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How Many Sides Does A Circle Have?

Believe it or not, this was a question asked by a primary teacher.  I guess I shouldn't be surprised, but in retrospect, I was stunned. Therefore, I decided this topic would make a great blog post.

The answer is not as easy as it may seem. A circle could have one curved side depending on the definition of "side!"  It could have two sides - inside and outside; however this is mathematically irrelevant. Could a circle have infinite sides? Yes, if each side were very tiny. Finally, a circle could have no sides if a side is defined as a straight line. So which one should a teacher use?

By definition a circle is a perfectly round 2-dimensional shape that has all of its points the same distance from the center. If asked then how many sides does it have, the question itself simply does not apply if "sides" has the same meaning as in a rectangle or square.

I believe the word "side" should be restricted to polygons (two dimensional shapes). A good but straight forward definition of a polygon is a many sided shape.  A side is formed when two lines meet at a polygon vertex. Using this definition then allows us to say:
  1.  A circle is not a polygon.
  2.  A circle has no sides.
One way a primary teacher can help students learn some of the correct terminology of a circle is to use concrete ways.  For instance,  the perimeter of a circle is called the circumference.  It is the line that forms the outside edge of a circle or any closed curve. If you have a circle rug in your classroom, ask the students is to come and sit on the circumference of the circle. If you use this often, they will know, but better yet understand circumference.

For older students, you might want to try drawing a circle by putting a pin in a board. Then put a loop of string around the pin, and insert a pencil into the loop. Keeping the string stretched, the students can draw a circle!

And just because I knew you wanted to know, when we divide the circumference by the diameter we get 3.141592654... which is the number π (Pi)!  How cool is that?

Only $1.90


If you are studying circles in your classroom, you might like this crossword puzzle. It is a great way for students to review vocabulary.


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.


The Epidemic of "Affluenza"

We live in a nation where we have so much to be thankful for. We enjoy a measure of wealth that billions in this world can only dream of and previous generations could not have even imagined. Is it possible that we have grown so accustomed to our affluence that we have lost the wonder of it? Is it possible that our affluence is harming us even as it blesses us?

Unfortunately, I think many in America are infected with the contagious and dangerous disease of "affluenza". How do I know? Because daily, I see people exhibiting the symptoms of the disease. One of the first symptoms is discontentment with what they have. As we possess more things, satisfaction and contentment declines. Many times wealth doesn't deliver joy, only emptiness.

Secondly, obsession is a symptom of affluenza.  I want more; I need more; I deserve more is advertised everyday on T.V.  If we already have a product, we are enticed to upgrade to the latest and newest version or to replace it altogether.

Ingratitude is another indicator of affluenza.  We have so much that we have no needs, just wants, and as we acquire those desires, we tend to forget the words, "Thank you." Finally "affluenza" results in a non-giving spirit.  We grudgingly give or give a meager amount to satisfy our conscious. Shouldn't our giving reflect our abundant blessings?

This Thanksgiving, take time to be thankful.  Share with those you love why you are thankful for them. Call someone you haven't seen for a while and tell them you are thankful for their love and friendship.  Invite someone who has no family to have dinner with your family. And don't forget to give thanks to God who gives us eternal life through His Son, Jesus Christ.



A Go Figure Debut for an Author Who Is New

Helal's TPT Store
Today’s post features a teacher from Ontario, Canada. Helal has taught third grade for three years, before taking maternity leave. Now she is staying home to spend some time with her toddler.

What Helal likes best about teaching is the impact she can make on her students. She loves seeing the students’ eyes light up when they finally get a new concept as well as building their character and being a positive memory to look back on in the future.

Helal describes her classroom as structured and organized. She finds that students really benefit from clear guidelines and communication; so, she makes sure that her schedules are up-to-date, that co-constructed criteria are displayed clearly and that students can always look at the walls to find examples and models of excellence.

In her spare time, Helal loves to write. In fact, she has written and published two children's books entitled: Nightly News with Safa and Zaid and the Gigantic Cloud. She also loves crafting; hence, the name of her Teachers Pay Teachers Store – Crafting Creatives. Her store currently contains 31 resources, most of which are reading resources, with five of them free. Her products range in subject matter from science, to math, to language, etc. to resource types (e.g. units, labels, etc.). Her overall goal with her resources is to help teachers inspire creativity in the classroom.

Free Resource
Her free item is a Bullying Poster. Its main objective is to help prevent bullying in your classroom and your school. The free file contains two versions/sizes of the same poster. One is 8.5” x 11” and the other is 17” x 22”.

Only $1.99
Her paid item is a fun cross-curricular activity that combines fractions and writing in one confidence boosting activity! First, students list four equal parts of themselves and then label them within a circle (being held by a superhero - they then design the superhero to look just like themselves)! Then, they use the story planner to tell the tale of how a villain tries to steal one of their superhero traits! This package includes a lined portion for the final story, a student re-visionary checklist and a rubric for teachers. Also included are superhero clip art images to help you decorate a bulletin board to show off student work, or create extra resources.

Helal also has a blog entitled My Everyday Classroom. If you visit her blog, you can get a free book study guide for the book she wrote called Zaid and the Gigantic Cloud. Her blog is not only professional, but it contains many interesting and relevant posts that I think you will find worth reading. Why not check it out?

Glyphs Are Really A Form of Graphing

$3.00
Sometimes I think that teachers believe a glyph is just a fun activity, but in reality glyphs are a non-standard way of graphing a variety of information to tell a story. It is a flexible data representation tool that uses symbols to represent different data. Glyphs are an innovative instrument that shows several pieces of data at once and requires a legend/key to understand the glyph. The creation of glyphs requires problem solving, communication, and data organization.

Remember coloring pages where you had to color in each of the numbers or letters using a key to color certain areas? How about coloring books that were filled with color-by-numbers? Believe it or not, those pages were a type of glyph.


For the Thanksgiving season I have created a Turkey Glyph. Not only is it a type of graph, but it is also an excellent activity for reading and following directions. Students finish a turkey using seven specific categories. At the end of the activity is a completed Turkey Glyph which the students are to "read" and answer the questions. Reading the completed glyph and interpreting the information represented is a skill that requires deeper thinking by the student. Students must be able to analyze the information presented in visual form. A glyph such as this one is very appropriate to use in the data management strand of mathematics.  If you are interested, just click under the resource cover page above.


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.

A Go Figure Debut for Part of a Trio Who Is New!

Today’s blog post features Sarah who retired last year after 33 years of teaching. Her Teachers Pay Teachers store is called Stem to Steam Trio.

Sarah taught special education, second grade and enrichment through STEM K-4! She enjoyed having the variety! She feels she has always been good at differentiating lessons, and she loves activities that are hands-on and engaging! Her husband (middle school STEM) and her sister (elementary art) are also teachers. They enjoy bouncing ideas off each other; thus the STEM to STEAM Trio was born.

Her family has a woodworking business called Anniversary Banks (anniversarybanks.com). She and her husband have a rather nice woodworking shop where they enjoy creating. Sarah also enjoys walking, gardening, reading, and traveling. And this year, she has even taken up golf! Of course playing with their dog, Snickers, and taking her for walks ranks up there! Now that she is home, Snickers follows her around like Mary's little lamb!
Only $4.00

Sarah finds making resources for TPT is a way to keep the creative juices flowing, while at the same time helping other teachers. There are 227 resources in their store, the majority being science and STEM related. One of these is a great science WebQuest. WebQuest activities challenge students to go on-line (usually to a specific website) to find answers to specific questions. The featured WebQuest about wild turkeys is entitled Wild Turkeys WebQuest. Students use some of the knowledge they have gained while completing the WebQuest to imagine, plan and create a turkey that balances on a branch. This STEM challenge is aligned to the Next Generation Science standards of engineering and design and includes worksheets guiding the student to locate certain information from the website along with specific instructions for the challenge. Materials for the challenge are things most teachers have.

Free Item
Out of the 227 resources, six are free. The freebie featured today is an award sampler. It contains three colorful awards that target 21st Century Skills! They are sure to make students of any age smile! They can be used at an end of the year celebration or given out sporadically throughout the year to reward and encourage students!

In addition, Sarah is a collaborator on the collaborative blog called Stem Activities for Kids. There you will find posts on STEM, science, technology, engineering and math. There are even STEM activities for little learners. This is one blog that you ought to take the time to visit!

Dump and Divide or Better Known As Converting Fractions to Decimals

When working with fractions, my remedial math college students are never quite sure which number to divide by. This same thing often occurred when I taught middle school and high school. So the question I had to answer was, "How can I help my students remember what number goes where?"


First, the student must understand and know the vocabulary for the three parts of a division problem. As seen in the problem above, each part is correctly named and identified.

Side Note: The symbol separating the dividend from the divisor in a long division problem is a straight vertical bar with an attached vinculum (you might have to look this word up) extending to the left, but it seems to have no established name of its own. Therefore, it can simply be called the "long division symbol" or the division bracket. I wish it were named something fancier, but sometimes plain and straight forward is the best!
Now let's look at a fraction that the student is asked to rewrite as a decimal. The fraction on your right is two-fifths and is read from top to bottom as two divided by five. That's easy enough, but when my students enter this into their calculators, many will put in the 5 first, and then press the
division sign, followed by the 2. Of course, they get the wrong answer. Now let's look at the dump and divide method.

First, dump the 2 into the calculator. Then press the division sign; then divide by 5. The answer is 0.4.

I am aware that many of students are not allowed to use calculators; so, let's look at how this method would work using the division bracket. We will use the same fraction of 2/5 and the same phrase, dump and divide.

First, take the numerator and dump it inside the division bracket. (Note: Use N side instead of inside so that numerator and N side both start with "N".) Now place the 5 outside of the long division bracket and divide. The answer is still .4.

Dump and Divide will also work when a division problem is written horizontally as a number sentence such as: 15 ÷ 3. First, reading left to right, dump 15 into the division bracket. Now place the 3 on the outside. Ask, "How many groups of three are in 15?" The answer is 5.

Try using Dump and Divide with your students, and then let me know how it works. You can e-mail by clicking on the page entitled Contact Me or just leave a comment.

Something Else to Think About:  Since many students do not know
their multiplication tables, reducing fractions is almost an
Divisibility Rules
impossible task. The divisibility rules, if learned and understood, can be an excellent math tool. The resource, Using Digital Root to Reduce Fractions, contains four easy to understand divisibility rules as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for the student. Download the preview to view the first divisibility rule plus three samples from the student check off list.


October - Is It "Fall" or "Autumn"?

It's finally October, one of my favorite months of the year. October means football (Ohio State, of course), cooler weather and gorgeous leaves. (It is also when my husband and I were married.) In October, we see the leaves turning colors, and the deciduous trees shedding their leaves.

Another name for fall is autumn, a rather odd name to me.  Through research, I discovered that the word autumn is from the Old French autumpne, automne, which came from the Latin autumnus. Autumn has been in general use since the 1960's and means the season that follows summer and comes before winter.
Fall is the most common usage among those in the United States; however, the word autumn is often interchanged with fall in many countries including the U.S.A. It marks the transition from summer into winter, in September if you live in the Northern Hemisphere or in March if you live in the Southern Hemisphere.  It also denotes when the days are noticeably shorter and the temperatures finally start to cool off. In North America, autumn is considered to officially start with the September equinox. This year that was September 23rd.
With all of that said, the leaves in our neighbor's yard have already begun to fall into ours which aggravates my husband because he is the one who gets to rake them. Maybe focusing on some activities using leaves will divert his attention away from the thought of raking to science investigations.  
Remember ironing leaves between wax paper?  We did that in school when I was a little girl (eons and eons ago).  Here is how to do it.
  1. Find different sizes and colors of leaves.
  2. Tear off two sheets about the same size of waxed paper.
  3. Set the iron on "dry".  No water or steam here!
  4. The heat level of the iron should be medium.
  5. Place leaves on one piece of the waxed paper.
  6. Lay the other piece on top.
  7. Iron away!
You can also use this activity to identify leaves.  According to my husband who knows trees, leaves and birds from his college studies, we "waxed" a maple leaf, sweet gum leaf, elm leaf, cottonwood leaf (the state tree of Kansas - they are everywhere), and two he doesn't recognize because they come from some unknown ornamental shrubs.

Maybe you would like to use leaves as a science investigation in your classroom.  I have one in my Teacher
Leaf Investigation
Pay Teachers store that is a six lesson science performance demonstration for the primary grades. The inquiry guides the primary student through the scientific method and includes 1) exploration time, 2) writing a good investigative question, 3) making a prediction, 4) designing a plan, 5) gathering the data, and 6) writing a conclusion based on the data. Be-leaf me, your students will have fun!

(A preview of the investigation is available. Just click on the title under the resource cover.) 

Using Bloom's 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.

Domino Math

Dots Fun for Everyone
It is believed dominoes evolved from dice. In fact, the numbers in a standard double-six set of dominoes represent all the rolls of two six-sided die. It is thought they originated in China around the 12th century. They have been used in a large variety of games for hundreds of years, and today, dominoes are played all over the world.

Games allow children to learn a great deal concerning mathematical concepts and number relationships. Often they are required to use critical thinking skills as well as varied math strategies to solve them. Since dominoes make a great manipulative for hands-on learning, I created a 29 page book of domino activities for grades 3-5 that are great for students who finish early or for introducing a new mathematical concept or for use at a math center. Using dominoes for a math practice center is a way to engage students in math center practice while giving them a chance to review math facts.

The activities and games vary in difficulty; so differentiated instruction is easy. The variety of pages allows you to choose the practice page that is just right for each student. This resource correlates well with the math series, Everyday Math, as well as with the CCSS standards.

Some of the domino activities in this resource use games while others will extend, enhance or introduce a new math concept. Since children are curious and inquisitive, plus some may have never seen dominoes, allow time for play and exploration before beginning any instruction. This is constructive as well as a productive use of class time. If they are not given this, most children will fool around and investigate during the teaching time. The activities include four digit place value, using the commutative property, problem solving, reducing proper and improper fractions and practicing multiplication and division facts. The games involve finding sums, using <, >, and = signs and ordering fractions.

To view examples from this resource as well as a complete Table of Contents, download the preview or FREE Version.

A Go Figure Debut for A Homeschool Teacher Who is New

Beverly's TPT Store
Beverly lives in Texas, and this will be her 9th year as a homeschool teacher. She has three children, a daughter who is 17 and two boys, ages 14 and 11. Beverly's favorite thing about homeschooling is getting to spend a lot of time with her children and being able to let them really pursue their interests. Her youngest, for example, is interested in computer programming. Beverly has some experience in that area so she found an online course, and she and her son learned the basics of Python together.  Her classroom is a little media room/playroom upstairs where she has a bulletin board, white board, book cases and a long table. At the moment, she is in the process of reorganizing it which includes painting the walls.

Before homeschooling, Beverly taught preschool. The children in her class were only two years of age; so, there wasn't a lot of formal teaching going on, but she absolutely enjoyed the experience.

Beverly has 99 products in her store which is called Terbet Lane; including six that are free. She has a lot of decor and brag tags as well as creative writing and math sets.

Only $6.00
One of her paid items is a set of  Halloween themed story element cards.  She uses these cards in her homeschool. She wanted to create something that would give her children a jumping off place to write creatively and that could change and be something different each time. Her middle son does not like to write and really needed the structure of using these cards to help him think of something to write and to give his writing direction.

To use the cards, have a child choose one card from each category: character, conflict and setting. They are required to use these elements in their story but can add any new characters, settings, etc. that they choose. Her children have really enjoyed using these cards and actually ask to do them. Yeah!

 
Free Resource
One of her free resources is a Halloween coloring page. With Halloween just a month away, this would be the     perfect freebie to download for those students who finish early or for those who just like to color.  (I still do!)

In addition, Beverly has a blog called Terbet Lane although it is pretty new. Her July post was about family fun. Genealogy is one  of her main hobbies, and this post contains some activities that are a fun way to involve younger kids and spark some interesting conversations about family.  I especially like her list of questions to ask grandparents because interviewing an older relative is a great way for kids to learn about their history while connecting with grandparents, aunts, and uncles. I hope you have time to check it out. Also take time to visit her store and see those 99 resources she has created!

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

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.