Win a $100 TPT Gift Certificate During the Two Day Teachers Pay Teachers Back to School Sale

Many of us are still preparing for the first day of school. Since most teachers are still decorating, copying worksheets, labeling supplies, etc., Teachers Pay Teachers is having a sale, starting Tuesday, August 5th and going through Wednesday, August 6th until 11:59 p.m. EST. This two day sale will give you time to purchase those items you still need for your classroom.  Many TPT sellers will offer sale discounts anywhere from 5% to 20% off.  Many of the resources in my store are on sale as well.  And to save you even more money, at checkout, you can receive another 5% discount IF the special code of BTS25 is entered. 

FREE

So here is your chance to purchase those products that are still on your Teachers Pay Teachers wish list. You don't even have to worry about shipping costs since the items are digital and available for download immediately after purchase. So grab that on-line shopping cart and rush on over to Teachers Pay Teachers to stock up.

By the way, when you stop by my store, be sure and download one of my free items that uses number tiles. Just click under the resource cover on your right.

In addition to sale prices, enter the TBOTEMC (The Best of Teacher Entrepreneurs Marketing Cooperative) $100 TPT Gift Card give away. Three $100 TPT Gift Cards will be awarded, and you could be the lucky winner of one! Just click on the link below for all of the details.

A Rafflecopter Giveaway

Problem Solving With Number Tiles in Middle School

Math Activities for Grades 5-8

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

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

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

A free version for each of my number tile resources is also available in my TPT store.  While visiting my store, take time to check out these additional Number Tile activities.

• The A, B, C's of Number Tiles - 26 Problem Solving Math Puzzles
• Geometric Math Puzzles –Using Number Tiles Learning to Learn About Plane Geometry
• Number Tiles – Hands On Math Activities for the Primary Grades

Using the Book, Anno's Mysterious Multiplying Jar, to Learn About Factorials

Let's look at one more book in this series of books that link literature and math. This book is more for those taking algebra as it as to do with factorials.  Factorial is a word that mathematicians use to describe a special kind of numerical relationship. Factorials are very simple things. They are just products, indicated by the symbol of an exclamation mark. The factorial function (symbol: !) means to multiply a series of descending natural numbers. For instance, "five factorial" is written as "5!" (a shorthand method) and means 5 × 4 × 3 × 2 × 1 = 120. Factorials are used in determining the numbers of combinations and permutations and in finding probability.

A Few Are Available
on Amazon

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

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

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

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

____   ____   ____   ____ 

Four Possible Spaces

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

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

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

• 
  Bubble Gum - Berry - Vanilla
• Bubble Gum - Vanilla - Berry

• Berry - Vanilla - Bubble Gum
• Berry - Bubble Gum - Vanilla

• Vanilla - Berry - Bubble Gum
• Vanilla - Bubble Gum - Strawberry

Or, if we use factorials, we arrive at the answer much faster: 3! = 3 × 2 × 1 = 6

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

Taking A Brief Break from Technology

For three weeks, I am setting aside all technology and work related items so I can spend some time to relax with my husband, my grandchildren and friends. In the meantime, if you really want something to read, choose one of my older posts. 

 

My blog posts will return on Wednesday, July 23, 2025.

A Day With No Math - Another Book that Links Math and Literature

Who wants to read about math? Who even likes it? Many, many times I have heard a parent of one of my students say, "I understand why my child cannot do math. I was never very good at math, either." Right! So you weren't good at reading; so, your child should be illiterate? So you don't like to play sports; so, PE should be optional? I don't think so.
 

My goal in life is to make people, students, adults, children, comfortable with math; to see its value; to learn to at least like it. After all, there isn't a day that goes by that you don't use math in some form. Did you read a clock today? Did you buy something with money? Did you go to the home improvement store to buy paint? Did you cook or keep score while you played a game? That is all math. Useful - right?


Ask yourself or your students, "What would happen if suddenly there were no numbers?" To find out, read A Day with No Math by Marilyn Kaye, published by Harcourt Brace Jaovanovich, Inc. in 1992. It is a great read aloud book. It's one I have used in workshops and in my own classroom with children, college students and adults. The book demonstrates how mathematics plays an important role in our daily lives and shows the reader how time, measurement, money and other mathematics are used everyday. The story helps kids to understand that math is a part of all aspects of our every day life and without it, our life would be such a mess. Try reading this if you hate math or even if you love it, and you will be surprised at how much math you really know. It will give you a different appreciation for math 

This is a book teachers will treasure to have in your classroom library. Currently, it's difficult to find, but Amazon seems to have a few copies 

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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$8.25

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!

_________________________________________________________

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!)

$3.50

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.

$3.50

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

$2.85

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.

$3.50

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