Ten Black Dots - Another Book that Links Math and Literature

I am an avid reader, and I love books that integrate math and literature. Lately, my blog has featured books that link the two.  

Available on Amazon
for $7.60

Today's featured book is Ten Black Dots by Donald Crews (Greenwillow Books, 1986).  This picture book is for grades PreK-2 and deals with numbers and operations. 

The book asks the question, What can you do with ten black dots?  Then the question is answered throughout the book by using  illustrations of everyday objects beginning with one dot and continuing up to ten. Simple rhymes accompany the pictures such as:

"Two dots can make the eyes of a fox, Or the eyes of keys that open locks."

Materials Needed: 

• Unifix cubes or Snap Cubes (multi-link cubes) as seen on the right
• Black circles cut from construction paper or black circle stickers
• Crayons
• Pencils
• Story paper
• Calculators -simple ones like you purchase for $1.00 at Walmart

Activities:

1)  Read the book a number of times to your class.  Let the students count the dots in each picture. On about the third reading, have the children use the snap cubes to build towers that equal the number of dots in each picture.

2)  Have the children think of different ways to make combinations, such as: How could we arrange four black dots?  (e.g. 1 and 3, 4 and 0, 2 and 2)  Have the children use black dots or snap cubes to make various combinations for each numeral from 2-10.

3)  This is a perfect time to work on rhyming words since the book is written in whimsical verse. Make lists of words so that the students will have a Word Wall of Rhyming Words for activity #4.

• How many words can we make that rhyme with:  sun?  fox?  face?  grow?  coat?  old?  rake?  rain?  rank?  tree?
• Except for the first letter, rhyming words do not have to be spelled the same.  Give some examples (fox - locks or see - me)

4)  Have the children make their own Black Dot books  (Black circle stickers work the best although you can use black circles cut from construction paper. I'm not a big fan of glue!)  Each child makes one page at a time.  Don't try to do this all in one day.  Use story paper so that the children can illustrate how they used the dots as well as write a rhyme about what they made.  Collate each book, having each child create a cover.

5)  Have the children figure out how many black dots are needed to make each book. (The answer is 55.)  This is a good time to introduce calculators and how to add numbers using the calculator.

If you can't find Ten Black Dots in your library, it is available on Amazon.

Another Book that Links Literature and Math

Available on Amazon
for about $5.99

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.”

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!

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Check this math resource entitled Number Tiles for Grades 5-8.  All the puzzles of this 26 page resource are solved in a similar way that magic squares are solved. The activities vary in levels of difficulty. Because the pages are not in any particular order, 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.

Anno's Counting Book - A Math Picture Book

School is out for some of you, and for others, the last day is very close. I thought I would recommend some books to read over the summer with your children or to use in your classroom next year. All the books link math and literature in some way.

Anno’s Counting Book by Mitsumasa Anno is one of the best math picture books for children that I have used with kindergartners and first graders. This wordless counting book shows a changing countryside through various times of the day and seasons. It introduces counting and number values from one to twelve. On each page, you can find several groups of items representing the illustrated number, such as 4 fish, 4 trees, and so on. The number is also represented by stacked cubes at the side of the illustration. The book contains one-to-one correspondence, groups and sets, and many other mathematical relationships. I purchased the Big Book version so that the entire class could easily see each picture.

Here are a couple of activities that you might try with the book.

1) “Read” the book to the children and discuss what is happening. The following questions will help the children to connect what is occurring in the book:

    a) What time of year is it when the story begins? Ends? How do you know?

    b) What are the seasons that you see throughout the book?

    c) How is the village changing?

    d) What kinds of transportation do you see?

    e) Compare and contrast what the children are doing in each scene.

2) Discuss what happens to the trees as the season change in the book. Are there different kinds of trees in the book? How do you know? (color of leaves, size, etc.)

    a) Have the students fold a 9” x 12” sheet of paper into fourths.

    b) Have them write the name of a season in each section. (summer, fall winter, spring)

    c) Have them draw the same tree in each section, but show how it looks in summer, fall, winter and spring.

                           

Math Task Cards - Creating Algebraic Equations Using Only Four Numbers

Are you ready to take your math skills to the next level? Solving algebraic math puzzles can help you hone your problem-solving skills, increase your analytical and critical thinking skills and boost your confidence in tackling difficult equations. Algebraic math puzzles are a great way to learn the fundamentals of algebra, strengthen your understanding of basic operations and apply core math concepts. 

Students need plenty of different opportunities to practice math in ways that both review and extend what they have learned. Because many of my remedial math college students (I call them mathphobics) lack problem solving skills or need practice, I use math task cards for them to complete individually or in pairs as an upfront focus activity. These math task cards rely on logic, mental math, and analytical skills and provide practice in building and creating equations while using PEMDAS. The goal is to expose students to various problem-solving strategies. 

Why task cards? Because task cards can target a specific math skill or concept while allowing the students to only focus on one problem at a time. This format prevents mathphobics from feeling overwhelmed and provides them a sense of accomplishment when a task is completed. Furthermore, the students are more engaged and often acquire a more in-depth understanding of the math concept. By trying new and different strategies and modifying their process, students will be more successful with each puzzle they solve.

The math task cards my students use contain two different math puzzles. The puzzles vary in difficulty from easy to challenging. Since there are easy, medium level and challenging puzzles, differentiation is made simple by choosing the level of difficulty appropriate for each student or team.

Each math puzzle is a square divided into four parts with a circle in the middle of the square. Each math puzzle contains four numbers, one in each corner of the square, with the answer in the circle. Using the four numbers, (each number must be used once) the student is to construct an equation that equals the answer contained in the circle. Students may use all four signs of operation (addition, subtraction, multiplication, division) or just one or two. In addition, each sign of operation may be used more than once. Parenthesis may be needed to create a true equation, and the Order of Operations (PEMDAS) must be followed.

Here is an example of what I mean. 

FREE

Were you able to figure out the puzzle, using all four numbers?

These task card or math puzzles can be used…

• At math centers
• As a math problem solving activity for students who finish early
• As enrichment work
• To give students extra practice with a math concept or skill
• As individual work
• In small groups
• As partner work

A free resource containing three such task cards is available at my TPT store.

I believe that math puzzles are key to getting students interested in mathematics, developing their skills, and creating an environment that makes learning enjoyable. So, let's unlock the door to learning with math puzzles and task cards!

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By the way, I didn't want to leave you without providing you with the answer to the above puzzle.  It is...

It Depends on the Angle - How to Distinguish between Complimentary and Supplementary Angles

My Basic Algebra Concepts class always does a brief chapter on geometry...my favorite to teach! We usually spend time working on angles and their definitions. My students always have difficulty distinguishing complimentary from supplementary angles. Since most of my students are visual learners, I had to come up with something that would help them to distinguish between the two.

The definition states that complementary angles are any two angles whose sum is 90°. (The angles do not have to be next to each other to be complementary.) As seen in the diagram on the left, a 30° angle + a 60° angle = 90° so they are complementary angles. Notice that the two angles form a right angle or 1/4 of a circle.

If I write the word complementary and change the first letter "C" into the number nine and I think of the letter "O" as the number zero, I have a memory trick my mathematical brain can remember.

Supplementary Angles are two angles whose sum is 180°. Again, the two angles do not have to be together to be supplementary, just so long as the total is 180 degrees. In the illustration on your right, a 110° angle + a 70° angle = 180°; so, they are supplementary angles. Together, they form a straight angle or 1/2 of a circle.

If I write the word supplementary and alter the "S" so it looks like an 8, I can mentally imagine 180°.

Since there are so many puns for geometric terms. I have to share a bit of geometry humor. (My students endure many geometry jokes!)

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You might be interested in a variety of hands-on ideas on how to introduce angles to your students. Check out Having Fun With Angles.  It explains how to construct different kinds of angles (acute, obtuse, right, straight) using items such as coffee filters, plastic plates, and your fingers. Each item or manipulative is inexpensive, easy to make, and simple for students to use. All of the activities are hands-on and work well for kinesthetic, logical, spatial, and/or visual learners.

Does Such a Thing as a Left Angle Exist?

Geometry is probably my favorite part of math to teach because it is so visual; plus the subject lends itself to doing many hands-on activities, even with my college students.  When our unit on points, lines and angles is finished, it is time for the unit test.  Almost every year I ask the following question:  What is a left angle?   Much to my chagrin, here are some of the responses I have received over the years NONE of which are true!

1)   A left angle is the opposite of a right angle.

2)  On a clock, 3:00 o'clock is a right angle, but 9:00 o'clock is a left angle.

3)  A left angle is when the base ray is pointing left instead of right.

    4)      A left angle is 1/2 of a straight angle, like when it is cut into two pieces, only it is the part on the left, not the part on the right.

5)      A left angle is 1/4 of a circle, but just certain parts. Here is what I mean.

Now you know why math teachers, at times, want to pull their hair out!  Just to set the record straight, in case any of my students are reading this, there is no such thing as a left angle!  No matter which way the base ray is pointing, any angle that contains 90^○ is called a right angle.

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If you would like some different hands-on ways to teach angles, you might look at the resource entitled, Angles: Hands-on Activities.  This resource explains how to construct different kinds of angles (acute, obtuse, right, straight) using items such as coffee filters, plastic plates, and your fingers. Each item or manipulative is inexpensive, easy to make, and simple for students to use. All of the activities are hands-on and work well for kinesthetic, logical, spatial, and/or visual learners.

                                      

A-MAY-Zing Word Crossword Puzzles

I love the beautiful month of May. Here in Kansas, the fresh cold winds are gone, as are the rains of early spring. May is known as the month of transition, the emerald birthstone, and holidays like Mother's Day, and Memorial Day. It is also recognized as Military Appreciation Month. Some other dates that hold significance are May 1st and May 5th.

May 1st is May Day, and marks the return of spring by the blossoming branches of the forsythia, or lilacs or daffodils popping their heads out of the ground, or the weather turning warmer. May 5th is Cinco de Mayo (The Fifth of May). This day celebrates the victory of the Mexican army over the French army at The Battle of Puebla in 1862. Did you know that no U.S. president has ever died in the month of May? In every other month of the year, at least one U.S. president has died.

Have you heard about these fun dates in May?

• May 1: School Principals’ Day
• May 2: World Tuna Day
• May 8: No Socks Day
• May 14 (second Wednesday in May): Root Canal Appreciation Day
• May 14: Dance Like a Chicken Day
• May 28: Slugs Return from Capistrano Day

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As I thought about May, I discovered that many words started with the word "MAY." In fact, after much research,  I found 20 different words. Using those words, I created two May-themed crossword puzzles, appropriate for grades 7-10. One crossword puzzle includes a word bank which makes it easier to solve while the more challenging one does not.  Even though the same vocabulary is used for each crossword, each grid is laid out differently; so, you have two distinct puzzles. I've also included the answer keys for both puzzles.

Here are some ideas on how you might use these puzzles.

1. Try giving the students the crossword with NO word bank to see how much they know.
2. Use the crossword with the word bank as a review of May and its traditions.
3. Use either crossword to work in pairs to complete the puzzle. Solving a crossword puzzle together is a great way to connect.
4. Copy it and make it available for those students who finish their work early.

Above all, just have fun!

Playing Math Games with older students

I currently teach remedial math students on the college level. These are the students who fail to pass the math placement test to enroll in College Algebra - that dreaded class that everyone must pass to graduate. The math curriculum at our community college starts with Basic Math, moves to Fractions, Decimals and Percents, and then to Basic Algebra Concepts. Most of my students are intelligent and want to learn, but they are deeply afraid of math. I refer to them as mathphobics.

We all have this type of student in our classrooms, whether it is middle school, high school, or college. When working with this type of student, it is important to bear in mind how all students learn. I always refer back to the Conceptual Development Model which states that a student must first learn at the concrete stage (use manipulatives) prior to moving to the pictorial stage, and in advance of the abstract level (the book). This means that lessons must include the use of different manipulatives. I use games a great deal because it is an easy way to introduce and use manipulatives without making the student feel like “a little kid.” I can also control the level of mathematical difficulty by varying the rules; thus, customizing the game to meet the instructional objectives my students are learning. However, as with any classroom activity, teachers should monitor and assess the effectiveness of the games.

When using games, other issues to think about are:

1) Excessive competition. The game is to be enjoyable, not a “fight to the death”.

2) Mastery of the mathematical concepts necessary for successful play. Mastery should be at an above average level unless teacher assistance is readily available when needed. A game should not be played if a concept has just been introduced.

3) Difficulty of the rules. If necessary, the rules should be modified or altered in order that the students will do well.

4) Physical requirements (students with special needs). These should be taken into account so that every player has an opportunity to win.

In addition to strengthening content knowledge, math games encourage students to develop such skills as staying on task, cooperating with others, and organization. Games also allow students to review mathematical concepts without the risk of being called “stupid”. Furthermore, students benefit from observing others solve and explain math problems using different strategies.

Games can also….

1. Pique student interest and participation in math practice and review.
2. Provide immediate feedback for the teacher. (i.e. Who is still having difficulty with a concept? Who needs verbal assurance? Why is a student continually getting the wrong answer?)
3. Encourage and engage even the most reluctant student.
4. Enhance opportunities to respond correctly.
5. Reinforce or support a positive attitude or viewpoint of mathematics.
6. Let students test new problem solving strategies without the fear of failing.
7. Stimulate logical reasoning.
8. Require critical thinking skills.
9. Allow the student to use trial and error strategies.

Mathematical games give the learner numerous opportunities to reinforce current knowledge and to try out strategies or techniques without the worry of getting the “wrong” answer. Games provide students of any age with a non-threatening environment for seeing incorrect solutions, not as mistakes, but as steps towards finding the correct mathematical solution.

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One math game my students truly enjoy playing is Bug Mania.  It provides motivation for the learner to practice addition, subtraction, and multiplication using positive and negative numbers. The games are simple to individualize since not every pair of students must use the same cubes or have the same objective. Since the goal for each game is determined by the instructor, the time required to play varies. It is always one that my students are anxious to play again and again!

Is FOIL to difficult for your students? Try Using the Box Method.

I tutored a student this summer who was getting ready to take Algebra II. He is a very visual, concrete person that needs many visuals to help him to succeed in math. We worked quite a bit on multiplying two binomials.

There are three different techniques you can use for multiplying polynomials. You can use the FOIL method, Box Method and the distributive property. The best part about it is that they are all the same, and if done correctly, will render the same answer!

Because most math teachers start with FOIL, I started there. The letters FOIL stand for First, Outer, Inner, Last. First means multiply the terms which occur first in each binomial. Then Outer means multiply the outermost terms in the product. Inner is for "inside" so those two terms are multiplied—second term of the first binomial and first term of the second). Last is multiplying the last terms of each binomial. My student could keep FOIL in his head, but couldn't quite remember what the letters represented, let alone which numbers to multiply; so, that method was quickly laid aside. 

I next tried the Box Method. Immediately, it made sense to him, and we were off to the races, so to speak. He continually got the right answer, and his confidence level continued to increase. Here is how the Box Method works.

First, you draw a 2 x 2 box. Second, write the binomials, one along the top of the box, and one binomial down the left hand side of the box. Let's assume the binomials are 2x + 4 and x + 3.

          (2x + 4) (x + 3)

Now multiply the top row by x; that is x times 2x and x times +4., writing the answers in the top row of the box, each in its own square.  After that, multiply  everything in the top row by +3, and write those answers in the second row of the box, each in its own square.

Looking at the box, circle the coefficients that have an x. They are located on the diagonal of the box.

To find the answer, write the term in the first square on the top row, add the terms on the diagonal, and write the number in the last square on the bottom row. Voila! You have your answer!

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Terrible at factoring trinomials (polynomials) in algebra? Then try this method which never fails! It is the one most students understand and grasp. This step-by-step guide teaches how to factor quadratic equations in a straightforward and uncomplicated way. It includes polynomials with common monomial factors, and trinomials with and without 1 as the leading coefficient. Some answers are prime. This simple method does not treat trinomials when a =1 differently since those problems are incorporated with “when a is greater than 1” problems.

Math Patterns to Investigate!

Some people say mathematics is the science of patterns which I think is a pretty accurate description. Not only do patterns take on many forms, but they occur in every part of mathematics. But then again patterns occur in other disciplines as well. They can be sequential, spatial, temporal, and even linguistic.

Recognizing number patterns is an important problem-solving skill. If you recognize a pattern when looking systematically at specific examples, that pattern can then be used to make things easier when needing a solution to a problem.

Mathematics is especially useful when it helps you to predict or make educated guesses, thus we are able to make many common assumptions based on reoccurring patterns. Let’s look at our first pattern below to see what we can discover.

What can you say about the multiplicand? (the number that is or is to be multiplied by another. In the problem 8 × 32, the multiplicand is 32.) Did you notice it is multiples of 9? What number is missing in the multiplier?

 
Now look at the product or answer. That’s an easy pattern to see! Use a calculator to find out what would happen if you multiplied 12,345,679 by 90, by 99 or by 108? Does another pattern develop or does the pattern end?

 
Here is a similar pattern that uses the multiples of 9. How is the multiplier in this pattern different from the ones in the problems above? Look at the first digit of each answer (it is highlighted). Notice how it increases by 1 each time. Now, observe the last digit of each answer. What pattern do you see there? Using a calculator, determine if the pattern continues or ends.

Recognizing, deciphering and understanding patterns are essential for several reasons. First, it aids in the development of problem solving skills. Secondly, patterns provide a clear understanding of mathematical relationships. Next, the knowledge of patterns is very helpful when transferred into other fields of study such as science or predicting the weather. But more importantly, understanding patterns provides the basis for comprehending Algebra since a major component of solving algebraic problems
is data analysis which, in turn, is related to the understanding of patterns. Without being able to recognize the development of patterns, the ability to be proficient in Algebra will be limited.

So everywhere you go today, look for patterns. Then think about how that pattern is related to mathematics. Better yet, share the pattern you see by making a comment on this blog posting.

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Check out the resource Pattern Sticks. It might be something you will want to use in your classroom.

A Negative number times a Negative Number Equals a Positive Number? Are You Kidding?

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. After all, 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.

On the left 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 number times a negative number equals a positive number!!!!

Isn't Mathematics Amazing?

Myths and Fun Facts about St. Patrick's Day

March 17th is St. Patrick’s Day; so, for fun, let’s explore some of the

myths surrounding this Irish holiday as well as a few fun facts.

Myths

1) St. Patrick was born in Ireland. Here is a surprise; St. Patrick isn’t Irish at all! He was really born in Britain, where as a teen, he was captured, sold into slavery, and shipped to Ireland.

2) St. Patrick drove all of the snakes out of Ireland. It’s

true there are none living in Ireland today, but according to scientists, none every did. You can’t chase something away that isn't there in the first place!

3) Since the leaves of a shamrock form a triad (a group of three), St. Patrick used it to describe the Trinity, the Father, the Son, and the Holy Spirit so that people could understand the Three in One. However, there is nothing in any literature or history to support this idea although it does make a great object lesson.

4) Legend says each of the four leaves of the clover means something. The first leaf is for hope; the second for faith; the third for love and the fourth leaf is for luck. Someone came up with this, but since a clover is just a plant, the leaves mean absolutely nothing.

5) Kissing the Blarney Stone will give you the eloquent power of winning or convincing talk. Once upon a time, visitors to this stone had to be held by the ankles and lowered head first over the wall surrounding the Blarney Stone to kiss it. Those attempting this were lucky not to receive the kiss of death.

Fun Facts

1) The tradition of wearing green originally was to promote Ireland otherwise known as "The Green Isle." After the British invasion of Ireland, few people wore green because it meant death. It would be like wearing red, white, and blue in the Middle East today. When the Irish immigrated to the U.S. because of the potato famine, few were accepted and most were scorned because of their Catholic beliefs. For fear of being ridiculed and mocked only a small number would wear green on St. Patrick’s Day. Those who didn't adorn green were pinched for their lack of Irish pride. This “pinching” tradition continues today.

2) Did you know that in 1962, Chicago, Illinois began dying the Chicago River green, using a vegetable dye? An environmentally safe dye is used in amounts that keep the river festively green for about four to five hours.

3) The Irish flag is green, white, and orange. The green represents the people of southern Ireland, and orange signifies the people of the north. White is the symbol of peace that brings the two groups together as a nation. 

4) A famous Irish dish is cabbage and corned beef which I love to eat!

It is estimated that there are about 10,000 regular three-leaf clovers for every one lucky four-leaf clover you might find. Those aren’t very good mathematical odds whether you are Irish or not!

Want some St. Patrick's Day activities for your classroom? 

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Check out these three resources.

• Pot of Gold Glyph for Grades 1-3
• March Fraction Word Puzzles for Grades 5-7
• St. Patrick's Day Crossword for Grades 6-8

Finding the Greatest Common Factor and Least Common Multiple

The most common method to find the greatest common factor (GCF) is to list all of the factors of each number, then list the common factors and choose the largest one.  Example: Find the GCF of 36 and 54.

1) The factors of 36 are: 1, 2, 3, 4, 6, 9, 12, 18, and 36.

2) The factors of 54 are: 1, 2, 3, 6, 9, 18, 27, and 54.

Therefore, the common factor(s) of 36 and 54 are: 1, 2, 3, 6, 9, 18.  Although the numbers in bold are all common factors of 36 and 54, 18 is the greatest common factor.

To find the lowest common multiple (LCM), students are asked to list all of the factors of the given numbers. Let's say the numbers are 9 and 12.  

1) The multiples of 9 are: 9, 18, 27, 36, 45, 54.

2) The multiples of 12 are: 12, 24,  36, 48, 60.

As seen above, the least common multiple for these two numbers is 36.  

We often instruct our students to first list the prime factors, then multiply the common prime factors to find the GCF. Often times, if just this rule is given, students become lost in the process. Utilizing a visual can achieve an understanding of any concept better than just a rule. A two circle Venn Diagram is such a visual and will allow students to follow the process as well as to understand the connection between each step. For example: Let’s suppose we have the numbers 18 and 12.

1) Using factor trees, the students list all the factors of each number.

2) Now they place all the common factors in the intersection of the two circles. In this case, it would be the numbers 2 and 3.

3) Now the students place the remaining factors in the correct big circle(s).

4) That leaves the 18 with a 3 all by itself in the big circle. The 12 has just a 2 in the big circle.

5) The intersection is the GCF; therefore, multiply 2 × 3 to find the GCF of  6.

6) To find the LCM, multiply the number(s) in the first big circle by the GCF (numbers in the intersection) times the number (s) in the second big circle.

3 × GCF × 2 = 3 × 6 × 2 = 36. The LCM is 36.

This is an effective method to use when teaching how to reduce fractions,

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I have turned this method into a resource for Teachers Pay Teachers. It is 16 pages and begins with defining the words factor, greatest common factor and least common multiple. What a factor tree is and how to construct and use a Venn Diagram as a graphic organizer is shown. Step-by-step examples are given as well as student practice pages. How to use a three circle Venn Diagram when given three different numbers is explained. Two pages of blank pages Venn Diagrams are included for classroom practice. To learn more, just click on the price under the resource cover on your right. A free version is also available.

Dividing Fractions Using KFC (Keeping, Flipping and Changing)

Ugh - It's time to teach the division of fractions. My experience has been that many students forget which fraction to flip and often, they forget to change the dreaded division sign to a multiplication sign. The other evening,  I was helping my 5th grade granddaughter with her homework. Really, she had completed it by herself, but she wanted me to check it. At the top of her paper were the letters "KFC". I asked her what they meant, and she replied, "Kentucky Fried Chicken." Now I have taught math for years and years, and I had never heard of that one!

She explained that the "K" stood for keep; "F" for flip, and "C" for change. Let's suppose the problem on the left was one of the problems on her homework paper.

First, she would Keep the first fraction. Next, she would Flip the second one, and then Change the division sign to a multiplication sign...like illustrated on the right. She would then cross cancel if possible (In this case it is).  Finally, she would multiply the numerator times the numerator and the denominator by the denominator to get the answer.

She was able to work all the division problems without any trouble by just remembering the letters KFC.

Yesterday, I was working in our college math lab when a student needed help. On the right is the problem he was having difficulty with. (For those of you who don't teach algebra or just plain hate it, I am sure this problem looks daunting and intimidating. Believe me, my student felt the same way!) 

First I had the student rewrite the problem with each fraction side by side with a division sign in between them like this.

Doesn't it look easier already? I then taught him KFC. You read that right! I did! (I figured if it worked for a 5th grader, it should work for him.) Surprisingly it made sense to him because he now had mnemonic device (an acronym) that he could easily recall. He rewrote the problem by Keeping the first fraction, Flipping the second, and Changing the division sign to a multiplication sign.

Now it was just a simple multiplication problem.  Had he been able to, he would have cross canceled, but in this case, he simply multiplied the numerator times the numerator and denominator by the denominator to get the answer.

So the next time you teach the division of fractions, or you come across a problem like the one above, don't panic!  Remember KFC, and try not to get hungry!