Palindromes in Words and Numbers - February 22, 2022 Is a Palindrome!

I was getting ready to pay for my meal at a buffet when I noticed the cashier's name tag.  It read "Anna" to which I replied, "Your name is a palindrome!"  The cashier just stared at me in disbelief.  I explained that a palindrome was letters that read the same backwards as forwards. Because you could read her name forwards and backwards, it qualified as a palindrome. She replied that she remembered a math teacher talking about those because of patterns (I love that math teacher), and she remembered the phrase "race car" was a palindrome.  We then began sharing palindromes that we knew such as radar, level and madam while my family waited impatiently in line. (Sometimes they have little patience with my math conversations.)

The word palindrome is derived from the Greek word palíndromos, which means "running back again". A palindrome can be a word, phrase or sentence which reads the same in both directions such as: "Eva, can I stab bats in a cave?" or "Was it a car or a cat I saw?" or "Rats live on no evil star."

But did you know there are also palindromic numbers?  A palindromic number is a number whose digits are the same if read in both directions (as seen on your left).   Whereas "1234" is not a palindromic number, because backwards it is "4321" which is not the same. (This year, the date 2-22-22 is a palindrome. Do you see why?)

Suppose a person starts with the number one and lists the palindromic numbers in order: 11, 22, 33, 44, 55...etc.  Can you continue the list? 

Did you notice that palindromic numbers are symmetrical?  Look carefully at the 17371 shown above.  It is symmetrical (when a figure can be folded along a line so the two halves match perfectly)on either side of the three whether read left to right or vice versa.

Palindromic numbers are very simple to generate from other numbers with the help of addition.

Try this:

1. Write down any number that has more than one digit. I will use 47.
2. Write down that number in reverse beneath the first number. (See illustration below.)
3. Add the two numbers together. (121)
4. 4. The sum of 121 is undeniably a palindrome.

Try an easy number first, such as 18.  At times you will need to use the first addition answer and repeat the process of reversing and adding. You will almost always get a palindrome answer within six steps.  Try one of these numbers 68 or 79.  Be careful because if you pick a number greater than 89, arriving at the palindromic answer will take more steps, but it will still work.  (See the two examples below.)

But don't try 196!  In fact, avoid it like the plague!   A computer has already gone through several thousand stages, and it still hasn't come up with a palindromic answer!

Example #1:

• Start with 75.
• Reverse 75 which makes 57.
• Add 75 and 57 and you get 132.  The answer 132 is not a palindrome.
• SO reverse 132, and it becomes 231.
• Add 132 and 231, and the answer is 363. 
• Since 363 is a palindrome, we are done!

Example #2:

• Begin with 255.
• Reverse 255 to get 552.
• Add 255 and 552. The answer is 807 which is not a palindrome.
• SO reverse 807 to get 708.
• Add 807 and 708. The answer of 5151 is not a palindrome.
• SO reverse 1515 to get 5151.
• Add 1515 and 5151 which is 6666.
• This is a palindrome; so, we are done!

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Let's Go Fly A Kite - Using the Correct Geometry Term for Diamond!

This was a comment I received from a fourth grade teacher, "Would you believe on the state 4th grade math test this year, they would not accept "diamond" as an acceptable answer for a rhombus, but they did accept "kite"!!!!!  Can you believe this? Since when is kite a shape name? Crazy."

First of all, there are NO diamonds in mathematics, but believe it or not, a kite is a geometric shape! The figure on the right is a kite. In fact, since it has four sides, it is classified as a quadrilateral. It has two pairs of adjacent sides that are congruent (the same length). The dashes on the sides of the diagram show which side is equal to which side. The sides with one dash are equal to each other, and the sides with two dashes are equal to each other.

A kite has just one pair of equal angles. These congruent angles are a light orange on the illustration on the left. A kite also has one line of symmetry which is represented by the dotted line. (A line of symmetry is an imaginary line that divides a shape in half so that both sides are exactly the same. In other words, when you fold it in half, the sides match.) It is like a reflection in a mirror.

The diagonals of the kite are perpendicular because they meet and form four right angles. In other words, one of the diagonals bisects or cuts the other diagonal exactly in half. This is shown on the diagram on the right. The diagonals are green, and one of the right angles is represented by the small square where the diagonals intersect.

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There you have it! Don't you think a geometric kite is very similar to the kites we use to fly as children? Well, maybe you didn't fly kites as a kid, but I do remember reading about Ben Franklin flying one! Anyway, as usual, the wind is blowing strong here in Kansas, so I think I will go fly that kite!

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This set of two polygon crossword puzzles features 16 geometric shapes with an emphasis on quadrilaterals and triangles. The words showcased in both puzzles are: congruent, equilateral, isosceles, parallelogram, pentagon, polygon, quadrilateral, rectangle, rhombus, right, scalene, square, trapezoid and triangle.  The purpose of these puzzles is to have students practice, review, recognize and use correct geometric vocabulary. Answer keys are included.

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There's A Place For Us! Teaching Place Value

When my college students (remedial math students) finish the first chapter in Fractions, Decimals, and Percents, we focus on place value. Over the years, I have come to the realization how vital it is to provide a careful development of the basic grouping and positional ideas involved in place value. An understanding of these ideas is important to the future success of gaining insight into the relative size of large numbers and in computing.  A firm grasp of this concept is needed before a student can be introduced to more than one digit addition, subtraction, multiplication, and division problems. It is important to stay with the concept until the students have mastered it. Often when students have difficulty with computation, the source of the problem can be traced back to a poor understanding of place value.

It was not surprising when I discovered that many of my students had never used base ten blocks to visually see the pattern of cube, tower, flat, cube, tower, flat.  When I built the thousands tower using ten one hundred cubes, they were amazed at how tall it was.  Comparing the tens tower to the thousands tower demonstrated how numbers grew exponentially.  Another pattern emerged when we moved to the left; each previous number was being multiplied by 10 to get to the next number.  We also discussed how the names of the places were also based on the pattern of:  name, tens, hundreds, name (thousands), ten thousands, hundred thousands, etc. 

I asked the question, "Why is our number system called base ten?"  I got the usual response, "Because we have ten fingers?"  Few were aware that our system uses only ten digits (0-9) to make every number in the base ten system.

We proceeded to look at decimals and discovered that as we moved to the right of the decimal point, each number was being divided by 10 to get to the next number. We looked at the ones cube and tried to imagine it being divided into ten pieces, then 100, then 1,000. The class decided we would need a powerful microscope to view the tiny pieces.  Again, we saw a pattern in the names of each place:  tenths, hundredths, thousandths, ten thousandths, hundred thousandths, millionths, etc.

I then got out the Decimal Show Me Boards.  (See illustration on the left.)  These are very simple to make. Take a whole piece of cardstock (8.5" x 11") and cut off .5 inches. Now cut the cardstock into fourths (2.75 inches).  Fold each fourth from top to bottom. Measure and mark the cardstock every two inches to create four equal pieces. Label the sections from left to right - tenths, hundredths, thousandths, ten thousandths. You can type up the names of the places which then can be cut out and glued onto the place value board.

Here are some examples of how I use the boards.  I might write the decimal number in words.  Then the students make the decimal using their show me boards by putting the correct numbers into the right place.  Pairs of students may create two different decimals, and then compare them deciding which one is greater.  Several students may make unlike decimals, and then order the decimals from least to greatest.  What I really like is when I say, "Show me", I can readily see who is having difficulty which allows me to spend some one-on-one time with that student.

Show Me Boards can also be made for the ones, tens, hundreds and thousands place.  Include as many places as you are teaching. I've made them up to the hundred thousands place by using legal sized paper. As you can see in the photo above, my two granddaughters love using them, and it is a good way for them to work on place value.

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A good way to practice any math skill is by playing a game. Your students might enjoy the No Prep place value game entitled: Big Number.  Seven game boards are included in this eleven page resource packet. The game boards vary in difficulty beginning with only two places, the ones and the tens.  Game Board #5 goes to the hundred thousands place and requires the learner to decide where to place six different numbers.  All the games have been developed to practice place value using problem solving strategies, reasoning, and intelligent practice.

Math Games! An Effective Way to Teach Math

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 smart 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 well 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 students feel like “little kids.” 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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If you want a challenging but fun and engaging math game, try Contact. It is a fun and attention-grabbing way for students to review basic math facts and to use critical thinking without doing another “drill and kill” activity.