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Earth Day is April 22nd. How Will You Celebrate?

Earth Day began in 1970, when Gaylord Nelson, a U.S. Senator from Wisconsin, wanted nation-wide teaching on the environment. He brought the idea to state governors, mayors of big cities, editors of college newspapers, and to Scholastic Magazine, which was circulated in U.S. elementary and secondary schools.

Eventually, the idea of Earth Day spread to many people across the country and is now observed each year on April 22nd. The purpose of the day is to encourage awareness of and appreciation for the earth's environment. It is usually celebrated with outdoor shows, where individuals or groups perform acts of service to the earth. Typical ways of observing Earth Day include planting trees, picking up roadside trash, and conducting various programs for recycling and conservation.

Symbols used by people to describe Earth Day include: an image or drawing of planet earth, a tree, a flower or leaves depicting growth or the recycling symbol. Colors used for Earth Day include natural colors such as green, brown or blue.

The universal recycling symbol as seen above is internationally recognized and used to designate recyclable materials. It is composed of three mutually chasing arrows that form a Mobius strip which, in math, is an unending single-sided looped surface. (And you wondered how I would get math in this article!?!) This symbol is found on products like plastics, paper, metals and other materials that can be recycled. It is also seen, in a variety of styles, on recycling containers, at recycling centers, or anywhere there is an emphasis on the smart use of materials and products.

Inspired by Earth Day, Trash to Treasure is a FREE resource. In it, you will discover how to take old, discarded materials and make them into new, useful, inexpensive products or tools for your classroom. Because these numerous activities vary in difficulty and complexity, they are appropriate for any PreK-3 classroom, and the visual and/or kinesthetic learners will love them.

To download the free version, just click under the cover page on your left.

Techniques for Remembering the Slope for Vertical and Horizontal Lines

I work in the Math Lab at the community college where I also teach. Last week, I had two College Algebra students who were having difficulty with slope.  They knew the equation y = mx + b, but were unsure when it came to horizontal or vertical lines. By the way, they were using their graphing calculators which I made them put away. (The book said no calculators.) I feel that if they construct the lines themselves, it puts a visual image into their brain much better than if the calculator does it for them. Sure enough, one of the sections in their math books gave the picture of the line from which they had to write the equation. They were amazed that I could just look at a graph and know the slope, give the equation, etc. When I taught high school math, my students couldn't use a graphing calculator until the middle of this particular chapter as I wanted them to physically draw the lines.

First, for those who have no idea what I am talking about, slope is rise over run.  Rise is how far a line goes up, and run is how far a line goes along.  At the right, the line goes up 3 and has a run 5; therefore, the slope is 3/5.  Rise/Run (Rise divided by Run) gives us the slope of the line.

When a line is horizontal, it has no rise, only a run. So the numerator would be zero (for no rise) and the denominator would be a number such as 5 for the run.  0 ÷ 5 = 0  This is true for any horizontal line.

A vertical line is different.  It has rise, but no run; therefore there would always be a number in the numerator, but always a zero in the denominator.  Since we cannot divide by zero, the slope is considered undefined. (I do use rise over run stating that a horizontal line might have 0/5 which is equal to 0 and that a vertical line might have 3/0 is undefined because we can't divide by zero. Our college algebra book uses O/K for okay and K/O for knock out which I like, but I still think the students need to know why.)

I wanted these two students to have a picture that would help them remember the difference.  I thought of a table for the horizontal line and asked them what would happen if the legs of the table were uneven.  They agreed that the table would have slope.  Therefore, the table would have a slope of zero if the legs were even.

I then went blank.  In other words, by creative juices stopped working, and I could not think of a picture that would help them visualize undefined. Since Teachers Pay Teachers has a forum,, I asked my fellow math teachers if they had any ideas.  Here is what some of them came up with.

The Enlightened Elephant suggested using a ski slope. She talks about skiing down a "cliff", which would not be possible (although some students try to argue that they could ski down a vertical cliff) and so the slope is "undefined" because it doesn't make sense to ski down a cliff.  Skiing on a horizontal line is possible so it's slope is zero,  She also talks about uphill (positive slope) and downhill (negative slope). 

Math by Lesley Elisabeth tells her students to use "HOY VUX" (rhymes with 'toy bucks')

             Horizontal - Zero (0) slope - y = ?   
             Vertical - Undefined slope - x = ?

All horizontal lines are =7 or = -3 etc., and all vertical lines are =1 or = 6, etc. Students forget this so the acronym HOY VUX helps them to remember. Once they've mastered the slope concept in Algebra I, for the rest of the school year, for Algebra II (especially equations of asymptotes - a line that continually approaches a given curve but does not meet it at any finite distance) and even in calculus classes for tangent lines, HOY VUX is just faster and more practical. 

Animated Algebra created a video lesson on the Slope Intercept.  She has a boy skateboard down a negative slope, literally right on the graph line. Karen then shows the same boy taking an escalator up on a line that has a positive slope. Later in the lesson, she rotates the line clockwise, each movement with a click, to show the corresponding slope number to link the line to the slope.  She includes lots of other visual cues to help students focus on and pay attention to the concepts.

When Dividing, Zero Is No Hero - Why We Can't Divide by Zero

Have you ever wondered why we can't divide by zero?  I remember asking that long ago in a math class, and the teacher's response was, "Because we just can't!"  I just love it when things are so clearly explained to me. So instead of a rote answer, let's investigate the question step-by-step.

The first question we need to answer is what does a does division mean?  Let's use the example problem on the right.
  1. The 6 inside the box means we have six items such as balls. (dividend) 
  2. The number 2 outside the box (divisor) tells us we want to put or separate the six balls into two groups. 
  3. The question is, “How many balls will be in each group?” 
  4. The answer is, “Three balls will be in each of the two groups.” (quotient)
                                      

Using the sequence above, let's look at another problem, only this time let's divide by zero.
  1. The 6 inside the box means we have six items like balls. (dividend) 
  2. The number 0 outside the box (divisor) tells us we want to put or separate the balls into groups into no groups. 
  3. The question is, “How many balls can we put into no groups?” 
  4. The answer is, “If there are no groups, we cannot put the balls into a group.” 
  5. Therefore, we cannot divide by zero because we will always have zero groups (or nothing) in which to put things. You can’t put something into nothing.
Let’s look at dividing by zero a different way. We know that division is the inverse (opposite) of multiplication; so………..
  1. In the problem 12 ÷ 3 = 4.  This means we can divide 12 into three equal groups with four in each group.
  2. Accordingly, 4 × 3 = 12.  Four groups with three in each group equals 12 things.
So returning to our problem of six divided by zero..... 
  1. If 6 ÷ 0 = 0....... 
  2. Then 0 × 0 should equal 6, but it doesn’t; it equals 0. So in this situation, we cannot divide by zero and get the answer of six.
We also know multiplication is repeated addition; so in the first problem of 12 ÷ 3, if we add three groups of 4 together, we should get a sum of 12. 4 + 4 + 4 = 12

As a result, in the second example of 6 ÷ 0, if six zeros are added together, we should get the answer of 6. 0 + 0 + 0 + 0 + 0 + 0 = 0 However we don’t. We get 0 as the answer; so, again our answer is wrong.
It is apparent that how many groups of zero we have is not important because they will never add up to equal the right answer. We could have as many as one billion groups of zero, and the sum would still equal zero. So, it doesn't make sense to divide by zero since there will never be a good answer. As a result, in the Algebraic world, we say that when we divide by zero, the answer is undefined. I guess that is the same as saying, "You can't divide by zero," but now at least you know why.

If you would like a free resource about this very topic, just click under the resource title page on your right.

Domino Math - Using Dominoes to Problem Solve and Practice Math Concepts

Dots Fun for Everyone
It is believed dominoes evolved from dice. In fact, the numbers in a standard double-six set of dominoes represent all the rolls of two six-sided die. It is thought they originated in China around the 12th century. They have been used in a large variety of games for hundreds of years, and today, dominoes are played all over the world.

Games allow children to learn a great deal concerning mathematical concepts and number relationships. Often, they are required to use critical thinking skills as well as varied math strategies to solve them. Since dominoes make a great manipulative for hands-on learning, I created a book of domino activities for grades 3-5 that are great for students who finish early or for introducing a new mathematical concept or for use at a math center. Using dominoes for a math practice center is a way to engage students while giving them a chance to review math facts.

The activities and three games vary in difficulty; so, differentiated instruction is easy. The variety of pages allows you to choose the practice page that is just right for each student. This resource correlates well with the CCSS standards.
Dots Fun

The activities in Dots Fun for Everyone (grades 3-5) include four digit place value, using the commutative property, problem solving, reducing proper and improper fractions and practicing multiplication and division facts. The games involve finding sums, using <, >, and = signs and ordering fractions.

These domino math activities in Dots Fun (primary grades) include recognizing sets, place value of two and four digit numbers, creating domino worms, gathering data, using the commutative property, and practicing addition and subtraction facts. The games involve matching, finding sums, and using greater than, less than, and equal signs. For these 13 activities and four games, you may use commercial sets dominoes or copy the blackline which is provided in the resource. This resource links closely with the CCSS standards. 

Some of the domino activities in these two resources use games while others will extend, enhance or introduce a new math concept. Since children are curious and inquisitive, plus some may have never seen dominoes, allow time for play and exploration before beginning any instruction. This is constructive as well as a productive use of class time. If they are not given this, most children will fool around and investigate during the teaching time.

To view examples from these resources as well as a complete Table of Contents, download the preview or FREE versions available at my TPT store.

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Even Today with Spell Check and Technology, Spelling is Important!

Noah Webster, an American lexicographer (one who compiles a dictionary) was the first person to write a dictionary of American English. It may have taken him more than 25 years to do so, but this book permanently altered the spelling of American English by offering a standardized way to spell and pronounce words. He learned 26 languages, including Anglo-Saxon and Sanskrit, in order to research the origins of our country's tongue. You may not know this, but Webster used the Bible as the foundation for his definitions.

Before his dictionary, Americans in different parts of the country spelled, pronounced and used words differently. To create uniformity, Noah used American spellings like "color" instead of the English "colour" and "music" instead " of "musick". He also added American words that didn't appear in English dictionaries like "skunk" and "squash". When he finished in 1828, Noah's dictionary contained 70,000 words.

During Webster's lifetime, American schools were anything but productive. Sometimes 70 children of all ages were crammed into one-room schoolhouses with no desks, poor books, and untrained teachers. The textbooks came from England. Noah thought Americans should learn from American books so he wrote a spelling book for children. Known for generations simply as The Blue-back Speller, millions of American children learned how to uniformly spell and pronounce words. Webster also established a system of rules to govern grammar, and reading. Clearly, he understood the power of words, their definitions, and the need for precise word usage in communication. Without a common oral and written language, he felt the country would remain divided.

Fast forward to today with the use of texting. Writing skills have turned into sentence fragments while spelling consists of numbers, symbols or abbreviations. These habits carry over when students are at school; consequently many really don’t know how to spell or write well. No longer can students punctuate correctly since text messages often contain run on sentences with no punctuation, In addition, with the constant use of lowercase letters, students fail to use capital letters where they should. How do I know? I teach at a community college where about 60% of our students are in remedial English which involves sentence structure, basic grammar and spelling. When assigning a written assignment, I must include how many words a good sentence contains and how many sentences are in an acceptable paragraph. Even these requirements do not guarantee a complete sentence.

It seems we have moved away from standard spelling to inventive spelling (an abbreviated, expedient form); yet customary spelling has not gone out of style. It is required at school, in business, at work and in just everyday life.  In addition, the correct spelling of words affects academic success. Students are frequently assessed on their skills in written language because it is considered a strong indication of their intelligence.

Spelling is an indication of a number of things when a person applies for a job.  When correct spelling is used, words are readable and communication is clear. This convinces a prospective employer that the job applicant has been well educated. It also tells them that they take care of detail and take pride in what they
present.  Let’s face it, university applications and job resumes littered with spelling errors don’t make it very far becuz badd spilleng is hrd two undrstnd wen yuu reed it.

Furthermore, good spelling streamlines communication. By following the identical rules for spelling words, we can all understand the text we read. Likewise, good spelling avoids confusion. In a way spelling is similar to football. It is up to the person passing the ball to make sure the receiver actually catches it. The same goes for spelling. If you write with intent and proper spelling, the receiver of that text will understand it.

As teachers and parents, we should care about the fundamental part good spelling plays in our language and everyday lives. We owe it to our students to give them the necessary skills and essential spelling tools for learning and communication so they can be successful.

If you liked this article and would like to purchase some useful spelling resources, check out these two games. Their purpose is to help and encourage students to practice spelling words in a non-threatening way while having fun learning to spell.


Ironing Coffee Filters to Make the Perfect Circle

When I teach angles or the properties of circles, I find that most children have difficulty cutting out a true circle (even with a blackline).  I have resorted to purchasing cheap coffee filters (not the cone shaped ones) and ironing them flat. You can iron several filters at one time, and once they are ironed, they form excellent ready-made circles. Here are some of the ways you can teach angles using these circles.
    Writing Formulas on the Coffee Filter Circle
  1. Introduce the fact that each and every circle contains 360 degrees.
  2. Have the students fold their coffee filter in half. Discuss that this is a straight angle. Ask, “How many degrees does it contain if it is one-half of a circle?” (180 degrees)
  3. Have the students fold the coffee filter one more time, into fourths. Talk about this angle being called a right angle and that it contains 90 degrees. Ask, "What fractional part of a circle is this?"
  4. Have the students use this fourth of a circle to locate places in the classroom where it will fit (e.g. the corner of their desk, a corner of a book, a corner of the board).
  5. Explain that these corners are right angles and without right angles, we would live in a crooked world. Nothing would be straight!
  6. With older students, have them write the parts of the circle and the formulas needed for solving problems about circles on the coffee filter circle.
Linking Math and Literature for Older Students

Read Sir Cumference and the First Round Table (A Math Adventure) by Cindy Neuschwander. This is a story about a clever knight of King Arthur’s named Sir Cumference. By using ideas offered by the knight’s wife, Lady Di of Ameter, and his son, Radius, King Arthur finds the perfect shape for his table. Basic geometric vocabulary involving circles (circumference, radius, and diameter) is introduced.
$3.50

Want more hands-on ideas for teaching angles? Check out Angles: Hands-On Geometry Activities.

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Pinterest and the Benefits of Using Tailwind to Increase Traffic to your Teachers Pay Teachers Store

Maybe you have heard of Tailwind and maybe you haven't. To set the record straight, it is not a wind blowing in the direction of travel of a vehicle or aircraft or a wind blowing from behind. It is a Pinterest and Instagram Marketing, Scheduling and Analytics Tool.

I was first introduced to it on the Teachers Pay Teachers blog. Since I use PinterestI am always pinning new ideas, teaching strategies, Ohio State stuff (Go Buckeyes!), ideas for my college classes, etc., I decided to try using Tailwind instead of individually pinning each resource from my TPT store or blog. Before deciding if the paid plan was right for me, I was able to schedule up to 100 pins on Pinterest for free . What was nice is that there was no time limit on that free trial! It allowed me to schedule up to 100 pins on Pinterest while I could watch my analytics and the number of repins my items were receiving before deciding if the paid plan was worth the money.

I saw several benefits right away! More traffic was coming to my Pinterest Boards as well as to my Teachers Pay Teachers StoreI therefore decided to invest in Tailwind.  I am aware that pinning everything all at once isn’t ideal; consequently, it is important to space pins out a little so I'm not overwhelming my followers. Pinterest has even suggested that too much pinning in a short time period could be viewed as “spamming”, potentially hurting my rankings in search results and feeds. With Tailwind, I have the ability to create my own posting schedule, and I can use interval pinning to space out my pins so "spamming" doesn't happen. I can set a time period between when each image that is pinned anywhere from two days to 90 days apart. Tailwind even gives me the best times and days to post.

Here are some ways I am currently using Tailwind:
  1. To schedule pins
  2. To know when the best time to pin by using the Smart Calendar
  3. To schedule multiple pins to different boards at the same time
  4. To use interval pinning
  5. To use Smartloops - They let you automatically repin the same type of content to specific boards without having to revisit them. 
  6. To use the Tailwind Analytics - to know which are my best pins and where my repins are going
  7. To find my best pins using Pin Inspector
  8. Create new pins from scratch using Tailwind Create - I use this all the time to take an old, outdated pin and remake it into a new pin. You just add your images, and it will create hundreds of pin designs that you can choose from and instantly schedule into your Tailwind queue.
  9. Use Ghostwriter AI to rewrite and personalize your pin descriptions
Even though I haven't tried these features, you can also:
  1. Create a personalized marketing plan.
  2. Create an email list so you can send and receive emails.
  3. Customize your profile.
I know this sounds like a long advertisement for Tailwind, but I am so excited about the many benefits and results of this service, I just had to share it with you, my readers. If you are interested in joining Tailwind, I have a Tailwind tribe called Math Counts where any teacher who teaches kindergarten through high school can post math resources for free. Each person in the tribe adds their own pins in the queue. Once you see the pins in the queue, you can add them to your scheduled pins in Tailwind so your resources keep getting re-pinned to different boards. Tailwind has said that you do not have to be a paying member of Tailwind to be in a Tribe. Here is the link if you are interested in joining Math Counts. Just copy and paste it into any search engine.

https://www.tailwindapp.com/tribe/join?h=eYO2Fvg6qtxnGj

If you decide to join my tribe (which costs you nothing), you will also get a free month ($15 credit) if you ever decide to join the Plus plan. Most bloggers only need Tailwind Plus, which costs $119 per year when paid annually. If you choose to pay monthly then it costs $19.99 per month, or $240 per year. Note that Tailwind’s free trial lets you schedule up to 100 pins and 30 Instagram posts to test the platform before paying.

Getting started with Tailwind is easy. In the members area are five training videos that walk you through everything step-by-step and in much more detail than this blog post. There’s also an in-depth FAQ section, and if you get stuck with anything, their customer service is responsive and helpful. All I can say is that it makes running my Pinterest account much easier; I can be more strategic in my Pinterest marketing efforts plus it saves me a ton of time!  I hope you will check it out.

Valentine Rebus Fun - Using Rebus Puzzles to Solve "Heart" Problems

$5.25
Many of my students love figuring out rebus puzzles. (a visual puzzle in which words are represented by combinations of pictures and individual letters.)  In a nut shell, they are essentially little pictures which cryptically represent a word, phrase, or saying.  Since Valentine's Day is just around the corner, I decided to have some fun and create 26 rebus puzzles for the month of February.

Hearts and Valentines is resource that features familiar expressions that contain the word "heart". (e.g. "From the Bottom of My Heart" or "Cross My Heart") Each illustration in this 13 page resource uses a picture or symbol to represent a common word or phrase.  Students must use logic and reasoning skills to solve the 26 rebuses. So that you don't have to figure out each one, the answers are included.

Each day during the month of February, put up one "Heart" illustration as a student focus activity, OR, if you choose, place two or three up at one time or all of them up at the same time. Students are to figure out which Heart expression the picture represents. It can be fun, but also a very challenging Valentine's Day activity!  Look at the following images and try to work out what they mean.

The first one is "a heart full of love." Were you able to figure it out?

The second one is a bit more challenging. The answer is "a heavy heart." Did you solve it on your own?

Challenge your students to make some of their own "heart" rebus puzzles. A few in this handout were created by middle school students who prove they can be very creative!

Completing a Glyph for Groundhog's Day, February 2, and Interpreting Data

On February 2nd in 1887, Groundhog Day, featuring a rodent meteorologist, was celebrated for the first time at Gobbler’s Knob in Punxsutawney, Pennsylvania. According to tradition, if a groundhog emerges from its hole on this day and sees its shadow, there will be six more weeks of winter weather! (YIKES!)  No shadow means an early spring. I'm hoping for the latter although our winter here in Kansas has been pretty mild.

No matter whether he sees his shadow or not, it is always fun for students to do special activities on Groundhog's Day.  In my Teachers Pay Teachers Store, I feature a Groundhog Day Glyph. Glyphs are really a form of graphing, and students need the practice. In addition, glyphs are an excellent activity for reading and following directions, and they involve problem solving, communication, and data organization. 

$3.00
This glyph has the students coloring or gluing different items on a groundhog based on information about themselves. Students are to finish the groundhog glyph using the eight categories listed below.

1) Head covering
2) In the Sky
3) Eyes
4) Around the Groundhog’s Neck
5) Flowers
6) Umbrella
7) Color the Groundhog
9) Name

Examples of the first three categories can be viewed on the preview version of the resource. So that each student has the same groundhog to start with, a printable outline is provided on page 4 of this six page activity. This handout also contains a page where the students are asked to identify the characteristics of someone who did their own groundhog glyph. An answer key is included. Kindergarten teachers can easily adapt this activity since the instructions include pictures.

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Measuring Snow - A Winter Craft for the NOT so Crafty, Like Me!

I am not a very crafty person; so, I am always looking for items that are easy to make that I can give to my grandchildren. One year, I gave them a snowman making kit that included buttons, a carrot, six rocks and two sticks. This year, I am giving them a Snow Measuring Tool.  Not only was it fun to use, but it also helped them to practice using a ruler. Here is how you can make one!
 
Here is the list of supplies you will need:  

1) A paint stick - free at most paint stores
2) A permanent marking pen
3) Something to glue at the top of the stick (You can make it, or be like me and purchase one from a craft store.)

First, using a ruler, mark off every inch along the paint stick. I was able to make nine marks. (Notice I used the plain side of the paint stick and not the side with all of the advertising.) Now write the inches beside each corresponding mark.

When that is completed, glue the item you have chosen at the top of the stick.  I really wanted to use a snowflake, but my local craft store didn't have any; so, I settled on using one of Santa's reindeer.  Which one, I'm not sure since it didn't come with a name.(Hint: My husband used Gorilla Glue so the reindeer wouldn't fall off.)

When it snows, venture outside and stick the Snow Measuring Tool into the snow and read the number of inches that have fallen. If it isn't exactly on an inch mark, then have your child estimate using fractional parts.

While you are measuring the snow, think about this saying: "Ten inches of snow equals one inch of rain." I am sure you have heard that claim as it is a commonly shared belief that seems to be repeated every time it snows a few feet. But, is the saying true? The immediate answer is: Sometimes.

When the temperature is around 30 degrees, one inch of liquid precipitation (rain) would fall as 10 inches of snow, presuming the storm is all snow. But, the amount of moisture in each snowflake differs depending on the temperature which in turn changes the snow to rain ratio.

For example, if a big January snowstorm occurred with colder temperatures (such as 25 degrees), the snow ratio would be closer to 15 inches of snow to one inch of rain. In fact, weathermen take this into account when forecasting how much snow a location will receive. There have been storms with snow closer to 20 degrees, moving the snow ratio closer to 20 to one. And, when it's warmer, say 35-40 degrees, the ratio moves to 5" of snow to 1" of rain.

So, after your children measure the snow in your yard with their Snow Measuring Tool, try converting the inches of snow into inches of rain based on the 10":1" ratio. By doing so, you may become your neighborhood's weather forecaster or even better, a first rate mathematician!

Your children might enjoy this snowman glyph. It's is an excellent winter activity for reading and following directions, and requires problem solving, communication, and data organization.

"SNOW" Much to Learn About Snow


Snow is much more than white, wet and cold. There are many unusual facts about snow that make it unique and one of the more complex types of precipitation.

  • Although snow appears white because of the countless tiny surfaces of each snowflake crystal reflecting most the wavelengths of light, snowflakes are actually colorless. Snow may take on other colors thanks to particulates (microscopic solids or liquid droplets) in the air or even from different strains of algae.
  • Many places around the world hold certain world records pertaining to snow. The most snow to fall in a 24-hour period occurred in 1921 in Silver Lake, Colorado. It received 76 inches of snow. That's over six feet!
  • Snowflakes come in many different shapes, and their sizes are determined by how many ice crystals connect together.
  • The largest snowflakes ever recorded fell in the state of Montana. The snowflakes were 15 inches in diameter.
  • The average snowflake falls at a speed of 3.1 miles per hour.
  • Snow that has been compacted after multiple melting and refreezing cycles is know as snow pack.
  • A snow storm describes a heavy snowfall that results in several inches of snowfall. A blizzard is classified as a snow storm combined with wind, which obscures visibility.
  • Snow can be heavy or light depending on its water content.
  • An avalanche occurs when snow that has accumulated on a mountain is disturbed by a thermal or physical impact, which causes the snow to rush downhill in a large mass. Preceding an avalanche is a phenomenon known as an avalanche wind caused by the approaching avalanche itself, which adds to it destructive potential.

If you find these snow facts interesting, try working a crossword where all of the words begin with the word "snow." This FREE resource includes two winter crossword puzzles; each with 25 words that all begin with “snow.” One crossword 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. Answers keys for both puzzles are included. Click under the title page to download your free copy.