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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.

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!

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.


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

Much Ado About Nothing

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!

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! 


Ben Franklin - A Math-a-Magician!

Available on Amazon
for about $3.99


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
Answers to Problems at the top of the Home page.

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.



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.

SMART Goals

My Math Study Skills class has just started chapter #5 on setting goals.  So many times my students will write goals such as "I will study more for math".  Sounds great, but this statement isn't a goal.  It is not specific or measurable, and I have no idea who is doing the goal.  Instead it should read something like this:  "I plan to set aside 15 minutes each Monday through Friday to study math."

Since mnemonic devices are a way to help students remember, I introduce the acronym (a word form created from the first letters of a series of words) SMART.

      A well written goal is learner oriented.  It emphasizes what the student is expected to do, not what the instructor will do.  It focuses on the outcome and not the learning activities that will lead to that outcome.  It uses clearly stated verbs that describe a definite action or behavior.  Finally, a well written goal describes an observable and measurable performance or end product.

     
      I keep this stair step visual in front of my students during the five weeks they are tracking their three math goals. It helps them to set-up attainable goals.
      When they accomplish a set goal, I have 
      noticed they feel more confident about math which, in turn, improves their self-esteem and helps the
      student to become a more internally motivated student.


I use a booklet called My Goal Tracker by Laura Candler, a top seller on Teachers Pay Teachers, which is free.  If you are interest in having your students set goals and keep track of how they are doing, I would suggest downloading this well laid out and easy to use booklet.



It's A Square Deal





My math-a-magical powers are back.  In my January 23rd post, I demonstrated how to easily divide by the decimal .25.  Today let's look put on our magician's hat and learn how to square any two digit number that ends in 5.

In the number below, look at the digit in the tens place (3) and count up one more.  In this case when we count up, the 3 becomes a 4.  Now multiply the two numbers together, 4 × 3 which equals 12.  Next do 52 which is 5 × 5 = 25.  Therefore, the answer to 352 is 1,225.


Let's try another one.  Remember, it must be a two digit number that ends in 5 for the magic trick to work. This time let's start with 65 and follow the exact same procedure used above.


Did you get the right answer?  Have your students try this using all the two digits numbers that end in 5, beginning with 15 and finishing with 95.  Since all math is based on patterns, ask hem to carefully look for a pattern and describe it.  I think you will be very surprised at what you find!

 

The Power of Math Tricks

Math tricks will never make you a great mathematician, but in the eyes of some, you can be a fantastic math-a-magician. My college students love it when I show them a trick they can then take home to amaze and impress their peers, parents, children or the best yet - their spouses. 

Remember when I demonstrated how to easily multiply by 11 in the post The Eleventh Hour?  Or how about the trick of multiplying by 12 in Quick Times?   Here is a new one I recently showed my students.

First, let's look at a problem where a number is divided by the decimal .25


The above example is really 9 x 4 which is 36, but why is this true?  Hopefully your students know that .25 is equivalent to 1/4; so this problem can be reworded as  9 divided by 1/4.  As seen below, when dividing a whole number by a fraction, we find the reciprocal of 1/4 and then multiply which gives us the answer of 36.
 

Based on the sample above, anytime a problem requires dividing by .25, simply multiply by four to get the correct answer.  Try these without using a calculator or paper and pencil.



Instead of using the reciprocal to divide fractions, I teach my students that this is the "cross" method. Simply look at the original problem and cross multiply as seen in the illustration below. 

Fractions for the
Confused
and Bewildered

First multiply the bottom right denominator with the top left numerator. (4 x 9)  Next multiply the bottom left denominator with the top right numerator, (1 x 1) and you get an answer of 36.  When doing the fractions this way, there is no confusion on the students' part about which fraction to invert. If you would like a more details on how to divide fractions this way, go to the post entitled: Don't Flip!

If you are interested in other alternative ways to teach the four operations of fractions, you can check out the resource on your right.


By the way, the answers to the above problems are  a) 24    b) 80    c) 72    d) 380.    How did you do?