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Free Holiday Resource

Free! Just Download
Christmas is upon us, and as teachers, we are always looking for fun, engaging activities for our students. Check out these free holiday lessons by The Best of Teacher Entrepreneurs Marketing Cooperative! This 26 page E-book includes a variety of activities for kindergarten through 12th grade from experienced Teachers Pay Teachers sellers. Since it is free, all you have to do is download it, and then use it! 

Here is a sample of some of the lessons included: 
  • Winter Love Color Sort
  • December Celebrations Sampler Unit
  • 3rd Grade Math for December
  • Dots Fun for Everyone: Math Activities that Use Dominoes
  • Christmas Thank You Notes
Wishing you the very best during this holiday season.
The Members of the Best of Teacher Entrepreneurs Marketing Cooperative.

Tools for Helping 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, 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. 1/4" grid paper
  2. Four co-ordinate graphs per page
  3. Full co-ordinate graph paper
  4. 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) = ax and Graph of f(x) = loga(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.

Using Free Milk Lid Jug Lids for Hands-On Math Activities

Milk Lid Math
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.
  • $3.00
    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 hands-on ideas with numerous activities listed under each one. These may be used with a whole group, small groups or as center activities.

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The ROOT of the Problem - What is Digital Root and How to Use It

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).

Here are several examples of finding Digital Root:

1) 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.

2) 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.

3) 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.

4) 201 = 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 presented by Kim Sutton. (If you have never been to one of her workshops - GO! It is well worth your time.) Anyway, 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!

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.

Using Lattice Multiplication in College - A Good or Bad Idea?

I work in the math lab at the college where we teach. All of our students have free access to this tutoring. This is one of my favorite things to do since I am working mostly one-on-one with students. Last week, I had a student who is taking Fundamentals of Algebra. It is a remedial class for those who do not pass the math test to take College Algebra. In this particular class, students are not allowed to use calculators. In other words, it helps if you know your math facts (and sadly, many don’t).

She had to multiply a three-digit number by a two-digit number in one of her word problems. Pretty easy, right? She kept telling me that she could only multiply using the “lettuce” form of multiplication. Many tutors in the lab had no idea what she was talking about; so, I asked her, “Do you mean lattice multiplication?” Once she demonstrated it to the group, I knew that she was using lattice multiplication.

For those of you unfamiliar with it, here is an illustration of what it looks like. 
Step #1

Draw a grid so that each digit has its own box. If it is a 3-digit by 3-digit problem, you will need a grid that is three by three. If it is a 3-digit problem by a 2-digit problem, you will need a grid that is three by two.

Step #2

Write the digits for the first factor going across the top of the lattice (grid), one digit per box. Write the second digits on the right going down the lattice, one number per box.

Step #3

Now, divide each box in half by drawing a diagonal line, starting in the top right corner and moving to the bottom left corner. Use a ruler if you like nice straight lines. You can have the diagonal lines continue so that they are outside of the grid boxes. This will help you with your answer!

Step #4

Work through the lattice and multiply each number together. Write the answers in the box. The number in the tens place goes in the upper part of the box, the number in the ones places goes in the lower part of the box. If there is not a number in the tens place, put a zero.

Step #5

To finish, you just add down the diagonal lines. Remember to regroup if necessary

Step #6

Finally, to figure out your product, read the numbers from the left of the grid around to the bottom of the grid.  In the example above, 789 x 461 = 363,729.

This form of multiplication dates back to the 1200s or before in Europe. It gets its name from the fact that to do the multiplication you fill in a grid which resembles a garden lattice, something you might find ivy growing on. Although it works, it is a pretty lengthy process for a college student, especially when she only has so much time to complete a test. 

The “vertical” method of multiplication is more efficient because most can write down and solve a 3-digit by 3-digit multiplication problem faster than many students can even draw the lattice. Unfortunately, I showed this student several other ways to multiply, but she failed to grasp their significance. She said that is how she learned in fifth grade, and she was too old to change! With all of that said, lattice multiplication will make it easier for her to transition from lattice multiplication of whole numbers to multiplication of polynomials. 

Using Glyphs to 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.

Just $3.00
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 fall or Halloween 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 pumpkin based on information about themselves. Students finish the pumpkin glyph using the seven categories listed below. 

1) Draw a hat on the pumpkin (girl or a boy?)

2) Color the stem for pets or no pets.

3) Draw eyes based on whether or not they wear glasses.

4) Write a Halloween greeting based on how many live in their house.

5) Do they like playing outside more than playing inside?

6) How do they get to school? (ride or walk?)

7) Pumpkins (How many letters in first name?)

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Bathroom Mathematics - The Changing Size of Toilet Paper

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. The 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 especially since it can be scarce in these COVID times.  Is nothing sacred in the world of mathematics?

Be-Leaf Me! Fall is Great! Using Leaves in Science Investigations

When my husband's Aunt Sue moved to Florida, she would send home some strange requests.  One year, she wanted us to send her a box of fall leaves.  Since Florida lacks deciduous trees, her students were unaware of the gorgeous colors produced by the trees up north.  The only problem with her request was that the leaves we sent would be dry and crumbling by the time she received them. What to do?

I solved the problem by ironing the leaves between two sheets of wax paper.  It was something I had learned in elementary school many, many years ago (back when the earth was cooling).  My granddaughters still collect leaves so we can do the activity together.  Here is how you do it.
  1. Find different sizes and colors of leaves.
  2. Tear off two sheets of waxed paper - about the same size.
  3. Set the iron on "dry".  No water or steam here!
  4. The heat level of the iron should be medium.
  5. Place leaves on one piece of the waxed paper.
  6. Lay the other piece on top.
  7. Iron away!
On the right, you will see what ours looked like when we were finished.

Only $5.25
You can also use this activity to identify leaves.  According to my husband who knows trees, leaves and birds from his college studies, we "waxed" a maple leaf, sweet gum leaf, elm leaf, cottonwood leaf (the state tree of Kansas), and two he doesn't recognize because they are some kind of ornamentals. So my suggestion is to get out there and start gathering leaves because your students, children and grandchildren will love me!

Do you want your students to have fun with leaves? Check out  a six lesson science performance demonstration for the primary grades which utilizes leaves. This inquiry guides the primary student through the scientific method of 1) exploration time, 2) writing a good investigative question, 3) making a prediction, 4) designing a plan, 5) gathering the data, and 6) writing a conclusion based on the data. A preview of the investigation is available. Just click on the title. After all you might have an unbe-leaf-able time!

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Five Assumptions About Distance Learning - Why It's NOT Working for Every Child!

My husband and I have taught together for over 80 years. We continue to do so because we love what we do; however, we are very disturbed by the terms Virtual Learning or Distance Learning or On-Line Learning. Now don't get the idea that because we are "old", we don't or can't use technology. Technology is part of our lives, especially in our teaching, but when we hear Distance Learning, many people make assumptions that simply aren't true.

Assumption  #1 Students Have Access to a Computer and to the Internet

Numerous students do not possess a computer nor do they have access to the Internet. For example, my daughter teaches immigrant children, and out of all her families, only two even own a phone. Even though the school may purchase each student a computer, without Internet access, it is worthless. This is a significant issue in rural and lower socioeconomic neighborhoods. Or when the computer goes home, it somehow disappears. A computer can buy much needed food or sadly, drugs for an addict.

I HATE learning at home!
Assumption  #2 Parents Create a Learning Environment for Their Children

It is obvious that people who believe this have not been in some of the children's homes that I've had the "privilege" of visiting. It's hard to study when chaos reigns or when parents are fighting or the child is expected to babysit younger siblings. Without set class times, children are often interrupted or distracted while studying. Learners with low motivation or unhealthy study habits often fall "through the cracks."

Assumption  #3 Parents Help Their Children Learn

I believe that most parents want to help their child, but because of work schedules, level of education, not speaking English, etc. many cannot.  And then there are others who simply won't because it takes too much of their time. Participation by the parent is really voluntary. In addition, if students have questions where the parents cannot help, s/he has to wait when the teacher is on-line to assist them.

Assumption  #4 Students Have an Intrinsic Desire to Learn

Maybe I've just taught too long to believe this one. Most students would rather skip the lessons and play whether it be outside or a game on the computer.  Rare is the child that can hardly wait for that video lesson to come on. After all, there really isn't any accountability since the teacher is just "in the computer." A student must be a self directed learner.

Assumption  #5 Computer Lessons are Better Because It's Technology

Unfortunately, there are many teachers uncomfortable with teaching on-line. I know because I teach at a community college, and when all in-person classes went to on-line classes in March, many teachers struggled to deliver valuable and worthwhile content. We lost many students because they disliked the on-line classes. I often sat three hours in the virtual math lab with not one student coming for help even though the in-person math lab was always full.

Presently, our college is struggling with enrollment because instead of taking on-line classes, the students have opted to take the semester off. On-line teaching cannot satisfy ALL educational needs and goals. (i.e. hands-on subjects such as public speaking, surgery, dental hygiene, science, sports, etc.) Just because it may be technologically possible to simulate a physical learning experience does not necessarily mean that it is the best way to teach it.

To summarize, I believe we are doing a disservice to most of our students when just on-line education takes place. Many have not been in school since March and have therefore lost three months of learning, not to mention meals, love, encouragement, etc. For countless students, school is their "safe" place. If face-to-face classes don't resume soon, our poorest children are the ones who will be the BIGGEST losers!.

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Back to School Ebook - Free Activities for Grades PreK-12

FREE to Download
We hope you and your students will enjoy our "Free Back to School Lessons" by The Best of Teacher Entrepreneurs Marketing Cooperative (TBOTEMC) for 2020. On each page, you will find links to a free resource as well as a priced resource created by a member of TBOTEMC.

Inside this 2020 edition of the "Free Back to School Lessons" you will find resources like:
  • Back to School Dot Paint Ten Frames FREEBIE
  • Syllable Count - Free
  • Getting to Know You Back to School Craft and Journal Activity Freebie
  • Christopher Columbus American History for Grades K-3 and Homeschool
  • ESL & ELD K-5 Curriculum Map - A Year Long Pacing Guide!
  • Blends Picture Sorts FREE Supports Jolly Phonics Distance Learning
  • Spelling Choices - Multiple Intelligences
  • Back to School Activity - Freebie
  • Dots Fun for Everyone: FREE Math Activities Plus a Game That Uses Dominoes
  • Boom Cards Pundles of Fun: Brain Teaser Logic Puzzles - FREE Sampler
  • FREE Response Process (Reading Skills) Assessment/Evaluation Grid
  • Conquering Confusing Words
PLUS much more! Altogether there are 35 pages of free items for grades PreK-12. All you have to do is download the free Ebook.

The Best of Teacher Entrepreneurs Marketing Cooperative is made up of teachers who work together to market their Teacher Pay Teachers products. Using the power of cross promotion, TBOTEMC  members are able to use their combined social media sites to its full potential. If you are a TPT seller, consider joining The Best of Teacher Entrepreneurs Marketing Cooperative and take your TPT store to the next level. For more information, email Victoria Leon at about TBOTEMC.  You'll be glad you did.

The Division House Mystery - What Is a Vinculum?

When writing division problems, we can use three different forms, a fraction, the division symbol (÷) or the division house. Why it has been called the division "house" has always been a mystery to me since most math symbols are named something that sounds important. I tried looking it up on-line, but never found a formal mathematical word. Well the mystery has been partially solved thanks to SamizdatMath on Teachers Pay Teachers who mentioned the word vinculum.

I decided to research "vinculum", and here is what I discovered. It is a Latin word that means to ‘bond’ or ‘tie’, and was first used by Michael Stifel in 1544 in Arithmetica integra. It is the horizontal line used to separate the numerator and denominator in a fraction. We also see it above the digital pattern that repeats in a repeating decimal or in geometry above two letters that represent a line segment.

Originally the line was placed under the items to be grouped. What today might be written 7(3x + 4) the early users of the vinculum would write 3x + 4. Today that line is placed over the items to be grouped. The line of a radical sign or the long division house is also called a vinculum.

The symbol is utilized to separate the dividend from the divisor, and is drawn as a right parenthesis with an attached vinculum (see illustration above) extending to the right. The vinculum shows that the digits of the dividend are to be kept together as they represent one whole number.

But when it is all said and done, the entire division "house" symbol seems to have no established name of its own. How mathematically sad! Consequently, it has simply be termed the "long division symbol," or sometimes the division "bracket" or division "house". So the next time you draw the symbol on the board, impress your students with the math word "vinculum"!

Can you find the vinculums in this cartoon?