Is zero an even number? Now, that's an "odd" question!

My daughter and her husband are heading to Las Vegas with his family to celebrate his parent's 50th wedding anniversary. I guess when you are at the roulette table, (never been there or done that) and you bet "even" and the little ball lands on 0 or 00, you lose. Yep, it's true; zero is not considered an even number on the roulette wheel, something you better know before you bet. This example is a non-mathematical, real-life situation where zero is neither odd or even. But in mathematics, by definition, zero is an even number. (An even number is any number that can be exactly divided by 2 with no remainder.) In other words, an odd number leaves a remainder of 1 when divided by 2 whereas an even number has nothing left over. Under this definition, zero is definitely an even number since 0 ÷ 2 = 0 has no remainder.

Zero also fits the pattern when you count which is the same as alternating even (E) and odd (O) numbers.

Most math books include zero as an even number; however, under special circumstances zero may be excluded. (For example, when defining even numbers to mean even NATURAL numbers.) Natural numbers are the set of counting numbers beginning with 1 {1, 2, 3, 4, 5....}; so, zero is not included. Consider the following simple illustrations. Let's put some numbers in groups of two and see what happens.

As you can see, even numbers such as 4 have no "odd man out" whereas odd numbers such as 3 always have one left over. Similarly, when zero is split into two groups, there is not a single star that does not fit into one of the two groups. Each group contains no stars or exactly the same amount. Consequently, zero is even.

Algebraically, we can write even numbers as 2n where n is an integer while odd numbers are written as 2n + 1 where "n" is an integer. If n = 0, then 2n = 2 x 0 = 0 (even) and 2n + 1 = 2 x 0 + 1 = 1 (odd). All integers are either even or odd. (This is a theorem). Zero is not odd because it cannot fit the form 2n + 1 where "n" is an integer. Therefore, since it is not odd, it must be even.

I know this seems much ado about nothing, but a great deal of discussion has surrounded this very fact on the college level. Some instructors feel zero is neither odd or even. (Yes, we like to debate things that seem obvious to others.)

Consider this multiple choice question. (It might just appear on some important standardized test.) Which answer would you choose and why?

Zero is…
           a) even
           b) odd
           c) all of the above
           d) none of the above

Mathematically, I see zero as the count of no objects, or in more formal terms, it is the number of objects in the empty set. Also, since zero is defined as an even number in most math textbooks, and is divisible by 2 with no remainder, then "a" is my answer.

Glyphs Can Help Students Gather Information, Interpret Data, and Follow Directions

What is a Glyph?

A glyph is a non-standard way of graphing a variety of information to tell a story. It is a flexible data representation tool that uses symbols to represent different data. Glyphs are an innovative instrument that shows several pieces of data at once and necessitates a legend/key to understand the glyph and require problem solving, communication, and data organization.

Remember coloring pages where you had to color in each of the numbers or letters using a key to color certain areas? Or how about coloring books that were filled with color-by-numbers? These color-by-number pages are a type of glyph. Some other activities we can call glyphs would be the paint-by-number kits, the water paints by color coded paint books, and in some cases, even model cars. Some of the model cars had numbers or letters attached to each piece that had to be glued together. These days, this could be considered a type of glyph.

What is the Purpose of a Glyph?

A glyph is a symbol that conveys information nonverbally. Glyphs may be used in many ways to get to know more about students and are extremely useful for students who do not possess the skill to write long, complex explanations. Reading a glyph and interpreting the information represented is a skill that requires deeper thinking. Students must be able to analyze the information presented in visual form. In other words, a glyph is a way to collect, display and analyze data. They are very appropriate to use in the CCSS data management strand (see standards below) of math. Glyphs actually a type of graph as well as a getting-to- know-you type of activity.

CCSS.Math.Content.1.MD.C.4 Organize, represent, and interpret data with up to three categories;

ask and answer questions about the total number of data points, how many in each category, and how many more or less are in one category than in another.

CCSS.Math.Content.2.MD.D.10 Draw a picture graph and a bar graph (with single-unit scale) to represent a data set with up to four categories. Solve simple put-together, take-apart, and compare problems using information presented in a bar graph.

For example, if the number of buttons on a gingerbread man tells how many people are in a family, the student might be asked to “Count how many people are in your family. Draw that many buttons on the gingerbread man." Since each child is different, the glyphs won't all look the same which causes the students to really look at the data contained in them and decide what the glyphs are showing.

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Holiday glyphs can be a fun way to gather information about your students. You can find several in my Teachers Pay Teachers store. My newest one is for Thanksgiving and involves reading and following directions while at the same time requiring problem solving, communication and data organization. The students color or put different items on a turkey based on information about themselves. Students finish the turkey glyph using the seven categories listed below.

1) Draw a hat on the turkey (girl or a boy?)
2) Creating a color pattern for pets or no pets.
3) Coloring the wings based on whether or not they wear glasses.
4) Writing a Thanksgiving greeting based on how many live in their house.
5) Do you like reading or watching TV the best?
6) How do they get to school. (ride or walk?)
7) Pumpkins (number of letters in first name)

My Students Are Having Difficulty Memorizing Those Dreaded Math Facts!

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

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

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

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

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

The ROOT of the Problem - Finding Digital Root to Reduce Fractions

When students skip count, they can easily say the 2's, 5's, and 10's which translates into easy memorization of those particular multiplication facts. Think what would happen if every primary teacher had their students practice skip counting by 3's, 4's, 6's, 7's, 8's and 9's! We would eradicate the drill and kill of memorizing multiplication and division facts.

Since many of my college students do not know their facts, I gravitate to the Divisibility Rules. Sadly, most have never seen or heard of them. I always begin with dividing by 2 since even numbers are understood by almost everyone. (Never assume a student knows what an even number is as I once had a college student who thought that every digit of a number must be even for the entire number to be even.) We then proceed to the rules for 5 and 10 as most students can skip count by those two numbers.

Finally, we learn about the digital root for 3, 6, and 9. This is a new concept but quickly learned and understood by the majority of my students. (See the definition below which is from A Simple Math Dictionary available on TPT).

Here are several examples of finding Digital Root:

a) 123 = 1 + 2 + 3 = 6. Six is the digital root for the number 123. Since 123 is an odd number, it is not divisible by 6. However, it is still divisible by 3.

b) 132 = 1 + 3 + 2 = 6. Six is the digital root for the number 132. Since 132 is an even number, it is divisible by 6 and by 3.

c) 198 = 1+ 9 + 8 = 18 = 1 + 8 = 9. Nine is the digital root for the number 198; so, 198 is divisible by 9 as well as by 3.

4d201 = 2 + 0 + 1 = 3. Three is the digital root for the number 201; so, 201 is divisible by 3.

The first time I learned about Digital Root was about eight years ago at a workshop. I was beside myself to think I had never learned Digital Root. Oh, the math classes I sat through, and the numbers I tried to divide by are too numerous to mention! It actually gives me a mathematical headache. And to think, not knowing Digital Root was the ROOT of my problem!

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A teacher resource on Using the Divisibility Rules and Digital Root is available at Teachers Pay Teachers. If you are interested, just click under the resource cover on your right.

Different Ideas on How to Teach Long Division

As presented in an earlier posting, there is another way to approach long division. However, since many of you are required to present it the long way, here are a couple of ideas to make it easier for your students.

First of all, have the students use graph paper. The squares help to keep the numbers aligned which seems to be a problem for many students. If you don't have graph paper, you can download free templates at Donna Young's Free Graph Paper and make your own. I like the idea of separating the problems with lines to ensure there is no cross over from one problem to another.

Secondly, try using the acronym (a mnemonic device) of Does McDonald's Sell Cheese Burgers. I have see this acronym many times on Pinterest, but usually the C is omitted.

Check means that after the student has subtracted, they should check to see if the remainder is smaller than the divisor. If it is equal to or larger than the divisor, then enough was not taken out of the dividend. This is a step often skipped when long division is taught; yet, if the student doesn't check and make the needed correction, the answer (quotient) will be wrong.

In order to learn division, the student must first have a good understanding of multiplication. Students don’t need to perfectly know all of the times tables, but a majority of the facts or having a reasonably quick strategy to figure out the answer is necessary.

Start by practicing division using the number series the students can easily skip count such as 2 and 5. Then gradually move up to nine. After that, move to division by double digit numbers using 10 since most students know how to skip count by 10. Once the concept is understood, teaching division will become more about guided practice to help your child to become comfortable with the division operation which, in reality, is a different kind of multiplication practice.

The Long and Short of It - Two different ways to do long Division

My remedial college math class is currently working on fractions. (Yes, even many college students don't understand them!) When we discussed how to change an improper fraction to a mixed numeral, long division came up. I showed the class a shortcut I was taught many years ago (approximately when the earth was cooling) and none, no not even one student, had seen it before. I wonder how many of you are unfamiliar with it as well? First let's look at long division and how most students are taught today. We will use 534 divided by 3.

Now if that doesn't make your head swim, I don't know what will. Everything written in the third column is what the student must mentally do to solve this problem. Then we wonder why students have trouble with this process. There is another way, and it is called short division for a reason. This is the way I learned it.......

I don't know about you, but I would rather have my students doing mental math to solve division problems than writing everything out in the long form. And the paper and frustration you will save will be astounding! So what will it be.....long division or short division?

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As a Side Note: Since many students do not know their multiplication tables, reducing fractions is almost an impossible task. The divisibility rules, if learned and understood, can be an excellent math tool. This resource contains four easy to understand divisibility rules and includes the rules for 1, 5, and 10 as well as the digital root rules for 3, 6, and 9. A clarification of what digital root is and how to find it is explained. Also contained in the resource is a dividing check off list for use by the student. If you are interested, just click under the resource title page.

Getting to Know You - A Back-to-School Get Acquainted Activity

School will soon be beginning for most of us. I teach on the college level, but I still feel the most important thing I can do is to make the students feel connected to one another so that they at least know one other person in the class. I always start each new class by playing a true/false game. I start off the first class by listing four items about myself, three that are true and one that is false. The students try to discover the false one. On a 3” × 5” card, I then have the students write four things about themselves, three true and one false from which we, as a class, try to find the false one. I then collect and save the cards.

At the next class meeting, I will choose 3-4 cards from which to read the true statements. As a class, we try to match the student to the card. It really helps the students to relax and have fun at the same time plus they get to know each other. I usually do this activity for a couple of weeks until I sense that the students are comfortable being in the group.

By the way, here are my four statements. Can you choose the false one?

I have 12 grandchildren, six of whom are adopted.
My husband asked me to marry me on our first date.
I am a big Jayhawk (Kansas University) fan. (We live in Kansas.)
I have been teaching for over 40 years.

Give up? You can find the answer on the page entitled Answers to Questions.

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You might be interested in this back to school item, a glyph entitled Back to School Glyph for grades K-3. The students color or put different items in a school yard based on information about themselves. This glyph is an excellent activity for reading and following directions, and requires problem solving, communication, and data organization.

Glyphs - A Form of Graphing - Completing a Back to School Glyph

Sometimes I think teachers believe glyphs are just fun activities, but in reality, glyphs are a non-standard way of graphing a variety of information to tell a story. It is a flexible data representation tool that uses symbols to represent different data. Glyphs are an innovative instrument that shows several pieces of data at once and requires a legend/key to understand the glyph. The creation of glyphs requires problem solving, communication as well as data organization.

Remember Paint by Number where you had to paint in each of the numbers or letters using a key to paint with the right color? How about coloring books that were filled with color-by-number pages? Believe it or not, both of these activities were a type of glyph.

For the fall, I have created a Back to School Glyph. Not only is it a type of graph, but it is also an excellent activity for reading and following directions. Students are to finish the Glyph using the nine categories

listed below.

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1) Draw a road to the school (girl or a boy?)
2) Trees (age?)
3) Flowers (pets or no pets?)
4) How did you get to school today? (transportation)
5) School Yard (what do you like to do best?)
6) The Sky (what grade are you in?)
7) School Flag (prefer outside or inside?)
8) Name (number of letters in first name)
9) School House (prefer books or T.V.?)

Reading the completed glyph and interpreting the information represented is a skill that requires deeper thinking by the student. Students must be able to analyze the information presented in visual form. A glyph such as this one is very appropriate to use in the data management strand of mathematics. If you are interested, just click under the resource cover page.

Various Tools to Help Students Graph Equations

I work in the math lab at the community college where I also teach. The math lab is staffed by only math instructors and offers free math tutoring to any of our students. We try to have many resources available for our students. When it comes to graphing, we have found that the computer can be very unfriendly. The graphs are often hard to see because they are small, and so finding points is next to impossible. We keep in stock some items that help our students.

First, we have graph paper that is always available. We keep an assortment of different kinds for our students:

1/4" grid paper
Four co-ordinate graphs per page
Full co-ordinate graph paper
Six small co-ordinate graphs per page

On the right, you see Markwan holding the example of #2.

We also have two-sided white boards. One side is blank while the opposite side contains a coordinate graph outline. Our students make good use of these. They like the fact that they can do the linear or quadratic equation on one side and then construct the graph on the other. (They don't have to draw the X-Y axes and tick marks for each problem or get out the ruler for accuracy.) Since the white boards are erasable, they can be used over and over again. On the left, Sam is "modeling" the white board. (Both young men wanted to be on my blog and were anxious for me to use their first names.)

Click Here for More Information

BUT my favorite item we have on hand are graphing sticky notes. I often use them in my math classes because students can take notes while drawing examples of graphs and then stick the example right into their math book. These post-it-notes are called MiniPLOTs®. They are a unique brand of Post-It Notes designed for math students, teachers and tutors. MiniPLOTs® are 3x3" paper pads with 50 coordinate grid, polar coordinates, or 3D solid shapes printed on each sheet. They are the perfect size for homework and tests. In addition, the company makes them for algebra, geometry, trigonometry, statistics and K-6 math (provides an innovative method of teaching students the basic multiplication and division factors in about six weeks).

These work great when I am grading math homework. When a student has graphed an equation wrong, I simply take a graphing sticky note, correctly graph the equation and stick it beside their incorrect answer. It's important that students see the correct answer so that the wrong one doesn't remain stuck in their heads!

The Math Lab also supplies a reference sheet entitled Graphs of Some Common Functions. It gives an example of the equation being graphed (i.e. f(x) = x ), a visual of the what the graph should look like, the domain, range, and symmetry origin. The students are free to use the laminated ones in the Math Lab, but can also take home a paper one to place in their math notebooks.

On the reverse side of this reference sheet are examples of: Graph of f(x) = a^xand Graph of f(x) = log_a(x). Besides a visual of the graph, it includes domain, range, decreasing on and horizontal or vertical asymptote.

Most students are visual learners and can see lines and curves and project how they behave intuitively. Their brains can easily understand, understand and recognize pictorially better than just remembering abstract equations. It is therefore important for students to construct and draw graphs so they can picture them in their minds. Hopefully some of these graphing tools will make constructing those graphs easier.

Securing Classroom Calculators so they don't walk off!

I teach at a community college which I love. I also spend three hours a week in the Math Lab which is a place where our students can come for math tutoring, to study or just to work in a group. It is staffed by math instructors. We try to have the supplies available that our students might need like a stapler, hole punch, white boards, pencils, scrap paper etc. We also have a set of scientific calculators which our students may borrow while in the Math Lab.

Most of our items tend to remain in the Math Lab. Of course, a few pencils disappear now and then, but generally, most supplies seem to stay put EXCEPT for the calculators. Now I must say, students who take these home do so unintentionally. They just pick it up, slip it in their backpack and head out the door. Fortunately, most students are honest and eventually return the calculators to us. The dilemma is we only have so many calculators; so, we want to make sure that if a student needs one, it is on hand. We needed to find a way to make sure the calculators didn’t walk off.

One of our team members came up with an innovative but simple solution.

She purchased small clip boards and attached the calculator to it by using Gorilla tape. The calculators are still accessible, but much too big or bulky to accidentally stick into a backpack. In addition, since they are on a clip board, they are easy to stand and display in the white board trays. At the end of the day, it is simple to count them to make sure none are missing. This idea has worked so well, that some of our math instructors are now using this method in their classrooms.

So if you teach math, and have a set of classroom calculators, why not give this idea a try?

The Changing Size of Toilet Paper - A Math Dilemma!

Which Roll is Today's Product?

Consumer’s Report featured an article about the number games of toilet paper. (Sounds like math to me!) Since I thought the article was interesting, I mentioned it to my husband who, being a science teacher, had to investigate. His motto: Never take anyone’s word for it.

So he marched to our bathroom and discovered that our toilet paper was smaller than the holder which had been there since 1989. (Yes, our house is old - like us). There was a little more than 1/2 inch showing on each side of the roll. To further investigate, my husband went to the trusty Internet. There he discovered the following facts.

1) Toilet paper was first manufactured in 1857. Before this, corncobs and many other "soft" items were used for this purpose.

Hey Elmer!
Look what's on sale
at Sears!

2) In the early American west, pages torn from newspapers or magazines were often used as toilet paper. The Sears catalog was commonly used for this purpose and even the Farmer's Almanac had a hole in it so it could be hung on a hook in the outhouse.

3) In 1935, Northern Tissue advertised "splinter free" toilet paper. (Yes, splinter free!) Early production procedures frequently left splinters embedded in the paper. And you thought cheap toilet paper was rough!

4) Toilet paper was originally manufactured in the shape of a square, 4.5" by 4.5" which was about the average size of a man's hand. The square made it handy to fold over a few times, but still be considered acceptable for sanitary use. Basically, this size was established because it worked, sort of like the 90 foot pitcher's mound or the ten foot basketball rim.

5) In the last ten years, the size of toilet paper has been reduced because manufacturers are trying to cut costs by trimming the sheet size. (Try placing one "square" in your hand now, and you will see what I mean.)

6) Most toilet paper producers have decreased the width of a roll from 4.5 inches to 4.2 inches (or something close to that).

7) Not only have many manufacturers diminished the size of the square (which is now a rectangle), but they have also placed fewer "squares" on a roll.

8) Unfortunately, it is not just the width of the roll that has been altered. The size of the cardboard tube in the middle now has a larger diameter, and that is not something you can easily compare in the store!

9) Typical sizes of popular brands which I had available to measure:

Kleenex Cottenelle - Standard: 4.5" x 4.0"
Angel Soft - Standard: 4.5" x 4.0"
Quilted Northern: 4.5" x 4.0"

What's really comical (or depressing) is that even though toilet paper is smaller and sometimes thinner and more transparent, it still costs the same as the old size. It is just like so many other products we purchase. No longer can we buy three pounds of coffee or a one pound can of beans. (I noticed the beans because I used them for students to feel how heavy 16 ounces was. They can now weighs 14 ounces!) Then there is the 1/2 gallon of ice cream which decreased overnight to 1.75 quarts and half gallon containers of Tropicana Orange Juice which suddenly became 59 ounces instead of 64! But toilet paper? I never thought they would play the number game with toilet paper. Is nothing sacred in the world of mathematics?

A Multiplication "Trick" To Know When You Are Multiplying by 11

Knowing how to do math does NOT require magic; although, sometimes working a problem can appear to be done magically. This week I want to talk about multiplying by eleven. Before I demonstrate the "trick", I have to get on my soap box for just a moment. In my humble opinion, all students should know their times tables through 12 even though the Common Core Standard for third grade says through 10 x 10. Remember, Common Core is the minimum or base line of what is to be learned. In Algebra, I insist that my students know the doubles through 25 x 25 and the square roots of those answers up to 625. It saves so much time when we are working with polynomials.

Now to our our amazing mathematical "trick". Let's look at the problem below which is 231 x 11.

First we write the problem vertically. Next, we bring down the number in the ones place which in this case is a one. Now we add the digits in the ones and tens place which is 3 + 1 and get the sum of four which is brought down into the answer.

Moving over to the hundreds place, we add that digit with the digit in the tens place 2 + 3 and get an answer of five which we bring down. Finally, we bring down the digit in the hundreds place which is a two. The answer to 231 x 11 is 2,541.

Now try 452 x 11 in your head. Did you get 4,972? Let's try one more. This time multiply 614 by 11. I'm waiting...... Is your answer 6,754?

Now it is time to make this process a little more difficult. What happens if we have to regroup or carry in one of these multiplication problems?

We will multiply 784 by 11. Notice that we start as we did before by just bringing down the number in the ones place. Next, we add 8 + 4 and get a sum of 12. We write down the 2 but carry or regroup the one. We now add 7 + 8 which is 15 and then add in the 1 we are carrying. That makes 16. We bring down the 6 but carry the 1 over. We have a 7 in the hundreds place, but must add in the one we are carrying to get a sum of 8. Thus our answer is 8,624.

Let's see if you can do these without paper or pencil. 965 x 11 768 x 11 859 x 11 After working the problems in your head, write down your answers and check them with a calculator. Try making up some four and five digit problems because this is a non-threatening way to have your students practice their multiplication facts. Have fun!

Hands-On Math Using FREE Milk Lid Jug Lids

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Start saving milk jug lids because there are countless hands-on math activities you can do in your classroom using this free manipulative. Here are just four of those ideas.

1) Sort the lids by various attributes such as:

Color
Snap-on or Twist-on
Label or No Label
Kind of edge (smooth or rough)

2) Let the students grab one handful of lids.

Ask the students to count the lids.
See if the students can write that number.

3) Make a pattern using two different colors of lids.

Identify the pattern using letters of the alphabet or numbers. The pattern above would be an A, A, B pattern or a 1, 1, 2 pattern.

Now ask the students to use more than two colors to make a pattern

Once more, have the students identify the pattern using alphabet letters or numbers.

4) Decide on a money value for each color of lid. (Example: Red lids are worth a nickel, blue lids are worth a dime, and white lids are worth a penny.) Put all of the lids into a bag and have the students draw out four lids. Have the students add up the total value of these four lids.

Use play money (coins) to have the students show the value of the lids.

Have the students practice writing money as either a part of a dollar or as cents.

Another idea is to have the students find all the combinations of lids that would equal a nickel or a dime or a quarter.

The resource, Milk Lid Math, contains over 15 hand-on ideas with numerous activities listed under each idea. The activities may be used with a whole group, small groups or as center activities.

A Recipe for a Homemade Frozen Treat for Those Hot Summer Days

June always brings the first day of summer. This year it was on June 21st. I'm not sure where you live, but I live in Kansas, and each day, it gets hotter and hotter! On a hot day, when you have been outside, there is nothing better than an ice cold treat. For years, I have made homemade Popsicles, first for my children and now for my grandchildren. I thought I would share the quick and easy recipe with you. (I know this might be considered the "far side" of math, but recipes do contain measurement and sometimes, even fractions!)

Popsicle Recipe - Will make 18

1 small package of Jello (any flavor)

Berry Blue is our favorite!

As you can see, four of my grandchildren like the Berry Blue.

1/2 cup sugar
2 cups boiling water
2 cups cold water

Boil the water. Add the boiling water to the sugar and the small package of Jello. Stir until all the Jello is dissolved. This takes about two minutes. Add the cold water and stir again.

Pour into three sets of Tupperware Popsicle Makers. If you don't have these (I'm not sure they are available anymore), use Popsicle molds found in stores. or use ice cube trays.

Place in the freezer until hardened. Eat and enjoy just like my grandchildren do!

Using Bloom's Taxonomy on a Geometry Test

As one of their assignments, my college students are required to create a practice test using pre-selected math vocabulary. This activity prompts them to review, look up definitions and apply the information to create ten good multiple choice questions while at the same time studying and assessing the material. Since I want the questions to be more than Level 1 (Remembering) or Level II (Understanding) of Bloom's Taxonomy, I give them the following handout to help them visualize the different levels. My students find it to be simple, self explanatory, easy to understand and to the point.

Level I - Remembering

What is this shape called?

Level II - Understanding

Circle the shape that is a triangle.

Level III - Applying

Enclose this circle in a square.

Level IV - Analyzing

What specific shapes were used to draw the picture on your right?

Level V - Evaluating

How is the picture on your right like a real truck? How is it different?

Level VI - Creating

Create a new picture using five different geometric shapes. (You may use the same shape more than once, but you must use five different geometric shapes.)

As teachers, we are only limited by our imagination as to the activities we ask our students to complete to help them prepare for a test. However, we still need to teach and provide information so the students can complete these types of tasks successfully. With the aid of the above chart, my students create well written practice tests using a variety of levels of Bloom's. When the task is completed, my students have also reviewed and studied for their next math exam. I consider that as time well spent!

FREE

If you would like a copy of the above chart in a similar but more detailed format, it is available on Teachers Pay Teachers as a FREE resource.

Also available is a simple math dictionary. This 30 page math dictionary for students uses easy and clear definitions as well as formulas and examples so that students can learn and understand new math words without difficulty or cumbersome language. Most definitions include diagrams and/or illustrations for the visual learner. Over 300 common math terms are organized alphabetically for quick reference.

Explaining the Difference Between Odd and Even Numbers

Sometimes we think everyone knows the difference between an odd and even number. When I was teaching my remedial math college class, we were learning the divisibility rules, the first of which is that every even number is divided by two. I wrote the number "546" on the board and asked the class if this was an odd or even number. I had one student who disagreed with the group answer of even. I asked him why he thought the number was odd, and he replied, "Because it has a "5" in it. " It was obvious this student got all the way through high school without a clear understanding of odd and even numbers. So the moral to this story is to be sure to discuss the difference between an even and an odd number with your students.

A good definition for an even number is that it can be put into groups of two without any left over, like giving each person a partner. But when you have an odd number of things and put them into groups of two, one will always be left out.

Try this approach. Make your hands into fists and place them side by side as seen in the illustration. Say a number. Now count, and as you count, put up one finger for each number said, alternating between hands, with fingers touching.

For instance, if you said “3”, you would count one, (left pointer fingerup) two, (right pointer finger up and touching the other pointer finger) three, (left middle finger up). Three is an odd number because one finger does not have a partner to touch.

Here is the sequence to use if the number given were "2". Two is an even number because each finger has a partner.

Repeat this several times, giving the students odd as well as even numbers. By always having a concrete visual (their fingers) will help the kinesthetic and visual learner to "see" the odds and evens.

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Activities such as this can be found in a math booklet entitled Number Tiles for The Primary Grades. It contains 17 different math problem solving activities that extend from simple counting, to even and odd numbers, to greater than or less than to solving addition and subtraction problems.

Different Ways to Write Tally Marks

Tally marks are the quickest way of keeping track of a group of five. One vertical line is made for each of the first four numbers; the fifth number is denoted by a diagonal line drawn across the previous four (i.e., from the top of the first line to the bottom of the fourth line). The diagonal fifth line cancels out the other four vertical lines making the entire set represent five.

Tally marks are also known as hash marks and can be defined in the unary numeral system. (A unary operation in a mathematical system is one element used to yield a single result, in this case a vertical line.) These marks are most useful in counting or tallying ongoing results, such as the score in a game or sport, as no intermediate results need to be erased or discarded. They also make it simple to add up the results by simply counting by 5’s. Here is an illustration of what I mean.

The value 1 is represented by | tally marks.
The value 2 is represented by | | tally marks.
The value 3 is represented by | | | tally marks.
The value 4 is denoted by |||| tally marks.
The value five is not denoted by | | | | | tally marks. For the number 5, draw four vertical lines (||||) with a diagonal (\) line through them.

I have seen many interesting ways to teach tally marks to younger children. Many teachers will use Popsicle sticks so that the students have a concrete hands-on way of making tally marks. Some have even tried pretzel sticks although there is a good chance some will disappear during the lesson.

But have you ever seen these kind of tally marks?

My husband, who teaches science, received this data collection paper from a student. The students were tossing coins marked TT, Tt, and tt to determine different genetic traits and tallying the results. The ones seen above are Japanese tally marks. (The student lived in Japan.) I was fascinated about how they were made so I asked him to have this student show me the sequence of how to draw the marks.

I'm not sure what they mean or why they are made this way, but if you look at the 2nd mark you will notice that it looks like a "T" for two. The fourth mark sort of looks like an "F" for four, but so does the third one. As you can see, each complete 正 character uses 5 strokes; so, a series of 正 would each represent 5, just like the English ones. However, to be honest, I am at a total lost to what this really means; so, I resorted to the internet. Here is what I learned.

Instead of lines, a certain Kanji character is used. In Japan, this mark reminds people of a sign for “masu” which was originally a square wooden box used to measure rice in Japan during the feudal period. Here is what the tally marks would look like if we compared the two systems.

The successive strokes of 正 (

) are used in China, Japan and Korea to designate tallies in votes, scores, points, sushi orders, and the like, much as

is used in Europe, Africa, Australia and North America. Tallies beyond five are written like this 正 with a line drawn underneath each group of five, followed by the remainder. For example, a tally of twelve

is written as 正正丅.

So the next time your visit Japan or go to a Japenese restaurant to order Sushi, look for the tally marks as the waiter takes your order.