### Faux Diamonds

In some preschool and kindergarten classes across the country, the geometric shape formerly known as a diamond is now being called a rhombus.  Why?  Does it matter?

To be honest, a diamond is not technically a mathematical shape whereas a rhombus is.  When someone says the word rhombus, you know they are referring to a quadrilateral that has all four sides the same length; the opposite sides are parallel, and the opposite angles are equal.  (Mathematical Warning: A rhombus is not thinner than a diamond, AND the plural form, rhombi, is not a dance performed on the program Dancing With the Stars.)

But what comes to mind when you hear the word diamond?  If you are a woman, you might envision a large sparkling gem setting on the ring finger of your left hand.  If you are a guy, you might think of a baseball infield. (The distance between each base is the same, making the shape a diamond.)  If you play cards, the word might bring to mind a suit of playing cards, OR you might recall a line in the song, Twinkle, Twinkle, Little Star.  Calling a rhombus a diamond is similar to calling a child a "kid" (could be a baby goat), or a home your "pad" (might be a notebook).  The first is an accurate term, the second one is not.

So how does this affect you as a teacher?  It doesn't, unless rhombus is on a local benchmark or state test.  But if you are an elementary grade teacher, please use the correct mathematical language because a middle school math teacher will thank you; a high school geometry teacher will sing your praises, (see song below) and a college math teacher, like me, will absolutely love you for it!

Rhombus, Rhombus, Rhombus
(sung to the "Conga" tune)
(The song where everyone is in a line with their hands on each other's shoulders)

Rhombus, rhombus, rhombus;
Rhombus, rhombus, rhombus
Once it was diamond;
Now it's called a rhombus.

### A Go Figure Debut for a Washingtonian Who's New

Our newest Go Figure Debut takes us west to the state of Washington where we meet a kindergarten teacher named Alex. (I love that name as it is my youngest granddaughter’s name.) This is her fourth year of teaching. She’s had the “privilege” of changing grades each year from 1st grade to 2nd grade, then to a multiage classroom of 1st - 3rd grade students. She now teaches kindergarten and is super thrilled because her heart truly lies with early primary students and beginning readers! She calls this a magical time!

She grew up in the same town where she teaches and actually teaches kindergarten at the identical school where her husband of seven years attended kindergarten! (I wonder how many stories she hears about long ago.) She and her husband have a dog named Tank who is a wild black lab. After seven years, she claims Tank is finally calming down, but she also admits that he gave them a “run for their money” by eating couch cushions, chewing up floors, etc. Nevertheless, they love him and think he is perfect.

Alex is very passionate about creating independent learners, even in the youngest of students. She is a fan and supporter of systems such as Cafe and The Daily 5 and actively uses these ideas in her classroom. She comes from a family of teachers. Her mother taught Montessori for many years, so she has traits in that area as well. Her students can often be found in many different areas of the room working on a variety of tasks at their level. She enjoys watching her students grow while becoming responsible for their own learning! Alex also loves to learn and claims she would go to school forever if student loans and tuition didn't exist. (Don't we wish?)

Alex currently has 107 products in her Teachers Pay Teachers store
 Free Item
called The Kindergarten Connection, and 16 of them are free downloads. If you teach preschool, kindergarten or first grade, here is a good place to find first rate, high quality resources for your classroom. One of her freebies is an Alphabet Match Christmas activity. The Christmas trees and gifts for matching capital and lowercase letters make a great literacy center for December!

I also want to highlight one of her "bundle" (several resources put together at a discount price) called Super Sight Words which is an interactive sight word book. The download includes a total of 40 books - one for each of the following Dolch Pre-Primer words:

a, and, away, big, blue, can, come, down, find, for, funny, go, help, here, I, in, is, it, jump, little, look, make, me, my, not, one, play, red, run, said, see, the, three, to, two, up, we, where, yellow, you

Each individual book features the following pages:
 Super Sight Words
1. Cover
2. Word hunt find and color
3. Trace the word
4. Write the word
5. Decorate the word (you can use torn paper, markers, anything you wish!)
6. Rainbow write the word (using dice to determine color)
7. Cut and paste the word
In addition, each book comes with a necklace, watch/bracelet and crown featuring that sight word! Students should love showing off their learning with these fun accessories!

So if you teach preschool, kindergarten or first grade, check out Alex's products.  I don't think you will be disappointed.  And if you have time, visit her blog (also called The Kindergarten Connection) for other creative ideas you can incorporate into your classroom.

### A Go Figure Debut for a Canadian Who is New

 Sandra

Today we move north, way north to British Columbia to meet a Teachers Pay Teachers seller. Sandra has been teaching for 22 years in the very urban and ESL environment of East Vancouver. She spent ten years in Kindergarten and the rest in first and second grades. For one of those years, she taught kindergarten in New Zealand which is a long ways from Canada!

Like many of us, her teaching style evolved over the years. She strongly believes in demonstrating and preserving respect for all of her students while having classroom structure and firm boundaries. She feels that children work best in an environment where they do hands-on activities in addition to being able to walk around and talk while learning. She recognizes that children learn to read when they read every day, and this is particularly true for struggling readers.

She has three university degrees…a B.A. in English Language (the history and development of the English tongue, not the literature developed from it), a B. Ed and a Masters of Education. What is even more impressive is that she has ten years of classical piano training from the Royal Conservatory of Toronto. This is why she describes herself as a classroom teacher who loves to teach music.

In her former life as a single, carefree woman, she traveled while having lots of adventures. Now she is a full-time teacher with two small children (oldest in kindergarten). Besides creating outstanding products for Teachers Pay Teachers, she loves paper crafting and scrap-booking as well as a playing and watching ice hockey. (Did she say playing?)

 Falling Leaves
 Thanksgiving Unit
Sandra currently has 54 products in her store that comprise various grade levels and subjects. Since it is October, her Falling Leaves (K-2) is a perfect resource, and it’s free. There are four mini books in this download that can be used for different levels of reading in a K/1 class. The books focus on interpreting the text, copying, signs of fall, drawing and coloring as well as using fine motor skills.

Sandra also has a 35 page Thanksgiving unit that includes two big books, a shape book, a pattern book, and a finger play poem with language and art activities. This unit is language based, and includes lessons on reading, writing, singing, finger plays and art.

I trust you will take a few minutes to visit her store and see the many first class resources she is offering.  You might also enjoy her blog which is called Sandra's Savvy Teaching Tips.  While you are there, why not become her newest follower?

### Learning Geometry Using Number Tiles

My college students will soon start the unit on plane geometry.  I love teaching geometry because it is so visual, but there are others who despise it because of the numerous new words to learn.  In fact, our plane geometry unit alone contains over 50 terms that must be learned as well as understood.

I have found that with my students, mathematical language is either a dead language (It should be buried and never resurrected!), a foreign language (It sounds like a different language from a far away country.), a nonsense language (It makes no sense to me - ever!) or a familiar, useful language. Many times, they are unduly frustrated because mathematical language has never been formally taught or applied to real life.  For example, many primary teachers will have their children sit on the circle when in fact, the children are sitting on the circumference of the circle.  What a wonderful, concrete way to introduce children to the concept of circumference!  Yet, this teaching moment is often missed, and circumference doesn't surface again until it is time to teach the chapter on circles.

 Plane Geometry + Number Tiles
Because I believe it is important to find different ways to introduce and practice math vocabulary, I created a new resource for Teachers Pay Teachers entitled: Geometric Math-A-Magical Puzzles.  It is a 48 page handout of puzzles that are solved like magic squares. Number tiles are positioned so that the total of the tiles on each line of the geometric shape add up to be the same sum. Most of the geometric puzzles have more than one answer; so, students are challenged to find a variety of solutions.

Before each set of activities, the geometry vocabulary used for that group of activities is listed. Most definitions include diagrams and/or illustrations. In this way, the students can learn and understand new math words without difficulty or cumbersome words. These activities vary in levels of difficulty. Because the pages are not arranged in any particular order, the students are free to skip around in the book. All of these activities are especially suitable for the visual and/or kinesthetic learner.

A ten page free mini download of this item is available if you want to try it with your students. Check it out!

### A Go Figure Debut for a Duo Who Are New!

This week, I would like to introduce two elementary teachers who work on the same fourth grade team in west Michigan.  (Uh-Oh! I'm an Ohio State fan; so that makes us rivals!)  Amy has been teaching for 12 years. She began her career teaching in a 3rd-5th grade classroom of students with emotional impairments. She is now in her fifth year as a fourth grade teacher and is loving every minute of it. She tries to find engaging and challenging ways for her students to apply their knowledge of the fourth grade standards every day! She claims to be "obsessed" (that's another word for fanatical) with curriculum. She tells me that she loves picking apart the standards and finding new and creative ways to deliver them.

Last year, Molly joined the fourth grade team Amy was on. Molly has been teaching for four years. She spent the previous two years in a fifth grade classroom just down the hall  from Amy. Molly is driven to provide higher level thinking activities that allow her students to apply the information she has presented over the course of a unit. She loves getting her students to think, and she is constantly looking for new and innovative ways to encourage them to think outside of the box.

Both Amy and Molly have a passion for creating and developing solid units of study that give students an opportunity to apply their knowledge and think deeply about their learning. Opening a Teachers Pay Teachers store together and sharing their work with other teachers has been their dream for some time now. They also have an interesting blog entitled Two Nutty Teachers Teachin' from the Same Tree.

They currently have 34 products in their store, five of which are free.  I especially like the free resource Experimenting with the Scientific Method. This eight page resource is a perfect tool for teaching the Scientific Method! While following the process, the students view the scientific method in action by performing an experiment using gummy bears. (Yeah - FOOD!) The product is formatted in a way so that it can be used individually or added into a Interactive Science Journal. By downloading this FREEBIE, you will receive suggestions on how to use the download, steps to run the experiment, and seven one-half-page graphic organizers for all the steps of the Scientific Method.

From their paid products I chose Fractions and Line Plots, A Three Season Collection.  These are season themed task cards (fall, winter and spring) that teach students how to create a line plot with fractions! The cards include real-world situations that your class can relate to in order to help them understand the steps involved in this difficult process. In the CCSS aligned 88 page product, Amy and Molly explain how they use this download in their own classrooms

Included in this product are the following:
1. Steps for Creating a Line Plot Handout
3. A set of questions to go with EACH task card (with answers)
4. Six example activities (with answers)
So take some time to check out these Two Nutty Teachers who seem to be doing amazing things together. They both thank you in advance and for taking part in their journey!

### Accentuate the Positive - Eliminate the Negative

Have you ever wondered why a negative number times a negative number equals a positive number?  As my mathphobic daughter would say, "No, Mom.  Math is something I never think about!"  Well, for all of us who tend to be left brained people, the question can be answered by using a pattern.  (Have you noticed a reoccurring theme in my articles?  All Math is Based on Patterns!

Let's examine 4 x -2 which means four sets of -2.  Using the number line above, start at zero and move left by twos - four times. Voila!  The answer is -8.  Locate -8 on the number line above.  Now try 3 x -2.  Again, begin at zero on the number line, but this time move left by twos - three times. Ta-dah!  We arrive at -6.  Therefore, 3 x -2 = -6.

Here is what the mathematical sequence looks like.  Moving down the sequence, observe that the farthest left hand column decreases by one each time, while the -2 remains constant. Simultaneously, the right hand answer column increases by 2 each time.  Therefore, based on this mathematical pattern, we can conclude that a negative times a negative equals a positive!!!!

Isn't mathematics amazing?

### Examining Multiple Choice Questions

According to Ron Berk (a keynote speaker and Professor Emeritus of Johns Hopkins University)the multiple choice question "holds world records in the categories of most popular, most unpopular, most misused, most loved and most hated" of all test questions.  Because of the many students teachers see each day and the little time teachers have to make tests and then grade them, multiple choice questions have become one of the favorite type of testing questions in education.  We see them on state assessments, national assessments, ACT tests, college tests, driving license tests, etc.  However, those who consistently use them aren't all that crazy about them and with good cause.

First of all, answering multiple choice questions doesn't teach students how to formulate answers; it teaches them how to select answers.  Many times choosing the right answer is more a literary skill rather than of content knowledge.  Multiple choice questions promote guessing, and if a guess is right, students get credit for something they didn't know.  Moreover, the instructor is deceived into thinking the student understands the concepts being tested.

Many multiple choice questions do not challenge students to think.  Instead they encourage the students to memorize.  In my opinion, test bank questions are the worst.  A simple analysis of this type of question in a variety of disciplines suggests that about 85% of the multiple choice questions test lower level knowledge, levels I (remembering) or II (understanding) of Bloom's Taxonomy.

When I first started at the community college where I teach math, the math assessments which the department used were mostly multiple choice.  I asked about testing our students using levels V or VI of Bloom's and the reaction I received was disheartening.  One instructor implied that our math students would be unable to answer such questions.  I guess his expectations were a great deal lower than mine. I am positive he hadn't read an article by two professors at Kansas State University.

According to Victoria Clegg and William Cashin of K.U., "Many college teachers believe the myth that the multiple choice question is only a superficial exercise - a multiple guess - requiring little thought and less understanding from the student.  It is true that many multiple choice items are superficial, but that is the result of poor test craftsmanship and not an inherent limitation of the item type.  A well designed multiple choice item can test high levels of student learning, including all six levels of Bloom's Taxonomy of cognitive objectives."  (Idea Paper No. 16, Sept. 1986)

So what are some things that make challenging multiple choice questions? Let's take a multiple choice test to help us answer that question.

Choose the best answer.  Which multiple choice question is the hardest to answer?

a) The one where it’s absolutely obvious that all choices are wrong answers.
b) The one where the question and/or answers are so badly written that two or more answers could be correct depending on how the student interprets the question.
c) The one where the list of possible answers are true or false; it depends on how the the student reads the question.
d) The one question where it is really two questions in one, but the options only answer one part of the question.
e) All of the above – except that there is no "all of the above" option given.

I trust you see the humor in this question.  Unfortunately, I have seen one or all of the above on math tests given in our department.

Now let's look at two different multiple choice questions from a mealworm test available on Teachers Pay Teachers.  The first one is pretty straight forward and requires little thinking on the part of the student. On Bloom's, it would represent a level I question - remembering.

 Mealworm Test
1) Which tool will help you best see the mealworm up close?

a) Ruler
b) Mirror
c) Hands Lens
d) Eyedropper

The next question from the same test requires the student to understand what a good scientific investigative question is. This would be a level IV question which is analyzing.

4) Which question can be answered by investigating?

a) Will the mealworm eat the fruit?
b) How far can a mealworm travel?
c) Will more mealworms go to the paper with an apple slice on it or to the one with no fruit on it?
d) Why do mealworms move?

How about this one from a butterfly test?  (also available on TPT)  What level of Bloom's does it represent?

3) How does the life cycle of a butterfly differ from the life cycle of a frog?

 Butterfly Test
a)     Only the butterfly has an egg.
b)     Only the butterfly has an adult stage.
c)      Only the frog has a tadpole.
d)     Only the frog has a pupa.

Again it is level IV because the student is asked to compare; yet, this test is for the grades 3-5 while the meal worm test is for grades 5-8.  I give you examples from both so you can see that challenging multiple choice questions can be written for most grade levels.

If you would like help writing good, challenging questions of all kinds, you might check out Bloom's Taxonomy Made Simple.  It is a five page handout that breaks the six levels of Bloom's down into workable, friendly parts, using the familiar story of The Three Little Pigs. Examples of good ideas of how to write assessment questions using all six levels of Bloom's are given.  For your practice, a follow-up activity of 16 questions is included.

### It's Another "Go Figure" Debut for Someone Who's New!

It’s time for another debut of someone who is a new seller on Teachers Pay Teachers. This time it is a primary teacher from North Carolina who, in reality, is a Rhode Islander at heart. Michaela’s Teachers Pay Teachers store is called K’s Kreations and presently contains 52 items. With over 80 votes, her overall ranking is a 4; so, immediately you know she has quality products.

She has taught first grade for six years and she tells me she loves it! (My first teaching assignment was first grade…a place where I have no intention of returning!) She’s had the opportunity to work with a diverse population of students, and she continually strives to create a class climate that is nurturing for everyone. Michaela believes that all students need plenty of opportunities to explore, collaborate, and develop self-confidence while learning. She utilizes as well as creates many hands-on activities and centers to do this.

 K's Kreations
Michaela began selling on TPT on January 1st of this year. She wanted to start off the new year with a new adventure! She loves to share ideas, mentor teachers, and come up with creative ways to engage students. The fact that thousands of teachers and students could be using her products makes her feel proud. What she loves most about TPT is the opportunity to reach out to many as well as the true feeling of belonging to a community on a larger scale.

Her store consists of games, math resources, literacy ideas as well as several seasonal items. She has eight items which you can download for free to get an idea of the products she creates. One of them is entitled. It is a whole group math activity that gets students up and moving. Students work with various partners to practice number sense and counting skills. Teams count how many objects (use counters or base ten blocks) each person has and determines who has the most/least.

 New Resource
Her newest product is entitled Fall Addition and Subtraction Homework for first grade for first grade and is a 50 page resource. It contains dozens of lesson ideas, a break down of the standards, and a variety of homework/quiz pages with answer keys. As a bonus, four Number Sense pages have been added to this packet to help meet the back-to-school needs.

I hope you will take a moment to check out all of her top quality products. Download one of her free items, and then take the time to leave her some feedback.  Michaela is anxious to improve and upgrade her resources as needed.

### Common Core Who-Dun-It Mystery

As a member of Teachers Pay Teachers, I often read and share on their Seller's Forum.  As the Common Core State Standards (CCSS) become more "common", many teachers are asking about things being omitted or totally left out.  Let's start this discussion with what the Common Core supposedly is.  CCSS is a state-led effort coordinated by the National Governors Association and the Council of Chief State School Officers. The standards establish common goals for reading, writing and math skills that students should develop from grades K-12.  Although classroom curriculum is left to the states (which actually had no input into the process), the standards emphasize critical thinking and problem solving and encourage thinking in-depth about fewer topics.

With that said, this is the way I perceive these standards.  When I started teaching, (I have been at it for 30+ years) the curriculum was a nice, juicy apple. Included were subjects like spelling, geography, history, and cursive writing.  In addition, areas such as effort and behavior were evaluated. I can't remember ever giving a state or national test, but I did have to teach art, music and P.E. The majority of the children went home for lunch where some adult was waiting for them.  Later, the arts were added to the curriculum and qualified teachers were hired to teach art, music and P.E. (Thank goodness!)

Then came the slicing of the apple.  One test was introduced and given each year.  (We gave the ITBS.)  Objectives were written that were different from the textbook, and more children were staying at school for lunch.  As time progressed, additional slices of the apple were removed as "before and after" school programs became necessary for children and free lunches became common place.  In addition, more than one test was required because now the district and the state wanted data.  History and geography became social studies, and phonics and spelling were replaced with the whole language approach.  I was directly a part of our district's benchmark test writing project where much money and time were devoted to the test that would reveal all, make teachers better, and students smarter.  Of course, none of those things occurred, and the money wasted could have been better spent on teachers who really make the difference in the classroom.  (By the way, all of those assessments are now gone.)

Many advocate that the CCSS will become the tool that can successfully turn around education.  Let's remember that these standards are merely the minimum each grade level is to master.  Think of the CCSS as the core of an apple; there is no "meat" on the core, just the left over part of the apple.  The basics are there; but teachers need to add the meat, but will they, especially since the high stake tests that are imminent will most likely only test the core?  My question is: How much testing will be required; how often, and at what expense in money and time?  And who will pay the price?

I did some of my own reading of the CCSS, particularly those for the "key" grades of K-3.  Yes, the CCSS requires the multiplication tables be taught through 10, but does that mean a teacher shouldn't go to 12?  I personally want all of my algebra students to know the doubles through 25 because it makes finding the square root so much easier.  Since the multiplication fact is in the student's head, no calculator is required!  I also observed that money is not mentioned anywhere in the common core for grades K-3 except in second grade.  Covering that standard is going to be a daunting task for 2nd grade teachers if students have never seen it before.  (I did find the money standard in grades 4th, 6th, and 7th, and after that, money was considered Consumer Science.)  In addition, there is no standard for patterns in kindergarten which I find quite disturbing since all math is based on patterns.  Time and introductory place value have also been deleted.  If students do not get these basic concepts in kindergarten, it is obvious they cannot grasp the more complex ones in later grades.

As I read the many different responses on the TPT Forum as well as various articles about the Common Core, I realized that many teachers are viewing them as the all-in-all.  If that is all that will be taught, education is in BIG trouble.  I would suggest reading a rather thought provoking and eye opening article by Carol Burris, principal of South Side High School in New York. She was named the 2010 New York State Outstanding Educator by the School Administrators Association of New York State, and she is co-author of the book Opening the Common Core.

My primary concern is that the Common Core will become so focused and fixated on a limited number of standards that little will be left of well-rounded education except a very inadequate and flimsy core. If a teacher only follows the common core and nothing more, students will miss important building blocks in between. I believe education is an exciting and engaging lifetime journey, not a final destination or a binding contract with any government. I trust and hope parents (families) are the constant in this equation (a math word!) while schools and teachers are the variable. (both will change over time) How can any test adequately measure that?

Here is a ten minute video which explains the Common Core so that
anyone can understand it. Check it out at: Common Core

### A Go Figure Debut for Someone Who's New

It's time to introduce another seller who is new to Teachers Pay Teachers. This time I have chosen a 6th grade math teacher. After visiting her store, I was impressed with what I saw, and knew that you, as one of my readers, would like her products as well.

Her name is LaDawn. She started her teaching career over 18 years ago. She has experience teaching 3rd grade, kindergarten, 5th grade and 6th grade. She currently teaches 6th grade math and she says she LOVES her job! Since she is a visual learner, that is the way she teaches her students. She uses the Promethean board on a daily basis and loves to incorporate music, comics, and videos to help her students grasp the math concept being taught. (This is my style of teaching, for sure!) She is married and has two children.

Her Teachers Pay Teachers store is called Math From My Angle. (I'm not sure what kind of angle she is, but I bet she is usually "right"!) She presently has 32 different products in her store. They are mostly math materials although I did see a cloud power point and a bingo game for the end of the year.

The quote that she uses at her store is: "The best teachers are those who show you where to look, but don’t tell you what to see." (Alexandra K. Trenfor) It appears from her store profile that this quote is something she truly believes.

A warm welcome to the world of Teachers Pay Teachers, LaDawn. We look forward to reviewing more of your products for my favorite subject in the whole world - MATH!

I have decided to post (this is an updated previous post from 2011) some questions about zero that my college students have asked me in class.  I will say this, "Zero can surely give you a severe headache unless one knows its properties."

Question #1 - Do you know why zero is an even number?    All mathematics is based on patterns.  Because of this, I know that an even plus an even number will always give me an even answer; an odd number added to an odd also gives me an even answer, and an odd number plus an even gives me an odd answer. In other words:    E + E = E     O + O = E     O + E = O﻿﻿﻿﻿﻿﻿﻿
﻿﻿﻿﻿﻿﻿﻿
The numbers 4 and -4 are both even numbers. If we add them together, their sum is zero.  Based on the math pattern of  E + E = E,  then zero has to be even as well.  If we substitute zero in other problems such as 1 + 0 = 1, it fits the O + E = O  rule just as 2 + 0 = 2 fits the E + E = E  rule.

In Algebra, even numbers can be written as 2 x n where n is an integer.  Odd numbers can be written in the form of  2 x n + 1.  If we have n represent 0, then  2 x n = 0 (even) and  2 x n + 1 = 1. (odd)

﻿﻿I say all of this to relate an actual incident that occurred in my classroom.  I wrote the number 934 on the white board, and commented that since it was even it was divisible by 2.  One of my students was perplexed because he did not understand how 934 could be even when it contained two odd numbers and only one even number.  He actually thought that all the digits of a number had to be even for the number to be even.  Funny?  Not really!  Amazingly, he had made it through 12 years of school without understanding Place value as it relates to even numbers. Unfortunately, I had assumed that everyone (especially my college students) knew what an even number was.  I no longer make assumptions about students and their math knowledge!

Question #2 - Is zero positive or negative? The definition for positive numbers is all numbers greater than zero, and the definition for negative numbers is all numbers less than zero. Therefore zero can be neither positive or negative.

Question #3 - Is zero a prime or composite number? To be a prime number, a number must have only two positive divisors, itself and one. Zero has an infinite number of divisors so it is not prime. A composite number can be written as a product of two factors, neither of which is itself. Since zero cannot be written as a product of two factors without including itself, zero, it is not composite.

Question #4 - Why can't you divide by zero? I love this question. Back in the dark ages when I asked it, I was always told, "Because I said so." Being an inquisitive student was not a blessing when I was growing up. Math teachers who knew all did not want to be questioned!!!! Anyway, I don't mind the question, and here is my practical answer.

First, we must understand division. Division means putting or separating a number of items into a number of specific groups or sets. When you divide, such as in the problem 12 divided by 2, you are really putting 12 things into two groups or two sets. Therefore, if you have the problem 8 divided by 0, it is impossible to put eight things into no groups. You cannot put something into nothing!
﻿﻿
Hopefully, this clears up a few things about zero.  I leave you with this math cheer.  (I always wanted to be a cheerleader!)

Zero, two, four, six, eight,

Who do we appreciate?

Even numbers! Even Numbers! Even Numbers!

### Can't Memorize Those Dreaded Math Facts!

Please note: This article was originally posted in March of 2011.  Because many parents question why their child cannot memorize math facts, I am posting the article again.

Many of my college students come to me without knowing their math facts. Some do, but most do not. Since we use calculators in the class, it really isn't an issue.  It just takes those students longer to do a test or their homework. One day, the students in my Basic Algebra Concepts class (a remedial math class) were playing a math game to practice adding and subtracting positive and negative numbers. We were using double die (see picture) where a small dice is located inside a larger dice. (I have to keep an eye on these because they tend to "disappear". The students love them!)  I noticed one of my students continually counting the dots on the die. He was unable to see the group of dots and know how many were in the set.  It was then that I realized he could not subitize sets. (to perceive at a glance the number of items presented)

Subitizing sets means that a person can look at a grouping or a set and identify how many there are without individually counting them.  (i.e. three fingers that are held up)  When a child is unable to do this, they cannot memorize math facts since memorizing is associating an abstract number with a concrete set.  Many teachers as well as parents fail to recognize the root cause of this memorization problem.  AND no amount of practicing, bribing, yelling, or pulling out your hair will change the situation.  So what can you do?

First of all, the problem must be identified.  Use a dice and see if the child must count each dot on each face. Try holding up fingers or laying out sets of candy (M&M's - yummy!) or using dominoes. Put five beans in a container, and ask the child how many are in the box. (They may count them the first few times.)  Take them out, and put them back in.  Ask the child again how many there are.  If, after several times, s/he is unable to recognize the set as a whole, then s/he cannot subitize sets.

How do you help such a child?  If you have small children at home, begin the subitization of sets by holding up various combinations of fingers.  My granddaughter just turned four; so, we worked on holding up two fingers on one hand and two fingers on the other; then one and three fingers, and of course, four fingers. I also like to use dominoes. They already have set groupings which can be identified, added, subtracted, and even multiplied. A dice is great because the child thinks you are playing a game, not doing math.  Roll one dice, and ask the child to identify the set of dots. Try the bean idea, but continue to change the number of beans in the box.  My granddaughters love the candy idea because they are allowed to eat them when we are done.  (All children need a little sugar now and then even though their parents try to control the intake.  I love being a Grandma!)

Gregory Tang has written two wonderful books for older children, The Grapes of Math and Math for All Seasons, which emphasize subitizing sets. At times, I even use them in my college classes!  I was fortunate to attend two of his workshops presented by Creative Mathematics. He not only has a sense of humor, but his books can be read again and again without a child becoming bored. Check them out!

### A "Go Figure" Debut for Someone Who's New!

We have many talented teachers who become sellers on Teachers Pay Teachers.  Presently TPT is over 61,000 strong; so, sometimes it is hard for a new store to get noticed. Every once in awhile, I will be introducing one of those new sellers to my blog readers.  I choose these particular stores because they contain high quality items, resources that are out of the ordinary and something that my husband or I can download and use in the classroom.

 Acorn's Store
Today I would like you to meet Acorn, a teacher from Dublin, Ireland who is "nuts" about science.  Barry has been a science and ICT (Information Communications Technology - a fancy term for computers!) teacher for 20 years. He has also been a career guidance teacher, and he is the author of two books. Barry describes his teaching style as "friendly", and adds that he is a passionate believer that if you can't make something easy to understand then you don't understand it.

I first found his store because I was intrigued with his free resource. I downloaded it for my husband who teaches science, and he was "hooked". Acorn's products are for grades 4-11 although my husband thinks they are perfect for middle schoolers. They are engaging and hold the students attention. Acorn uses humor interspersed with necessary science knowledge that students are required to know. Check out his freebie entitled: Ionic & Covalent Bonding. It's an animated journey into chemistry and is the love story that was never told. In other words, it is a Chemical Romance. A short "fast draw" animation including cheesy science jokes demonstrates how ionic bonding occurs between sodium and chlorine.

Welcome to the world of Teachers Pay Teachers, Acorn. We look forward to reviewing more of your unique products!

### Ben Franklin - A Math-a-Magician!

Did you know that Benjamin Franklin created many inventions, including the Magic Square? (A magic square is a box of numbers arranged so that any line of numbers adds up to be the same number, including the diagonals!) Richard Walz has written a historical fiction book about this. It is fun for the students to read while at the same time it gives them a great deal of historical information. It also contains many activities that can be used along with the book.

History shows that Franklin served as clerk for the Pennsylvania Assembly. Uninterested in the meetings, Ben would doodle on a piece of paper to pass the time. In 1771, he stated, "I was at length tired with sitting there to hear debates, in which, as clerk, I could take no part, and which were often so un-entertaining that I was induc'd to amuse myself with making magic squares or circles" (Franklin 1793). So being bored, Ben wrote down numbers in a box divided into squares, and then pondered how the numbers added up in rows and columns...and thus the Magic Square was born. In fact, he studied and composed some amazing magical squares, even going so far as to declare one square “the most magically magical of any magic square made by any magician.”

I love to use Magic Squares in my classroom. The construction and analysis of magic squares provides practice in mental arithmetic, operations with numbers, geometry, and measurement plus it encourages logical reasoning and creativity, all in a game-like setting.  Furthermore, they are a powerful tool for teaching students basic addition skills since each row, column, and diagonal must add up to be the same sum.

One effective way to use a Magic Square is to omit a few of the numbers from the boxes, then have students try to figure out which numbers are missing. To find these numbers, first the students will have to calculate the magic sum. A Magic Square also provides an engaging way to develop mental math skills. Try using magic squares as a warm-up at the beginning of math class or as a math center activity. In addition, students might also want to create their own Magic Squares and then have their classmates solve them.

Below is a magic square for you to solve. You are to arrange the digits 1-9 in the squares below
so that each column, row and diagonal adds up to 15. Can you do it?

To find a solution to this magic square puzzle, look under

Your students can make their own Magic Squares by following these steps.  Begin by using a box divided into nine squares.  (There are larger ones, but as they grow in size so does the difficulty.)

1. The first numeral is placed in the top row, center column.
2. An attempt is always make to place the next numeral in the square above and to the right of numeral last placed.  All the rest of the rules tell you what to do when rule #2 cannot be satisfied.
3. If, in the placement of the next numeral according to rule #2, the numeral falls above the limits of the magic square, place the numeral in the bottom square of the next column to the right of the last placed numeral.
4. If, in the normal placement of the next numeral, it falls to the right of the limits of the magic square, that numeral is placed in the left-hand square of the row above the last placed numeral.
5. If the cell above and to the right is filled, place the numeral in the cell immediately below the last square filled.
6. Using this method, filling the upper right-hand cell completes a sequence of moves. Then this happens, the next numeral is placed in the cell immediately below the upper right hand corner square.
Do these steps sound absolutely confusing?  Maybe the pictures below will help to clarify the rules.

Now have your students try this.  Using the blank nine squared Magic Square seen above, use the numerals 11, 12, 13, 14, 15, 16, 17, 18 and 19 to make each row (horizontal, diagonally & vertical) add up to 45. Ask your students if they see a pattern between this new Magic Square and the first one.  (Ten has simply been added to each digit.)  You might also try making your own at Make Your Own Magic Squares.

This post has only scratched the surface of Magic Squares, but isn't that like most things in math?  I trust your students will give Magic Squares a try while having fun doing it!

Check out my newest product entitled The A, B, C's of Number Tiles.  All of the 26 letter puzzles of this 42 page handout are solved in a similar way that magic squares are solved. The activities vary in levels of difficulty. Because the pages are arranged alphabetically, and are not in any particular order based on difficulty, the students are free to skip around in the book. Since the students do not write in the book, the math-a-magical puzzles can be copied and laminated so that they can be used from year to year.

### When Dividing, Zero Is No Hero

Have you ever wondered why we can't divide by zero?  I remembering 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 that 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.

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.
 Free - Why We Can't Divide by Zero

### The Disturbing Saga of Math Cannibals!!!

I have just started to teach Basic Algebra Concepts to my college mathphobics.  This is where the "rubber meets the road" as they say.  The biggest hurdle for my students is understanding positive and negative numbers.  Multiplying and dividing seem to be no problem, but addition and subtraction are another story.  To state that subtracting a positive number is the same as adding a negative number is considered hieroglyphics to many.  Since many of my students are visual/kinesthetic learners, I needed a strategy that would connect the abstract to the concrete. ﻿

I took film canisters (a Trash to Treasure idea!) and filled them with two sided beans. One side of the bean is red (negative), and the other side is white (positive). Suppose the students have the problem -5 + 2.  They would get out five red beans and two white ones as illustrated on the left. Then the fun begins because suddenly the beans become "cannibalistic".  The red ones begin to "eat" the white ones and vice versa. (In reality, the students are matching each red bean with a white one and moving them aside; see illustration on the right.)  After each bean has been “eaten” by the opposing color, three red beans remain.  As a result, the answer to the problem of -5 + 2 is -3.

 Don't stew; study!
If the problem were -2 - 6, the students would lay out two red beans and six red beans.  Since all the beans are the same color and no bean desires to "eat" anyone on their team, the student simply counts all of the red beans.  So  -2 - 6 = - 8.

What happens with a problem such as 5 + -3?  At the beginning, I have the students get out five white beans and three red ones; then match them resulting in the answer of 2.  Unfortunately, in our Algebra book, the double signs vanish by about the third page of the chapter; so, the students must recognize what to do.

The first option is to insert a + sign such as in the problem – 4 – 2 = -4 - +2.  This allows them to see that, in reality, they are subtracting a positive number.

However, what do they do with -4 - -2?   I instruct them to circle the two signs, and use the multiplication rule for a negative times a negative to change the double minus signs into a plus sign as seen in the illustration on the left. They can then proceed to use their beans to solve the problem.  This may seem unusual, but it makes sense to my mathphobics.

You might ask, "How long do the students use the beans? It’s interesting, but all of my students put them away, just at different times.  A few only need them for the first assignment whereas others need them for many.  I once had a special education student who was mainstreamed into my regular PreAlgebra class.  He was the last one to rely on the beans, but he did eventually put them away.  The important thing was he had a picture in his head that he could use over and over again.  Incidentally, he passed the class with a “C”, completing all of the same work the other students did.

Need a game instead of a worksheet to practice adding and subtracting positive and negative numbers?  Try Bug Mania or Roll and Calculate.  Just click on the name of the game.