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TABLE OF CONTENTS

This presentation was tested only with few latest versions of Internet Explorer, Firefox and Chrome.

Welcome to Big Number Workshop, a DIY course of early math, which nurtures understanding. I assume, your have taught your child count to 10 and you came here to teach billions. OK, let's navigate a plane.

Watch how the pattern develops. The final fit of building four "spinners" next to each other must be easy for 6-year old, but placing the paddles of the first spinner requires the cool skill of counting.

The classical Battleship game is an engaging way to practice. You only need to print so-called graph paper.


The FORWARD link is at the bottom right.

Click on the gray square to make it blue. Get any suitable toy (or just a sheet of graph paper) and challenge your student to find the square you clicked. Click again to make the blue square gray. You can build simple patterns this way, just make sure to present them one piece at a time. You've seen an example on the previous page.

In FIREPEGS, you can build moving patterns of colors. Your student can replicate them using a toy or another screen.


To go forward, you may also click on the header.
Use the ruler on top of it to jump to any page.
The table of contents is on page 00.
Links to learn more are on page 22.

If your student is familiar with patterns, here is a little challenge.

If a part of a patterns appears at once, you may build it in any way you like, but the order of the parts must be followed.

Now challenge you student to turn the pattern of squares into this.

The picture shows 4 sets: a set of elephants, a set of books, a set of spoons and a set of apples. What do these sets have in common?

They all have the same quantity: five. The quantity of elephant is equal to the quantity of spoons, etc. To prove, we match elephants and spoons one to one. This way we compare two quantities finding which one is bigger. The other quantity is smaller, but if none is bigger, the quantities are equal.

From time to time, ask you student this question again. Prepare sets of some items. Eliminate the commonalities until only the quantity is left. If you offer counting bears and Dienes cubes and the answer is they are made of plastic, throw in a set of steel balls. Do it until the quantity is the only commonality left.

In the second or the third grade, American students can't answer what numbers stand for. Remind them: numbers stand for quantity, and a quantity can be only discovered matching one to one. As long as the kids keep this in mind, they understand what they are doing.

To count, we must be able to separate the objects, serialize them and (usually) recognize them as the members of some class. We count in space, in time and within given spatial and temporal constrains.

In practice, we are usually concerned with the quantities of the members of a class, and to define classes we abstract. The set of all the objects on the picture has a quantity, but the objects belong to the set as long as they lay within the picture borders. Elephants, however, belong to the class of elephants because they are elephants.

Every book on the picture is different, and it belongs to a class of the same name books. To think of the quantity of books - any books - we have to generalize. Similarly, we can add apples to oranges, but only if we consider them fruits. Multiplying, we produce a new class. Tracking classes is utterly important for using arithmetic.

Every students is unique, and it belongs to a class of its own. Speaking about the quantity of students, we leave behind their personalities.

The previous and this page outline several lessons what can be played out using various household objects.

Why bring the elephants if we want to tell how many of them we have seen last week? We can capture the quantity of the elephants using apples as counting tokens. Or we can use elephants to capture the quantity of the apple seeds.

During the season, we can match every hurricane to a counting token. Later, then the hurricanes are long gone, we will have their quantity captured.

A line of tokens is good for capturing quantities: We can instantly compare them.

If the bars are not growing smoothly, your computer may be busy doing something else.

We can capture quantities using the words like one-two-three, but not for a long time. The counting words tend to run away from our memory even if we support them with vocalization and gesticulation. Tokens stand. Writings stand too, but we don't count in writing, we only memorize the results. Tokens and other computing devices keep quantities and allows to change them many times.

One-two-free are mental tokens. They belong to the ordered set of the counting words. It it important to remember both words and their order. Like before, we match the mental tokens to the objects we count, but we do it in the order of the words, never skipping any of them or using more than once. If we do it right, we find the word or phrase for our quantity.

Using tokens, we can capture quantities without counting words, but practically every kid knows those words anyway. Let's teach more of them.

Speaking about toys, I did not mention the board. The board must holds the counting tokens in place, providing spatial organization. If a line of tokens gets out of hands, we can cut it and place the pieces on the board side by side.

I am using so-called Lauri's boards and pegs, just because I couldn't find anything better yet. A line of 74 Lauri's pegs will be approximately 7'2" long - that's longer than most household tables.

The board provides an opportunity to play with the quantities up to one hundred and learn the respective counting words.

The lines of tokens on the picture is cut into pieces of 3 and 5. The script drops one yellow and one red token at a time. The quantity of the yellow and the red tokens is the same.

Cutting lines into pieces make it easy to tell the quantities and to compare them. We tell the number of the cuts and the remaining loose tokens. For example, eight tens and two is obviously bigger than two tens and eight.

Remark, however, that ten is not the same as 10. If we cut by three, one cut and zero tokens is 10, but it makes three!

Patterns provide ample material for counting (later they can introduce addition and multiplication). How many kinds of shapes are here? How many tokens in every kind? How many colors and how many tokens of every color? Are the quantities of the tokens of the different colors equal? Build this pattern and take it apart moving peg by peg to a counting board.

To sort out this pattern, please click below to select red, blue, green or yellow. How the different color layouts are different? Click on REPLAY to see them all at once.

Click here to get another simple pattern.

Seriously, can 6-year olds learn to count to 100? Of course they can. They can count to much bigger numbers. Even 4-year olds can. Just give them what to count, and a reason why.

Practicing counting and learning the names of the quantities from zero to one hundred, you lead you student to the positional representation, in which one token stands for one hundred of them. When we count one hundred tokens in the positional decimal representation, we leave ninety nine tokens behind. The following pages will show you how exponential positional counting work.

Back to exponential positional counting. First, as soon as we accumulated 10 counting tokens, we must exchange them for one more powerful token. Usually, but not necessary, the base quantity 10 is equal to ten. We exchange ten ones for one ten, ten tens for one hundred, etc. Multiplication by the same base number makes our counting exponential.

Strictly following exponentiality, we always know how many tokens went into a more powerful one: Once we established the exchange rate, we keep it the same, and once we have enough tokens, we always exchange.

To tell the power of tokens, we can make them different - e.g. having different colors - but to simplify the logistics, we use the positional representation putting more powerful tokens to the left.

Hence, we build short lines of tokens. The quantity of the tokens in every line grows up until it reaches the exchange rate. At this moment, we exchange.

What is the quantity of thousands in six thousand seven hundred twenty one? Six? Correct. How many hundreds? Seven? Wrong. If every thousand has ten hundreds, there are sixty seven hundreds here. And six hundred seventy two tens.

In other words, 6721 ones, 672 tens, 67 hundreds and 6 thousands. Without this knowledge firmly established, how could we subtract?

The questions like this teach to use counting words and appreciate the digits.

For further practicing, please find and print out the digits from any irrational number - it can be pi, e, square root of 2, etc. And, for everything sacred, don't call them decimals. 6721 is a decimal number too.

Meanwhile, don't stop at one hundred. Keep counting! Follow a simple rule: When you get one token too many, pass it to the left and clean up the overflowed position. If the token passed to the left happened to become one too many again, pass it to the left and immediately clean the position again.

Click to pause the counter at any time. Click to make it slower or faster while it's running.

Counting after after 100 is easy: it's just more of the same except for few new words and images. Clocks, books, measuring tapes, movies are ready to help.

The counter you just watched works in real time because it counts the ticks of the computer clock. It adds one token per tick. If the tick triggers an overflow, a complex event takes place: blinking red, passing a token to the left, cleaning up and repeating this as many times as required. When the work is done, the counter has made only 1 step forward.

The biggest number in 10-digit counter is 9,999,999,999. At the rate of one count per second, it will take 317 years to reach.

Passing to the left and cleaning up doesn't have to happen when we step over nine. The forbidden number, at which the building up stops - the base - can be anything bigger than one. Remember, ten and 10 are not the same!

The bottom counter goes 0, 1, 2, 3, 4, 5, 10..., which is known as base-6, or heximal. The top counter goes 0, 1, 10..., which is base-2, or binary. Zero here means no tokens. Bases smaller than 10 are convenient for learning long operations.

Pause, slower and faster are with you.

In positional counting, cleaning up is the best part. Getting rid of accumulated quantity by exchanging it for more powerful tokens instead of hauling it on forever—this is what makes a positional system so helpful. The counters reveals its two-dimensional soul. Every position is a separate number line. A regular number—a string of digits—obscures this fact.

The kids who understood quantity start solving small problems one by one. This makes them ready to learn big numbers. Positional counting is the gateway to this magnificent world.

This and the following green pages make a presentation in presentation. The subject is worth digging deeper.

The main Table of Contents and the ruler point to this page. Below is the local Table of Contents. Use it, or just click on FORWARD.

1  Introduction: Count 3 Times Starting From 5
2  Introduction: Count 8 Times Starting From 37
3  Introduction: Count 9 Times Starting From 9994
4  Introduction: How much is 983?
5  Review the Introduction and Discover Addition
6  Enter and Add Two 9-digit Numbers
7  About the Adding App

Playing with counting, let your student understand that the positional system is the same everywhere. Count up and down starting from different quantities and naming them as you go. Prevent and destroy the fear of big numbers instigated at school.

Next exercise is to count forward a number of times. Count through a given quantity of 1s and stop when all of them are spent. Adding 3 to 5, we take 1 from the number above the position and put another token on top of the stack until we have no 1s to add. Playing with your toy, give the student a bunch of tokens, then form a line of tokens to be added.

Click to start counting. You can also do it step by step.

On this step, we are going to count up 8 times starting from 37. Or, if it matters, we will count 8 times by 10 starting from 370.

Click to start counting. You can also do it step by step.

An overflow is indicated with a red light on top of the 9th token. The rule is the same as before: clean the position and pass the token to the left. In case of another overflow, do it again.

The yellow light marks the position, in which the counting is taking place. On overflow it blinks red and splits. Another light travels to the left and blinks red if there is no room in this position, too. When the overflow is settled, the counting resumes at the initial light.

We are about to count up 9 times starting from 9994(or 9 times by 100 starting from 999400).

Click to start counting. You can also do it step by step.

On the 6th count we encountered an overflow. The yellow light turned red and spawned another yellow light to the left, where we tried to count up again. No luck, another overflow. Then again and again. Finally, we found the place to rest the overflow token. Our attention returned to the initial position under the yellow light, where we found 3 more counts and performed them.

Congratulations, you just finished counting forward 983 times starting from 999375. You did it in 3 steps, and it was incredibly easy. Care to do it again?

Click to start counting. You can also do it step by step.

Another name for what you have done is addition:

999375 + 983 = 1000358
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
1111111111111111111111111111111111111111
11111111111111111111111

Let me stress, adding 983 you counted forward 983 times. Please think about counting 983 one by one. Arithmetic does this for us. And it was not even multiplication. Or was it?

Long addition is positional counting broken into short and easy pieces. On this step, you can add any two 9-digit numbers. Here is how.

Click on a token to make it the top token. Click on the top token to zero the position. Click on the red index to copy the addend to the digital register, and enter the other addend.

Start adding or do it step by step.

Learning long operations with tokens is easier in many ways. For example, the adding app, which you just tried, uses only two numbers and builds the result on top of one of them. It doesn't rely on any knowledge of "math facts," and it handles overflows immediately. Without deferred carries, one can instantly see how much is left to add, and what we are going to add it to. Best of all, this algorithm is reversible. It's equally good for subtraction.

Unlike the counter, the adder app does not have to run in real time. It slows down on overflows. Step-by-step mode allows to see every move.

This is the last green page.

EXtending the knowledge of long addition to digits on paper may happen immediately or take few more lessons. The children are motivated. In writing, long addition is hard to learn, but easy to perform.

Like counting words, digits work if we remember their order. Nothing in 7 reminds 1+1+1+1+1+1+1, but we can count through 0,1,2,3,4,5,6,7 and tell the result.

Remember how we navigated the board? Now is the time to to make and use charts of addition and practice adding very big numbers instead of single digits.

In writing, the digit 0 in writing is the same as an empty space on the board. In fact, 0 is a framed empty space. We need 0 as a placeholder: most of us are poor writers and a number may have many 0s in a row.

If transition from tokens to digits is not going well, you can use only one token in every position and walk it along. The quantity on the board is 630055. Some positions are empty, some have 0s. By the way, if we consider negative values as holes, the empty spaces instead of 0s are perfectly fine.

Outside of math, every number must be a number of something. What counting process could possibly make such values?

How do we match the same quantity to different number of digits - 9 or 10? Why do we count 0 if 0 is nothing? Time for a bonus lesson: Stick and Stake.

Imagine you point at the bottom token and say zero. Can you point at the next token and say one? You can if words are token names. Otherwise, zero isn't a token. It's nothing.

Imagine now you point right below the bottom token. OK, it's 0. No tokens under this one. Then, the point between this token and the next token is 1, etc.

Our innate mental ability to deal with quantities is very poor. We are probably not much better computers than many other animals. However, we have learned to do much more than them.

Our troubles with quantities starts with the fact that we can't perceive and remember them. Computing devices like the one I describe here are enormously helpful because they hold quantities. Then. they allow to use the result of the previous operation in the current one. In computing such devices are called accumulators.

Numerous accumulating include voting, producing, fund raising, wealth building, weight losing and running virtually any business, from a household to a bank. Teach them your student. Start from performing singular long additions and move to adding several long numbers at once. Normally, kids appreciate us being serious with them.

to make STEM subjects learnable and their students teachable.

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© 2016 Georgiy Kuznetsov