Gnomes and Gnumbers: A Mathematical Tale

    Once upon a time, in the deepest, darkest recesses of the earth, there lived a clan of four greedy gnomes. The first was as blue as a crisp, clear sky.  The second was the shade of a red apple in autumn.  The third was as yellow as the sun and the fourth was greener than a blade of grass.

    Now gnomes are a curious folk.  Burrowing deep in the ground, they build hidden cities that guard the treasures of the earth.  Of course you already knew that, but these gnomes were particularly curious folk.  Not content to work with their fellow earth-dwellers, they wanted to make a name for themselves and build up a treasure greater than any the world had ever known.  So these four miserly gnomes dug and burrowed and tunneled until they reached the middle of the earth where they believed their treasure would remain protected.

    Days and weeks and months went by and the four gnomes came to realize that they had nothing to do but count and sort their mounds and mounds of jewels all the day long. It wasn't long before the gnomes began arguing among themselves about the true owner of the treasure.

    "It is all mine!"  squeaked the small yellow gnome as he scooped great piles of rainbow-colored gems toward himself.  "I am the one who found the way to the middle of the earth.  The jewels belong to me."

  After many days of bickering and quarreling, each of the four gnomes had clearly argued their reason for being the rightful owner of the treasure.  Finally, the red gnome offered a solution.

    "Perhaps if we divide the gems evenly among ourselves..." he timidly suggested. (Having kept some of his wits about him, he realized that the only way for peace and quiet to return to their cavernous home was to share the treasure equally.)

    "But how shall we know if we have divided the treasure evenly?" asked the green gnome.

      The gnomes fell silent, each trying to think of a way to be absolutely certain that their pile was equal to the piles of the other gnomes.

    After what seemed like hours upon hours, the yellow gnome suggested that they take a trip back up through the winding tunnels to the surface of the earth in search of someone with wisdom who could solve their problem.  The others reluctantly agreed and soon they were on their way.

    The gnomes made their way through a tunnel which led to a secret hollow in a magnificent, ancient tree.  Climbing out of the hollow, they tripped over the gnarled feet of the tree and looked around hoping to find someone to direct them to the dwelling place of a wise one who would offer a solution to their problem.  Suddenly a sparkle of brilliant color caught the eye of the green gnome.

    "What is that?" he asked in a whisper.

    The gnomes turned and together discovered a pile of the most beautiful gemstones.

    "They are mine!"  shouted the green gnome.  "I spotted them first!"

    Before he had a chance to dive upon the pile of gems, the red gnome held up his wee hand and interrupted.  "Let us use sticks to count how many gems there are and then we shall share them. Surely this is a great treasure which we must carefully guard."  Then reaching up to the tree above him, he pulled down on a branch and snapped it off.

    "Huuuummmphh!"

"What was that?" asked the blue gnome.

    The other gnomes ignored the deep and unusual sound and went about their work of counting.

    After spreading the gems on the ground, the gnomes laid down the branch next to the first stone.  And when they added a second gem they broke the branch in two.  For the third stone they broke one of the two branches again and saw that they had three stones and three sticks. Then a fourth gem was moved but the sticks were becoming hard to keep track of and they stopped to find a solution.

    "Let us use two of the pieces of the branch to make the shape of the Letter V.  That looks like the "V" between our thumb and other fingers when we hold up our hand.  That will help us to remember the number five"  suggested one of the gnomes.

    Placing one of the leftover sticks before the V-shaped pair, the gnomes counted four.

    "One before five is four."

    The gnomes continued to count the gems in this way.  The V-shaped formation showed five and then adding a stick after it made six.

    "One after five is six."  The gnomes were rather pleased with themselves.

    Pulling a seventh gem from the group, they decided to break the single stick in half to make seven.  One of those was broken again to make eight.

    "This is getting confusing again," sighed the red gnome.  "What can we do to count so many gemstones?

     "I know," boasted the greedy yellow gnome.  "Let's cross two sticks together, like our arms crossed in front of our chest.  This will remind us that the gems are all mine!"

     The other three were not happy with his solution, but not being able to come up with a better one, they agreed.  It was true, when both hands are crossed together they counted ten fingers. The sticks in the shape of a V were rearranged to form an X.

    "One before ten is nine" said the yellow gnome.

    Again the gnomes continued moving the gems and counting.  The X-shaped sticks by themselves represented ten gems and then one more was added in front of it.

    "One after ten is eleven.  And two after ten is twelve."

    The gnomes were tired and sat down to rest.

    "We have so many more gems to count.  We will most definitely need another branch" announced the yellow gnome.

    And just as he raised his arm to grab the nearest branch..."Huuuummmphh!"  An enormous woody hand swooped down and scooped him up high above the green ceiling of the forest.

LESSON PLANS

According to the age and ability of your child, this lesson could be used over a period of one week or up to one month.  All work should be done in a Main Lesson Book with both stick and block crayons.  Recommended resources and suggested sources are linked below. The lesson plans are suggested for two different age groups.  The ages are only offered as a general estimate but, of course, the parent will be able to best identify the work which will most benefit her own child.   

The original art work can be downloaded and reprinted for use as a visual in telling this story.

The Greedy Gnomes
Download greedy_gnomes.PDF

The Ancient Tree
Download ancient_tree.PDF

Tree Folk
Download treefolk2.jpg

If you are using this math story with a 5 or 6 year old (or a child with little or no previous math experience), you may want to consider introducing one Roman Numeral per day.  Before you tell this story, go outside with your child and find a stick to use for story time.  The child can break it apart as the gnomes break their stick.  (Rather than breaking a stick, using toothpicks is another option.)  The illustration offered above of the gemstones, sticks and Roman Numerals is a sample page of a Main Lesson Book.  For a young child you will want to demonstrate drawing this page and allow your child to imitate you. 

After the Roman Numerals I through XII are introduced, you will want to spend a few days writing the numbers 1 through 12 on another page of the Main Lesson Book.  Ask your child to draw a different colored gemstone between each number to keep them from running together into one huge number. 

As you work through this introduction you should spend some part of your lesson playing counting games.  Find creative ways of reinforcing number recognition.  Model numbers one through twelve out of modeling beeswax or instead, cut them in salt dough with number cookie cutters, which can be baked and later painted. 

Add movement to your child's learning and take time to play number games.  Toss a bean bag back and forth while taking turns counting the next number.  (Say "One" and toss the bean bag to your child.  He or she will respond "Two" and throw it back to you as you say "Three" and so on.)  The early learning game of Walking the Line can offer another fun exercise in counting.  Mark a line with masking tape or use a grout line on a tile flour.  As your child walks forward, together count forward to twelve and as he or she walks backwards, count the numbers backwards with your child from twelve to one.

If you are using this math story with a 7 or 8 year old (or a child who has a basic understanding of numbers and familiarity with one or two of the four basic math processes), you may be able to tell this story in one day. Have your child complete the entire Main Lesson Book entry of Roman Numerals I through XII on the same or following day.  (As with a younger child, some children will need to imitate your example.  Don't hesitate to draw with your child.) On the following day ask your child to narrate the story as you look at yesterday's drawing in the Main Lesson Book. 

According to the ability of your child, have him write (in crayon) the numbers one through one hundred in his Main Lesson Book.  (You can spread this exercise out of several days if necessary.) Ask him to change colors every tenth number. (For example, 1 through 9 are green, 10-19 are red, 20-29 are blue, etc...) This will help in preparation for learning place value.  Have him separate each number by drawing a gemstone between each number to keep the numbers from running together. 

One day, or if necessary over the course of several days, have your child model the numbers one through twelve using beeswax or use cookie cutters and salt dough and bake. 

SUGGESTED RESOURCES AND RECOMMENDED SUPPLIERS

Buy the best art supplies that you are able to afford.  The following are suggestions and substitutions should be made according to the preferences and resources of each individual family.

• Main Lesson Book

Traditional Main Lesson Books with onions skins separating the pages (to resist crayon smudges between pages) are available from A Toy Garden. (You may want to consider buying two if you choose this option.) Another suggestion is to use an artist's sketch book or simply collect pages in a binder or album, separated by plastic page protectors. 

• Beeswax Crayons

Nothing compares to beeswax crayons.  The smell of sweet honey and the glide of application make using them an enjoyable sensory experience.  Unlike Crayola and other popular brands of crayons, the color of beeswax crayons can be blended on the page.  The pages in the Main Lesson Book should be drawn with stick crayons and the backgrounds should be shaded with block crayons.  Encourage your child to never leave white on her page.  Each entry is a work of art and should be approached as so.  These art supplies are special and the child should be encouraged to treat them with care. We recommend either Stone or Stockmar which is available from Paper, Scissors, Stone.

• Handcraft Supplies

Spend time with your child in making his own set of math gnomes. Use wooden bases and natural wool felt.  Again, the natural fibers are a sensory experience that enrich the child's learning and train the senses.  Handcraft supplies are available from A Child's Dream Come True (Scroll to the bottom of this page for the wooden dwarf bodies and this page for plant-dyed, wool felt. If these supplies exceed your budget, the pictures can serve as a help in finding less expensive alternatives at your local craft store.)

• Blackboard and Pastels

Using artist's pastels and a quality blackboard allows the written experience of math to tend toward the beautiful and creative.  Unlike ordinary colored chalk, artist's pastels glide on the surface. Excellent quality blackboards and pastels are available from Paper, Scissors, Stone.

• Modeling Beeswax

Modeling beeswax is a treat for the senses.  From the sweet fragrance of honey to its response to the warmth of the hands, this medium adds much to the learning experience.  We recommend Stockmar Modeling Beeswax which is available from Paper, Scissors, Stone.

An alternative is to use an easy salt dough recipe and cut out numbers and the process signs with cookie cutters. 

• Glass Gemstones

Children learn abstract concepts best when they are able to visualize them and use hands-on manipulatives.  At your local craft store you can purchase bags of glass gemstones. You will need at least one hundred stones for this year's math work. Be sure to keep this choking hazard away from small children. 

• Sorting Baskets

The gnomes will need baskets to sort their gems.  Magic Cabin sells miniature baskets as part of their tree fort collection.  You will need twenty baskets (two sets from Magic Cabin) to work on place value work with the gemstones.

• Tube Watercolors,Watercolor Brushes and Paper, and a Watercolor Board

Tube watercolors are a pleasure to work with and quality paper and brushes make a huge difference.  Using a watercolor board offers a smooth surface on which to work and a convenient spot for a work of art to dry.  These boards are also easily wiped clean and protect kitchen tables from stains. 

Paper, Scissors, Stone sells Stone watercolors which are of excellent quality.  Simply squeeze a small amount (the size of a dime) in the bottom of a small mason jar and fill with an inch of water.  Don't mix the paint.  Use the tinted water for a lighter color effect and lightly touch the brush to the small mound of paint ("the worm") at the bottom of the jar for a deeper hue.

• Small Bean Bags

Ordinary bean bags can be used for number tossing games.  Counting, skip counting and math facts can be practiced while having fun with this movement.  You can splurge with an herbal-filled set or enjoy a handcraft project with your child and knit your own using the  simple patterns in Melanie Falick's Kids Knitting.  Perhaps you can come up with a creative way to glue, sew or knit the different math process symbols on different bean bags. 

• Index Cards

Last but not least, just an ordinary set of unlined index cards available at any office supply store.

Be sure to check back regularly as the Gnome and Gnumbers Album on the left sidebar is updated with Main Lesson Book samples.

Gnomes and Gnumbers: Lesson Two

In which EVEN gnomes make an ODD discovery.

    The yellow gnome held his eyes closed tightly as the woody hand lifted him up, high above the treetops.  Suddenly everything was still and quiet except for the steady sound of a groaning wind that was blowing his pointed hat directly behind his head.  Mustering up enough courage to open just one eye, the gnome saw before him a tree that looked like a man.  Or was it a man who looked like a tree?  As these questions danced in his head, the Tree Man spoke,

    “You have taken that which does not belong to you” he groaned in his deep, woodsy voice. 

    “I…I’m...I’m sure I don’t know what you mean, thank you,” replied the gnome, hoping that being polite would help matters.

    The gnome was completely confused by the Tree Man’s strange words and, wondering what had happened to his three friends far below, he ignored the woeful words of the Tree Man and begged to be returned to the ground.

    “Please, put me down on the ground with my friends.  I am a gnome, an earth-dweller, and it is not natural for me to be suspended at such great heights.”

    With that the gnome heard the familiar groan that had sounded when the branch had been snapped from the tree. 

    “Huuuummmphh!  And neither is it natural for my ancient branches to be broken into bits” replied the Tree Man.

    With these words he lowered his enormous wooden hand and soon the gnome was safely back on the ground with his three companions. Suddenly the Tree Man took a step forward into a clearing and the four gnomes could plainly see his enormous stature.

    Before they had a chance to speak, the Tree Man addressed them,

    “I am Old Dismas.  I have lived in these woods since it came to be.  When the earth was still young, a wizard cast an enchantment on my wood,

Whosoever shall harm and tear thee asunder, shall forever count until he hath discovered the secret of numbers.

  With those words, Old Dismas left the gnomes in the clearing.  The earth groaned under the weight of his steps and the quaking sound of the forest floor echoed throughout the woods. 

    "What are we to do?  Oh dear!  Oh me!  We've  brought a terrible enchantment upon ourselves!" cried the blue gnome.

    "It sounds as if we will have to find something to count.  The Ancient Tree said that the enchantment will not break until we discover the secret of numbers.  What shall we count?"

    The gnomes confusedly looked around until they again spotted the mountain of gemstones.

    "The gemstones!  That's what we shall count" shouted the red gnome, with much excitement.

    The four gnomes approached the huge mountain of gemstones and noticed that there were small baskets scattered around the ground. Each of them grabbed one and collected as many gems as he could fit in his basket. 

    After their baskets were filled to the brim, the green gnome asked, "Now what do we do?"

    "How about we break a branch off that tree over there and use the sticks to count our gemstones" replied the blue gnome, with a look of enthusiasm.

    "No!" shouted the other three. "Don't you remember?  That's what got us in this mess in the first place!"

    The four gnomes didn't know what to do, so they emptied their baskets of gemstones and began spreading them on the ground in order to count.  The blue and green gnomes decided to sort their gemstones into pairs of two, but soon the blue gnome became frustrated.

    "Well now, this is rather ODD.  Every time I try to sort my gems in pairs, I always find one left over."

    The green gnome continued his work of sorting and soon matched each gemstone perfectly in pairs.  Upon seeing his success, the blue gnome grumbled,

    "Now, that is not EVEN fair. You do not have any left over.  How is it that when I match my gemstones in pairs I always have one left over and you are able to match every last one?"

   Soon the crunching and snapping of twigs could be heard again beneath the feet of Old Dismas.  As he was approaching the clearing, he overheard the debate and offered an explanation.

     "Yes, it is ODD when a group has one left over and for this reason we call those numbers ODD. How many gemstones have you counted?" asked Old Dismas.

    "Eleven" replied the blue gnome.  "Is eleven an ODD number?"

    "Yes" answered the Ancient Tree.  "And how many gemstones have you counted?" he then asked the green gnome.

    "Twelve.  Twelve gemstones" answered the gnome.  "Twelve is an EVEN number then because all of my gems have a partner.  That's it! I've discovered the secret of numbers!"  The silly green gnome began dancing around his sorted gemstones.

    "Not so fast," laughed Old Dismas. "The secret is far deeper than the discovery of ODD and EVEN.  But don't loose hope.  There is someone here to help you."

    And suddenly a King dressed in purple was standing before them.

To Be Continued...

LESSON PLANS

This lesson continues the story from Lesson 1 and introduces the concept of odd and even numbers.  If your child has not yet mastered the material presented in Lesson 1, wait before continuing with Lesson 2.  Regardless, remember to practice and review Roman Numerals throughout the week.  If your child is interested, introduce further Roman Numerals such as L equals 50 and C equals 100.  Use your child's work in the Main Lesson Book as a springboard for review.  By creating a Main Lesson Book, your child is writing and illustrating his own Math book!

If you are continuing with a 5 to 6 year old, rely heavily on the manipulatives (the gemstones) to teach this concept.  Talk to your child about the things around her that naturally exist in pairs:  two eyes, two ears, two feet, two socks or shoes.  Then proceed to group the gemstones in pairs of two.  Fill up the miniature baskets with gemstones (keep it under 10 for a young child) and ask him to sort them in pairs.  If one is left over you can make a game of saying, like the blue gnome, "Well, isn't this ODD!"  Point out that when all the gemstones have a partner, they are all EVEN. Make an Odd Number and Even Number page in your child's Main Lesson Book modeled after the sample page in the story.  Draw with your child if he is having trouble expressing these concepts in writing and allow your child to spend a few days on his page if necessary. Young children learn best by imitation.

If you're working with a 7 to 8 year old, repeat the above activity and review it each day.  Create and add the Even and Odd Number pages in the Main Lesson Book. Encourage your child to decorate his page with a border that reflects the story or concept learned.  Remember, these pages are to be approached as works of art. When your child has grasped the concepts of even and odd, continue the work by using the blackboard. Write a random number and ask your child to tell you if it is even or odd.  Allow her to use the gemstones to find the answer.

Continue to combine movement with your child's learning.  Games of bean bag toss are a fun way to practice numbers.  This week, play games of Even and Odd toss.  As you toss the bean bag to your child say, "One!" and as your child catches it, he responds, "Three!"  Continue through the odd numbers as he is able and then try the even numbers. Weather permitting, take your lessons outside.  Use stones or acorns and sidewalk chalk to make number pictures on the ground.

Beginning this week we are going to concentrate on a different number each week.  Don't skip this activity, even if your child knows his numbers well and can work basic math equations. We're going to be considering the concrete reality behind the very abstract idea of numbers.  This week we'll begin with Number 1. 

Consider the world and  its Creator.  What exists that expresses oneness? Some answers could be one God, one sun, one moon, one nose, etc... (Remember these answers are taken from the experience of a young child.  There is no need to correct him and point out that the sun is a star and there are many in the universe, etc...)  Ask your child to make a Number 1 page in his Main Lesson Book.

GNOMES AND GNUMBERS STORYBOOK

It is highly encouraged that you print out these math stories and artwork and read-aloud to your child in a comfortable setting and for this reason we are offering them in PDF format for free download.  For the story files, the larger font size was chosen so that the child (or an older sibling) can go back and read the stories on his own.  Keep them in a 3-ring binder, each page stored in it's own page protector so the child can flip through without worrying about ripping pages. Include the artwork throughout the story and soon your child will have a storybook of his own.

You can make this storybook as simple or detailed as you wish.  The cover page shown here can be printed for use in a binder or perhaps you will want to get creative and make a felt relief on an album using ideas from a book such as Felt Wee Folk.

Gnomes and Gnumbers Cover Art
Download gnomes_cover_page.pdf

Gnomes and Gnumbers: A Mathematical Tale (Story One)
Download gnomes_and_gnumbers_lesson_one.pdf

Gnomes and Gnumbers:  In which EVEN gnomes make an ODD discovery. (Story Two)
Download gnomes_and_gnumbers_lesson_two.PDF

Old Dismas and the Yellow Gnome-Art (Lesson Two)
Download old_dismas_and_yellow_gnome_art.PDF

Old Dismas-Art (Lesson Two)
Download old_dismas_art.PDF

Odd Numbers-Main Lesson Book Sample
Download odd_numbers_mlb.PDF

And for fun you can download a coloring page of Old Dismas and the Yellow Gnome.
Download old_dismas_coloring_page.pdf

About the name Old Dismas:  This name was chosen in honor of the repentant thief on the Cross, St. Dismas. The character in this tale not only exhibits wisdom to those who struggle with the passion of greed, but offers it with patience and love.  For this reason it seemed appropriate for the wise Tree Man of our story to share a name with the grace-filled man who from a tree offered us the timeless example, "Lord, remember me when you come into your Kingdom" (Luke 23:42).

Like the thief will I confess thee: Remember me, O Lord, in thy Kingdom.
-Prayer Before Communion, St. John Chrysostom

Gnomes and Gnumbers: Lesson Three

In which the gnomes learn the VALUE of their PLACE.

   By this time the clearing was littered with baskets thrown here and there.  Some were filled with gemstones and others had been toppled over.  The mountain of gemstones was still high above the gnomes, but gems were now scattered everywhere, some in even numbered piles and some mismatched in odd groupings.  The gnomes knew that they must continue counting in order to find the secret of numbers, but they were getting more and more frustrated the longer they tried to count.  And while they were happy that Old Dismas had explained the difference between odd and even numbers, they still knew that there were far more gemstones than they were able to count on their own.

    The gnomes were surprised when they suddenly saw a purple-clad king come out from behind the great trunk legs of Old Dismas. 

    "There is someone here for you to meet," the Ancient Tree announced to the gnomes. 

    The king stepped out in the clearing. He was an odd looking fellow, almost as wide as he was tall. He was dressed in purple robes that seemed to reflect every color imaginable and, instead of a crown, he wore a pointed cap trimmed in gold.

    Careful not to trip over the gemstones and baskets, he began, 

    "I am King Equals, King of all Numeria.  It was my ancestor who cast the enchantment on the Ancient Tree long, long ago...the enchantment that you have brought upon yourselves because of your greed.  And he who cast the enchantment was a great wizard king, one who knew the secret of numbers that you so eagerly seek to discover."

    The gnomes didn't know what to say.  Who was this King Equals and what did he want from them?  Finally the red gnome replied,

    "We know the enchantment.  Old Dismas explained it to us.  And we know we must count until we discover the secret of numbers.  But these gemstones are far too many for us and when we count we soon become confused.  Can you help us?"

    King Equals nodded his head and pulled a magic wand out from beneath his purple cape. 

    "You're a wizard too?" gasped the green gnome. 

    "Well, of course.  And I can offer you a very simple way to solve your problem.  If you want to find the solution to keeping track of the numbers you count, you must repeat this verse with me,

   Regal old yellow gnomes buy itchy vests.

Simply repeat this verse and the solution will present itself to you."

     The gnomes were convinced that King Equals was a fool, but they soon decided that since they had no better plan there was no harm in trying the verse, as silly as it seemed.  Together they began repeating the strange words,

    "Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests..."

    Suddenly a radiant light broke through the tops of the trees and shone upon the mountain of gemstones.  And then there was a most beautiful sight.  The reflection of the gemstones created a brilliant rainbow and a a prism of color filled the sky.  Red, orange,  yellow, green, blue,  indigo, and violet. 

    "Use these bands of color to help you keep track of your numbers as they grow. But remember, only nine of each kind can fit in one band" instructed King Equals.

    Still awestruck by the exquisite color surrounding them, the gnomes were confused by the words, "Only nine of each kind can fit in one band."  And anticipating their confusion, the King began to demonstrate.

    "Do you see this first band of red?  We call this band Units.  Each gemstone represents one unit.  And we can fill the red band with nine of them."

    King Equals picked up a basket and filled it with 9 sparkling gemstones and carefully laid them in the red band of color.

    "But there are many more than nine gemstones, Your Majesty.  How can this possibly help us?  We have a mountain of gemstones to count" remarked the confused green gnome.

    "Ah, but don't you see these other bands?  There are many places within which you may count.  When we fill up the red Units we move on to the orange band. This band is called Tens."

    "Oh! I know what ten is.  We counted ten and made up a symbol for it in the shape of an X using branches from...Oh.  Sorry Dismas. I'm sure it was not pleasant to have your branches broken off" admitted the yellow gnome with great regret.

    The Ancient Tree lowered his heavy head and offered a gentle smile.

    King Equals continued his explanation, "After 9 gemstones fill the Units band we must continue by grouping them together in families of ten." 

    As he said this, the King carefully lined up the gemstones in groups of ten and placed them in the orange band of color.

    "Now remember, only nine of each kind can fit in one band.  So once we have nine groups of ten in the Tens place, we must move on to the yellow band of color. We call this the Hundreds."

     Beginning to understand the sorting game, the gnomes began gathering gemstones once again in their baskets.  When they filled the Hundreds with nine groups of one hundred gemstones, they moved on to the green. King Equals went on to explain that the green band was called Thousands and the numbers continued far past that.  There were blue Ten Thousands, indigo Hundred Thousands and a place in the violet band for a number called Millions. The gnomes continued their work, and stopped only to notice that King Equals had opened his purple cape to reveal a funny little symbol made up of two lines on his chest. 

    "Excuse me, Your Majesty, but what is that symbol you wear on your chest?  Is it some sort of royal family crest?" asked the red gnome.

    "Oh, it is much more than that.  Much more.  But that story if for another time.  For now, continue your counting and fill this rainbow with gemstones.  Soon you will learn the purpose of the symbol I wear upon my chest" answered the King.

    The gnomes whispered among themselves, "Well, I sure hope that mysterious symbol can help us discover the secret of numbers!"

    Of course, you know that Ancient Trees can hear the softest, most faintest sound in the forest.  And hearing the musings of the silly gnomes, Old Dismas let out a laugh that shook the forest floor.

LESSON PLANS

Supplies

There is no need to move on to this lesson until the concepts introduced in the previous two lessons have been mastered.  Move at your child's pace. The Gnomes and Gnumbers lessons are archived in chronological order on this page for your convenience.

You will need to assemble a Rainbow Tray for this lesson, as well as collect a set of numerals.  If you already have a set of moveable numerals like the ones shown in the picture above, use them.  If not, and purchasing a set is not in your budget, spend one afternoon baking a set with your child.  The easy salt dough recipe is here and a set of cutters can be purchased. Another option is to model the numerals with modeling beeswax (Suppliers are linked on the right sidebar.) or Sculpey Clay.

If you already own a Base Ten set, you can make use of it for place value work.  If not, you can download the Ten Bar and Hundred Flat files below.  These were made using Do-A-Dot markers on graph paper. Donna Young's site offers graph paper files for download if your child would like to make a set of base ten gemstones.  The graph paper used here is the 10 x 14 size. 

Downloads

You can download the story and artwork by clicking on the links below. 

Lesson Three Story

Download gnomes_and_gnumbers.Lesson 3.pdf

King Equals Picture

Download king_equals_art.PDF

King Equals Coloring Page

Download king_equals_coloring_page.pdf

Main Lesson Book Sample, Rainbow

Download Rainbow.pdf

Gemstone Ten Bars (Print out 2 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_ten_bars.pdf

Gemstone Hundred Flat (Print out 9 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_hundred_flat.PDF

Presenting the Rainbow

We have chosen to present the rainbow in its true order of colors rather than the simplified version used in many children's songs and books.  Red, Orange, Yellow, Green, Blue, Indigo, and Violet.  Have fun with the mnemonic device created by Marybeth Foss, Regal old yellow gnomes buy itchy vests, for memorizing the colors of the rainbow and their order. 

After reading the story and presenting this lesson on place value, have your child use the gemstones to practice identifying the place value of each numeral in a larger number.  Depending on the age and ability of the child, you will want to go as high as he or she is able.  A younger child may only understand up to the hundreds place, while an older child will enjoy discovering a million.

Ask your child to illustrate a place value page in the Main Lesson Book.  The sample rainbow page offered above can be used as a model for your child. 

Place Value Activities

There are three place value activities that you can play this week using the Rainbow Tray and gemstones.  First start by filling a small basket with gemstones.  For instance, if your basket is filled with 23 gemstones, your child will count nine gemstones and, reminding him that only nine of each kind can be placed in each band, will then cross over to the Tens color band.  The remaining 20 should be grouped in two sets of ten gemstones. Then show your child how to swap out 10 gemstones for a ten bar. Continue in this way until your child understands the concept.

Next, make a 2 or 3-digit number (or larger for an older child) using the moveable numerals. For instance, arrange the numerals so that they show 45.  Ask you child to build this number with the gemstone materials or a Base 10 set next to the Rainbow Tray and then place the numerals in the correct strips of color on the tray.

Another fun activity is to sort gemstones in different sizes of cups.  Begin with 10 small bathroom-sized cups and ask your child to fill each with 10 gemstones.  When ten small cups are filled, they can be poured into a larger 8 ounce cup, representing 100 (pictured here).  Ten of these 8-ounce cups can then be poured into a large 32-ounce cup to represent 1,000.  This tactile experience gives the child the opportunity to see and feel these numbers.  The goal is to bring the abstract into the realm of the concrete.

Fun and Movement

Add movement by playing games of rainbow bean bag toss.  Say "Red" and toss your child a bean bag.  He should respond by saying, "Orange" and then toss the bag back to you. Continue in order to memorize the sequence of the colors in the rainbow.

Review

Don't forget to review the concepts covered in the previous lessons. Together read your child's entries in the Main Lesson Book and discuss each page.  Practice Roman Numerals at least one day this week and use the bean bags to play a game of toss, using odd and even counting as the rhythm.   

If you feel that your child would benefit from math sheet review, there is a wonderful math worksheet generator online called The Math Worksheet Site. You can get a subscription for only $2.50 per month. (The price is reduced depending on the length of your subscription.)  The subscription gives you unlimited use of the site and allows you to create custom math worksheets for your child.  The site is very easy to use. Consider creating sheets on Roman Numerals, odd and even counting, and place value for your child.

Number Study

This week we will continue our study of numbers by concentrating on the Number Two.  Ask your child to consider the world around him and find thing that exist in pairs of two.  Possible responses might be two eyes, two ears, two lights (the sun and moon), two feet, etc...

Gemstone Fun for the Little Ones

Nicholas Foss came up with a fun activity for the little ones.  Using rainbow-colored craft sticks, he sorted the different colored gemstones on each stick and made the pattern of a rainbow.  Since these were not glued down, a young child can repeat this activity many times.

Form Drawing

This week we are adding the element of form drawing.  This exercise in creating rhythmic patterns prepares a young child for handwriting and improves handwriting for the older child.  When presenting these running forms, allow your child to first practice in a shallow tray of sand or cornmeal.  Tracing the form with a finger in the air also helps the child in learning to imitate the pattern.  Your child should use stick crayons when drawing the forms.  Later in this main lesson we will be painting the geometric shapes and practice in form drawing will help in preparing the child for this activity. 

One running form will be introduced each week for a period of nine weeks and we will be using the art of story telling to capture the attention and interest.  Read the following story to your child as an introduction to the first running form.  If your child has not had much practice in drawing, it may be a good idea to spend some time practicing standing forms such as squares, triangles, circles, rectangles and stars. Draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon they will learn the movement and pattern and be able to work independently.

Our form drawing tale will be centered on the story of King Equals and how he brought peace to the Kingdom of Numeria.  Read this story to your child and practice drawing the form before making a main lesson book page (a sample page is shown here).  Perhaps after practicing the form each day, the child can draw it in the Main Lesson Book on Friday. Encourage your child to draw this form with a continuous and uninterrupted movement.  Have fun with the colors.  You can use these forms to practice memorizing the order of the colors of the rainbow too.

The Story of the Great King Equals

Once upon a time there was a sad prince who lived in a great stone castle in a divided kingdom.  For many years his ancestors ruled the kingdom in peace; but one day, in the furthest part of the kingdom, a great treasure was discovered in hidden cave.  Ever since that day the people of the kingdom argued and quarreled over the question of who was the rightful owner of the cave and the great treasure.  As the prince looked out through the parapets of the great stone castle, he longed for the day when his family's great kingdom would be one again.

At this point show your child the sample page of the running form pictured above. (Download great_king_equals.Story 1.PDF

) These lines resemble the parapet of
a castle wall.  Practice drawing this form each day, retelling the
story each time.  By the end of the week your child should be able to
narrate the story himself as he works on mastering this form.   

Living Books

Now is a good time to add a few mathematically themed read-alouds. Living math goes hand-in-hand with Living Books. Here are a few titles for this week:

How Much is a Million

Millions of Cats

One Hundred is a Family

Roman Numerals I to MM

If You Made a Million

Gnomes and Gnumbers: Lesson Four

In which the green gnome learns that counting is a PLUS.

     Having learned that with the help of a beautiful rainbow it was easy to recognize even the largest of numbers, the four gnomes went back to their work of trying to unlock the secret of numbers. 

    "We now know that some numbers are odd, while others are even," recalled the yellow gnome. "And we also know that no matter how large a number is, we can figure it out with the help of the bands of the rainbow.  But the enchantment has not yet been broken.  We obviously haven't figured out the secret of numbers.  What else could there be?"

    As soon as the yellow gnome finished speaking, the green gnome walked to the base of the mountain of gemstones, picked up the largest basket he could find and filled it to the top.   Then taking the gemstones out, one by one, he lined them on the ground and began,

    "One gemstone and one gemstone and one gemstone..."  he continued for quite some time.

    Curious and confused, the other three gnomes sat down and watched their green friend continue.  Soon the silence was broken by the woodsy "Huuuuumph!" of Old Dismas.

    "Addition."

    The green gnome didn't hear the Ancient Tree and continued, "...and one gemstone and one gemstone."

    Old Dismas spoke up, "Did you know that what you are doing has a name?"

    Suddenly the green gnome stopped.  He had been certain that what he was doing was something very new and had been discovered by him alone. 

    "Addition.  What you are doing is called Addition" repeated Old Dismas. "You see how you are adding the gemstones, one gemstone and one gemstone and so on and so forth?  That is called Addition.  How many gemstones do you now have?" asked Old Dismas.

    "Well, let's see.  I...I don't know.  One gemstone and one gemstone and one gemstone..."  The green gnome realized that while he could continue to add one gemstone after another to his long line, he was no closer to knowing how many he had counted.

    Walking over to the green gnome, King Equals once again reached inside his cape and pulled out his magic wand. 

    "I have a gift for you that will help you to answer your question."  The purple king waved his wand and pointed it towards the chest of the little green gnome.  Suddenly a strange symbol appeared. It was made up of two lines, one going up and down and the other going side to side.  And while the little lines went in opposite directions, they crossed exactly in the middle.

    "Whenever you want to find out how many, just remember to use this sign between each number.  And when you have finished, call on me and you will find your answer" finished King Equals.

    The green gnome looked again at his long line of gemstones and this time he traced his new sign in the earth between each gemstone. "One gemstone + one gemstone.  King Equals?"

    The king approached the gnome and answered, "Two gemstones.  One gemstone and one gemstone equals two gemstones."

    The green gnome continued, "Two gemstones + one gemstone.  Equals?" 

    King Equals answered, "Two gemstones and one more gemstone is the same as three gemstones."

    The little green gnome soon understood that the funny little sign joined the gemstones together and made them one.  Not only that, but King Equals' answer told him the number that they made. 

    "Oh!  I get it.  And three gemstones and one gemstone is four gemstones!"  The green gnome was so excited.  He merrily danced around his work.

    "Very good, my friend.  You have understood well.  And because you have figured out this mystery of counting, I give you a new name. From this time forth you will be known in all Numeria as Plus."  King Equals tapped the shoulder of of the happy gnome, bestowing upon him his new name.

    "I guess I've figured it out then.  Addition is the secret of numbers" announced Plus, rising from a ridiculously low bow.

    "Not so fast," replied Old Dismas, chuckling to himself.  "You are one step closer, that is true; but there is still much more to learn.  And until then, I suggest that you and your friends practice using your new symbol, for it will help you in unlocking this great secret."

    The red and yellow gnomes rushed to Plus' side, ready to play the new counting game.  But not the blue gnome.  No, he wasn't interested in the game of Plus.  Instead, the blue gnome quietly tip-toed around to the back of the mountain of gemstones.

    "I have a better way" he laughed to himself as he began filling the pockets of his cape with handfuls of colorful gemstones.

LESSON PLANS

So far we've learned about Roman Numerals, odd and even numbers, place value and the process of addition. At this point the story of the four greedy gnomes stops for a few weeks in order to allow the children an opportunity to absorb the material.  During the next three weeks new ideas on teaching the concept of addition and extension activities will be posted, along with the next three installment of the Form Drawing stories and the next three assignments in our Number Study. 

Each week we will go more in-depth with the concept of addition, but before we continue in this study it is important to make sure that your child understands the purpose and function of adding and making sums.  Explain that the addition symbol shows that we are joining the two things together that are on either side of it and that the symbol for equals simply means "is the same as."  Ask your child to make a page in his Main Lesson Book modeled after the one shown above.  Concentrate this week on adding 0 and 1 to a number.  Depending on the age and ability of your child, you can also make a +1 page in the Main Lesson Book (0+1 through 11+1 is enough).  Allow your child to use the gemstones for practice. Fill one basket with a small amount of gems and a second basket with one.  Ask your child to count the gems together to find the answer. You can illustrate these problems on the blackboard or let your child use Do-A-Dots to illustrate addition with gemstones on paper. 

Add rhythm and movement by playing addition bean bag toss.  Say, "One plus one is..." and toss the bean bag to your child.  The child should respond with the sum, "Two!" and then say, "Two plus one is..."  and toss the bean bag back to you. Continue in this way as long as your child is enjoying the game. (Remember to play games of bean bag toss to practice counting forward and backward from 0-100, odd and even counting, and the colors of the rainbow.)

For extra practice you can design your own worksheets using The Math Worksheet Site's math sheet generator.  Now is a good time to begin enjoying resources such a The Quarter Mile Math or Calculadder.

ADDITION STORY DOWNLOADS

Lesson Four Story Download lesson_four.pdf

Plus Gnome Artwork Download Plus.pdf
 

FORM DRAWING

This week we continue our lessons in Form Drawing by telling the second part of The Story of the Great King Equals. Tell this story to your child as many times as necessary until he can successfully draw the new running form.  By the end of the week, add this form to the Main Lesson Book.  Keep in mind that these forms should be drawn with stick crayons in one continuous stroke. You can download the story with picture here (Download form_drawing_2.pdf

). First review last week's story and then continue with the following:

Looking through the parapets of the great stone castle, the prince saw in the distance an immense range of mountains.  "I will depart from this divided kingdom and travel to the mountains, in search of a wise man who can tell me how to unite my family's kingdom once again" decided the purple prince.  And with much hope, the prince left his home in the great stone castle and set off on his horse over the misty mountains.

The running form in the drawing this week is similar in appearance to a mountain range.  Point this out to your child and then practice drawing the form with him. Encourage your child to use these forms when making page borders in their Main Lesson Book.

NUMBER STUDY

This week we will also continue our study of numbers by considering the number three.  Ask your child to think of things that are three.  St. Patrick's 3-leaf clover that signifies the Three Persons of the Holy Trinity, and a triangle are three examples.  Have your child make a number three page for her Main Lesson Book.

LIVING BOOKS

RAINBOW Last week we used the rainbow as a way to separate place and assign value to numbers.  This week you should review place value.  And with the rainbow in mind, now would be a fun time to learn more about them. Read The Magic School Bus Makes a Rainbow: A Book About Color,  Don Freeman's A Rainbow of My Own or All the Colors of the Rainbow. Spend some time learning about what rainbows are and how they are formed. You can also have fun with color and rainbows.  And you may want to read the story of Noah in a children's Bible too.

Continue spending time reading math-themed living books.  Last week we suggested a few titles.  If you have not yet read them with your child, go back to that list and read through.  Now is a good time to review the things we've learned in the past four weeks and reading the concepts in a living picture book will help reinforce the concepts.

Each week during our four-week addition unit, more theme-related books will be suggested. The following books are this week's suggestions for living books on the topic of addition. Try to read one each day.

Mission Addition

Twelve Ways to Get Eleven

Each Orange Had Eight Slices

Hershey's Kisses Addition Book

Math For All Seasons

Let's Have SUM Fun with PLUS!

The exciting mathematical tale of Gnomes and Gnumbers will continue in January 2008, but that doesn't mean our work is done. Far from it! From now until then we have much work to do with our new friend Plus. Our clever green friend has made an important discovery and for the following four weeks, assuming that you will be breaking for the weeks of both Thanksgiving and Christmas, we offer the following extensions to further the study of the process of Addition.

For the child who is just beginning her study of addition, you will want to spend the next month working with simple addition equations and memorizing the addition tables.  Begin with the +1 table and continue up until +9. The use of manipulatives, whether the gnomes' gemstones or a Base Ten set, is an essential element in presenting an abstract concept in a concrete way.

Older children who already have a solid understanding will benefit from the Whole to Parts exercises listed below and can increase their skills by working on Addition with Regrouping topics using a Base Ten set.  If you do not already own a Base Ten set, a gemstone base ten set for download was offered in Lesson Three of Gnomes and Gnumbers. If you have not already, you may want to download those files and prepare them.

REVIEW

Don't forget to review the concepts we've covered in the first four lessons of Gnomes and Gnumbers.  For Roman Numeral practice, write a Roman Numeral on a piece of paper or blackboard and ask your child to write or say the number that it represents.  On another day give your child a few different numbers and ask her to write them on paper or the blackboard in Roman Numeral form.

Along with games of bean bag toss, review odd and even numbers by filling baskets with different amounts of gemstones and asking your child to sort the gems in groups of twos.  Remind her that if one is left over, the number is odd.  An alternative review activity is to write numbers on paper or the blackboard and ask your child to circle all odd numbers in one color and all even numbers in a different color.  When analyzing larger numbers, be sure to remind your child that the numeral in the units place is the only numeral that we consider when determining whether or not a number is odd or even.

Continue to review the concept of place value by using the rainbow tray.  Make different numbers by placing the numerals within the bands of the rainbow.  Ask your child to identify the place value of the numerals.  For instance, if you demonstrate the number 1, 325 on the rainbow tray, ask your child questions such as "Which number is in the hundred's place?" or "In what place is the three?" You can extend the questions to include, "If a two is in the ten's place, what is the value of the two?"  The answer is two tens or twenty.  Continue the activity from Lesson Three of building numbers with the Base Ten set or gemstones and asking your child to place the correct numerals on the rainbow tray.  You can reverse this activity by making a number on the tray and asking your child to build it with the manipulatives. (Review Lesson Three for more explanation on place value activities.)

ADDITION EXTENSIONS

Addition Strip Board

The Addition Strip Board, a great aid in comprehending and memorizing the addition tables, is a traditional Montessori resource.  You can purchase a wooden board and strips for a discounted price at Alison's Montessori or you can download a free PDF from Montessori Materials. (Click on "Addition Strip Board" under the heading "Addition Material.")  For maximum durability, we suggest printing on card stock and mounting the print-outs on foam board. As beautiful as the traditional set is, this more economical paper alternative is just as effective as using the more expensive wooden set. You will also want to print out this free set of Addition Equation Slips (Click on "Addition Combination Slips.") or purchase the matching wooden set from Alison's.

To introduce the board to a child, ask him to randomly choose an addition equation slip.  In the picture here, the equation "4+3=" was chosen.  The equation was built by lining the blue 4 strip on the first line, followed by the red 3 strip.  With both strips lined up together, it is obvious by checking the number line that the sum is 7.  The number seven tile was placed after the equation and the sum was recorded on graph paper.

Using graph paper for recording equations helps the child line up the numbers in the equation.  (It is even more helpful for advanced topics such as multiple-digit addition or long division.) For free graph paper you can visit Donna Young's site.  There you will find many different sizes of graph paper for download.  For a young child, the larger squares are best.  If you would prefer a bound main lesson book filled with graph paper you can find one at Paper, Scissors, Stone. (Scroll down to find the "Arithmetic Book.")

Spend the next few weeks introducing and memorizing the addition tables +1 through +9.  You can use the equation slips to introduce each table, as well as for daily practice.  For instance, if your child is going to be working on the +4 table, collect the equation slips that express the +4 equations.  Ask your child to build the equations on the strip board.  For further practice, the equations slips can be used as flashcards as your child works on memorizing the addition tables.  And if he can't remember the sum, always ask him to build the equation on the board.

Don't forget to add fun and movement by playing games of addition bean bag toss.  Using a small bean bag, toss the bag to your child as you say "One plus one equals..."  Your child should respond with the sum, "Two," as he tosses it back to you.  Continue up to 1+9.  Use this game as a memory aid for all 9 addition tables. 

Addition with Regrouping

Using a Base Ten set, you can teach your older child how to add larger numbers that require regrouping or carrying. Consider printing multiple-digit addition equation sheets from The Math Worksheet Site or write your own on graph paper.  Use the Base Ten materials to demonstrate the process, reminding your child of the counsel of King Equals, "Only nine of each kind can fit in one band."

Begin with a problem that requires regrouping such as "35+68=" and build each number with the gemstone materials or your Base Ten set.

Pointing
out that we only add together those of like kind and that we always
start with the Units, combine the Units into one pile.

Counting the Units we find that there are 13 cubes. Remind your child of the wise counsel of King Equals, "Only nine of each kind can fit in one band."  Thirteen is more than nine and cannot fit in the red units band of our place value rainbow.  We're going to have to do some trading.  Get out a Ten Bar and line up ten of the thirteen units against it.  Trade then Ten Bar for the ten Units.

Remind your child that we can only add those of like kind, therefore the Ten Bar must be added to the pile of Tens and not the Units.  We now only have three Units and move on to adding the Tens together.

"Only nine of each kind can fit in one band."  Since we have ten we'll have to trade again. Get out a Hundred Flat and trade the ten Ten Bars for one Hundred. 

We've stayed true to King Equal's instruction and now have three Units and one Hundred.  Ask your child to use the numerals and place them in the correct bands of the rainbow. (You can bake these numerals, buy a set of moveable numerals or write them on the blackboard or paper in the different colors of each band.)

This example is one in which there is a zero in the Ten's place.  It is important to point out that when we trade the different values and end up with nothing in one of the places, we need to make sure that we use a zero to hold the place.  Demonstrate with this example that if you didn't use the zero to hold the Ten's Place, the number would no longer be one hundred and three, but thirteen.  The child can clearly see that more than 13 is represented by the blocks.

When recording these problems we recommend using graph paper. When carrying numbers above the equation, it is much easier to keep track of the Tens or Hundreds that we've carried over if we can line them up in the square above the Ten's or Hundred's Place. (Sources for graph paper are listed above.)

Whole to Parts

The following exercise is recommended for all children, even those who have experience working with addition equations.

Rather than simply memorize addition equations and their sums, it is important to study numbers in a holistic way by considering them from the whole to the parts. What is the number 12?  It can be many things, such as 6+6 or 9+3; but 6+6 can only ever be 12, just as 9+3 can only ever be 12.  The focus in this study is to consider what makes up the WHOLE number and for this reason, even a child who has memorized the addition tables can benefit from this exercise.  An understanding of the whole of numbers serves in building a stronger foundation for more advanced work in the future. 

Over the course of the next month spend a separate day on each of the numbers, 1 through 12. Consider each number as a whole and from that starting point, pick out its parts.  There are two ways to have fun with this exercise.  The first is to draw Number Ladders.

By drawing a simple ladder we can imagine that the number we are considering is walking up its rungs.  Each rung represents a different way to make up that number.  Twelve can be eleven and one or it can be five and seven.  The ladder shows the many parts of the number.  Point out the patterns that numbers make.  Going down the left side of the ladder we see that the numbers increase from zero to twelve, while on the right side the numbers decrease from twelve to zero. 

Ask you child to make a number ladder for the numbers one to twelve.  When he has finished this over a series of many days, consider the twelve number ladders together and compare them. For instance, the eight ladder has many more rungs than the two ladder.  Pointing out these patterns and differences transforms an abstract concept into a concrete reality.

A book such as 12 Ways to Get 11 is a great example of expressing the parts of numbers in pictoral form.  Make a number picture for each of the twelve numbers.  Perhaps seven is a scene of a house with one door, two windows, two trees, one sun, and one chimney. (1+2+2+1+1=7) If your child enjoys this activity, consider using the blackboard for repeated play or maybe he'll want to make his own number picture book.

Number Study: Four

This week we will continue our study of numbers by considering the number four.  Ask your child to think of things that are four.  The four inspired authors of the Gospels, Saints Matthew, Mark, Luke and John;  the four cardinal directions, north, south, east and west; and the four sides of a square or rectangle are all possible examples.  Have your child make a Number Four page for his Main Lesson Book.

Making Numbers

Shaping the numerals out of modeling clay is a beneficial tactile experience for all learners, but especially for those with learning challenges such as dyslexia.  Later this week Elizabeth will be posting about modeling the numbers as well as modeling representations of number sentences. 

When considering an equation, such as four plus two, it is helpful for a child to model the numeral four, an addition symbol, the numeral two and an equals symbol and answer the question by rolling out small balls to demonstrate the equation.  Four small balls grouped with two small clay balls adds up to six.  The equation is visually represented and the child has been offered the experience of forming the numerals with her own hands.  Just as the exercise of Number Study gives the child an opportunity to contemplate the reality of the number in the created world, so too the exercise of making numbers offers the actual experience of forming the individual numbers and feeling its shape.

Encourage an older child to participate in this activity by pairing him up with a younger sibling.  In teaching and directing a younger child, the older child not only increases his understanding, but demonstrates it. 

Form Drawing

Continue the lessons in Form Drawing by reading the next installment in the tale "The Story of the Great King Equals."  Who is King Equals? Where did he come from?  And how did he get that funny symbol?  The answers to these questions and more will be discovered in this tale. After reading this week's portion of the story, draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon she will learn the movement and pattern and be able to work independently. Encourage your child to draw this form using a stick crayon with a continuous and uninterrupted motion.   

Descending toward the base of the misty mountains, the Purple Prince looked out upon the vast land.  As far as the eye could see were rolling hills covering the land.  Spurring his loyal horse to continue on their journey, with great difficulty the purple prince rode over large hills. He was greatly relieved to discover a stretch of small hills.  At last he came to a series of medium-sized hills.  Finishing this leg of the journey and dismounting his horse, the purple prince was dismayed at what lay before him.  Would he ever find a wise man to help him unite his kingdom?

Download story with sample of running form here. (Download form_drawing_three.pdf

)

Living Books

One Monkey Too Many (Adding 1, Counting)

12 Ways to Get 11 (Whole to Parts, Addition)

So Many Cats! (Counting to 12)

Anno's Counting Book (Counting)

 

 

In which EVEN gnomes make an ODD discovery.

    The yellow gnome held his eyes closed tightly as the woody hand lifted him up, high above the treetops.  Suddenly everything was still and quiet except for the steady sound of a groaning wind that was blowing his pointed hat directly behind his head.  Mustering up enough courage to open just one eye, the gnome saw before him a tree that looked like a man.  Or was it a man who looked like a tree?  As these questions danced in his head, the Tree Man spoke,

    “You have taken that which does not belong to you” he groaned in his deep, woodsy voice. 

    “I…I’m...I’m sure I don’t know what you mean, thank you,” replied the gnome, hoping that being polite would help matters.

    The gnome was completely confused by the Tree Man’s strange words and, wondering what had happened to his three friends far below, he ignored the woeful words of the Tree Man and begged to be returned to the ground.

    “Please, put me down on the ground with my friends.  I am a gnome, an earth-dweller, and it is not natural for me to be suspended at such great heights.”

    With that the gnome heard the familiar groan that had sounded when the branch had been snapped from the tree. 

    “Huuuummmphh!  And neither is it natural for my ancient branches to be broken into bits” replied the Tree Man.

    With these words he lowered his enormous wooden hand and soon the gnome was safely back on the ground with his three companions. Suddenly the Tree Man took a step forward into a clearing and the four gnomes could plainly see his enormous stature.

    Before they had a chance to speak, the Tree Man addressed them,

    “I am Old Dismas.  I have lived in these woods since it came to be.  When the earth was still young, a wizard cast an enchantment on my wood,

Whosoever shall harm and tear thee asunder, shall forever count until he hath discovered the secret of numbers.

  With those words, Old Dismas left the gnomes in the clearing.  The earth groaned under the weight of his steps and the quaking sound of the forest floor echoed throughout the woods. 

    "What are we to do?  Oh dear!  Oh me!  We've  brought a terrible enchantment upon ourselves!" cried the blue gnome.

    "It sounds as if we will have to find something to count.  The Ancient Tree said that the enchantment will not break until we discover the secret of numbers.  What shall we count?"

    The gnomes confusedly looked around until they again spotted the mountain of gemstones.

    "The gemstones!  That's what we shall count" shouted the red gnome, with much excitement.

    The four gnomes approached the huge mountain of gemstones and noticed that there were small baskets scattered around the ground. Each of them grabbed one and collected as many gems as he could fit in his basket. 

    After their baskets were filled to the brim, the green gnome asked, "Now what do we do?"

    "How about we break a branch off that tree over there and use the sticks to count our gemstones" replied the blue gnome, with a look of enthusiasm.

    "No!" shouted the other three. "Don't you remember?  That's what got us in this mess in the first place!"

    The four gnomes didn't know what to do, so they emptied their baskets of gemstones and began spreading them on the ground in order to count.  The blue and green gnomes decided to sort their gemstones into pairs of two, but soon the blue gnome became frustrated.

    "Well now, this is rather ODD.  Every time I try to sort my gems in pairs, I always find one left over."

    The green gnome continued his work of sorting and soon matched each gemstone perfectly in pairs.  Upon seeing his success, the blue gnome grumbled,

    "Now, that is not EVEN fair. You do not have any left over.  How is it that when I match my gemstones in pairs I always have one left over and you are able to match every last one?"

   Soon the crunching and snapping of twigs could be heard again beneath the feet of Old Dismas.  As he was approaching the clearing, he overheard the debate and offered an explanation.

     "Yes, it is ODD when a group has one left over and for this reason we call those numbers ODD. How many gemstones have you counted?" asked Old Dismas.

    "Eleven" replied the blue gnome.  "Is eleven an ODD number?"

    "Yes" answered the Ancient Tree.  "And how many gemstones have you counted?" he then asked the green gnome.

    "Twelve.  Twelve gemstones" answered the gnome.  "Twelve is an EVEN number then because all of my gems have a partner.  That's it! I've discovered the secret of numbers!"  The silly green gnome began dancing around his sorted gemstones.

    "Not so fast," laughed Old Dismas. "The secret is far deeper than the discovery of ODD and EVEN.  But don't loose hope.  There is someone here to help you."

    And suddenly a King dressed in purple was standing before them.

To Be Continued...

LESSON PLANS

This lesson continues the story from Lesson 1 and introduces the concept of odd and even numbers.  If your child has not yet mastered the material presented in Lesson 1, wait before continuing with Lesson 2.  Regardless, remember to practice and review Roman Numerals throughout the week.  If your child is interested, introduce further Roman Numerals such as L equals 50 and C equals 100.  Use your child's work in the Main Lesson Book as a springboard for review.  By creating a Main Lesson Book, your child is writing and illustrating his own Math book!

If you are continuing with a 5 to 6 year old, rely heavily on the manipulatives (the gemstones) to teach this concept.  Talk to your child about the things around her that naturally exist in pairs:  two eyes, two ears, two feet, two socks or shoes.  Then proceed to group the gemstones in pairs of two.  Fill up the miniature baskets with gemstones (keep it under 10 for a young child) and ask him to sort them in pairs.  If one is left over you can make a game of saying, like the blue gnome, "Well, isn't this ODD!"  Point out that when all the gemstones have a partner, they are all EVEN. Make an Odd Number and Even Number page in your child's Main Lesson Book modeled after the sample page in the story.  Draw with your child if he is having trouble expressing these concepts in writing and allow your child to spend a few days on his page if necessary. Young children learn best by imitation.

If you're working with a 7 to 8 year old, repeat the above activity and review it each day.  Create and add the Even and Odd Number pages in the Main Lesson Book. Encourage your child to decorate his page with a border that reflects the story or concept learned.  Remember, these pages are to be approached as works of art. When your child has grasped the concepts of even and odd, continue the work by using the blackboard. Write a random number and ask your child to tell you if it is even or odd.  Allow her to use the gemstones to find the answer.

Continue to combine movement with your child's learning.  Games of bean bag toss are a fun way to practice numbers.  This week, play games of Even and Odd toss.  As you toss the bean bag to your child say, "One!" and as your child catches it, he responds, "Three!"  Continue through the odd numbers as he is able and then try the even numbers. Weather permitting, take your lessons outside.  Use stones or acorns and sidewalk chalk to make number pictures on the ground.

Beginning this week we are going to concentrate on a different number each week.  Don't skip this activity, even if your child knows his numbers well and can work basic math equations. We're going to be considering the concrete reality behind the very abstract idea of numbers.  This week we'll begin with Number 1. 

Consider the world and  its Creator.  What exists that expresses oneness? Some answers could be one God, one sun, one moon, one nose, etc... (Remember these answers are taken from the experience of a young child.  There is no need to correct him and point out that the sun is a star and there are many in the universe, etc...)  Ask your child to make a Number 1 page in his Main Lesson Book.

GNOMES AND GNUMBERS STORYBOOK

It is highly encouraged that you print out these math stories and artwork and read-aloud to your child in a comfortable setting and for this reason we are offering them in PDF format for free download.  For the story files, the larger font size was chosen so that the child (or an older sibling) can go back and read the stories on his own.  Keep them in a 3-ring binder, each page stored in it's own page protector so the child can flip through without worrying about ripping pages. Include the artwork throughout the story and soon your child will have a storybook of his own.

You can make this storybook as simple or detailed as you wish.  The cover page shown here can be printed for use in a binder or perhaps you will want to get creative and make a felt relief on an album using ideas from a book such as Felt Wee Folk.

Gnomes and Gnumbers Cover Art
Download gnomes_cover_page.pdf

Gnomes and Gnumbers: A Mathematical Tale (Story One)
Download gnomes_and_gnumbers_lesson_one.pdf

Gnomes and Gnumbers:  In which EVEN gnomes make an ODD discovery. (Story Two)
Download gnomes_and_gnumbers_lesson_two.PDF

Old Dismas and the Yellow Gnome-Art (Lesson Two)
Download old_dismas_and_yellow_gnome_art.PDF

Old Dismas-Art (Lesson Two)
Download old_dismas_art.PDF

Odd Numbers-Main Lesson Book Sample
Download odd_numbers_mlb.PDF

And for fun you can download a coloring page of Old Dismas and the Yellow Gnome.
Download old_dismas_coloring_page.pdf

About the name Old Dismas:  This name was chosen in honor of the repentant thief on the Cross, St. Dismas. The character in this tale not only exhibits wisdom to those who struggle with the passion of greed, but offers it with patience and love.  For this reason it seemed appropriate for the wise Tree Man of our story to share a name with the grace-filled man who from a tree offered us the timeless example, "Lord, remember me when you come into your Kingdom" (Luke 23:42).

Like the thief will I confess thee: Remember me, O Lord, in thy Kingdom.
-Prayer Before Communion, St. John Chrysostom

Gnomes and Gnumbers: Lesson Three

In which the gnomes learn the VALUE of their PLACE.

   By this time the clearing was littered with baskets thrown here and there.  Some were filled with gemstones and others had been toppled over.  The mountain of gemstones was still high above the gnomes, but gems were now scattered everywhere, some in even numbered piles and some mismatched in odd groupings.  The gnomes knew that they must continue counting in order to find the secret of numbers, but they were getting more and more frustrated the longer they tried to count.  And while they were happy that Old Dismas had explained the difference between odd and even numbers, they still knew that there were far more gemstones than they were able to count on their own.

    The gnomes were surprised when they suddenly saw a purple-clad king come out from behind the great trunk legs of Old Dismas. 

    "There is someone here for you to meet," the Ancient Tree announced to the gnomes. 

    The king stepped out in the clearing. He was an odd looking fellow, almost as wide as he was tall. He was dressed in purple robes that seemed to reflect every color imaginable and, instead of a crown, he wore a pointed cap trimmed in gold.

    Careful not to trip over the gemstones and baskets, he began, 

    "I am King Equals, King of all Numeria.  It was my ancestor who cast the enchantment on the Ancient Tree long, long ago...the enchantment that you have brought upon yourselves because of your greed.  And he who cast the enchantment was a great wizard king, one who knew the secret of numbers that you so eagerly seek to discover."

    The gnomes didn't know what to say.  Who was this King Equals and what did he want from them?  Finally the red gnome replied,

    "We know the enchantment.  Old Dismas explained it to us.  And we know we must count until we discover the secret of numbers.  But these gemstones are far too many for us and when we count we soon become confused.  Can you help us?"

    King Equals nodded his head and pulled a magic wand out from beneath his purple cape. 

    "You're a wizard too?" gasped the green gnome. 

    "Well, of course.  And I can offer you a very simple way to solve your problem.  If you want to find the solution to keeping track of the numbers you count, you must repeat this verse with me,

   Regal old yellow gnomes buy itchy vests.

Simply repeat this verse and the solution will present itself to you."

     The gnomes were convinced that King Equals was a fool, but they soon decided that since they had no better plan there was no harm in trying the verse, as silly as it seemed.  Together they began repeating the strange words,

    "Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests..."

    Suddenly a radiant light broke through the tops of the trees and shone upon the mountain of gemstones.  And then there was a most beautiful sight.  The reflection of the gemstones created a brilliant rainbow and a a prism of color filled the sky.  Red, orange,  yellow, green, blue,  indigo, and violet. 

    "Use these bands of color to help you keep track of your numbers as they grow. But remember, only nine of each kind can fit in one band" instructed King Equals.

    Still awestruck by the exquisite color surrounding them, the gnomes were confused by the words, "Only nine of each kind can fit in one band."  And anticipating their confusion, the King began to demonstrate.

    "Do you see this first band of red?  We call this band Units.  Each gemstone represents one unit.  And we can fill the red band with nine of them."

    King Equals picked up a basket and filled it with 9 sparkling gemstones and carefully laid them in the red band of color.

    "But there are many more than nine gemstones, Your Majesty.  How can this possibly help us?  We have a mountain of gemstones to count" remarked the confused green gnome.

    "Ah, but don't you see these other bands?  There are many places within which you may count.  When we fill up the red Units we move on to the orange band. This band is called Tens."

    "Oh! I know what ten is.  We counted ten and made up a symbol for it in the shape of an X using branches from...Oh.  Sorry Dismas. I'm sure it was not pleasant to have your branches broken off" admitted the yellow gnome with great regret.

    The Ancient Tree lowered his heavy head and offered a gentle smile.

    King Equals continued his explanation, "After 9 gemstones fill the Units band we must continue by grouping them together in families of ten." 

    As he said this, the King carefully lined up the gemstones in groups of ten and placed them in the orange band of color.

    "Now remember, only nine of each kind can fit in one band.  So once we have nine groups of ten in the Tens place, we must move on to the yellow band of color. We call this the Hundreds."

     Beginning to understand the sorting game, the gnomes began gathering gemstones once again in their baskets.  When they filled the Hundreds with nine groups of one hundred gemstones, they moved on to the green. King Equals went on to explain that the green band was called Thousands and the numbers continued far past that.  There were blue Ten Thousands, indigo Hundred Thousands and a place in the violet band for a number called Millions. The gnomes continued their work, and stopped only to notice that King Equals had opened his purple cape to reveal a funny little symbol made up of two lines on his chest. 

    "Excuse me, Your Majesty, but what is that symbol you wear on your chest?  Is it some sort of royal family crest?" asked the red gnome.

    "Oh, it is much more than that.  Much more.  But that story if for another time.  For now, continue your counting and fill this rainbow with gemstones.  Soon you will learn the purpose of the symbol I wear upon my chest" answered the King.

    The gnomes whispered among themselves, "Well, I sure hope that mysterious symbol can help us discover the secret of numbers!"

    Of course, you know that Ancient Trees can hear the softest, most faintest sound in the forest.  And hearing the musings of the silly gnomes, Old Dismas let out a laugh that shook the forest floor.

LESSON PLANS

Supplies

There is no need to move on to this lesson until the concepts introduced in the previous two lessons have been mastered.  Move at your child's pace. The Gnomes and Gnumbers lessons are archived in chronological order on this page for your convenience.

You will need to assemble a Rainbow Tray for this lesson, as well as collect a set of numerals.  If you already have a set of moveable numerals like the ones shown in the picture above, use them.  If not, and purchasing a set is not in your budget, spend one afternoon baking a set with your child.  The easy salt dough recipe is here and a set of cutters can be purchased. Another option is to model the numerals with modeling beeswax (Suppliers are linked on the right sidebar.) or Sculpey Clay.

If you already own a Base Ten set, you can make use of it for place value work.  If not, you can download the Ten Bar and Hundred Flat files below.  These were made using Do-A-Dot markers on graph paper. Donna Young's site offers graph paper files for download if your child would like to make a set of base ten gemstones.  The graph paper used here is the 10 x 14 size. 

Downloads

You can download the story and artwork by clicking on the links below. 

Lesson Three Story

Download gnomes_and_gnumbers.Lesson 3.pdf

King Equals Picture

Download king_equals_art.PDF

King Equals Coloring Page

Download king_equals_coloring_page.pdf

Main Lesson Book Sample, Rainbow

Download Rainbow.pdf

Gemstone Ten Bars (Print out 2 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_ten_bars.pdf

Gemstone Hundred Flat (Print out 9 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_hundred_flat.PDF

Presenting the Rainbow

We have chosen to present the rainbow in its true order of colors rather than the simplified version used in many children's songs and books.  Red, Orange, Yellow, Green, Blue, Indigo, and Violet.  Have fun with the mnemonic device created by Marybeth Foss, Regal old yellow gnomes buy itchy vests, for memorizing the colors of the rainbow and their order. 

After reading the story and presenting this lesson on place value, have your child use the gemstones to practice identifying the place value of each numeral in a larger number.  Depending on the age and ability of the child, you will want to go as high as he or she is able.  A younger child may only understand up to the hundreds place, while an older child will enjoy discovering a million.

Ask your child to illustrate a place value page in the Main Lesson Book.  The sample rainbow page offered above can be used as a model for your child. 

Place Value Activities

There are three place value activities that you can play this week using the Rainbow Tray and gemstones.  First start by filling a small basket with gemstones.  For instance, if your basket is filled with 23 gemstones, your child will count nine gemstones and, reminding him that only nine of each kind can be placed in each band, will then cross over to the Tens color band.  The remaining 20 should be grouped in two sets of ten gemstones. Then show your child how to swap out 10 gemstones for a ten bar. Continue in this way until your child understands the concept.

Next, make a 2 or 3-digit number (or larger for an older child) using the moveable numerals. For instance, arrange the numerals so that they show 45.  Ask you child to build this number with the gemstone materials or a Base 10 set next to the Rainbow Tray and then place the numerals in the correct strips of color on the tray.

Another fun activity is to sort gemstones in different sizes of cups.  Begin with 10 small bathroom-sized cups and ask your child to fill each with 10 gemstones.  When ten small cups are filled, they can be poured into a larger 8 ounce cup, representing 100 (pictured here).  Ten of these 8-ounce cups can then be poured into a large 32-ounce cup to represent 1,000.  This tactile experience gives the child the opportunity to see and feel these numbers.  The goal is to bring the abstract into the realm of the concrete.

Fun and Movement

Add movement by playing games of rainbow bean bag toss.  Say "Red" and toss your child a bean bag.  He should respond by saying, "Orange" and then toss the bag back to you. Continue in order to memorize the sequence of the colors in the rainbow.

Review

Don't forget to review the concepts covered in the previous lessons. Together read your child's entries in the Main Lesson Book and discuss each page.  Practice Roman Numerals at least one day this week and use the bean bags to play a game of toss, using odd and even counting as the rhythm.   

If you feel that your child would benefit from math sheet review, there is a wonderful math worksheet generator online called The Math Worksheet Site. You can get a subscription for only $2.50 per month. (The price is reduced depending on the length of your subscription.)  The subscription gives you unlimited use of the site and allows you to create custom math worksheets for your child.  The site is very easy to use. Consider creating sheets on Roman Numerals, odd and even counting, and place value for your child.

Number Study

This week we will continue our study of numbers by concentrating on the Number Two.  Ask your child to consider the world around him and find thing that exist in pairs of two.  Possible responses might be two eyes, two ears, two lights (the sun and moon), two feet, etc...

Gemstone Fun for the Little Ones

Nicholas Foss came up with a fun activity for the little ones.  Using rainbow-colored craft sticks, he sorted the different colored gemstones on each stick and made the pattern of a rainbow.  Since these were not glued down, a young child can repeat this activity many times.

Form Drawing

This week we are adding the element of form drawing.  This exercise in creating rhythmic patterns prepares a young child for handwriting and improves handwriting for the older child.  When presenting these running forms, allow your child to first practice in a shallow tray of sand or cornmeal.  Tracing the form with a finger in the air also helps the child in learning to imitate the pattern.  Your child should use stick crayons when drawing the forms.  Later in this main lesson we will be painting the geometric shapes and practice in form drawing will help in preparing the child for this activity. 

One running form will be introduced each week for a period of nine weeks and we will be using the art of story telling to capture the attention and interest.  Read the following story to your child as an introduction to the first running form.  If your child has not had much practice in drawing, it may be a good idea to spend some time practicing standing forms such as squares, triangles, circles, rectangles and stars. Draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon they will learn the movement and pattern and be able to work independently.

Our form drawing tale will be centered on the story of King Equals and how he brought peace to the Kingdom of Numeria.  Read this story to your child and practice drawing the form before making a main lesson book page (a sample page is shown here).  Perhaps after practicing the form each day, the child can draw it in the Main Lesson Book on Friday. Encourage your child to draw this form with a continuous and uninterrupted movement.  Have fun with the colors.  You can use these forms to practice memorizing the order of the colors of the rainbow too.

The Story of the Great King Equals

Once upon a time there was a sad prince who lived in a great stone castle in a divided kingdom.  For many years his ancestors ruled the kingdom in peace; but one day, in the furthest part of the kingdom, a great treasure was discovered in hidden cave.  Ever since that day the people of the kingdom argued and quarreled over the question of who was the rightful owner of the cave and the great treasure.  As the prince looked out through the parapets of the great stone castle, he longed for the day when his family's great kingdom would be one again.

At this point show your child the sample page of the running form pictured above. (Download great_king_equals.Story 1.PDF

) These lines resemble the parapet of
a castle wall.  Practice drawing this form each day, retelling the
story each time.  By the end of the week your child should be able to
narrate the story himself as he works on mastering this form.   

Living Books

Now is a good time to add a few mathematically themed read-alouds. Living math goes hand-in-hand with Living Books. Here are a few titles for this week:

How Much is a Million

Millions of Cats

One Hundred is a Family

Roman Numerals I to MM

If You Made a Million

Gnomes and Gnumbers: Lesson Four

In which the green gnome learns that counting is a PLUS.

     Having learned that with the help of a beautiful rainbow it was easy to recognize even the largest of numbers, the four gnomes went back to their work of trying to unlock the secret of numbers. 

    "We now know that some numbers are odd, while others are even," recalled the yellow gnome. "And we also know that no matter how large a number is, we can figure it out with the help of the bands of the rainbow.  But the enchantment has not yet been broken.  We obviously haven't figured out the secret of numbers.  What else could there be?"

    As soon as the yellow gnome finished speaking, the green gnome walked to the base of the mountain of gemstones, picked up the largest basket he could find and filled it to the top.   Then taking the gemstones out, one by one, he lined them on the ground and began,

    "One gemstone and one gemstone and one gemstone..."  he continued for quite some time.

    Curious and confused, the other three gnomes sat down and watched their green friend continue.  Soon the silence was broken by the woodsy "Huuuuumph!" of Old Dismas.

    "Addition."

    The green gnome didn't hear the Ancient Tree and continued, "...and one gemstone and one gemstone."

    Old Dismas spoke up, "Did you know that what you are doing has a name?"

    Suddenly the green gnome stopped.  He had been certain that what he was doing was something very new and had been discovered by him alone. 

    "Addition.  What you are doing is called Addition" repeated Old Dismas. "You see how you are adding the gemstones, one gemstone and one gemstone and so on and so forth?  That is called Addition.  How many gemstones do you now have?" asked Old Dismas.

    "Well, let's see.  I...I don't know.  One gemstone and one gemstone and one gemstone..."  The green gnome realized that while he could continue to add one gemstone after another to his long line, he was no closer to knowing how many he had counted.

    Walking over to the green gnome, King Equals once again reached inside his cape and pulled out his magic wand. 

    "I have a gift for you that will help you to answer your question."  The purple king waved his wand and pointed it towards the chest of the little green gnome.  Suddenly a strange symbol appeared. It was made up of two lines, one going up and down and the other going side to side.  And while the little lines went in opposite directions, they crossed exactly in the middle.

    "Whenever you want to find out how many, just remember to use this sign between each number.  And when you have finished, call on me and you will find your answer" finished King Equals.

    The green gnome looked again at his long line of gemstones and this time he traced his new sign in the earth between each gemstone. "One gemstone + one gemstone.  King Equals?"

    The king approached the gnome and answered, "Two gemstones.  One gemstone and one gemstone equals two gemstones."

    The green gnome continued, "Two gemstones + one gemstone.  Equals?" 

    King Equals answered, "Two gemstones and one more gemstone is the same as three gemstones."

    The little green gnome soon understood that the funny little sign joined the gemstones together and made them one.  Not only that, but King Equals' answer told him the number that they made. 

    "Oh!  I get it.  And three gemstones and one gemstone is four gemstones!"  The green gnome was so excited.  He merrily danced around his work.

    "Very good, my friend.  You have understood well.  And because you have figured out this mystery of counting, I give you a new name. From this time forth you will be known in all Numeria as Plus."  King Equals tapped the shoulder of of the happy gnome, bestowing upon him his new name.

    "I guess I've figured it out then.  Addition is the secret of numbers" announced Plus, rising from a ridiculously low bow.

    "Not so fast," replied Old Dismas, chuckling to himself.  "You are one step closer, that is true; but there is still much more to learn.  And until then, I suggest that you and your friends practice using your new symbol, for it will help you in unlocking this great secret."

    The red and yellow gnomes rushed to Plus' side, ready to play the new counting game.  But not the blue gnome.  No, he wasn't interested in the game of Plus.  Instead, the blue gnome quietly tip-toed around to the back of the mountain of gemstones.

    "I have a better way" he laughed to himself as he began filling the pockets of his cape with handfuls of colorful gemstones.

LESSON PLANS

So far we've learned about Roman Numerals, odd and even numbers, place value and the process of addition. At this point the story of the four greedy gnomes stops for a few weeks in order to allow the children an opportunity to absorb the material.  During the next three weeks new ideas on teaching the concept of addition and extension activities will be posted, along with the next three installment of the Form Drawing stories and the next three assignments in our Number Study. 

Each week we will go more in-depth with the concept of addition, but before we continue in this study it is important to make sure that your child understands the purpose and function of adding and making sums.  Explain that the addition symbol shows that we are joining the two things together that are on either side of it and that the symbol for equals simply means "is the same as."  Ask your child to make a page in his Main Lesson Book modeled after the one shown above.  Concentrate this week on adding 0 and 1 to a number.  Depending on the age and ability of your child, you can also make a +1 page in the Main Lesson Book (0+1 through 11+1 is enough).  Allow your child to use the gemstones for practice. Fill one basket with a small amount of gems and a second basket with one.  Ask your child to count the gems together to find the answer. You can illustrate these problems on the blackboard or let your child use Do-A-Dots to illustrate addition with gemstones on paper. 

Add rhythm and movement by playing addition bean bag toss.  Say, "One plus one is..." and toss the bean bag to your child.  The child should respond with the sum, "Two!" and then say, "Two plus one is..."  and toss the bean bag back to you. Continue in this way as long as your child is enjoying the game. (Remember to play games of bean bag toss to practice counting forward and backward from 0-100, odd and even counting, and the colors of the rainbow.)

For extra practice you can design your own worksheets using The Math Worksheet Site's math sheet generator.  Now is a good time to begin enjoying resources such a The Quarter Mile Math or Calculadder.

ADDITION STORY DOWNLOADS

Lesson Four Story Download lesson_four.pdf

Plus Gnome Artwork Download Plus.pdf
 

FORM DRAWING

This week we continue our lessons in Form Drawing by telling the second part of The Story of the Great King Equals. Tell this story to your child as many times as necessary until he can successfully draw the new running form.  By the end of the week, add this form to the Main Lesson Book.  Keep in mind that these forms should be drawn with stick crayons in one continuous stroke. You can download the story with picture here (Download form_drawing_2.pdf

). First review last week's story and then continue with the following:

Looking through the parapets of the great stone castle, the prince saw in the distance an immense range of mountains.  "I will depart from this divided kingdom and travel to the mountains, in search of a wise man who can tell me how to unite my family's kingdom once again" decided the purple prince.  And with much hope, the prince left his home in the great stone castle and set off on his horse over the misty mountains.

The running form in the drawing this week is similar in appearance to a mountain range.  Point this out to your child and then practice drawing the form with him. Encourage your child to use these forms when making page borders in their Main Lesson Book.

NUMBER STUDY

This week we will also continue our study of numbers by considering the number three.  Ask your child to think of things that are three.  St. Patrick's 3-leaf clover that signifies the Three Persons of the Holy Trinity, and a triangle are three examples.  Have your child make a number three page for her Main Lesson Book.

LIVING BOOKS

RAINBOW Last week we used the rainbow as a way to separate place and assign value to numbers.  This week you should review place value.  And with the rainbow in mind, now would be a fun time to learn more about them. Read The Magic School Bus Makes a Rainbow: A Book About Color,  Don Freeman's A Rainbow of My Own or All the Colors of the Rainbow. Spend some time learning about what rainbows are and how they are formed. You can also have fun with color and rainbows.  And you may want to read the story of Noah in a children's Bible too.

Continue spending time reading math-themed living books.  Last week we suggested a few titles.  If you have not yet read them with your child, go back to that list and read through.  Now is a good time to review the things we've learned in the past four weeks and reading the concepts in a living picture book will help reinforce the concepts.

Each week during our four-week addition unit, more theme-related books will be suggested. The following books are this week's suggestions for living books on the topic of addition. Try to read one each day.

Mission Addition

Twelve Ways to Get Eleven

Each Orange Had Eight Slices

Hershey's Kisses Addition Book

Math For All Seasons

Let's Have SUM Fun with PLUS!

The exciting mathematical tale of Gnomes and Gnumbers will continue in January 2008, but that doesn't mean our work is done. Far from it! From now until then we have much work to do with our new friend Plus. Our clever green friend has made an important discovery and for the following four weeks, assuming that you will be breaking for the weeks of both Thanksgiving and Christmas, we offer the following extensions to further the study of the process of Addition.

For the child who is just beginning her study of addition, you will want to spend the next month working with simple addition equations and memorizing the addition tables.  Begin with the +1 table and continue up until +9. The use of manipulatives, whether the gnomes' gemstones or a Base Ten set, is an essential element in presenting an abstract concept in a concrete way.

Older children who already have a solid understanding will benefit from the Whole to Parts exercises listed below and can increase their skills by working on Addition with Regrouping topics using a Base Ten set.  If you do not already own a Base Ten set, a gemstone base ten set for download was offered in Lesson Three of Gnomes and Gnumbers. If you have not already, you may want to download those files and prepare them.

REVIEW

Don't forget to review the concepts we've covered in the first four lessons of Gnomes and Gnumbers.  For Roman Numeral practice, write a Roman Numeral on a piece of paper or blackboard and ask your child to write or say the number that it represents.  On another day give your child a few different numbers and ask her to write them on paper or the blackboard in Roman Numeral form.

Along with games of bean bag toss, review odd and even numbers by filling baskets with different amounts of gemstones and asking your child to sort the gems in groups of twos.  Remind her that if one is left over, the number is odd.  An alternative review activity is to write numbers on paper or the blackboard and ask your child to circle all odd numbers in one color and all even numbers in a different color.  When analyzing larger numbers, be sure to remind your child that the numeral in the units place is the only numeral that we consider when determining whether or not a number is odd or even.

Continue to review the concept of place value by using the rainbow tray.  Make different numbers by placing the numerals within the bands of the rainbow.  Ask your child to identify the place value of the numerals.  For instance, if you demonstrate the number 1, 325 on the rainbow tray, ask your child questions such as "Which number is in the hundred's place?" or "In what place is the three?" You can extend the questions to include, "If a two is in the ten's place, what is the value of the two?"  The answer is two tens or twenty.  Continue the activity from Lesson Three of building numbers with the Base Ten set or gemstones and asking your child to place the correct numerals on the rainbow tray.  You can reverse this activity by making a number on the tray and asking your child to build it with the manipulatives. (Review Lesson Three for more explanation on place value activities.)

ADDITION EXTENSIONS

Addition Strip Board

The Addition Strip Board, a great aid in comprehending and memorizing the addition tables, is a traditional Montessori resource.  You can purchase a wooden board and strips for a discounted price at Alison's Montessori or you can download a free PDF from Montessori Materials. (Click on "Addition Strip Board" under the heading "Addition Material.")  For maximum durability, we suggest printing on card stock and mounting the print-outs on foam board. As beautiful as the traditional set is, this more economical paper alternative is just as effective as using the more expensive wooden set. You will also want to print out this free set of Addition Equation Slips (Click on "Addition Combination Slips.") or purchase the matching wooden set from Alison's.

To introduce the board to a child, ask him to randomly choose an addition equation slip.  In the picture here, the equation "4+3=" was chosen.  The equation was built by lining the blue 4 strip on the first line, followed by the red 3 strip.  With both strips lined up together, it is obvious by checking the number line that the sum is 7.  The number seven tile was placed after the equation and the sum was recorded on graph paper.

Using graph paper for recording equations helps the child line up the numbers in the equation.  (It is even more helpful for advanced topics such as multiple-digit addition or long division.) For free graph paper you can visit Donna Young's site.  There you will find many different sizes of graph paper for download.  For a young child, the larger squares are best.  If you would prefer a bound main lesson book filled with graph paper you can find one at Paper, Scissors, Stone. (Scroll down to find the "Arithmetic Book.")

Spend the next few weeks introducing and memorizing the addition tables +1 through +9.  You can use the equation slips to introduce each table, as well as for daily practice.  For instance, if your child is going to be working on the +4 table, collect the equation slips that express the +4 equations.  Ask your child to build the equations on the strip board.  For further practice, the equations slips can be used as flashcards as your child works on memorizing the addition tables.  And if he can't remember the sum, always ask him to build the equation on the board.

Don't forget to add fun and movement by playing games of addition bean bag toss.  Using a small bean bag, toss the bag to your child as you say "One plus one equals..."  Your child should respond with the sum, "Two," as he tosses it back to you.  Continue up to 1+9.  Use this game as a memory aid for all 9 addition tables. 

Addition with Regrouping

Using a Base Ten set, you can teach your older child how to add larger numbers that require regrouping or carrying. Consider printing multiple-digit addition equation sheets from The Math Worksheet Site or write your own on graph paper.  Use the Base Ten materials to demonstrate the process, reminding your child of the counsel of King Equals, "Only nine of each kind can fit in one band."

Begin with a problem that requires regrouping such as "35+68=" and build each number with the gemstone materials or your Base Ten set.

Pointing
out that we only add together those of like kind and that we always
start with the Units, combine the Units into one pile.

Counting the Units we find that there are 13 cubes. Remind your child of the wise counsel of King Equals, "Only nine of each kind can fit in one band."  Thirteen is more than nine and cannot fit in the red units band of our place value rainbow.  We're going to have to do some trading.  Get out a Ten Bar and line up ten of the thirteen units against it.  Trade then Ten Bar for the ten Units.

Remind your child that we can only add those of like kind, therefore the Ten Bar must be added to the pile of Tens and not the Units.  We now only have three Units and move on to adding the Tens together.

"Only nine of each kind can fit in one band."  Since we have ten we'll have to trade again. Get out a Hundred Flat and trade the ten Ten Bars for one Hundred. 

We've stayed true to King Equal's instruction and now have three Units and one Hundred.  Ask your child to use the numerals and place them in the correct bands of the rainbow. (You can bake these numerals, buy a set of moveable numerals or write them on the blackboard or paper in the different colors of each band.)

This example is one in which there is a zero in the Ten's place.  It is important to point out that when we trade the different values and end up with nothing in one of the places, we need to make sure that we use a zero to hold the place.  Demonstrate with this example that if you didn't use the zero to hold the Ten's Place, the number would no longer be one hundred and three, but thirteen.  The child can clearly see that more than 13 is represented by the blocks.

When recording these problems we recommend using graph paper. When carrying numbers above the equation, it is much easier to keep track of the Tens or Hundreds that we've carried over if we can line them up in the square above the Ten's or Hundred's Place. (Sources for graph paper are listed above.)

Whole to Parts

The following exercise is recommended for all children, even those who have experience working with addition equations.

Rather than simply memorize addition equations and their sums, it is important to study numbers in a holistic way by considering them from the whole to the parts. What is the number 12?  It can be many things, such as 6+6 or 9+3; but 6+6 can only ever be 12, just as 9+3 can only ever be 12.  The focus in this study is to consider what makes up the WHOLE number and for this reason, even a child who has memorized the addition tables can benefit from this exercise.  An understanding of the whole of numbers serves in building a stronger foundation for more advanced work in the future. 

Over the course of the next month spend a separate day on each of the numbers, 1 through 12. Consider each number as a whole and from that starting point, pick out its parts.  There are two ways to have fun with this exercise.  The first is to draw Number Ladders.

By drawing a simple ladder we can imagine that the number we are considering is walking up its rungs.  Each rung represents a different way to make up that number.  Twelve can be eleven and one or it can be five and seven.  The ladder shows the many parts of the number.  Point out the patterns that numbers make.  Going down the left side of the ladder we see that the numbers increase from zero to twelve, while on the right side the numbers decrease from twelve to zero. 

Ask you child to make a number ladder for the numbers one to twelve.  When he has finished this over a series of many days, consider the twelve number ladders together and compare them. For instance, the eight ladder has many more rungs than the two ladder.  Pointing out these patterns and differences transforms an abstract concept into a concrete reality.

A book such as 12 Ways to Get 11 is a great example of expressing the parts of numbers in pictoral form.  Make a number picture for each of the twelve numbers.  Perhaps seven is a scene of a house with one door, two windows, two trees, one sun, and one chimney. (1+2+2+1+1=7) If your child enjoys this activity, consider using the blackboard for repeated play or maybe he'll want to make his own number picture book.

Number Study: Four

This week we will continue our study of numbers by considering the number four.  Ask your child to think of things that are four.  The four inspired authors of the Gospels, Saints Matthew, Mark, Luke and John;  the four cardinal directions, north, south, east and west; and the four sides of a square or rectangle are all possible examples.  Have your child make a Number Four page for his Main Lesson Book.

Making Numbers

Shaping the numerals out of modeling clay is a beneficial tactile experience for all learners, but especially for those with learning challenges such as dyslexia.  Later this week Elizabeth will be posting about modeling the numbers as well as modeling representations of number sentences. 

When considering an equation, such as four plus two, it is helpful for a child to model the numeral four, an addition symbol, the numeral two and an equals symbol and answer the question by rolling out small balls to demonstrate the equation.  Four small balls grouped with two small clay balls adds up to six.  The equation is visually represented and the child has been offered the experience of forming the numerals with her own hands.  Just as the exercise of Number Study gives the child an opportunity to contemplate the reality of the number in the created world, so too the exercise of making numbers offers the actual experience of forming the individual numbers and feeling its shape.

Encourage an older child to participate in this activity by pairing him up with a younger sibling.  In teaching and directing a younger child, the older child not only increases his understanding, but demonstrates it. 

Form Drawing

Continue the lessons in Form Drawing by reading the next installment in the tale "The Story of the Great King Equals."  Who is King Equals? Where did he come from?  And how did he get that funny symbol?  The answers to these questions and more will be discovered in this tale. After reading this week's portion of the story, draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon she will learn the movement and pattern and be able to work independently. Encourage your child to draw this form using a stick crayon with a continuous and uninterrupted motion.   

Descending toward the base of the misty mountains, the Purple Prince looked out upon the vast land.  As far as the eye could see were rolling hills covering the land.  Spurring his loyal horse to continue on their journey, with great difficulty the purple prince rode over large hills. He was greatly relieved to discover a stretch of small hills.  At last he came to a series of medium-sized hills.  Finishing this leg of the journey and dismounting his horse, the purple prince was dismayed at what lay before him.  Would he ever find a wise man to help him unite his kingdom?

Download story with sample of running form here. (Download form_drawing_three.pdf

)

Living Books

One Monkey Too Many (Adding 1, Counting)

12 Ways to Get 11 (Whole to Parts, Addition)

So Many Cats! (Counting to 12)

Anno's Counting Book (Counting)

 

 

In which the gnomes learn the VALUE of their PLACE.

   By this time the clearing was littered with baskets thrown here and there.  Some were filled with gemstones and others had been toppled over.  The mountain of gemstones was still high above the gnomes, but gems were now scattered everywhere, some in even numbered piles and some mismatched in odd groupings.  The gnomes knew that they must continue counting in order to find the secret of numbers, but they were getting more and more frustrated the longer they tried to count.  And while they were happy that Old Dismas had explained the difference between odd and even numbers, they still knew that there were far more gemstones than they were able to count on their own.

    The gnomes were surprised when they suddenly saw a purple-clad king come out from behind the great trunk legs of Old Dismas. 

    "There is someone here for you to meet," the Ancient Tree announced to the gnomes. 

    The king stepped out in the clearing. He was an odd looking fellow, almost as wide as he was tall. He was dressed in purple robes that seemed to reflect every color imaginable and, instead of a crown, he wore a pointed cap trimmed in gold.

    Careful not to trip over the gemstones and baskets, he began, 

    "I am King Equals, King of all Numeria.  It was my ancestor who cast the enchantment on the Ancient Tree long, long ago...the enchantment that you have brought upon yourselves because of your greed.  And he who cast the enchantment was a great wizard king, one who knew the secret of numbers that you so eagerly seek to discover."

    The gnomes didn't know what to say.  Who was this King Equals and what did he want from them?  Finally the red gnome replied,

    "We know the enchantment.  Old Dismas explained it to us.  And we know we must count until we discover the secret of numbers.  But these gemstones are far too many for us and when we count we soon become confused.  Can you help us?"

    King Equals nodded his head and pulled a magic wand out from beneath his purple cape. 

    "You're a wizard too?" gasped the green gnome. 

    "Well, of course.  And I can offer you a very simple way to solve your problem.  If you want to find the solution to keeping track of the numbers you count, you must repeat this verse with me,

   Regal old yellow gnomes buy itchy vests.

Simply repeat this verse and the solution will present itself to you."

     The gnomes were convinced that King Equals was a fool, but they soon decided that since they had no better plan there was no harm in trying the verse, as silly as it seemed.  Together they began repeating the strange words,

    "Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests....Regal old yellow gnomes buy itchy vests..."

    Suddenly a radiant light broke through the tops of the trees and shone upon the mountain of gemstones.  And then there was a most beautiful sight.  The reflection of the gemstones created a brilliant rainbow and a a prism of color filled the sky.  Red, orange,  yellow, green, blue,  indigo, and violet. 

    "Use these bands of color to help you keep track of your numbers as they grow. But remember, only nine of each kind can fit in one band" instructed King Equals.

    Still awestruck by the exquisite color surrounding them, the gnomes were confused by the words, "Only nine of each kind can fit in one band."  And anticipating their confusion, the King began to demonstrate.

    "Do you see this first band of red?  We call this band Units.  Each gemstone represents one unit.  And we can fill the red band with nine of them."

    King Equals picked up a basket and filled it with 9 sparkling gemstones and carefully laid them in the red band of color.

    "But there are many more than nine gemstones, Your Majesty.  How can this possibly help us?  We have a mountain of gemstones to count" remarked the confused green gnome.

    "Ah, but don't you see these other bands?  There are many places within which you may count.  When we fill up the red Units we move on to the orange band. This band is called Tens."

    "Oh! I know what ten is.  We counted ten and made up a symbol for it in the shape of an X using branches from...Oh.  Sorry Dismas. I'm sure it was not pleasant to have your branches broken off" admitted the yellow gnome with great regret.

    The Ancient Tree lowered his heavy head and offered a gentle smile.

    King Equals continued his explanation, "After 9 gemstones fill the Units band we must continue by grouping them together in families of ten." 

    As he said this, the King carefully lined up the gemstones in groups of ten and placed them in the orange band of color.

    "Now remember, only nine of each kind can fit in one band.  So once we have nine groups of ten in the Tens place, we must move on to the yellow band of color. We call this the Hundreds."

     Beginning to understand the sorting game, the gnomes began gathering gemstones once again in their baskets.  When they filled the Hundreds with nine groups of one hundred gemstones, they moved on to the green. King Equals went on to explain that the green band was called Thousands and the numbers continued far past that.  There were blue Ten Thousands, indigo Hundred Thousands and a place in the violet band for a number called Millions. The gnomes continued their work, and stopped only to notice that King Equals had opened his purple cape to reveal a funny little symbol made up of two lines on his chest. 

    "Excuse me, Your Majesty, but what is that symbol you wear on your chest?  Is it some sort of royal family crest?" asked the red gnome.

    "Oh, it is much more than that.  Much more.  But that story if for another time.  For now, continue your counting and fill this rainbow with gemstones.  Soon you will learn the purpose of the symbol I wear upon my chest" answered the King.

    The gnomes whispered among themselves, "Well, I sure hope that mysterious symbol can help us discover the secret of numbers!"

    Of course, you know that Ancient Trees can hear the softest, most faintest sound in the forest.  And hearing the musings of the silly gnomes, Old Dismas let out a laugh that shook the forest floor.

LESSON PLANS

Supplies

There is no need to move on to this lesson until the concepts introduced in the previous two lessons have been mastered.  Move at your child's pace. The Gnomes and Gnumbers lessons are archived in chronological order on this page for your convenience.

You will need to assemble a Rainbow Tray for this lesson, as well as collect a set of numerals.  If you already have a set of moveable numerals like the ones shown in the picture above, use them.  If not, and purchasing a set is not in your budget, spend one afternoon baking a set with your child.  The easy salt dough recipe is here and a set of cutters can be purchased. Another option is to model the numerals with modeling beeswax (Suppliers are linked on the right sidebar.) or Sculpey Clay.

If you already own a Base Ten set, you can make use of it for place value work.  If not, you can download the Ten Bar and Hundred Flat files below.  These were made using Do-A-Dot markers on graph paper. Donna Young's site offers graph paper files for download if your child would like to make a set of base ten gemstones.  The graph paper used here is the 10 x 14 size. 

Downloads

You can download the story and artwork by clicking on the links below. 

Lesson Three Story

Download gnomes_and_gnumbers.Lesson 3.pdf

King Equals Picture

Download king_equals_art.PDF

King Equals Coloring Page

Download king_equals_coloring_page.pdf

Main Lesson Book Sample, Rainbow

Download Rainbow.pdf

Gemstone Ten Bars (Print out 2 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_ten_bars.pdf

Gemstone Hundred Flat (Print out 9 copies on card stock.  Laminate before cutting, if desired.)

Download gnomes_hundred_flat.PDF

Presenting the Rainbow

We have chosen to present the rainbow in its true order of colors rather than the simplified version used in many children's songs and books.  Red, Orange, Yellow, Green, Blue, Indigo, and Violet.  Have fun with the mnemonic device created by Marybeth Foss, Regal old yellow gnomes buy itchy vests, for memorizing the colors of the rainbow and their order. 

After reading the story and presenting this lesson on place value, have your child use the gemstones to practice identifying the place value of each numeral in a larger number.  Depending on the age and ability of the child, you will want to go as high as he or she is able.  A younger child may only understand up to the hundreds place, while an older child will enjoy discovering a million.

Ask your child to illustrate a place value page in the Main Lesson Book.  The sample rainbow page offered above can be used as a model for your child. 

Place Value Activities

There are three place value activities that you can play this week using the Rainbow Tray and gemstones.  First start by filling a small basket with gemstones.  For instance, if your basket is filled with 23 gemstones, your child will count nine gemstones and, reminding him that only nine of each kind can be placed in each band, will then cross over to the Tens color band.  The remaining 20 should be grouped in two sets of ten gemstones. Then show your child how to swap out 10 gemstones for a ten bar. Continue in this way until your child understands the concept.

Next, make a 2 or 3-digit number (or larger for an older child) using the moveable numerals. For instance, arrange the numerals so that they show 45.  Ask you child to build this number with the gemstone materials or a Base 10 set next to the Rainbow Tray and then place the numerals in the correct strips of color on the tray.

Another fun activity is to sort gemstones in different sizes of cups.  Begin with 10 small bathroom-sized cups and ask your child to fill each with 10 gemstones.  When ten small cups are filled, they can be poured into a larger 8 ounce cup, representing 100 (pictured here).  Ten of these 8-ounce cups can then be poured into a large 32-ounce cup to represent 1,000.  This tactile experience gives the child the opportunity to see and feel these numbers.  The goal is to bring the abstract into the realm of the concrete.

Fun and Movement

Add movement by playing games of rainbow bean bag toss.  Say "Red" and toss your child a bean bag.  He should respond by saying, "Orange" and then toss the bag back to you. Continue in order to memorize the sequence of the colors in the rainbow.

Review

Don't forget to review the concepts covered in the previous lessons. Together read your child's entries in the Main Lesson Book and discuss each page.  Practice Roman Numerals at least one day this week and use the bean bags to play a game of toss, using odd and even counting as the rhythm.   

If you feel that your child would benefit from math sheet review, there is a wonderful math worksheet generator online called The Math Worksheet Site. You can get a subscription for only $2.50 per month. (The price is reduced depending on the length of your subscription.)  The subscription gives you unlimited use of the site and allows you to create custom math worksheets for your child.  The site is very easy to use. Consider creating sheets on Roman Numerals, odd and even counting, and place value for your child.

Number Study

This week we will continue our study of numbers by concentrating on the Number Two.  Ask your child to consider the world around him and find thing that exist in pairs of two.  Possible responses might be two eyes, two ears, two lights (the sun and moon), two feet, etc...

Gemstone Fun for the Little Ones

Nicholas Foss came up with a fun activity for the little ones.  Using rainbow-colored craft sticks, he sorted the different colored gemstones on each stick and made the pattern of a rainbow.  Since these were not glued down, a young child can repeat this activity many times.

Form Drawing

This week we are adding the element of form drawing.  This exercise in creating rhythmic patterns prepares a young child for handwriting and improves handwriting for the older child.  When presenting these running forms, allow your child to first practice in a shallow tray of sand or cornmeal.  Tracing the form with a finger in the air also helps the child in learning to imitate the pattern.  Your child should use stick crayons when drawing the forms.  Later in this main lesson we will be painting the geometric shapes and practice in form drawing will help in preparing the child for this activity. 

One running form will be introduced each week for a period of nine weeks and we will be using the art of story telling to capture the attention and interest.  Read the following story to your child as an introduction to the first running form.  If your child has not had much practice in drawing, it may be a good idea to spend some time practicing standing forms such as squares, triangles, circles, rectangles and stars. Draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon they will learn the movement and pattern and be able to work independently.

Our form drawing tale will be centered on the story of King Equals and how he brought peace to the Kingdom of Numeria.  Read this story to your child and practice drawing the form before making a main lesson book page (a sample page is shown here).  Perhaps after practicing the form each day, the child can draw it in the Main Lesson Book on Friday. Encourage your child to draw this form with a continuous and uninterrupted movement.  Have fun with the colors.  You can use these forms to practice memorizing the order of the colors of the rainbow too.

The Story of the Great King Equals

Once upon a time there was a sad prince who lived in a great stone castle in a divided kingdom.  For many years his ancestors ruled the kingdom in peace; but one day, in the furthest part of the kingdom, a great treasure was discovered in hidden cave.  Ever since that day the people of the kingdom argued and quarreled over the question of who was the rightful owner of the cave and the great treasure.  As the prince looked out through the parapets of the great stone castle, he longed for the day when his family's great kingdom would be one again.

At this point show your child the sample page of the running form pictured above. (Download great_king_equals.Story 1.PDF

) These lines resemble the parapet of
a castle wall.  Practice drawing this form each day, retelling the
story each time.  By the end of the week your child should be able to
narrate the story himself as he works on mastering this form.   

Living Books

Now is a good time to add a few mathematically themed read-alouds. Living math goes hand-in-hand with Living Books. Here are a few titles for this week:

How Much is a Million

Millions of Cats

One Hundred is a Family

Roman Numerals I to MM

If You Made a Million

Gnomes and Gnumbers: Lesson Four

In which the green gnome learns that counting is a PLUS.

     Having learned that with the help of a beautiful rainbow it was easy to recognize even the largest of numbers, the four gnomes went back to their work of trying to unlock the secret of numbers. 

    "We now know that some numbers are odd, while others are even," recalled the yellow gnome. "And we also know that no matter how large a number is, we can figure it out with the help of the bands of the rainbow.  But the enchantment has not yet been broken.  We obviously haven't figured out the secret of numbers.  What else could there be?"

    As soon as the yellow gnome finished speaking, the green gnome walked to the base of the mountain of gemstones, picked up the largest basket he could find and filled it to the top.   Then taking the gemstones out, one by one, he lined them on the ground and began,

    "One gemstone and one gemstone and one gemstone..."  he continued for quite some time.

    Curious and confused, the other three gnomes sat down and watched their green friend continue.  Soon the silence was broken by the woodsy "Huuuuumph!" of Old Dismas.

    "Addition."

    The green gnome didn't hear the Ancient Tree and continued, "...and one gemstone and one gemstone."

    Old Dismas spoke up, "Did you know that what you are doing has a name?"

    Suddenly the green gnome stopped.  He had been certain that what he was doing was something very new and had been discovered by him alone. 

    "Addition.  What you are doing is called Addition" repeated Old Dismas. "You see how you are adding the gemstones, one gemstone and one gemstone and so on and so forth?  That is called Addition.  How many gemstones do you now have?" asked Old Dismas.

    "Well, let's see.  I...I don't know.  One gemstone and one gemstone and one gemstone..."  The green gnome realized that while he could continue to add one gemstone after another to his long line, he was no closer to knowing how many he had counted.

    Walking over to the green gnome, King Equals once again reached inside his cape and pulled out his magic wand. 

    "I have a gift for you that will help you to answer your question."  The purple king waved his wand and pointed it towards the chest of the little green gnome.  Suddenly a strange symbol appeared. It was made up of two lines, one going up and down and the other going side to side.  And while the little lines went in opposite directions, they crossed exactly in the middle.

    "Whenever you want to find out how many, just remember to use this sign between each number.  And when you have finished, call on me and you will find your answer" finished King Equals.

    The green gnome looked again at his long line of gemstones and this time he traced his new sign in the earth between each gemstone. "One gemstone + one gemstone.  King Equals?"

    The king approached the gnome and answered, "Two gemstones.  One gemstone and one gemstone equals two gemstones."

    The green gnome continued, "Two gemstones + one gemstone.  Equals?" 

    King Equals answered, "Two gemstones and one more gemstone is the same as three gemstones."

    The little green gnome soon understood that the funny little sign joined the gemstones together and made them one.  Not only that, but King Equals' answer told him the number that they made. 

    "Oh!  I get it.  And three gemstones and one gemstone is four gemstones!"  The green gnome was so excited.  He merrily danced around his work.

    "Very good, my friend.  You have understood well.  And because you have figured out this mystery of counting, I give you a new name. From this time forth you will be known in all Numeria as Plus."  King Equals tapped the shoulder of of the happy gnome, bestowing upon him his new name.

    "I guess I've figured it out then.  Addition is the secret of numbers" announced Plus, rising from a ridiculously low bow.

    "Not so fast," replied Old Dismas, chuckling to himself.  "You are one step closer, that is true; but there is still much more to learn.  And until then, I suggest that you and your friends practice using your new symbol, for it will help you in unlocking this great secret."

    The red and yellow gnomes rushed to Plus' side, ready to play the new counting game.  But not the blue gnome.  No, he wasn't interested in the game of Plus.  Instead, the blue gnome quietly tip-toed around to the back of the mountain of gemstones.

    "I have a better way" he laughed to himself as he began filling the pockets of his cape with handfuls of colorful gemstones.

LESSON PLANS

So far we've learned about Roman Numerals, odd and even numbers, place value and the process of addition. At this point the story of the four greedy gnomes stops for a few weeks in order to allow the children an opportunity to absorb the material.  During the next three weeks new ideas on teaching the concept of addition and extension activities will be posted, along with the next three installment of the Form Drawing stories and the next three assignments in our Number Study. 

Each week we will go more in-depth with the concept of addition, but before we continue in this study it is important to make sure that your child understands the purpose and function of adding and making sums.  Explain that the addition symbol shows that we are joining the two things together that are on either side of it and that the symbol for equals simply means "is the same as."  Ask your child to make a page in his Main Lesson Book modeled after the one shown above.  Concentrate this week on adding 0 and 1 to a number.  Depending on the age and ability of your child, you can also make a +1 page in the Main Lesson Book (0+1 through 11+1 is enough).  Allow your child to use the gemstones for practice. Fill one basket with a small amount of gems and a second basket with one.  Ask your child to count the gems together to find the answer. You can illustrate these problems on the blackboard or let your child use Do-A-Dots to illustrate addition with gemstones on paper. 

Add rhythm and movement by playing addition bean bag toss.  Say, "One plus one is..." and toss the bean bag to your child.  The child should respond with the sum, "Two!" and then say, "Two plus one is..."  and toss the bean bag back to you. Continue in this way as long as your child is enjoying the game. (Remember to play games of bean bag toss to practice counting forward and backward from 0-100, odd and even counting, and the colors of the rainbow.)

For extra practice you can design your own worksheets using The Math Worksheet Site's math sheet generator.  Now is a good time to begin enjoying resources such a The Quarter Mile Math or Calculadder.

ADDITION STORY DOWNLOADS

Lesson Four Story Download lesson_four.pdf

Plus Gnome Artwork Download Plus.pdf
 

FORM DRAWING

This week we continue our lessons in Form Drawing by telling the second part of The Story of the Great King Equals. Tell this story to your child as many times as necessary until he can successfully draw the new running form.  By the end of the week, add this form to the Main Lesson Book.  Keep in mind that these forms should be drawn with stick crayons in one continuous stroke. You can download the story with picture here (Download form_drawing_2.pdf

). First review last week's story and then continue with the following:

Looking through the parapets of the great stone castle, the prince saw in the distance an immense range of mountains.  "I will depart from this divided kingdom and travel to the mountains, in search of a wise man who can tell me how to unite my family's kingdom once again" decided the purple prince.  And with much hope, the prince left his home in the great stone castle and set off on his horse over the misty mountains.

The running form in the drawing this week is similar in appearance to a mountain range.  Point this out to your child and then practice drawing the form with him. Encourage your child to use these forms when making page borders in their Main Lesson Book.

NUMBER STUDY

This week we will also continue our study of numbers by considering the number three.  Ask your child to think of things that are three.  St. Patrick's 3-leaf clover that signifies the Three Persons of the Holy Trinity, and a triangle are three examples.  Have your child make a number three page for her Main Lesson Book.

LIVING BOOKS

RAINBOW Last week we used the rainbow as a way to separate place and assign value to numbers.  This week you should review place value.  And with the rainbow in mind, now would be a fun time to learn more about them. Read The Magic School Bus Makes a Rainbow: A Book About Color,  Don Freeman's A Rainbow of My Own or All the Colors of the Rainbow. Spend some time learning about what rainbows are and how they are formed. You can also have fun with color and rainbows.  And you may want to read the story of Noah in a children's Bible too.

Continue spending time reading math-themed living books.  Last week we suggested a few titles.  If you have not yet read them with your child, go back to that list and read through.  Now is a good time to review the things we've learned in the past four weeks and reading the concepts in a living picture book will help reinforce the concepts.

Each week during our four-week addition unit, more theme-related books will be suggested. The following books are this week's suggestions for living books on the topic of addition. Try to read one each day.

Mission Addition

Twelve Ways to Get Eleven

Each Orange Had Eight Slices

Hershey's Kisses Addition Book

Math For All Seasons

Let's Have SUM Fun with PLUS!

The exciting mathematical tale of Gnomes and Gnumbers will continue in January 2008, but that doesn't mean our work is done. Far from it! From now until then we have much work to do with our new friend Plus. Our clever green friend has made an important discovery and for the following four weeks, assuming that you will be breaking for the weeks of both Thanksgiving and Christmas, we offer the following extensions to further the study of the process of Addition.

For the child who is just beginning her study of addition, you will want to spend the next month working with simple addition equations and memorizing the addition tables.  Begin with the +1 table and continue up until +9. The use of manipulatives, whether the gnomes' gemstones or a Base Ten set, is an essential element in presenting an abstract concept in a concrete way.

Older children who already have a solid understanding will benefit from the Whole to Parts exercises listed below and can increase their skills by working on Addition with Regrouping topics using a Base Ten set.  If you do not already own a Base Ten set, a gemstone base ten set for download was offered in Lesson Three of Gnomes and Gnumbers. If you have not already, you may want to download those files and prepare them.

REVIEW

Don't forget to review the concepts we've covered in the first four lessons of Gnomes and Gnumbers.  For Roman Numeral practice, write a Roman Numeral on a piece of paper or blackboard and ask your child to write or say the number that it represents.  On another day give your child a few different numbers and ask her to write them on paper or the blackboard in Roman Numeral form.

Along with games of bean bag toss, review odd and even numbers by filling baskets with different amounts of gemstones and asking your child to sort the gems in groups of twos.  Remind her that if one is left over, the number is odd.  An alternative review activity is to write numbers on paper or the blackboard and ask your child to circle all odd numbers in one color and all even numbers in a different color.  When analyzing larger numbers, be sure to remind your child that the numeral in the units place is the only numeral that we consider when determining whether or not a number is odd or even.

Continue to review the concept of place value by using the rainbow tray.  Make different numbers by placing the numerals within the bands of the rainbow.  Ask your child to identify the place value of the numerals.  For instance, if you demonstrate the number 1, 325 on the rainbow tray, ask your child questions such as "Which number is in the hundred's place?" or "In what place is the three?" You can extend the questions to include, "If a two is in the ten's place, what is the value of the two?"  The answer is two tens or twenty.  Continue the activity from Lesson Three of building numbers with the Base Ten set or gemstones and asking your child to place the correct numerals on the rainbow tray.  You can reverse this activity by making a number on the tray and asking your child to build it with the manipulatives. (Review Lesson Three for more explanation on place value activities.)

ADDITION EXTENSIONS

Addition Strip Board

The Addition Strip Board, a great aid in comprehending and memorizing the addition tables, is a traditional Montessori resource.  You can purchase a wooden board and strips for a discounted price at Alison's Montessori or you can download a free PDF from Montessori Materials. (Click on "Addition Strip Board" under the heading "Addition Material.")  For maximum durability, we suggest printing on card stock and mounting the print-outs on foam board. As beautiful as the traditional set is, this more economical paper alternative is just as effective as using the more expensive wooden set. You will also want to print out this free set of Addition Equation Slips (Click on "Addition Combination Slips.") or purchase the matching wooden set from Alison's.

To introduce the board to a child, ask him to randomly choose an addition equation slip.  In the picture here, the equation "4+3=" was chosen.  The equation was built by lining the blue 4 strip on the first line, followed by the red 3 strip.  With both strips lined up together, it is obvious by checking the number line that the sum is 7.  The number seven tile was placed after the equation and the sum was recorded on graph paper.

Using graph paper for recording equations helps the child line up the numbers in the equation.  (It is even more helpful for advanced topics such as multiple-digit addition or long division.) For free graph paper you can visit Donna Young's site.  There you will find many different sizes of graph paper for download.  For a young child, the larger squares are best.  If you would prefer a bound main lesson book filled with graph paper you can find one at Paper, Scissors, Stone. (Scroll down to find the "Arithmetic Book.")

Spend the next few weeks introducing and memorizing the addition tables +1 through +9.  You can use the equation slips to introduce each table, as well as for daily practice.  For instance, if your child is going to be working on the +4 table, collect the equation slips that express the +4 equations.  Ask your child to build the equations on the strip board.  For further practice, the equations slips can be used as flashcards as your child works on memorizing the addition tables.  And if he can't remember the sum, always ask him to build the equation on the board.

Don't forget to add fun and movement by playing games of addition bean bag toss.  Using a small bean bag, toss the bag to your child as you say "One plus one equals..."  Your child should respond with the sum, "Two," as he tosses it back to you.  Continue up to 1+9.  Use this game as a memory aid for all 9 addition tables. 

Addition with Regrouping

Using a Base Ten set, you can teach your older child how to add larger numbers that require regrouping or carrying. Consider printing multiple-digit addition equation sheets from The Math Worksheet Site or write your own on graph paper.  Use the Base Ten materials to demonstrate the process, reminding your child of the counsel of King Equals, "Only nine of each kind can fit in one band."

Begin with a problem that requires regrouping such as "35+68=" and build each number with the gemstone materials or your Base Ten set.

Pointing
out that we only add together those of like kind and that we always
start with the Units, combine the Units into one pile.

Counting the Units we find that there are 13 cubes. Remind your child of the wise counsel of King Equals, "Only nine of each kind can fit in one band."  Thirteen is more than nine and cannot fit in the red units band of our place value rainbow.  We're going to have to do some trading.  Get out a Ten Bar and line up ten of the thirteen units against it.  Trade then Ten Bar for the ten Units.

Remind your child that we can only add those of like kind, therefore the Ten Bar must be added to the pile of Tens and not the Units.  We now only have three Units and move on to adding the Tens together.

"Only nine of each kind can fit in one band."  Since we have ten we'll have to trade again. Get out a Hundred Flat and trade the ten Ten Bars for one Hundred. 

We've stayed true to King Equal's instruction and now have three Units and one Hundred.  Ask your child to use the numerals and place them in the correct bands of the rainbow. (You can bake these numerals, buy a set of moveable numerals or write them on the blackboard or paper in the different colors of each band.)

This example is one in which there is a zero in the Ten's place.  It is important to point out that when we trade the different values and end up with nothing in one of the places, we need to make sure that we use a zero to hold the place.  Demonstrate with this example that if you didn't use the zero to hold the Ten's Place, the number would no longer be one hundred and three, but thirteen.  The child can clearly see that more than 13 is represented by the blocks.

When recording these problems we recommend using graph paper. When carrying numbers above the equation, it is much easier to keep track of the Tens or Hundreds that we've carried over if we can line them up in the square above the Ten's or Hundred's Place. (Sources for graph paper are listed above.)

Whole to Parts

The following exercise is recommended for all children, even those who have experience working with addition equations.

Rather than simply memorize addition equations and their sums, it is important to study numbers in a holistic way by considering them from the whole to the parts. What is the number 12?  It can be many things, such as 6+6 or 9+3; but 6+6 can only ever be 12, just as 9+3 can only ever be 12.  The focus in this study is to consider what makes up the WHOLE number and for this reason, even a child who has memorized the addition tables can benefit from this exercise.  An understanding of the whole of numbers serves in building a stronger foundation for more advanced work in the future. 

Over the course of the next month spend a separate day on each of the numbers, 1 through 12. Consider each number as a whole and from that starting point, pick out its parts.  There are two ways to have fun with this exercise.  The first is to draw Number Ladders.

By drawing a simple ladder we can imagine that the number we are considering is walking up its rungs.  Each rung represents a different way to make up that number.  Twelve can be eleven and one or it can be five and seven.  The ladder shows the many parts of the number.  Point out the patterns that numbers make.  Going down the left side of the ladder we see that the numbers increase from zero to twelve, while on the right side the numbers decrease from twelve to zero. 

Ask you child to make a number ladder for the numbers one to twelve.  When he has finished this over a series of many days, consider the twelve number ladders together and compare them. For instance, the eight ladder has many more rungs than the two ladder.  Pointing out these patterns and differences transforms an abstract concept into a concrete reality.

A book such as 12 Ways to Get 11 is a great example of expressing the parts of numbers in pictoral form.  Make a number picture for each of the twelve numbers.  Perhaps seven is a scene of a house with one door, two windows, two trees, one sun, and one chimney. (1+2+2+1+1=7) If your child enjoys this activity, consider using the blackboard for repeated play or maybe he'll want to make his own number picture book.

Number Study: Four

This week we will continue our study of numbers by considering the number four.  Ask your child to think of things that are four.  The four inspired authors of the Gospels, Saints Matthew, Mark, Luke and John;  the four cardinal directions, north, south, east and west; and the four sides of a square or rectangle are all possible examples.  Have your child make a Number Four page for his Main Lesson Book.

Making Numbers

Shaping the numerals out of modeling clay is a beneficial tactile experience for all learners, but especially for those with learning challenges such as dyslexia.  Later this week Elizabeth will be posting about modeling the numbers as well as modeling representations of number sentences. 

When considering an equation, such as four plus two, it is helpful for a child to model the numeral four, an addition symbol, the numeral two and an equals symbol and answer the question by rolling out small balls to demonstrate the equation.  Four small balls grouped with two small clay balls adds up to six.  The equation is visually represented and the child has been offered the experience of forming the numerals with her own hands.  Just as the exercise of Number Study gives the child an opportunity to contemplate the reality of the number in the created world, so too the exercise of making numbers offers the actual experience of forming the individual numbers and feeling its shape.

Encourage an older child to participate in this activity by pairing him up with a younger sibling.  In teaching and directing a younger child, the older child not only increases his understanding, but demonstrates it. 

Form Drawing

Continue the lessons in Form Drawing by reading the next installment in the tale "The Story of the Great King Equals."  Who is King Equals? Where did he come from?  And how did he get that funny symbol?  The answers to these questions and more will be discovered in this tale. After reading this week's portion of the story, draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon she will learn the movement and pattern and be able to work independently. Encourage your child to draw this form using a stick crayon with a continuous and uninterrupted motion.   

Descending toward the base of the misty mountains, the Purple Prince looked out upon the vast land.  As far as the eye could see were rolling hills covering the land.  Spurring his loyal horse to continue on their journey, with great difficulty the purple prince rode over large hills. He was greatly relieved to discover a stretch of small hills.  At last he came to a series of medium-sized hills.  Finishing this leg of the journey and dismounting his horse, the purple prince was dismayed at what lay before him.  Would he ever find a wise man to help him unite his kingdom?

Download story with sample of running form here. (Download form_drawing_three.pdf

)

Living Books

One Monkey Too Many (Adding 1, Counting)

12 Ways to Get 11 (Whole to Parts, Addition)

So Many Cats! (Counting to 12)

Anno's Counting Book (Counting)

 

 

The exciting mathematical tale of Gnomes and Gnumbers will continue in January 2008, but that doesn't mean our work is done. Far from it! From now until then we have much work to do with our new friend Plus. Our clever green friend has made an important discovery and for the following four weeks, assuming that you will be breaking for the weeks of both Thanksgiving and Christmas, we offer the following extensions to further the study of the process of Addition.

For the child who is just beginning her study of addition, you will want to spend the next month working with simple addition equations and memorizing the addition tables.  Begin with the +1 table and continue up until +9. The use of manipulatives, whether the gnomes' gemstones or a Base Ten set, is an essential element in presenting an abstract concept in a concrete way.

Older children who already have a solid understanding will benefit from the Whole to Parts exercises listed below and can increase their skills by working on Addition with Regrouping topics using a Base Ten set.  If you do not already own a Base Ten set, a gemstone base ten set for download was offered in Lesson Three of Gnomes and Gnumbers. If you have not already, you may want to download those files and prepare them.

REVIEW

Don't forget to review the concepts we've covered in the first four lessons of Gnomes and Gnumbers.  For Roman Numeral practice, write a Roman Numeral on a piece of paper or blackboard and ask your child to write or say the number that it represents.  On another day give your child a few different numbers and ask her to write them on paper or the blackboard in Roman Numeral form.

Along with games of bean bag toss, review odd and even numbers by filling baskets with different amounts of gemstones and asking your child to sort the gems in groups of twos.  Remind her that if one is left over, the number is odd.  An alternative review activity is to write numbers on paper or the blackboard and ask your child to circle all odd numbers in one color and all even numbers in a different color.  When analyzing larger numbers, be sure to remind your child that the numeral in the units place is the only numeral that we consider when determining whether or not a number is odd or even.

Continue to review the concept of place value by using the rainbow tray.  Make different numbers by placing the numerals within the bands of the rainbow.  Ask your child to identify the place value of the numerals.  For instance, if you demonstrate the number 1, 325 on the rainbow tray, ask your child questions such as "Which number is in the hundred's place?" or "In what place is the three?" You can extend the questions to include, "If a two is in the ten's place, what is the value of the two?"  The answer is two tens or twenty.  Continue the activity from Lesson Three of building numbers with the Base Ten set or gemstones and asking your child to place the correct numerals on the rainbow tray.  You can reverse this activity by making a number on the tray and asking your child to build it with the manipulatives. (Review Lesson Three for more explanation on place value activities.)

ADDITION EXTENSIONS

Addition Strip Board

The Addition Strip Board, a great aid in comprehending and memorizing the addition tables, is a traditional Montessori resource.  You can purchase a wooden board and strips for a discounted price at Alison's Montessori or you can download a free PDF from Montessori Materials. (Click on "Addition Strip Board" under the heading "Addition Material.")  For maximum durability, we suggest printing on card stock and mounting the print-outs on foam board. As beautiful as the traditional set is, this more economical paper alternative is just as effective as using the more expensive wooden set. You will also want to print out this free set of Addition Equation Slips (Click on "Addition Combination Slips.") or purchase the matching wooden set from Alison's.

To introduce the board to a child, ask him to randomly choose an addition equation slip.  In the picture here, the equation "4+3=" was chosen.  The equation was built by lining the blue 4 strip on the first line, followed by the red 3 strip.  With both strips lined up together, it is obvious by checking the number line that the sum is 7.  The number seven tile was placed after the equation and the sum was recorded on graph paper.

Using graph paper for recording equations helps the child line up the numbers in the equation.  (It is even more helpful for advanced topics such as multiple-digit addition or long division.) For free graph paper you can visit Donna Young's site.  There you will find many different sizes of graph paper for download.  For a young child, the larger squares are best.  If you would prefer a bound main lesson book filled with graph paper you can find one at Paper, Scissors, Stone. (Scroll down to find the "Arithmetic Book.")

Spend the next few weeks introducing and memorizing the addition tables +1 through +9.  You can use the equation slips to introduce each table, as well as for daily practice.  For instance, if your child is going to be working on the +4 table, collect the equation slips that express the +4 equations.  Ask your child to build the equations on the strip board.  For further practice, the equations slips can be used as flashcards as your child works on memorizing the addition tables.  And if he can't remember the sum, always ask him to build the equation on the board.

Don't forget to add fun and movement by playing games of addition bean bag toss.  Using a small bean bag, toss the bag to your child as you say "One plus one equals..."  Your child should respond with the sum, "Two," as he tosses it back to you.  Continue up to 1+9.  Use this game as a memory aid for all 9 addition tables. 

Addition with Regrouping

Using a Base Ten set, you can teach your older child how to add larger numbers that require regrouping or carrying. Consider printing multiple-digit addition equation sheets from The Math Worksheet Site or write your own on graph paper.  Use the Base Ten materials to demonstrate the process, reminding your child of the counsel of King Equals, "Only nine of each kind can fit in one band."

Begin with a problem that requires regrouping such as "35+68=" and build each number with the gemstone materials or your Base Ten set.

Pointing
out that we only add together those of like kind and that we always
start with the Units, combine the Units into one pile.

Counting the Units we find that there are 13 cubes. Remind your child of the wise counsel of King Equals, "Only nine of each kind can fit in one band."  Thirteen is more than nine and cannot fit in the red units band of our place value rainbow.  We're going to have to do some trading.  Get out a Ten Bar and line up ten of the thirteen units against it.  Trade then Ten Bar for the ten Units.

Remind your child that we can only add those of like kind, therefore the Ten Bar must be added to the pile of Tens and not the Units.  We now only have three Units and move on to adding the Tens together.

"Only nine of each kind can fit in one band."  Since we have ten we'll have to trade again. Get out a Hundred Flat and trade the ten Ten Bars for one Hundred. 

We've stayed true to King Equal's instruction and now have three Units and one Hundred.  Ask your child to use the numerals and place them in the correct bands of the rainbow. (You can bake these numerals, buy a set of moveable numerals or write them on the blackboard or paper in the different colors of each band.)

This example is one in which there is a zero in the Ten's place.  It is important to point out that when we trade the different values and end up with nothing in one of the places, we need to make sure that we use a zero to hold the place.  Demonstrate with this example that if you didn't use the zero to hold the Ten's Place, the number would no longer be one hundred and three, but thirteen.  The child can clearly see that more than 13 is represented by the blocks.

When recording these problems we recommend using graph paper. When carrying numbers above the equation, it is much easier to keep track of the Tens or Hundreds that we've carried over if we can line them up in the square above the Ten's or Hundred's Place. (Sources for graph paper are listed above.)

Whole to Parts

The following exercise is recommended for all children, even those who have experience working with addition equations.

Rather than simply memorize addition equations and their sums, it is important to study numbers in a holistic way by considering them from the whole to the parts. What is the number 12?  It can be many things, such as 6+6 or 9+3; but 6+6 can only ever be 12, just as 9+3 can only ever be 12.  The focus in this study is to consider what makes up the WHOLE number and for this reason, even a child who has memorized the addition tables can benefit from this exercise.  An understanding of the whole of numbers serves in building a stronger foundation for more advanced work in the future. 

Over the course of the next month spend a separate day on each of the numbers, 1 through 12. Consider each number as a whole and from that starting point, pick out its parts.  There are two ways to have fun with this exercise.  The first is to draw Number Ladders.

By drawing a simple ladder we can imagine that the number we are considering is walking up its rungs.  Each rung represents a different way to make up that number.  Twelve can be eleven and one or it can be five and seven.  The ladder shows the many parts of the number.  Point out the patterns that numbers make.  Going down the left side of the ladder we see that the numbers increase from zero to twelve, while on the right side the numbers decrease from twelve to zero. 

Ask you child to make a number ladder for the numbers one to twelve.  When he has finished this over a series of many days, consider the twelve number ladders together and compare them. For instance, the eight ladder has many more rungs than the two ladder.  Pointing out these patterns and differences transforms an abstract concept into a concrete reality.

A book such as 12 Ways to Get 11 is a great example of expressing the parts of numbers in pictoral form.  Make a number picture for each of the twelve numbers.  Perhaps seven is a scene of a house with one door, two windows, two trees, one sun, and one chimney. (1+2+2+1+1=7) If your child enjoys this activity, consider using the blackboard for repeated play or maybe he'll want to make his own number picture book.

Number Study: Four

This week we will continue our study of numbers by considering the number four.  Ask your child to think of things that are four.  The four inspired authors of the Gospels, Saints Matthew, Mark, Luke and John;  the four cardinal directions, north, south, east and west; and the four sides of a square or rectangle are all possible examples.  Have your child make a Number Four page for his Main Lesson Book.

Making Numbers

Shaping the numerals out of modeling clay is a beneficial tactile experience for all learners, but especially for those with learning challenges such as dyslexia.  Later this week Elizabeth will be posting about modeling the numbers as well as modeling representations of number sentences. 

When considering an equation, such as four plus two, it is helpful for a child to model the numeral four, an addition symbol, the numeral two and an equals symbol and answer the question by rolling out small balls to demonstrate the equation.  Four small balls grouped with two small clay balls adds up to six.  The equation is visually represented and the child has been offered the experience of forming the numerals with her own hands.  Just as the exercise of Number Study gives the child an opportunity to contemplate the reality of the number in the created world, so too the exercise of making numbers offers the actual experience of forming the individual numbers and feeling its shape.

Encourage an older child to participate in this activity by pairing him up with a younger sibling.  In teaching and directing a younger child, the older child not only increases his understanding, but demonstrates it. 

Form Drawing

Continue the lessons in Form Drawing by reading the next installment in the tale "The Story of the Great King Equals."  Who is King Equals? Where did he come from?  And how did he get that funny symbol?  The answers to these questions and more will be discovered in this tale. After reading this week's portion of the story, draw with your child and allow her to imitate you.  Don't hesitate to draw a form for a young child, allowing her to trace your work.  Soon she will learn the movement and pattern and be able to work independently. Encourage your child to draw this form using a stick crayon with a continuous and uninterrupted motion.   

Descending toward the base of the misty mountains, the Purple Prince looked out upon the vast land.  As far as the eye could see were rolling hills covering the land.  Spurring his loyal horse to continue on their journey, with great difficulty the purple prince rode over large hills. He was greatly relieved to discover a stretch of small hills.  At last he came to a series of medium-sized hills.  Finishing this leg of the journey and dismounting his horse, the purple prince was dismayed at what lay before him.  Would he ever find a wise man to help him unite his kingdom?

Download story with sample of running form here. (Download form_drawing_three.pdf

)

Living Books

One Monkey Too Many (Adding 1, Counting)

12 Ways to Get 11 (Whole to Parts, Addition)

So Many Cats! (Counting to 12)

Anno's Counting Book (Counting)