even became a center piece of the party. Also, the party goers thought it was extremely yummy!

Recycled Butterflies

Two of my grandchildren are in kindergarten and of course, everything is new and exciting to them.  They came home one day with egg carton caterpillars.  I know most of us have made one of these in our lifetime, but to these two, they were the best craft ever!

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

But, here is the good part!  They got to make a cocoon out of a toilet paper cylinder.  They covered it by gluing on white cotton balls.  Then the made a butterfly out of tissue paper and a small plastic bag tie.  They 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 34 pages 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.
Only $7.00

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 34 page resource entitled Trash to Treasure.

Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK - 3rd classroom, and the visual and/or kinesthetic learners will love them.

A Go Figure Debut for a High School English Teacher Who Is New!

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During college, Literary Roses worked as a writing tutor at a university writing center. This experience solidified her desire to become a teacher. Currently, she is a high school English teacher and has been for ten years. She teaches Shakespeare, poetry, plays, writing, and a many other different literary works.

Literary Roses loves to interact with her students! In some students she sees a similarity to those stressed out college students she tutored who were striving to learn with difficulty. She is aware that not all students will grasp what comes so easily to others, and their struggling makes her want to work to help them connect with difficult concepts and skills. Even though teaching is very challenging, she states that she values her profession and believes in its worth.

Only $19.99

The resources in her Teachers Pay Teachers store reflect her passion for literature. All of her products are geared towards eleventh and twelfth graders. One of her 112 products is entitled Elie Wiesel’s Night: Common Core Curriculum Unit.

This unit contains many power points that answer the ten most frequent questions students have regarding the Holocaust, and includes graphs and photos to aid in comprehension. It also teaches concepts such as the dehumanization of Holocaust prisoners, the symbolism of the young Pipel, personification and irony with so much more discussed in this unit. It has 113 ratings with one person in particular saying,

"This is the most incredible, creative unit I have ever located on Night!
It is one of my favorite autobiographies to teach, and my students will benefit from these incredible resources and activities. I've been teaching English for over 20 years,
and I have to say this is the most extraordinary, first-rate novel unit I have ever seen.
I cannot thank you enough. :-)"

Free Resource

Literary Roses also has over 20 free resources in her store. Introduction to the Romantic Period is just one of them. It is an 11 slide power point that explains the Romantic Period to students. It includes the causes for the shift in ideas and the characteristics reflected in the literature. If you are a high school English teacher, you might want to download it.

In addition to teaching and creating resources, Literary Roses enjoys shopping, running, reading, and being with her church family. Her two children keep her quite busy, but she believes being a mom and a teacher are truly works of the heart! Why not take a few minutes and check out the quality resources in her store? 

Plant Mathematics and Fibonacii

Oak Leaves
We continue to look at Fibonacci numbers and how nature continually exhibits this pattern. As stated in my last post, this number pattern can be linked to ordinary things we see every day such as the branching in trees, the arrangement of leaves on a stem, the flowering of an artichoke, or the fruitlets of a pineapple. BUT were you aware that scores of plants, including the elm or linden trees, grow their leaves, twigs and branches placed exactly half way (1/2) around the stem from each other?

Similarly, plants, like beech trees, have leaves located 1/3 of a revolution around the stem from the previous leaves. In the same way, plants like the oak tree have leaves positioned at 2/5 of a rotation. Plants like the holly continue this pattern at 3/8, while larches (conifers) are next at 5/13. The sequence extends on and on. Looking at these fractions side by side (see below) do you see a number pattern in the numerators?

Likewise pay attention to the precision of a similar pattern in the denominators. Interestingly, in the numerators and denominators, if you add the two sequential numbers together, you create a Fibonacci series where all numbers in the series are the sum of the two preceding numbers.

Mathematicians recognize this unique pattern as the Fibonacci sequence. Since patterns such as this one are commonplace in botany as well as other areas of science, they are regularly studied so we can better understand the relationship between mathematics and our world. In my opinion, such mathematical precision and accuracy can only be the product of an intelligent Designer. "Through Him all things were made; without Him nothing was made that has been made." (John 1:3 - NIV)