Renee M. Newman Student # 23393

August 9, 1998

EDU 503: Instruction Design & Classroom Techniques

La Salle University - Master of Science in Special Education

Dyscalculia: Instructional Design & Classroom Techniques-

Lesson 1: K-3 Number Basics

© 1998 Renee M. Newman

Introduction: American, Educated, Foreigner to Math ………………………………………………………1.

AN EXPLAINATION FOR THE INNUMERACY EPIDEMIC

The Art of Teaching.......................................................................................................2.

Introducing Math to the Foreigner

Hooks on which to Hang our Mathematical Hats………………………………………………………………3.

To persons with dyscalculia, math remains a foreign language. It surrounds them, parades before their eyes and whirls about their ears, but they can make little sense of it. Because of this predicament the foreigners to math are relegated to the sidelines, mere spectators. Sometimes they are invited to play the game, seduced by clever promises of simplicity and profit, and in their ignorance and gullibility they hand over money for unprofitable investment schemes.

Financial and economic news hurries by, incomprehensible, like so much blurred scenery pulled by the windows of a bullet train. Even the details of their own financial lives have acquired a mysteriousness that confounds them. Personal management of so many essential details remains marginal, year after year. The meaning of fundamental concepts- credit, debit, discount, points, interest, yield, dividend, gross, and net- elusively hangs in an inhibiting fog.

They work hard and earn a fair share but their purses are chronically empty. There never seems enough for obligations, and debt mounts. Eventually, money becomes a devilish snake- tempting them into purchases and payments that enslave them. They continue dangerously, recklessly, on this course until frustrated into bankruptcy. If fortunate enough to have a financial savior, they are bailed out, sometimes repeatedly, and never suffer this plight.

It is no different than the predictive state of illiterates who make up 80% of the prison population. (Weger 1989, 36) The foreigner to math is lucky that there are no debtor's prisons. For all 103 million U.S. households, bankruptcy rates increased 13% in the decade between 1985 and 1995 (Francese 1997) and then increased an astounding 29% in the one year between 1995 and 1996! In 1980, 97% of all bankruptcies, or 288,000, were by non-businesses. By January 1998, personal bankruptcies soared to 1,152,000- a rate that quadrupled in just 18 years. (USA Today 1997)

1.

2.

The handwriting is on the wall. American bankruptcies will continue to multiply. Why? Just look at the grim statistics: Almost 93% of America's 17-year-olds graduate without proficiency in multi-step problem solving and algebra. (NCES 1997, 123-124) An alarming 1 of every 4.5 American adults, or 22%, cannot perform simple arithmetic. (NCES 1997, 416) Cambridge professor Mahesh Sharma estimates that 6% of all children have a bonafide math learning disability, known as developmental dyscalculia. Very few ever get appropriate help with their problems.(CTLM 1989, 86)

Only 56% of exiting 17-year-olds can compute decimals, fractions, and percentages. Over 46% cannot recognize geometric figures, solve simple equations, or use moderately complex math reasoning. The United States Department of Education has found that 93% of high school graduates cannot solve problems involving fractions or percentages. They cannot solve 2-step problems involving variables, or identify equal algebraic equations, or solve linear equations and inequalities. An alarming 93% cannot synthesize and learn from varied specialized reading content. An amazing 91% cannot infer relationships and draw conclusions using detailed scientific information. (USDE 1991)

What becomes of these graduates? They become the adults for whom math remains a foreign language.

We have a mysterious situation. These graduates remain clueless despite 12 or 13 continuous years of systematic math instruction delivered by professional teachers. The curriculum was researched, proven scientifically sound, and approved by state and local experts. So what happened?

It is this author's suspicion is that these students were not taught appropriately. The language and science and discipline of math was not sufficiently anchored to their native language, or integrated and applied meaningfully to their daily experiences, and thus, was never assimilated.

Along the way, no one took the time to see that the prerequisite hooks were sufficiently installed. The hooks, or grade appropriate math instruction- no matter how finely crafted- fell out of memory when the weight of additional instruction was applied.

A hook can be in crumbly plaster, or shallowly embedded in the underlying lath, but any carpenter knows that serious fasteners must be anchored deep into solid support members- the wall studs- the permanent framework of lasting understanding and memory.

The goal of this paper is to provide practical, no cost, solid, properly anchored hooks upon which to hang our mathematical hats. Those hats become increasingly weighty as mathematical study advances. Because we cannot afford repairs and remodeling, we must be sure the hooks in the halls of our memory remain durable- lasting our lifetimes.

The author believes there is no better way to do this than to begin presenting (teaching) and integrating mathematical concepts, matter-of-factly, from birth.

The scope of this paper will encompass only the most basic concepts of numeration. Although these basic concepts can and should be presented sooner, the explanations contained here are appropriate for the child at least 6 years of age, although the ideas are appropriate for the Kindergarten through 4^th grade child.

3.

When teaching any subject, it is always best to know your students well- their likes, dislikes, interests, hobbies, favorite things, common and unique experiences, etc. The more you know about them, the more adeptly and securely you will drive their mathematical hooks. The associations and conclusions you help them draw will be genuine, vivid, meaningful, and ultimately woven into their experiences.

That is the teacher's goal- to get the student to "own" the information- to help them learn it well enough to teach it, remember it for all time, and to use it meaningfully, quickly, easily, appropriately, and profitably.

4.

Although "number sense" is intrinsic to humans, arithmetic and mathematics are acquired through this sense. Beginning with the natural and enjoyable inclination to count, children develop the foundations for number manipulations. Not every child will uncover the mysteries of mathematics without frustration or difficulty. But, as is required for so many necessary tasks- dressing, buttoning, the tying of shoes, - curiosity, desire, patience, and persistence are required. (Sharma 1990, 25)

Smart parents begin early, exposing their children to the characteristics and language of math, diligently pointing out and conversing about numbers and their uses in every day life.

"Math is a bonafide second language," says Mahesh Sharma, Head Professor of Education at Cambridge College. The six linguistic elements of mathematics must be deliberately taught: symbols, concepts, vocabulary, syntax, voice, and translation.

The object is to facilitate sufficient mastery of the language of mathematics so that each student can think and learn math independently. (Sharma 1990, 15, 23)

The language of mathematics is used to communicate ideas, properties, relationships, and behavior concerning defined actions, objects, symbols, sets or collections, classifications, organization, quantity, size, shape, order, space, time, form, velocity, speed, distance, magnitude, etc. Each math idea has three components: (1.) linguistic, (2.) conceptual, and (3.) procedural or skill. Each component requires deliberate specific and differentiated instructional attention. Math learning should be interactive. A concept should be introduced using concrete objects, and with logical language emphasis that is tied to everyday quantitative language expressions. (Sharma 1990, 23)

5.

THE 6 LINGUISTIC ELEMENTS OF MATHEMATICS -

1. symbols
2. concepts
3. vocabulary
4. syntax
5. voice
6. translation

(Sharma 1990, 23)

1. follow sequential directions
2. understand and apply classification systems
3. order, organize, and sequence
4. have command of spatial orientation and spatial organization
5. understand and employ estimation
6. visually cluster objects
7. recognize and extend patterns
8. visualize
9. think deductively
10. think inductively.

(Sharma 1990, 24)

6.

When thinking in terms of mathematics skills, several prerequisite skills are necessary before a student can acquire mastery of even simple math procedures. This framework is not formally taught in school or addressed in textbooks. Without these skills, any math learning that takes place is inevitably transitory. (Sharma 1990, 24)

The student must be able to: (1.) follow sequential directions; (2.) understand and apply classifications; (3.) order, organize, and sequence; (4.) have command of spatial orientation and spatial organization; (5.) understand and employ estimation; (6.) visually cluster objects; (7.) recognize and extend patterns; (8.) visualize; (9.) think deductively; and (10.) think inductively. (Sharma 1990, 24)

These are pre-math skills that are learned from birth and during the preschool years. Children learn these skills through normal explorations, exchanges with others, ordinary daily experiences, and play. When the caregiver notices that the infant or toddler is not making expected connections and advances in physical and cognitive skills, deliberate and immediate effort should be made to determine the causes of developmental delay. Despite popular opinion urging parents to relax and wait for their child to catch up developmentally, this author stresses the importance of taking immediate action to remedy the problem.

For each child, keep a medical and developmental diary. This can be a notebook, in which you note dates and pregnancy details, birth details, notes on early infant health, the results of all tests, the dates and details of every exam, advice and professional comments, baby firsts, remarkable events, your thoughts, suspicions, and concerns, etc. Take the notebook with you to each doctor visit. Record data, weight, height, health status, medications prescribed, dosage details, adverse reactions, results of blood and urine tests, etc. Insert pictures of the child at significant points in his development. It is important to note language development, as well.

7.

Do not rely on the doctor's office to be the sole keeper of your child's records Ask and note definitions of terms you do not understand. Take charge of your child's health by asking questions until you clearly understand all aspects. In addition to regularly scheduled "baby well visits" with a competent pediatrician, regular vision, and hearing testing should be done. When developmental delays are noticed, seek a thorough evaluation by an occupational therapist (OT).

Obtain the exceptional book, by Burton L. White, titled The First Three Years of Life, (ISBN # 0-13-319161-3). White clearly outlines each developmental stage and offers simple suggestions for optimizing the child's development at each stage. White's advice is founded on years of research at Harvard University's child development center White shows how to produce superior intelligence by making good use of the first three years of life.

As Dr. White verifies, the better a mother's nutrition during pregnancy, the better the baby's health and intelligence. As soon after conception as possible, a mother should follow the recommendations in the book What to Eat When You're Expecting (ISBN#: 0-89480-015-9). A mother will also immeasurably benefitr from the companion book, What to Expect While You're Expecting, (ISBN#: 0-89480-769-2). Both books are by Workman Publishing.

It is strongly recommended by White, and numerous other experts, that mothers breast feed their babies as long as possible- up to 2 years. Nursing insures proper immunity, warding off ear infections and allergies during the first years, when language fundamentals are acquired. (Newman 1995, 18)

After nursing, continue with good nutrition and health habits. Avoid refined sugars, which suppress the immune system, damage the pancreas over time, and cause illness and sometimes hyperactivity. (Newman 1995, 18)

8.

9.

Good speech at home is paramount. Never speak baby-talk to your baby. Eventually, the child has the capacity to understand a sophistocated vocabulary and sentense structure.

This will become evident when the child begins to Skill gaps must be identified and filled promptly. At the first sign of frustration with mathematics, perform a thorough investigation. "Simply correcting the child’s mistakes and relying on constant repetition is unlikely to be effective." (Miles 1987, 1) When remediating math deficiencies, supply appropriate models for math concepts. Explain that there are several ways to find the answer to a problem, but some are better, simpler, faster, and smarter than others. Allow the child to use his inefficient coping mechanisms, but deliberately call attention to and develop appropriate, more organized, and efficient strategies for organizing, remembering, and manipulating math facts and concepts. (Sharma 1990, 26)

Ideally, each math concept should be introduced at the intuitive level, progress to the concrete, then to the pictorial or representational, then to the abstract, then to applications, and finally culminate at the communication level. (Sharma 1990, 23)

This process is outlined in the following table.

Every math concept should be considered using both types of thinking. Quantitative approaches use standard deductive reasoning, sequential, procedural and algebraic algorithms. Qualitative approaches use "visual, spatial, inductive, and pattern recognition strategies." (Sharma 1990, 22)

Once the teacher is sure that prerequisite skills and sufficient cognitive understanding are present, a new concept should be introduced in the following sequence:

Math is really a language apart from typical English. To read it and write it, one must know the secret code. It has a unique set of symbols that have specific meaning. It has rules that must be followed. It has a vocabulary that must be understood. But once these fundamentals are mastered, the user has command of the language and freedom to participate in the possibilities it affords. Without the language of mathematics there would be no computers, phones, TVs, roads, bridges, cars, air travel, money, businesses, or even civilization. In fact, the very ideas of counting, measuring, saving and planning are the basis of human organization. One cannot decide that they can live without it. That is impossible.

In order to understand the language of math, we must have a good understanding of how each part works. The words in the language of math are made up of parts, some of which are familiar to English words you already know. By connecting the new math words with English words you already know, you will be able to remember and understand them. By knowing the meaning of the parts, you will be able to figure out the meanings of new words by remembering the meaning of parts you have learned.

Let us start with the idea of roots. We know that all plants have roots that grow deep into the ground and extract moisture and nutrition from the soil. They also anchor a tree securely in the ground, giving it strength against strong winds that might other wise knock it over. We can say that the roots are the most important part of the plant. When you cut down a pine tree to use as a Christmas tree, you cut the trunk, detaching the tree from its roots. The tree soon dies and dries up.

Words have roots, also. The root of a word, called a root word, is the most important part of the word. The root gives the word its essential meaning. Other parts of words are attached to the root word and alter or change its meaning a little. These other parts that are affixed to the root word are called Affixes (af’ fix es). Affix means to attach something. For example. "Affix these stamps to each letter before mailing. Fix one stamp in the right top corner of each envelope."

Some common affixes that you know are pre- and suf-. Here are some examples:

In America, our number system is called the decimal system. Our whole math code is organized around the number 10. That is why it is called the decimal system. Deci- and dec- are prefixes that mean 10. Every time you see these prefixes, think "10." Here are some examples that you have heard of:

Our system of money and counting is called a base 10 system or decimal system. We use 10 numerals, or number symbols, to express all of our number ideas. A symbol is a quick sign that stands for or represents an idea. A sign that says STOP, is a symbol for the idea of bringing a vehicle to a complete stop before proceeding forward. When you see the shape of the sign you automatically know what it means. The same is true of the 10 numerical symbols, or digits used in the language of mathematics. Each number or symbol or digit represents an idea.

We combine these 10 numerals to express all of our number ideas.

The place where we put the symbols when we write them is very important. Just like the location of a sign or symbol tells you where the idea it represents applies (imagine a no smoking symbol on a table in a restaurant.). The symbol’s location tells you that there is no smoking allowed at that table. Where number symbols appear tells you where the idea of the number applies. We call the spaces where numbers "sit" places. Each place has a meaning. Known as its "place value."

Place value is not really hard to understand. Think of seats in the family car or van. The front seats are the most comfortable and offer the best view. They have the highest "place value." The front seats are the most desirable or valuable seats. The most important people- parents, grandparents, or other grown-ups usually occupy them. The back seats have lower place value. They are worth less and are less desirable. Usually less important people sit there- children and pets. The middle seat has the lowest place value because it is uncomfortable to be sandwiched in the middle, on the hump, without a window view. Usually the smallest, weakest occupant gets stuck in the middle seat.

When only the driver is in the vehicle, he always gets the best place- the driver’s seat. He only takes up only one place or seat. When we write any of the ten digits or numerals by themselves, they only take up one place, too. We call the space a single digit or number takes up the one’s place. Let’s try it to make sure:

This is how place value works. Think of the pot in this picture as the decimal point. All of the numbers on the rainbow side of the pot are places for whole dollars. All of the places on the right side of the pot are places for t he coins that do not yet add up to a whole dollar.

But you grandma gives you one dollar for playing quietly while she takes a nap. You take the one-dollar bill and put it in your bank. Now you record the amount of money you have. It looks like this: $1 (dollar bill). 0 (dimes) 0 (pennies). [$1.00]

When we count, we are keeping track of increases. We are adding things up. The new number we say is a symbol for the idea of the total amount we have counted or added up so far.

When you count your fingers, you start with n0thing, at 0 ("zer0"). Then you touch your first (1^st) finger, your left pinky, and say, " 1(one)." Then you touch your next or second (2^nd) finger and say, " 2 (two)." Then you touch your third (3^rd) finger and say, " 3 (three)." Then you touch your fourth (4^th) finger and say, " 4 (four)." Next you touch your fifth (5^th) finger which is your left thumb and say, " 5 (five)."

At this point you are half done counting all of your ten fingers, you are 5/10^th s ("five-tenths") of the way finished because you have counted 5 out of 10 fingers. You are done counting 1 of your 2 hands. The symbolic way to write the idea of half is to write 1 out of 2 hands or ½ or 5 out of 10 fingers or 5/10^th s or .50 or 50% ("per cent").

Wow! There is more than one way to state an idea in the language of math, just like in English. There are several ways to express an idea. You can say: "Half the milk is left in my glass." Or: "The glass is half full." Or : "That glass of milk is half empty." Or: "That glass of milk is filled to the middle." Or: "Mom filled my glass up half way with milk."

In Math language you could say: "Only 50% of the milk is left in my glass." Or: "My glass is ½ full." Or: "The glass of milk is ½ empty." Or "Mom filled my glass up 5/10^th s of the way." Or "Mom filled my glass to .50." Or "Mom gave me a ½ glass of milk." Let’s get back to your hands, since you only need ½ of them (or one of your two) to drink a glass of milk!

You continue counting on your second (2^nd) hand, your right hand. You touch your right thumb and say, " 6 (six)," because this is the sixth (6^th) finger you’ve added or counted. You touch your seventh (7^th) finger and say, "7 (seven)". Then you touch your eighth (8^th) finger and say, *8 (eight)." Then you touch your ninth (9^th) finger and say, "9 (nine)." Finally you touch your last and final finger, your right pinky, the tenth (10^th) finger, and say, "10 (ten.)."

Now you are done. You are finished. You have counted 10 of 10 fingers, or 10/10. You have counted all of them: 100% ("per cent") of them. You have also counted 2 out of 2 hands: 2/2 or all or 100% ("per cent") of your hands. Written in math language, the counting or adding up of your fingers looks like this:

Now you were lucky because last night there was a big birthday party for you and your parents told everyone that you were eager to fill your new piggy. So everyone gave you a little gift to put inside. Look at the arrows below. Watch how the dollars add up. First Grandma gave you 1 dollar. (0+1=1) Then Auntie gave you $1. (one dollar bill), (1+1=2) then Uncle gave you $1. (2+1=3), then Cousin gave you $1. (3+1=4), then Dad gave you 2 dollars (4+1+1=6 OR 4+2=6). Then Mom gave you two dollars. (6+1+1=8 OR 6+2=8) Then your sister gave you one dollar. (8+1=9) Then your brother gave you a dollar. (9+1=10).

You have stuffed each of the dollar bills into your piggy bank and it is very full. Remember the object of the game with numbers and money is to trade in your less desirable or less valuable money for ones with more place value. It is known as trading up. Think of it as finally becoming important enough to sit up front in a more desirable or valuable place.

You went from having $9.00 (nine or 9 one dollar bills. 0 dimes and 0 pennies), to having 10 (ten) single or one dollar bills. Look at how many places the number 10 takes up: 1 0 The number is made out of two (2) digits or numerals: one (1) and zero (0). It takes up 2 places! Look at the picture above. The first number on the rainbow side of the decimal point is called the one’s place, because it denotes or tells how many 1 dollar bills or singles you have.

You can now trade in your ten (10) one dollar bills ($1.) for one (1) ten dollar bill ($10.). In the ten’s place you record a one (1) because you now have one (1) ten dollar bill. In the one’s place you record a zero (0) because you have no one dollar bills anymore. It is more desirable to have a $10. bill than ten singles. One $10. Bill taks up less room in your pocket or bank. It is more valuable than a single dollar. Now you put the $10. bill in your bank and record the total amount of money in there as: $10 . 00

(1- $10. Bill + 0-$1. Bills 0 dimes + 0 pennies)

0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

.

$1.00 $2.00 $3.00 $4.00 $5.00 $6.00 $700 $8.00 $9.00 $10.00 $11.00 $12.00 $13.00 $14.00 $15.00 $16.00

Wow. You have $10. 00 (ten dollarss or a ten dollar bill) in your bank. Your friend gives you another $1. Bill. Now you record the total in your bank as 1. Ten dollar bill, 1 One dollar bill. 0 dimes, 0 pennies [$11 00}

The decimal system increases to the left by multiples (x) of 10. X 10

Multiples and multiply by means to add the same number to itself a certain number of times.

1 penny multiplied (x) by 10 or added to itself 10 times = 10 ¢ or 1 dime. 1¢+1¢+1¢+1¢+1¢+1¢+1¢+1¢+1¢+1¢=10¢

1 dime multiplied (x) by 10 or added to itself 10 times = 100¢ or $1. Dollar 10¢+10¢+10¢+10¢+10¢+10¢+10¢+10¢+10¢+10¢=100¢

1 dollar multiplied (x) by 10 or added to itself 10 times = $10. Dollars

$.1. + $.1. +$.1. +$.1. +$.1. +$.1. +$.1. +$.1. +$.1. +$.1. = $10.

1 ten dollar bill multiplied (x) by 10 or added to itself 10 times = $100.

$10. + $10. +$10. +$10. +$10. +$10. +$10. +$10. +$10. +$10. = $100.

1 one hundred dollar bill multiplied by (x) 10 or added to itself 10 times = $1,000.

$100. + $100. +$100. +$100. +$100. +$100. +$100. +$100. +$100. +$100. = $1,000.

1 one thousand dollar bill multiplied (x) by 10 or added to itself 10 times = $10,000. (ten thousand)

$1,000. + $1,000. +$1,000. +$1,000. +$1,000. +$1,000. +$1,000. +$1,000. +$1,000. +$1,000. = $10,000.

1 ten thousand dollar bill multiplied by (X) 10 or added to itself 10 times = $100,000.One hundred thousand.

$ 10,000. + $ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. +$ 10,000. =$100,000.

1 hundred thousand dollars multiplied by (x) 10 = 1 million $1,000,000.

$100,000. + $100,000. +$100,000. +$100,000. +$100,000. +$100,000. +$100,000. +$100,000. +$100,000. +$100,000. = $1,000,000.

1 million multiplied by 10 (x) or added to itself 10 times = $10,000,000. Or ten million.

10 million multiplied by (x) 10 or added to itself 10 times = $100,000,000. One hundred million.

100 million multiplied by (x) 10, or added to itself 10 times = $1,000,000,000. One Billion

1 billion multiplied by (x) 10, or added to itself 10 times = $10,000,000,000. Ten billion.

10 billion multiplied by (x) 10, or added to itself = $100, 000,000,000. One hundred billion.

100 billion multiplied by (x) 10, or added to itself 10 times = $1,000,000,000,000. One trillion.

1 trillion multiplied by (x) 10, or added to itself 10 times = $10,000,000,000.,000. Ten Trillion.

10 trillion multiplied by (x) 10, or added to itself 10 times = $100,000,000,000,000. One hundred trillion.

100 trillion multiplied by (x) 10, or added to itself 10 times = $1,000,000,000,000,000,000,000,000. 1 zillion

.01 One cent. = 1 penny = 1¢ = 1¢/100¢ in a dollar X 10=

. 10 One dime = 10 cents = 10¢ = 10 ¢/100 ¢or 1/10^th of a dollar. X 10=

1. One dollar = 100 cents = 100¢ = 100¢/100¢= 1 whole dollar.= $1. X 10 =

10. Ten dollars =$10. ........................................................................................................................... X 10 =

100 One Hundred dollars = $100. X 10=

1,000. One thousand dollars = $1,000. X 10=

10,000. Ten thousand dollars = $10,000. X 10 =

100,000. One hundred thousand dollars = $100,000. .................................................................... X 10 =

1,000,000. One million dollars = $1,000,000........................................................................................ .X 10 =

10,000,000. Ten million dollars = $10,000,000. . X 10 =

100,000,000. One hundred million dollars = $100,000,000. X 10 =

1,000,000,000. One billion dollars = $1,000,000,000. X 10 =

10,000,000,000. Ten billion dollars = $10,000,000,000. X 10 =

100,000,000,000. One hundred billion dollars = $100,000,000,000. X 10 =

1,000,000,000,000. One trillion dollars = $1,000,000,000,000.. X 10 =

10,000,000,000,000. Ten trillion dollars = $10,000,000,000,000. X 10 =

100,000,000,000,000. One hundred trillion dollars = $100,000,000,000,000. X 10 =

1,000,000,000,000,000. One zillion dollars = $1,000.000,000,000,000. ...........................................................infinity

16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32

.

Each quart is 1/4 or 1 quarter of a gallon.

TELLING TIME:

The smaller hand or arrow points to the hour of the day.

The larger hand or arrow points to the minutes being counted.

When the small hand and the big hand are both on 12, it is 12 o'clock.

When the big hand points to the three, it is 12:15, or "quarter after 12," or "15 minutes after 12."

When the big hand gets to the 6, it is 12:30, or "half past 12," or "30 minutes after 12," or a " half hour after 12.:

When the big hand is on the 9, it is 12:45, or "12 and 3 quarters," or "quarter to ," or "45 minutes after 12," or 15 minutes before 1."

When the big hand reaaaches the 12 again, the smaller hand will have advanced to the, marking an additional hour. It will then be 1 0'clock.

The pattern is: 15 minutes + 15 minutes = 30 minutes ( a half of an hour, 2/4 = 1/2).

30 minutes + 15 minutes = 45 minutes. (3/4 "three quarters" of an hour.)

45 minutes + 15 minutes = 60 minutes (a whole hour, 4/4, or 4 quarters of an hour).

Many sporting games have their time periods divided into 4 equal parts, or quarters. There may be a short break after each quarter, but after the 2^nd quarter is over, it's time for the 1/2 time show. This is the break that marks that half of the game time is over. It also gives the players time to rest, and the fans time to eat, get up and stretch, eat, drink, and go to the bathroom. The quarters are spoken of as follows: "First quarter, Second quarter, Third quarter, and Fourth quarter. 1^st 2^nd 3^rd 4^th

A compass has 4 quarters:.

Each of the 4 quarters is also called a QUADRANT. (quad rant)

Some cities, like Washington, D.C., are divided into quarters, or 4 equal parts. After each street name, initials appear that tell what section or quarter of the city that the address is found in.

You go to the grocery store with your dad and he says, "We only have $20. To spend, so don't beg me for anything."

Then he says, "Here," and hands you the $20. Bill. He says "You know, you're getting smarter every day. I think it's time for you to be in charge of some things. Here is a list of things we need. As you choose the the items and put them into the grocery cart, I want you to add them up, making sure you do not go over $20., or else we will have to put something back."

"Let me just show you a few things about prices first. Stores mark prices to trick you into thinking that things cost less than they actually do. Look at cereal. The big box of Lucky Charms has a price of $4.89. That means that it costs 4 dollars + 89/100 pennies or cents needed to make another dollar. One more penny and it would cost $4.90. One more dime and it would cost $5. Dollars for that box of cereal."

"If we put this box in our cart, then we round the $4.89 price up to $5.00 and say, 'Well, we've already spent $5. Of our $20. '"

1 dozen eggs

1 loaf of bread

1 box of cereal

1 jar of peanut butter

1 jar of of jelly

Your dad asks, "Shall we keep the Lucky Charms?"

You say, "Yes," and put it in the cart. Next you get the bread. The kind your family eats costs $1.89 so you round it up to $2. And put it in the basket. So far your purchases add up this way: 5 + 2 = 7. Wow! You had no idea it would cost that much for so little.

Next you see the peanutbutter. You grab the brand you always get. It is $2.89, so you round that up to an even $3. And add: 7 + 3 = 10. Half of your money is spent and you still have 5 more items to get.

Now you choose Raspberry jelly. It is also $2.89, so you round it up to $3. And add that to your running total. 10 + 3 = 13. Now you get a gallon of milk, that is $2.69, so you round it to $3., too, and add it to 13. 13 + 3 = 16. Next you grab a big chunk of cheddar cheese. It is $5.89. You round it up to $6. And add it to 16 + 6 = 22. Oops! You're over $20! And you still have to get eggs! You will have to put the big cheese back and choose a smaller one. To make figuring easier, you get the jumbo eggs first. The dozen costs $1.19, so you round that down to $1. First you subtract the $6. For the cheese. $22. - $6. = $16. Now you add 1 for the eggs. 16 + 1 = 17. How much do you have left to spend on cheese? To find out, you count from 17 to 20: 18, 19, 20. That's $3. For cheese.

You see a smaller chunk for $4.89 but that is more than $3. You see an even smaller chunk for $3.89, but it is still over $3. The only smaller chunk is $1.89, so you have to choose that one. You round it to $2. And add to 17. 17 + 2 = 19. You have checked off all of the items on your list: cheese, eggs, milk, cereal, bread, peanut butter, and jelly. Now you proceed to the checkout.

Now use your calculator to add up your purchases. You estimated that they would cost $19. How close were you?

Cheese $1.89 $2.00

Milk $2.69 $3.00

Eggs $1.19 $1.00

Bread $1.89 $2.00

Peanut butter $2.89 $3.00

Jelly $2.89 $3.00

The actual total of your purchases comes to $18.33. You hand the cashier your $20. Bill. Now quickly figure how much change you should get back. Count up from 18.33 to 20.

From 33 to 100: Count by 10's to determine how many tens or dimes you should get back.: 43, 53, 63, 73, 83, 93 (7 dimes or 70 cents)

Now count from 3 to 10 to determine how many pennies you get. 4,5,6,7,8,9,10. That's 7 pennies or cents.

70 + 7 = 77 cents.

77 + 33 = 1.00

18. + .77 + .33 = 19.

Now count form 19 to 20 to determine how many dollar bills you have coming back.

20. That's 1.

1.dollar + .77 = $1.77 or One dollar and 77 cents.

$1.00 .25 .25 .10 .05 .02

$1.00 $1.25 $1.50 $1.60 $1.65 $1.67

As the cashier hands the money to you, she will count it back to you, adding the change to the total of your purchases: $18.33. up to the $20. you gave her.

If the number in the 100ths place is 5,6,7,8,or 9, you change it to a 10, which means a dime.

So you record the dime, not in the 100ths or pennies's place, but in the dime's column or tenths place.

Again, we begin rounding the pennies or hundredth's place first, then round the tenth's or dime's place, then the dollar or one's place.

A WHOLE number is also called a NATURAL number. A WHOLE or NATURAL number is any counting number, 0 and above. Think of it like this. When you order a pizza you NATURALLY get a WHOLE pizza! When a delivery boy counts the number of pizza's he delivers on a Saturday night, he starts counting at 0. When he first starts his shift, he has delivered no or zero 0 pizzas. But as the night goes on, he can add them up, beginning with the first 1. Then 2,3,4,5,6,7,8,9,10,11,12,13,14,15,16, 17, 18, 19, 20, 21, 22, 23, 24, 25,26, 27, 28, 29, 30, 31, 32, 33, 34, 35.

Think of numbers that are fractions or parts of wholes as UNNATURAL NUMBERS. For instance, .5 means half of a whole. But have you ever heard of a HALF of a PERSON coming to school? No, of course not!. That is NOT NATURAL! Have you ever heard of a restaurant delivering a HALF of a pizza? No! That is not natural, either.

NATURAL or WHOLE numbers are recorded on the LEFT of the decimal point.

When no decimal point is shown, watch out for certain words that tell you that a number is UNNATURAL or PART of a WHOLE number, and must then appear to the RIGHT of the decimal point.